diff --git a/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusGeometry.lean b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusGeometry.lean
index ba13223..d57b89f 100644
--- a/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusGeometry.lean
+++ b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusGeometry.lean
@@ -4,19 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.MainNaturality
-import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.DoubleCosetOrbitGeometry
 
-set_option autoImplicit false
+import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.MainTransferNormOrbitMk
 
-/-!
-# Transfer--norm Frobenius geometry
 
-For a finite Galois extension and an intermediate field, this module builds the
-Frobenius-side subgroup and orbit equivalences used in transfer--norm
-naturality.  The reusable orbit and double-coset constructions are isolated in
-`DoubleCosetOrbitGeometry`.
--/
+set_option autoImplicit false
+
 
 universe u
 
@@ -35,424 +28,9 @@ section transferFrobeniusGeometry
 
 variable {G : Type u} [Group G] [TopologicalSpace G]
 
-/-- The absolute group of an intermediate field, identified with its
-literal copy inside the absolute group of the base field. -/
-noncomputable def transferNormNaturalityIntermediateAbsoluteEquiv
-    (K K' : ClosedSubgroup G)
-    (hK'K : K'.toSubgroup ≤ K.toSubgroup) :
-    K'.toSubgroup ≃* extensionSubgroup K K' hK'K :=
-  MulEquiv.ofBijective
-    ((Subgroup.inclusion hK'K).codRestrict
-      (extensionSubgroup K K' hK'K) (fun k' => k'.2))
-    ⟨fun _ _ h => Subtype.ext (congrArg (fun z => z.1.1) h), by
-      rintro ⟨k, hk'⟩
-      let k' : K'.toSubgroup := ⟨k.1, hk'⟩
-      exact ⟨k', Subtype.ext rfl⟩⟩
-
-/-- The absolute intermediate-field equivalence evaluates by the underlying transfer map. -/
-@[simp]
-theorem transferNormNaturalityIntermediateAbsoluteEquiv_apply
-    (K K' : ClosedSubgroup G)
-    (hK'K : K'.toSubgroup ≤ K.toSubgroup)
-    (k' : K'.toSubgroup) :
-    ((transferNormNaturalityIntermediateAbsoluteEquiv K K' hK'K k').1 : G) = k'.1 :=
-  rfl
-
-/-- Normality of `L | K` restricts to every intermediate field `K'`. -/
-theorem transferNormNaturality_intermediateExtension_normal
-    (K K' L : ClosedSubgroup G)
-    (hLK' : L.toSubgroup ≤ K'.toSubgroup)
-    (hK'K : K'.toSubgroup ≤ K.toSubgroup)
-    [hLnormal : (extensionSubgroup K L (hLK'.trans hK'K)).Normal] :
-    (extensionSubgroup K' L hLK').Normal := by
-  have hcomap : extensionSubgroup K' L hLK' =
-      (extensionSubgroup K L (hLK'.trans hK'K)).comap
-        (Subgroup.inclusion hK'K) := by
-    ext k'
-    rw [Subgroup.mem_comap, mem_extensionSubgroup_iff,
-      mem_extensionSubgroup_iff]
-    rfl
-  rw [hcomap]
-  exact hLnormal.comap (Subgroup.inclusion hK'K)
 
 namespace DegreeData
 
-/-- The restriction map on the infinite Frobenius quotients is injective
-when the top field is unchanged. -/
-theorem transferNormNaturalityFrobeniusTowerMap_injective
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
-    Function.Injective
-      (D.finiteReciprocityNaturalityFrobeniusTowerMap
-        E.base.field E.field.field L L
-        (hL.trans E.below) hL E.below le_rfl) := by
-  intro x y
-  refine QuotientGroup.induction_on x ?_
-  intro k'
-  refine QuotientGroup.induction_on y ?_
-  intro l' h
-  apply QuotientGroup.eq.mpr
-  have hmem :
-      (Subgroup.inclusion E.below k')⁻¹ * Subgroup.inclusion E.below l' ∈
-        D.extensionInertiaWithin E.base.field L (hL.trans E.below) :=
-    QuotientGroup.eq.mp h
-  constructor
-  · apply (mem_extensionSubgroup_iff E.field.field L hL (k'⁻¹ * l')).2
-    have hG := (mem_extensionSubgroup_iff E.base.field L
-      (hL.trans E.below)
-      ((Subgroup.inclusion E.below k')⁻¹ *
-        Subgroup.inclusion E.below l')).1 hmem.1
-    simpa using hG
-  · have hI := hmem.2
-    change D.degree (((Subgroup.inclusion E.below k')⁻¹ *
-      Subgroup.inclusion E.below l' : E.base.field.toSubgroup) : G) = 1 at hI
-    change D.degree ((k'⁻¹ * l' : E.field.field.toSubgroup) : G) = 1
-    exact hI
-
-/-- The copy of `G(\widetilde L/K')` inside
-`G(\widetilde L/K)`.  This is the subgroup `H` used in the classical
-double-coset proof of transfer--norm naturality. -/
-def transferNormNaturalityFrobeniusIntermediateSubgroup
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
-    Subgroup (E.base.field.toSubgroup ⧸
-      D.extensionInertiaWithin E.base.field L (hL.trans E.below)) :=
-  (D.finiteReciprocityNaturalityFrobeniusTowerMap
-    E.base.field E.field.field L L
-    (hL.trans E.below) hL E.below le_rfl).range
-
-/-- The subgroup above is also the image of `G_K'` under the quotient
-projection `G_K → G(\widetilde L/K)`. -/
-theorem transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
-    D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL =
-      (extensionSubgroup E.base.field E.field.field E.below).map
-        (QuotientGroup.mk'
-          (D.extensionInertiaWithin E.base.field L
-            (hL.trans E.below))) := by
-  ext q
-  constructor
-  · rintro ⟨x, rfl⟩
-    refine QuotientGroup.induction_on x ?_
-    intro k'
-    refine ⟨Subgroup.inclusion E.below k', ?_, rfl⟩
-    exact k'.2
-  · rintro ⟨k, hk', rfl⟩
-    let k' : E.field.field.toSubgroup := ⟨k.1, hk'⟩
-    refine ⟨QuotientGroup.mk k', ?_⟩
-    change QuotientGroup.mk (Subgroup.inclusion E.below k') =
-      QuotientGroup.mk k
-    rfl
-
-/-- Quotient projection maps the literal absolute subgroup belonging to
-`K'` onto its copy `H` inside `G(\widetilde L/K)`. -/
-noncomputable def transferNormNaturalityIntermediateToFrobeniusSubgroup
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
-    extensionSubgroup E.base.field E.field.field E.below →*
-      D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL := by
-  refine ((QuotientGroup.mk'
-    (D.extensionInertiaWithin E.base.field L (hL.trans E.below))).comp
-      (extensionSubgroup E.base.field E.field.field E.below).subtype).codRestrict
-        (D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL) ?_
-  intro m
-  change QuotientGroup.mk m.1 ∈
-    D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL
-  rw [D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
-    E L hL]
-  exact ⟨m.1, m.2, rfl⟩
-
-/-- The map from the intermediate quotient onto the Frobenius subgroup is surjective. -/
-theorem transferNormNaturalityIntermediateToFrobeniusSubgroup_surjective
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
-    Function.Surjective
-      (D.transferNormNaturalityIntermediateToFrobeniusSubgroup
-        E L hL) := by
-  intro h
-  have hh : h.1 ∈
-      (extensionSubgroup E.base.field E.field.field E.below).map
-      (QuotientGroup.mk'
-        (D.extensionInertiaWithin E.base.field L (hL.trans E.below))) := by
-    rw [← D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
-      E L hL]
-    exact h.2
-  obtain ⟨m, hm, hval⟩ := hh
-  refine ⟨⟨m, hm⟩, ?_⟩
-  apply Subtype.ext
-  unfold transferNormNaturalityIntermediateToFrobeniusSubgroup
-  simpa only [MonoidHom.codRestrict_apply, MonoidHom.comp_apply,
-    Subgroup.subtype_apply] using hval
-
-/-- The intermediate-to-Frobenius map has the stated value on each representative. -/
-@[simp]
-theorem transferNormNaturalityIntermediateToFrobeniusSubgroup_apply
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
-    (m : extensionSubgroup E.base.field E.field.field E.below) :
-    (D.transferNormNaturalityIntermediateToFrobeniusSubgroup
-      E L hL m).1 =
-        (QuotientGroup.mk m.1 : E.base.field.toSubgroup ⧸
-          D.extensionInertiaWithin E.base.field L (hL.trans E.below)) := by
-  rfl
-
-/-- The canonical coset equivalence from `G_K/G_Σ` to
-`G(\widetilde L/K)/Γ` intertwines the two copies of the `K'`-action. -/
-theorem frobeniusFixedCosetClosureEquiv_equivariant
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
-    (σ : D.FrobeniusElements E.base L (hL.trans E.below))
-    (m : extensionSubgroup E.base.field E.field.field E.below)
-    (x : E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field
-      (D.frobeniusFixedField E.base L (hL.trans E.below) σ)
-      (D.frobeniusFixedField_le E.base L (hL.trans E.below) σ)) :
-    D.frobeniusFixedCosetClosureEquiv E.base L (hL.trans E.below) σ (m • x) =
-      (D.transferNormNaturalityIntermediateToFrobeniusSubgroup
-        E L hL m) •
-        D.frobeniusFixedCosetClosureEquiv E.base L
-          (hL.trans E.below) σ x := by
-  refine Quotient.inductionOn' x ?_
-  intro k
-  change QuotientGroup.mk (QuotientGroup.mk (m.1 * k)) =
-    QuotientGroup.mk
-      ((D.transferNormNaturalityIntermediateToFrobeniusSubgroup
-        E L hL m).1 * QuotientGroup.mk k)
-  rw [D.transferNormNaturalityIntermediateToFrobeniusSubgroup_apply]
-  rfl
-
-/-- Restriction from the infinite Frobenius quotient onto the finite
-Galois quotient is surjective. -/
-theorem transferNormNaturalityExtensionRestriction_surjective
-    (D : DegreeData G)
-    (K L : ClosedSubgroup G)
-    (hLK : L.toSubgroup ≤ K.toSubgroup)
-    [hLnormal : (extensionSubgroup K L hLK).Normal] :
-    Function.Surjective (D.extensionRestriction K L hLK) := by
-  intro q
-  refine QuotientGroup.induction_on q ?_
-  intro k
-  exact ⟨QuotientGroup.mk k, rfl⟩
-
-/-- The kernel of restriction to `G(L/K)` is contained in the subgroup
-coming from `G(\widetilde L/K')`. -/
-theorem transferNormNaturalityExtensionRestriction_ker_le_intermediate
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
-    (D.extensionRestriction E.base.field L (hL.trans E.below)).ker ≤
-      D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL := by
-  rw [D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
-    E L hL]
-  intro q hq
-  revert hq
-  refine QuotientGroup.induction_on q ?_
-  intro k hk
-  change D.extensionRestriction E.base.field L (hL.trans E.below)
-      (QuotientGroup.mk k) = 1 at hk
-  rw [D.extensionRestriction_mk] at hk
-  have hkL : k ∈
-      extensionSubgroup E.base.field L (hL.trans E.below) := by
-    exact QuotientGroup.eq_one_iff k |>.1 hk
-  have hkK' : k ∈
-      extensionSubgroup E.base.field E.field.field E.below := by
-    apply (mem_extensionSubgroup_iff
-      E.base.field E.field.field E.below k).2
-    exact hL ((mem_extensionSubgroup_iff E.base.field L
-      (hL.trans E.below) k).1 hkL)
-  exact ⟨k, hkK', rfl⟩
-
-/-- `H` has finite index in `G(\widetilde L/K)`, with no normality
-assumption on the intermediate extension `K'/K`. -/
-theorem transferNormNaturalityFrobeniusIntermediateFiniteIndex
-    (D : DegreeData G) [IsTopologicalGroup G]
-    (R : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ R.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup R.base.field L (hL.trans R.below)).Normal]
-    [hL'normal : (extensionSubgroup R.field.field L hL).Normal] :
-    (D.transferNormNaturalityFrobeniusIntermediateSubgroup R L hL).FiniteIndex := by
-  rw [D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
-    R L hL]
-  let I := D.extensionInertiaWithin R.base.field L (hL.trans R.below)
-  let M := extensionSubgroup R.base.field R.field.field R.below
-  have hIM : I ≤ M := by
-    intro k hk
-    apply (mem_extensionSubgroup_iff
-      R.base.field R.field.field R.below k).2
-    exact hL ((mem_extensionSubgroup_iff R.base.field L
-      (hL.trans R.below) k).1 hk.1)
-  let p := QuotientGroup.mk' I
-  have hker : p.ker ≤ M := by
-    simpa [p] using hIM
-  let : M.FiniteIndex := Subgroup.finiteIndex_of_finite_quotient
-  rw [Subgroup.finiteIndex_iff,
-    M.index_map_eq (QuotientGroup.mk'_surjective I) hker]
-  exact Subgroup.FiniteIndex.index_ne_zero
-
-/-- The copy of `G(\widetilde L/K')` is closed in
-`G(\widetilde L/K)`. -/
-theorem transferNormNaturalityFrobeniusIntermediate_isClosed
-    (D : DegreeData G) [IsTopologicalGroup G] [CompactSpace G] [T2Space G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
-    IsClosed (D.transferNormNaturalityFrobeniusIntermediateSubgroup
-      E L hL : Set
-        (E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
-          (hL.trans E.below))) := by
-  let : CompactSpace E.field.field.toSubgroup :=
-    isCompact_iff_compactSpace.mp E.field.field.isClosed'.isCompact
-  let : IsClosed
-      (D.extensionInertiaWithin E.field.field L hL :
-        Set E.field.field.toSubgroup) :=
-    D.extensionInertiaWithin_isClosed E.field L hL
-  let : IsClosed (D.extensionInertiaWithin E.base.field L
-      (hL.trans E.below) : Set E.base.field.toSubgroup) :=
-    D.extensionInertiaWithin_isClosed E.base L (hL.trans E.below)
-  let f := D.finiteReciprocityNaturalityFrobeniusTowerMapContinuous
-    E.base.field E.field.field L L
-    (hL.trans E.below) hL E.below le_rfl
-  change IsClosed (Set.range f)
-  have hrange : Set.range f = Set.range f.toContinuousMap := by
-    ext y
-    constructor <;> rintro ⟨x, rfl⟩ <;> exact ⟨x, rfl⟩
-  rw [hrange]
-  simpa only [Set.image_univ] using
-    (isCompact_univ.image f.continuous).isClosed
-
-/-- The transfer-orbit index set
-`⟨σ⟩ \ G(\widetilde L/K) / H` is canonically the norm double-coset
-index set `G_K' \ G_K / G_Σ`.  The equivalence is inversion of double
-cosets, followed by passage from powers of `σ` to their closure `Γ` and
-the canonical identification `G_K/G_Σ ≃ G(\widetilde L/K)/Γ`. -/
-noncomputable def transferNormNaturalityTransferNormOrbitEquiv
-    (D : DegreeData G) [IsTopologicalGroup G] [CompactSpace G] [T2Space G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
-    (σ : D.FrobeniusElements E.base L (hL.trans E.below)) :
-    Quotient (orbitRel (Subgroup.zpowers σ.1)
-      ((E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
-        (hL.trans E.below)) ⧸
-          D.transferNormNaturalityFrobeniusIntermediateSubgroup
-            E L hL)) ≃
-      Quotient (orbitRel
-        (extensionSubgroup E.base.field E.field.field E.below)
-        (E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field
-          (D.frobeniusFixedField E.base L (hL.trans E.below) σ)
-          (D.frobeniusFixedField_le E.base L
-            (hL.trans E.below) σ))) := by
-  let P := E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
-    (hL.trans E.below)
-  let H := D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL
-  let M := extensionSubgroup E.base.field E.field.field E.below
-  let Γ := D.frobeniusClosure E.base L (hL.trans E.below) σ
-  let S := D.frobeniusFixedField E.base L (hL.trans E.below) σ
-  let hSK := D.frobeniusFixedField_le E.base L (hL.trans E.below) σ
-  let f : M →* H := D.transferNormNaturalityIntermediateToFrobeniusSubgroup
-    E L hL
-  let e := D.frobeniusFixedCosetClosureEquiv
-    E.base L (hL.trans E.below) σ
-  letI : H.FiniteIndex :=
-    D.transferNormNaturalityFrobeniusIntermediateFiniteIndex E L hL
-  have hHclosed : IsClosed (H : Set P) :=
-    D.transferNormNaturalityFrobeniusIntermediate_isClosed E L hL
-  have hΓ :
-      (closedSubgroupGenerated ({σ.1} : Set P)).toSubgroup = Γ.toSubgroup := by
-    simp [Γ, DegreeData.frobeniusClosure]
-  let eΓ : P ⧸ (closedSubgroupGenerated ({σ.1} : Set P)).toSubgroup ≃
-      P ⧸ Γ.toSubgroup := Subgroup.quotientEquivOfEq hΓ
-  have heΓ (h : H)
-      (x : P ⧸ (closedSubgroupGenerated ({σ.1} : Set P)).toSubgroup) :
-      eΓ (h • x) = h • eΓ x := by
-    refine Quotient.inductionOn' x ?_
-    intro p
-    rfl
-  let eΓorbit := orbitQuotientEquivOfSurjectiveEquivariant
-    (MonoidHom.id H) Function.surjective_id eΓ heΓ
-  have hf : Function.Surjective f :=
-    D.transferNormNaturalityIntermediateToFrobeniusSubgroup_surjective
-      E L hL
-  have he (m : M)
-      (x : E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field S hSK) :
-      e (m • x) = f m • e x := by
-    exact D.frobeniusFixedCosetClosureEquiv_equivariant
-      E L hL σ m x
-  let eAction := orbitQuotientEquivOfSurjectiveEquivariant f hf e he
-  exact (orbitQuotientSwapEquiv (Subgroup.zpowers σ.1) H).trans
-    ((orbitQuotientClosedCyclicEquiv H hHclosed σ.1).trans
-      (eΓorbit.trans eAction.symm))
-
-/-- The transfer-norm orbit equivalence sends quotient representatives to their norm orbits. -/
-@[simp]
-theorem transferNormNaturalityTransferNormOrbitEquiv_mk
-    (D : DegreeData G) [IsTopologicalGroup G] [CompactSpace G] [T2Space G]
-    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
-    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
-    [hLnormal :
-      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
-    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
-    (σ : D.FrobeniusElements E.base L (hL.trans E.below))
-    (k : E.base.field.toSubgroup) :
-    D.transferNormNaturalityTransferNormOrbitEquiv E L hL σ
-        (Quotient.mk'' (QuotientGroup.mk
-          (QuotientGroup.mk k : E.base.field.toSubgroup ⧸
-            D.extensionInertiaWithin E.base.field L (hL.trans E.below)) :
-          (E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
-            (hL.trans E.below)) ⧸
-              D.transferNormNaturalityFrobeniusIntermediateSubgroup
-                E L hL)) =
-      Quotient.mk'' (QuotientGroup.mk k⁻¹ :
-        E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field
-          (D.frobeniusFixedField E.base L (hL.trans E.below) σ)
-          (D.frobeniusFixedField_le E.base L
-            (hL.trans E.below) σ)) := by
-  unfold transferNormNaturalityTransferNormOrbitEquiv
-  simp only [Equiv.trans_apply, orbitQuotientSwapEquiv_mk,
-    orbitQuotientClosedCyclicEquiv_mk,
-    orbitQuotientEquivOfSurjectiveEquivariant_mk,
-    orbitQuotientEquivOfSurjectiveEquivariant_symm_mk,
-    Subgroup.quotientEquivOfEq_mk]
-  apply congrArg Quotient.mk''
-  exact (D.frobeniusFixedCosetClosureEquiv E.base L
-    (hL.trans E.below) σ).symm_apply_apply (QuotientGroup.mk k⁻¹)
-
 /-- On the classical chosen transfer representative `t`, the preceding
 equivalence is literally the norm orbit represented by `t⁻¹`. -/
 theorem transferNormNaturalityTransferNormOrbitEquiv_apply
diff --git a/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusRestriction.lean b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusRestriction.lean
new file mode 100644
index 0000000..4f8dc8c
--- /dev/null
+++ b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusRestriction.lean
@@ -0,0 +1,142 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.MainTransferFrobeniusSubgroup
+
+
+set_option autoImplicit false
+
+
+universe u
+
+namespace ClassFormation
+
+open KummerTheory
+open CyclicCohomology
+open CategoryTheory
+
+noncomputable section
+
+open scoped BigOperators
+open MulAction
+
+section transferFrobeniusGeometry
+
+variable {G : Type u} [Group G] [TopologicalSpace G]
+
+
+namespace DegreeData
+
+/-- Restriction from the infinite Frobenius quotient onto the finite
+Galois quotient is surjective. -/
+theorem transferNormNaturalityExtensionRestriction_surjective
+    (D : DegreeData G)
+    (K L : ClosedSubgroup G)
+    (hLK : L.toSubgroup ≤ K.toSubgroup)
+    [hLnormal : (extensionSubgroup K L hLK).Normal] :
+    Function.Surjective (D.extensionRestriction K L hLK) := by
+  intro q
+  refine QuotientGroup.induction_on q ?_
+  intro k
+  exact ⟨QuotientGroup.mk k, rfl⟩
+
+
+theorem transferNormNaturalityExtensionRestriction_ker_le_intermediate
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
+    (D.extensionRestriction E.base.field L (hL.trans E.below)).ker ≤
+      D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL := by
+  rw [D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
+    E L hL]
+  intro q hq
+  revert hq
+  refine QuotientGroup.induction_on q ?_
+  intro k hk
+  change D.extensionRestriction E.base.field L (hL.trans E.below)
+      (QuotientGroup.mk k) = 1 at hk
+  rw [D.extensionRestriction_mk] at hk
+  have hkL : k ∈
+      extensionSubgroup E.base.field L (hL.trans E.below) := by
+    exact QuotientGroup.eq_one_iff k |>.1 hk
+  have hkK' : k ∈
+      extensionSubgroup E.base.field E.field.field E.below := by
+    apply (mem_extensionSubgroup_iff
+      E.base.field E.field.field E.below k).2
+    exact hL ((mem_extensionSubgroup_iff E.base.field L
+      (hL.trans E.below) k).1 hkL)
+  exact ⟨k, hkK', rfl⟩
+
+/-- `H` has finite index in `G(\widetilde L/K)`, with no normality
+assumption on the intermediate extension `K'/K`. -/
+theorem transferNormNaturalityFrobeniusIntermediateFiniteIndex
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (R : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ R.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup R.base.field L (hL.trans R.below)).Normal]
+    [hL'normal : (extensionSubgroup R.field.field L hL).Normal] :
+    (D.transferNormNaturalityFrobeniusIntermediateSubgroup R L hL).FiniteIndex := by
+  rw [D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
+    R L hL]
+  let I := D.extensionInertiaWithin R.base.field L (hL.trans R.below)
+  let M := extensionSubgroup R.base.field R.field.field R.below
+  have hIM : I ≤ M := by
+    intro k hk
+    apply (mem_extensionSubgroup_iff
+      R.base.field R.field.field R.below k).2
+    exact hL ((mem_extensionSubgroup_iff R.base.field L
+      (hL.trans R.below) k).1 hk.1)
+  let p := QuotientGroup.mk' I
+  have hker : p.ker ≤ M := by
+    simpa [p] using hIM
+  let : M.FiniteIndex := Subgroup.finiteIndex_of_finite_quotient
+  rw [Subgroup.finiteIndex_iff,
+    M.index_map_eq (QuotientGroup.mk'_surjective I) hker]
+  exact Subgroup.FiniteIndex.index_ne_zero
+
+/-- The copy of `G(\widetilde L/K')` is closed in
+`G(\widetilde L/K)`. -/
+theorem transferNormNaturalityFrobeniusIntermediate_isClosed
+    (D : DegreeData G) [IsTopologicalGroup G] [CompactSpace G] [T2Space G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
+    IsClosed (D.transferNormNaturalityFrobeniusIntermediateSubgroup
+      E L hL : Set
+        (E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
+          (hL.trans E.below))) := by
+  let : CompactSpace E.field.field.toSubgroup :=
+    isCompact_iff_compactSpace.mp E.field.field.isClosed'.isCompact
+  let : IsClosed
+      (D.extensionInertiaWithin E.field.field L hL :
+        Set E.field.field.toSubgroup) :=
+    D.extensionInertiaWithin_isClosed E.field L hL
+  let : IsClosed (D.extensionInertiaWithin E.base.field L
+      (hL.trans E.below) : Set E.base.field.toSubgroup) :=
+    D.extensionInertiaWithin_isClosed E.base L (hL.trans E.below)
+  let f := D.finiteReciprocityNaturalityFrobeniusTowerMapContinuous
+    E.base.field E.field.field L L
+    (hL.trans E.below) hL E.below le_rfl
+  change IsClosed (Set.range f)
+  have hrange : Set.range f = Set.range f.toContinuousMap := by
+    ext y
+    constructor <;> rintro ⟨x, rfl⟩ <;> exact ⟨x, rfl⟩
+  rw [hrange]
+  simpa only [Set.image_univ] using
+    (isCompact_univ.image f.continuous).isClosed
+
+
+end DegreeData
+end transferFrobeniusGeometry
+end
+end ClassFormation
diff --git a/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusSubgroup.lean b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusSubgroup.lean
new file mode 100644
index 0000000..d46fda8
--- /dev/null
+++ b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferFrobeniusSubgroup.lean
@@ -0,0 +1,249 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.MainNaturality
+import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.DoubleCosetOrbitGeometry
+
+
+set_option autoImplicit false
+
+
+universe u
+
+namespace ClassFormation
+
+open KummerTheory
+open CyclicCohomology
+open CategoryTheory
+
+noncomputable section
+
+open scoped BigOperators
+open MulAction
+
+section transferFrobeniusGeometry
+
+variable {G : Type u} [Group G] [TopologicalSpace G]
+
+/-- The absolute group of an intermediate field, identified with its
+literal copy inside the absolute group of the base field. -/
+noncomputable def transferNormNaturalityIntermediateAbsoluteEquiv
+    (K K' : ClosedSubgroup G)
+    (hK'K : K'.toSubgroup ≤ K.toSubgroup) :
+    K'.toSubgroup ≃* extensionSubgroup K K' hK'K :=
+  MulEquiv.ofBijective
+    ((Subgroup.inclusion hK'K).codRestrict
+      (extensionSubgroup K K' hK'K) (fun k' => k'.2))
+    ⟨fun _ _ h => Subtype.ext (congrArg (fun z => z.1.1) h), by
+      rintro ⟨k, hk'⟩
+      let k' : K'.toSubgroup := ⟨k.1, hk'⟩
+      exact ⟨k', Subtype.ext rfl⟩⟩
+
+/-- The absolute intermediate-field equivalence evaluates by the underlying transfer map. -/
+@[simp]
+theorem transferNormNaturalityIntermediateAbsoluteEquiv_apply
+    (K K' : ClosedSubgroup G)
+    (hK'K : K'.toSubgroup ≤ K.toSubgroup)
+    (k' : K'.toSubgroup) :
+    ((transferNormNaturalityIntermediateAbsoluteEquiv K K' hK'K k').1 : G) = k'.1 :=
+  rfl
+
+/-- Normality of `L | K` restricts to every intermediate field `K'`. -/
+theorem transferNormNaturality_intermediateExtension_normal
+    (K K' L : ClosedSubgroup G)
+    (hLK' : L.toSubgroup ≤ K'.toSubgroup)
+    (hK'K : K'.toSubgroup ≤ K.toSubgroup)
+    [hLnormal : (extensionSubgroup K L (hLK'.trans hK'K)).Normal] :
+    (extensionSubgroup K' L hLK').Normal := by
+  have hcomap : extensionSubgroup K' L hLK' =
+      (extensionSubgroup K L (hLK'.trans hK'K)).comap
+        (Subgroup.inclusion hK'K) := by
+    ext k'
+    rw [Subgroup.mem_comap, mem_extensionSubgroup_iff,
+      mem_extensionSubgroup_iff]
+    rfl
+  rw [hcomap]
+  exact hLnormal.comap (Subgroup.inclusion hK'K)
+
+namespace DegreeData
+
+/-- The restriction map on the infinite Frobenius quotients is injective
+when the top field is unchanged. -/
+theorem transferNormNaturalityFrobeniusTowerMap_injective
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
+    Function.Injective
+      (D.finiteReciprocityNaturalityFrobeniusTowerMap
+        E.base.field E.field.field L L
+        (hL.trans E.below) hL E.below le_rfl) := by
+  intro x y
+  refine QuotientGroup.induction_on x ?_
+  intro k'
+  refine QuotientGroup.induction_on y ?_
+  intro l' h
+  apply QuotientGroup.eq.mpr
+  have hmem :
+      (Subgroup.inclusion E.below k')⁻¹ * Subgroup.inclusion E.below l' ∈
+        D.extensionInertiaWithin E.base.field L (hL.trans E.below) :=
+    QuotientGroup.eq.mp h
+  constructor
+  · apply (mem_extensionSubgroup_iff E.field.field L hL (k'⁻¹ * l')).2
+    have hG := (mem_extensionSubgroup_iff E.base.field L
+      (hL.trans E.below)
+      ((Subgroup.inclusion E.below k')⁻¹ *
+        Subgroup.inclusion E.below l')).1 hmem.1
+    simpa using hG
+  · have hI := hmem.2
+    change D.degree (((Subgroup.inclusion E.below k')⁻¹ *
+      Subgroup.inclusion E.below l' : E.base.field.toSubgroup) : G) = 1 at hI
+    change D.degree ((k'⁻¹ * l' : E.field.field.toSubgroup) : G) = 1
+    exact hI
+
+/-- The copy of `G(\widetilde L/K')` inside
+`G(\widetilde L/K)`.  This is the subgroup `H` used in the classical
+double-coset proof of transfer--norm naturality. -/
+def transferNormNaturalityFrobeniusIntermediateSubgroup
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
+    Subgroup (E.base.field.toSubgroup ⧸
+      D.extensionInertiaWithin E.base.field L (hL.trans E.below)) :=
+  (D.finiteReciprocityNaturalityFrobeniusTowerMap
+    E.base.field E.field.field L L
+    (hL.trans E.below) hL E.below le_rfl).range
+
+/-- The subgroup above is also the image of `G_K'` under the quotient
+projection `G_K → G(\widetilde L/K)`. -/
+theorem transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
+    D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL =
+      (extensionSubgroup E.base.field E.field.field E.below).map
+        (QuotientGroup.mk'
+          (D.extensionInertiaWithin E.base.field L
+            (hL.trans E.below))) := by
+  ext q
+  constructor
+  · rintro ⟨x, rfl⟩
+    refine QuotientGroup.induction_on x ?_
+    intro k'
+    refine ⟨Subgroup.inclusion E.below k', ?_, rfl⟩
+    exact k'.2
+  · rintro ⟨k, hk', rfl⟩
+    let k' : E.field.field.toSubgroup := ⟨k.1, hk'⟩
+    refine ⟨QuotientGroup.mk k', ?_⟩
+    change QuotientGroup.mk (Subgroup.inclusion E.below k') =
+      QuotientGroup.mk k
+    rfl
+
+/-- Quotient projection maps the literal absolute subgroup belonging to
+`K'` onto its copy `H` inside `G(\widetilde L/K)`. -/
+noncomputable def transferNormNaturalityIntermediateToFrobeniusSubgroup
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
+    extensionSubgroup E.base.field E.field.field E.below →*
+      D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL := by
+  refine ((QuotientGroup.mk'
+    (D.extensionInertiaWithin E.base.field L (hL.trans E.below))).comp
+      (extensionSubgroup E.base.field E.field.field E.below).subtype).codRestrict
+        (D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL) ?_
+  intro m
+  change QuotientGroup.mk m.1 ∈
+    D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL
+  rw [D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
+    E L hL]
+  exact ⟨m.1, m.2, rfl⟩
+
+/-- The map from the intermediate quotient onto the Frobenius subgroup is surjective. -/
+theorem transferNormNaturalityIntermediateToFrobeniusSubgroup_surjective
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal] :
+    Function.Surjective
+      (D.transferNormNaturalityIntermediateToFrobeniusSubgroup
+        E L hL) := by
+  intro h
+  have hh : h.1 ∈
+      (extensionSubgroup E.base.field E.field.field E.below).map
+      (QuotientGroup.mk'
+        (D.extensionInertiaWithin E.base.field L (hL.trans E.below))) := by
+    rw [← D.transferNormNaturalityFrobeniusIntermediateSubgroup_eq_map
+      E L hL]
+    exact h.2
+  obtain ⟨m, hm, hval⟩ := hh
+  refine ⟨⟨m, hm⟩, ?_⟩
+  apply Subtype.ext
+  unfold transferNormNaturalityIntermediateToFrobeniusSubgroup
+  simpa only [MonoidHom.codRestrict_apply, MonoidHom.comp_apply,
+    Subgroup.subtype_apply] using hval
+
+/-- The intermediate-to-Frobenius map has the stated value on each representative. -/
+@[simp]
+theorem transferNormNaturalityIntermediateToFrobeniusSubgroup_apply
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
+    (m : extensionSubgroup E.base.field E.field.field E.below) :
+    (D.transferNormNaturalityIntermediateToFrobeniusSubgroup
+      E L hL m).1 =
+        (QuotientGroup.mk m.1 : E.base.field.toSubgroup ⧸
+          D.extensionInertiaWithin E.base.field L (hL.trans E.below)) := by
+  rfl
+
+/-- The canonical coset equivalence from `G_K/G_Σ` to
+`G(\widetilde L/K)/Γ` intertwines the two copies of the `K'`-action. -/
+theorem frobeniusFixedCosetClosureEquiv_equivariant
+    (D : DegreeData G) [IsTopologicalGroup G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
+    (σ : D.FrobeniusElements E.base L (hL.trans E.below))
+    (m : extensionSubgroup E.base.field E.field.field E.below)
+    (x : E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field
+      (D.frobeniusFixedField E.base L (hL.trans E.below) σ)
+      (D.frobeniusFixedField_le E.base L (hL.trans E.below) σ)) :
+    D.frobeniusFixedCosetClosureEquiv E.base L (hL.trans E.below) σ (m • x) =
+      (D.transferNormNaturalityIntermediateToFrobeniusSubgroup
+        E L hL m) •
+        D.frobeniusFixedCosetClosureEquiv E.base L
+          (hL.trans E.below) σ x := by
+  refine Quotient.inductionOn' x ?_
+  intro k
+  change QuotientGroup.mk (QuotientGroup.mk (m.1 * k)) =
+    QuotientGroup.mk
+      ((D.transferNormNaturalityIntermediateToFrobeniusSubgroup
+        E L hL m).1 * QuotientGroup.mk k)
+  rw [D.transferNormNaturalityIntermediateToFrobeniusSubgroup_apply]
+  rfl
+
+
+end DegreeData
+end transferFrobeniusGeometry
+end
+end ClassFormation
diff --git a/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferNormOrbitEquiv.lean b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferNormOrbitEquiv.lean
new file mode 100644
index 0000000..5ce3d85
--- /dev/null
+++ b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferNormOrbitEquiv.lean
@@ -0,0 +1,107 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.MainTransferFrobeniusRestriction
+
+
+set_option autoImplicit false
+
+
+universe u
+
+namespace ClassFormation
+
+open KummerTheory
+open CyclicCohomology
+open CategoryTheory
+
+noncomputable section
+
+open scoped BigOperators
+open MulAction
+
+section transferFrobeniusGeometry
+
+variable {G : Type u} [Group G] [TopologicalSpace G]
+
+
+namespace DegreeData
+
+/-- The transfer-orbit index set
+`⟨σ⟩ \ G(\widetilde L/K) / H` is canonically the norm double-coset
+index set `G_K' \ G_K / G_Σ`.  The equivalence is inversion of double
+cosets, followed by passage from powers of `σ` to their closure `Γ` and
+the canonical identification `G_K/G_Σ ≃ G(\widetilde L/K)/Γ`. -/
+noncomputable def transferNormNaturalityTransferNormOrbitEquiv
+    (D : DegreeData G) [IsTopologicalGroup G] [CompactSpace G] [T2Space G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
+    (σ : D.FrobeniusElements E.base L (hL.trans E.below)) :
+    Quotient (orbitRel (Subgroup.zpowers σ.1)
+      ((E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
+        (hL.trans E.below)) ⧸
+          D.transferNormNaturalityFrobeniusIntermediateSubgroup
+            E L hL)) ≃
+      Quotient (orbitRel
+        (extensionSubgroup E.base.field E.field.field E.below)
+        (E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field
+          (D.frobeniusFixedField E.base L (hL.trans E.below) σ)
+          (D.frobeniusFixedField_le E.base L
+            (hL.trans E.below) σ))) := by
+  let P := E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
+    (hL.trans E.below)
+  let H := D.transferNormNaturalityFrobeniusIntermediateSubgroup E L hL
+  let M := extensionSubgroup E.base.field E.field.field E.below
+  let Γ := D.frobeniusClosure E.base L (hL.trans E.below) σ
+  let S := D.frobeniusFixedField E.base L (hL.trans E.below) σ
+  let hSK := D.frobeniusFixedField_le E.base L (hL.trans E.below) σ
+  let f : M →* H := D.transferNormNaturalityIntermediateToFrobeniusSubgroup
+    E L hL
+  let e := D.frobeniusFixedCosetClosureEquiv
+    E.base L (hL.trans E.below) σ
+  letI : H.FiniteIndex :=
+    D.transferNormNaturalityFrobeniusIntermediateFiniteIndex E L hL
+  have hHclosed : IsClosed (H : Set P) :=
+    D.transferNormNaturalityFrobeniusIntermediate_isClosed E L hL
+  have hΓ :
+      (closedSubgroupGenerated ({σ.1} : Set P)).toSubgroup = Γ.toSubgroup := by
+    simp only [Γ, DegreeData.frobeniusClosure, Set.range_const]
+  let eΓ : P ⧸ (closedSubgroupGenerated ({σ.1} : Set P)).toSubgroup ≃
+      P ⧸ Γ.toSubgroup := Subgroup.quotientEquivOfEq hΓ
+  have heΓ (h : H)
+      (x : P ⧸ (closedSubgroupGenerated ({σ.1} : Set P)).toSubgroup) :
+      eΓ (h • x) = h • eΓ x := by
+    refine Quotient.inductionOn' x ?_
+    intro p
+    change eΓ (QuotientGroup.mk (h.val * p)) = h • eΓ (QuotientGroup.mk p)
+    simp only [eΓ, Subgroup.quotientEquivOfEq_mk]
+    rfl
+  let eΓorbit := orbitQuotientEquivOfSurjectiveEquivariant
+    (MonoidHom.id H) Function.surjective_id eΓ heΓ
+  have hf : Function.Surjective f :=
+    D.transferNormNaturalityIntermediateToFrobeniusSubgroup_surjective
+      E L hL
+  have he (m : M)
+      (x : E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field S hSK) :
+      e (m • x) = f m • e x := by
+    exact D.frobeniusFixedCosetClosureEquiv_equivariant
+      E L hL σ m x
+  let eAction := orbitQuotientEquivOfSurjectiveEquivariant f hf e he
+  exact (orbitQuotientSwapEquiv (Subgroup.zpowers σ.1) H).trans
+    ((orbitQuotientClosedCyclicEquiv H hHclosed σ.1).trans
+      (eΓorbit.trans eAction.symm))
+
+
+end DegreeData
+
+end transferFrobeniusGeometry
+end
+
+end ClassFormation
diff --git a/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferNormOrbitMk.lean b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferNormOrbitMk.lean
new file mode 100644
index 0000000..0e23056
--- /dev/null
+++ b/Lean4/ClassFieldTheory/AbstractClassFieldTheory/Reciprocity/Construction/MainTransferNormOrbitMk.lean
@@ -0,0 +1,74 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Construction.MainTransferNormOrbitEquiv
+
+
+set_option autoImplicit false
+
+
+universe u
+
+namespace ClassFormation
+
+open KummerTheory
+open CyclicCohomology
+open CategoryTheory
+
+noncomputable section
+
+open scoped BigOperators
+open MulAction
+
+section transferFrobeniusGeometry
+
+variable {G : Type u} [Group G] [TopologicalSpace G]
+
+
+namespace DegreeData
+
+/-- The transfer-norm orbit equivalence sends quotient representatives to their norm orbits. -/
+@[simp]
+theorem transferNormNaturalityTransferNormOrbitEquiv_mk
+    (D : DegreeData G) [IsTopologicalGroup G] [CompactSpace G] [T2Space G]
+    (E : FiniteResidueAbstractExtension D) (L : ClosedSubgroup G)
+    (hL : L.toSubgroup ≤ E.field.field.toSubgroup)
+    [hLnormal :
+      (extensionSubgroup E.base.field L (hL.trans E.below)).Normal]
+    [hL'normal : (extensionSubgroup E.field.field L hL).Normal]
+    (σ : D.FrobeniusElements E.base L (hL.trans E.below))
+    (k : E.base.field.toSubgroup) :
+    D.transferNormNaturalityTransferNormOrbitEquiv E L hL σ
+        (Quotient.mk'' (QuotientGroup.mk
+          (QuotientGroup.mk k : E.base.field.toSubgroup ⧸
+            D.extensionInertiaWithin E.base.field L (hL.trans E.below)) :
+          (E.base.field.toSubgroup ⧸ D.extensionInertiaWithin E.base.field L
+            (hL.trans E.below)) ⧸
+              D.transferNormNaturalityFrobeniusIntermediateSubgroup
+                E L hL)) =
+      Quotient.mk'' (QuotientGroup.mk k⁻¹ :
+        E.base.field.toSubgroup ⧸ extensionSubgroup E.base.field
+          (D.frobeniusFixedField E.base L (hL.trans E.below) σ)
+          (D.frobeniusFixedField_le E.base L
+            (hL.trans E.below) σ)) := by
+  unfold transferNormNaturalityTransferNormOrbitEquiv
+  simp only [Equiv.trans_apply, orbitQuotientSwapEquiv_mk,
+    orbitQuotientClosedCyclicEquiv_mk,
+    orbitQuotientEquivOfSurjectiveEquivariant_mk,
+    orbitQuotientEquivOfSurjectiveEquivariant_symm_mk,
+    Subgroup.quotientEquivOfEq_mk]
+  apply congrArg Quotient.mk''
+  exact (D.frobeniusFixedCosetClosureEquiv E.base L
+    (hL.trans E.below) σ).symm_apply_apply (QuotientGroup.mk k⁻¹)
+
+
+end DegreeData
+
+end transferFrobeniusGeometry
+end
+
+end ClassFormation
diff --git a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/AdeleBaseChange.lean b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/AdeleBaseChange.lean
index fa7f215..d8d918e 100644
--- a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/AdeleBaseChange.lean
+++ b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/AdeleBaseChange.lean
@@ -7,15 +7,9 @@ Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 import ClassFieldTheory.AlgebraicNumberTheory.Idele.BaseChange
 import ClassFieldTheory.AlgebraicNumberTheory.Completion.AdicCompletionComparison
 
-set_option autoImplicit false
 
-/-!
-# Scalar extension from relative to ordinary adeles
+set_option autoImplicit false
 
-This file upgrades the relative-to-ordinary idele comparison to the
-underlying adele rings.  The additive structure is needed to transport
-determinant norms in a field tower.
--/
 
 open scoped NumberField TensorProduct RestrictedProduct
 open NumberField IsDedekindDomain
diff --git a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/Idele/NormTopology/ExtensionBehavior.lean b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/Idele/NormTopology/ExtensionBehavior.lean
index 2e624da..f838ee6 100644
--- a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/Idele/NormTopology/ExtensionBehavior.lean
+++ b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/Idele/NormTopology/ExtensionBehavior.lean
@@ -6,6 +6,7 @@ Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 
 import ClassFieldTheory.AlgebraicNumberTheory.Idele.NormTopology.FiniteNormArithmetic
 
+
 set_option autoImplicit false
 
 /-!
@@ -40,12 +41,20 @@ private theorem infinitePlaceCompletionMap_isometry
     Isometry
       (NumberField.LiesOver.completionMap
         (v := v₀) (w := W)) := by
-  unfold NumberField.LiesOver.completionMap
-  exact
-    (InfinitePlace.Completion.isometryEquivCompletion W).symm.isometry.comp
-      ((UniformSpace.Completion.isometry_mapRingHom
-          (InfinitePlace.LiesOver.isometry_algebraMap W v₀)).comp
-        (InfinitePlace.Completion.isometryEquivCompletion v₀).isometry)
+  apply AddMonoidHomClass.isometry_of_norm
+    (NumberField.LiesOver.completionMap (v := v₀) (w := W))
+  intro x
+  refine InfinitePlace.Completion.induction_on v₀ x
+    (isClosed_eq
+      (continuous_norm.comp NumberField.LiesOver.continuous_completionMap)
+      continuous_norm) ?_
+  intro a
+  rw [NumberField.LiesOver.completionMap_coe]
+  have h := (InfinitePlace.LiesOver.isometry_algebraMap W v₀).norm_map_of_map_zero
+      (map_zero (algebraMap (WithAbs v₀.1) (WithAbs W.1))) a
+  simp only [InfinitePlace.Completion.norm_coe, WithAbs.norm_eq_apply_ofAbs,
+    WithAbs.equiv_apply, InfinitePlace.coe_apply] at h ⊢
+  with_reducible exact h
 
 omit [NumberField L] in
 /-- Mapping a unit along an infinite-place completion map preserves its
diff --git a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/Ideal.lean b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/Ideal.lean
index d626883..6255279 100644
--- a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/Ideal.lean
+++ b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/Ideal.lean
@@ -4,23 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.AlgebraicNumberTheory.RayClass.Topology
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.LocallyCompact
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.PrincipalNorm
-import ValuedFieldTheory.Valuation.AbsoluteValue.Theory.AbsoluteValues
-import Mathlib.NumberTheory.NumberField.Completion.InfinitePlace
-import Mathlib.Topology.Algebra.IsOpenUnits
-import Mathlib.Topology.Algebra.Ring.Compact
 
-set_option autoImplicit false
+import ClassFieldTheory.AlgebraicNumberTheory.RayClass.IdealApproximation
 
-/-!
-# Ideals prime to a ray-class modulus
 
-This file defines the subgroup of fractional ideals prime to a modulus,
-connects it with the corresponding finite-idele higher-unit conditions, and
-develops the approximation maps used in ray-class ideal constructions.
--/
+set_option autoImplicit false
+
 
 open scoped NumberField WithZero Classical
 open NumberField IsDedekindDomain
@@ -32,690 +21,6 @@ variable {K : Type*} [Field K] [NumberField K]
 
 namespace RayClass
 
-/-- Fractional ideals having valuation zero at every finite prime in the
-support of the modulus. -/
-def primeToModulusIdeals (m : Modulus K) :
-    Subgroup (FractionalIdealGroup K) where
-  carrier := {I | ∀ v, v ∈ m.finitePart.support →
-    FractionalIdeal.count K v
-      (I : FractionalIdeal (nonZeroDivisors (𝓞 K)) K) = 0}
-  one_mem' v _ := FractionalIdeal.count_one K v
-  mul_mem' {I J} hI hJ v hv := by
-    rw [Units.val_mul,
-      FractionalIdeal.count_mul K v (Units.ne_zero I) (Units.ne_zero J),
-      hI v hv, hJ v hv, add_zero]
-  inv_mem' {I} hI v hv := by
-    rw [Units.val_inv_eq_inv_val, FractionalIdeal.count_inv K v,
-      hI v hv, neg_zero]
-
-@[simp]
-theorem mem_primeToModulusIdeals_iff
-    (m : Modulus K) (I : FractionalIdealGroup K) :
-    I ∈ primeToModulusIdeals m ↔
-      ∀ v, v ∈ m.finitePart.support →
-        FractionalIdeal.count K v
-          (I : FractionalIdeal (nonZeroDivisors (𝓞 K)) K) = 0 :=
-  Iff.rfl
-
-/-- A finite prime outside the support of `m`, regarded as an element of
-the group of fractional ideals prime to `m`. -/
-def primeToModulusIdeal
-    (m : Modulus K)
-    (v : HeightOneSpectrum (𝓞 K))
-    (hv : v ∉ m.finitePart.support) :
-    primeToModulusIdeals m :=
-  ⟨FractionalIdealGroup.prime v, by
-    intro w hw
-    have hwv : w ≠ v := by
-      intro h
-      exact hv (h ▸ hw)
-    change
-      FractionalIdeal.count K w
-          (v.asIdeal :
-            FractionalIdeal (nonZeroDivisors (𝓞 K)) K) =
-        0
-    exact
-      FractionalIdeal.count_maximal_coprime
-        K w hwv.symm⟩
-
-/-- Coercing a prime outside the modulus support recovers its prime
-fractional ideal. -/
-@[simp]
-theorem primeToModulusIdeal_coe
-    (m : Modulus K)
-    (v : HeightOneSpectrum (𝓞 K))
-    (hv : v ∉ m.finitePart.support) :
-    (primeToModulusIdeal m v hv :
-        FractionalIdealGroup K) =
-      FractionalIdealGroup.prime v :=
-  rfl
-
-/-- Finite ideles satisfying the higher-unit condition at every prime in
-the support of the modulus. -/
-def finitePrimeToModulusSubgroup (m : Modulus K) :
-    Subgroup (FiniteIdeleGroup K) where
-  carrier := {a | ∀ v, v ∈ m.finitePart.support →
-    a v ∈ localHigherUnitGroup v (m.finitePart v)}
-  one_mem' v _ := (localHigherUnitGroup v (m.finitePart v)).one_mem
-  mul_mem' ha hb v hv :=
-    (localHigherUnitGroup v (m.finitePart v)).mul_mem (ha v hv) (hb v hv)
-  inv_mem' ha v hv :=
-    (localHigherUnitGroup v (m.finitePart v)).inv_mem (ha v hv)
-
-/-- Ideles satisfying the infinite positivity and finite higher-unit
-conditions of a modulus. -/
-def idelePrimeToModulusSubgroup (m : Modulus K) :
-    Subgroup (IdeleGroup K) :=
-  m.infiniteCongruenceSubgroup.prod
-    (finitePrimeToModulusSubgroup m)
-
-theorem localHigherUnitGroup_le_integralUnits
-    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ) :
-    localHigherUnitGroup v n ≤
-      (v.adicCompletionIntegers K).units := by
-  intro x hx
-  rw [mem_localHigherUnitGroup_iff] at hx
-  obtain ⟨y, rfl, _⟩ := hx
-  exact y.property
-
-theorem fractionalIdeal_mem_primeToModulusIdeals
-    (m : Modulus K) (a : IdeleGroup K)
-    (ha : a ∈ idelePrimeToModulusSubgroup m) :
-    IdeleGroup.fractionalIdeal a ∈ primeToModulusIdeals m := by
-  intro v hv
-  change FractionalIdeal.count K v
-      (((FractionalIdealGroup.factorization (K := K))
-        (FiniteIdeleGroup.valuationVector a.2) :
-          FractionalIdealGroup K) :
-        FractionalIdeal (nonZeroDivisors (𝓞 K)) K) = 0
-  rw [FractionalIdealGroup.count_factorization,
-    FiniteIdeleGroup.valuationVector_apply]
-  apply (FiniteIdeleGroup.localOrder_eq_zero_iff v (a.2 v)).2
-  exact localHigherUnitGroup_le_integralUnits v (m.finitePart v) (ha.2 v hv)
-
-/-- The fractional-ideal map restricted to ideles prime to a modulus. -/
-def primeToIdealMap (m : Modulus K) :
-    idelePrimeToModulusSubgroup m →*
-      primeToModulusIdeals m where
-  toFun a :=
-    ⟨IdeleGroup.fractionalIdeal a,
-      fractionalIdeal_mem_primeToModulusIdeals m a a.property⟩
-  map_one' := by
-    apply Subtype.ext
-    exact map_one _
-  map_mul' a b := by
-    apply Subtype.ext
-    exact map_mul _ _ _
-
-/-- A finite idele with a prescribed valuation vector away from the
-support of a modulus and value one on its support. -/
-def valuationVectorSectionPrimeTo
-    (m : Modulus K)
-  (e : HeightOneSpectrum (𝓞 K) →₀ ℤ) :
-    FiniteIdeleGroup K :=
-  ⟨fun v =>
-      if v ∈ m.finitePart.support then 1
-      else FiniteIdeleGroup.chosenLocalOrderSection v (e v), by
-    filter_upwards
-      [m.finitePart.support.eventually_cofinite_notMem,
-        e.support.eventually_cofinite_notMem] with v hvm he
-    simp only [hvm, ↓reduceIte]
-    apply (FiniteIdeleGroup.localOrder_eq_zero_iff v _).1
-    rw [FiniteIdeleGroup.localOrder_chosenLocalOrderSection,
-      Finsupp.notMem_support_iff.mp he]⟩
-
-theorem valuationVector_valuationVectorSectionPrimeTo
-    (m : Modulus K)
-    (e : HeightOneSpectrum (𝓞 K) →₀ ℤ)
-    (he : ∀ v, v ∈ m.finitePart.support → e v = 0) :
-    FiniteIdeleGroup.valuationVector
-        (valuationVectorSectionPrimeTo m e) =
-      Multiplicative.ofAdd e := by
-  apply Multiplicative.ext
-  ext v
-  rw [FiniteIdeleGroup.valuationVector_apply]
-  by_cases hv : v ∈ m.finitePart.support
-  · change
-      (FiniteIdeleGroup.localOrder v
-        (if v ∈ m.finitePart.support then 1
-          else FiniteIdeleGroup.chosenLocalOrderSection v (e v))).toAdd =
-        e v
-    rw [ite_eq_left hv, map_one]
-    exact (he v hv).symm
-  · change
-      (FiniteIdeleGroup.localOrder v
-        (if v ∈ m.finitePart.support then 1
-          else FiniteIdeleGroup.chosenLocalOrderSection v (e v))).toAdd =
-        e v
-    rw [ite_eq_right hv,
-      FiniteIdeleGroup.localOrder_chosenLocalOrderSection]
-
-theorem primeToIdealMap_surjective (m : Modulus K) :
-    Function.Surjective (primeToIdealMap m) := by
-  intro I
-  let e : HeightOneSpectrum (𝓞 K) →₀ ℤ :=
-    FractionalIdealGroup.countVector (I : FractionalIdealGroup K)
-  have he : ∀ v, v ∈ m.finitePart.support → e v = 0 := by
-    intro v hv
-    exact I.property v hv
-  let a : IdeleGroup K :=
-    (1, valuationVectorSectionPrimeTo m e)
-  have ha : a ∈ idelePrimeToModulusSubgroup m := by
-    constructor
-    · exact m.infiniteCongruenceSubgroup.one_mem
-    · intro v hv
-      change
-        (if v ∈ m.finitePart.support then 1
-          else FiniteIdeleGroup.chosenLocalOrderSection v (e v)) ∈
-            localHigherUnitGroup v (m.finitePart v)
-      rw [ite_eq_left hv]
-      exact (localHigherUnitGroup v (m.finitePart v)).one_mem
-  refine ⟨⟨a, ha⟩, ?_⟩
-  apply Subtype.ext
-  apply FractionalIdealGroup.ext_count
-  intro v
-  change FractionalIdeal.count K v
-      (((FractionalIdealGroup.factorization (K := K))
-        (FiniteIdeleGroup.valuationVector
-          (valuationVectorSectionPrimeTo m e)) :
-          FractionalIdealGroup K) :
-        FractionalIdeal (nonZeroDivisors (𝓞 K)) K) =
-      FractionalIdeal.count K v
-        ((I : FractionalIdealGroup K) :
-          FractionalIdeal (nonZeroDivisors (𝓞 K)) K)
-  rw [valuationVector_valuationVectorSectionPrimeTo m e he,
-    FractionalIdealGroup.count_factorization]
-  exact FractionalIdealGroup.countVector_apply I v
-
-theorem ideleCongruenceSubgroup_le_primeTo
-    (m : Modulus K) :
-    m.ideleCongruenceSubgroup ≤
-      idelePrimeToModulusSubgroup m := by
-  intro a ha
-  exact ⟨ha.1, fun v _ => ha.2 v⟩
-
-/-- The congruence subgroup, viewed inside the subgroup of ideles prime
-to the modulus. -/
-def congruenceSubgroupInPrimeTo (m : Modulus K) :
-    Subgroup (idelePrimeToModulusSubgroup m) :=
-  m.ideleCongruenceSubgroup.subgroupOf
-    (idelePrimeToModulusSubgroup m)
-
-theorem primeToIdealMap_ker (m : Modulus K) :
-    (primeToIdealMap m).ker =
-      congruenceSubgroupInPrimeTo m := by
-  ext a
-  constructor
-  · intro ha
-    have hintegral :
-        (a : IdeleGroup K) ∈
-          IdeleGroup.integralAtFinitePlaces (K := K) := by
-      rw [← IdeleGroup.fractionalIdeal_ker,
-        MonoidHom.mem_ker]
-      exact congrArg Subtype.val
-        (MonoidHom.mem_ker.mp ha)
-    constructor
-    · exact a.property.1
-    · intro v
-      by_cases hv : v ∈ m.finitePart.support
-      · exact a.property.2 v hv
-      · rw [Finsupp.notMem_support_iff.mp hv,
-          localHigherUnitGroup_zero]
-        exact hintegral v
-  · intro ha
-    apply MonoidHom.mem_ker.mpr
-    apply Subtype.ext
-    change IdeleGroup.fractionalIdeal (a : IdeleGroup K) = 1
-    rw [← MonoidHom.mem_ker,
-      IdeleGroup.fractionalIdeal_ker]
-    intro v
-    exact localHigherUnitGroup_le_integralUnits v (m.finitePart v) (ha.2 v)
-
-/-- The quotient of ideles prime to a modulus by the congruence subgroup,
-identified with fractional ideals prime to the modulus. -/
-def quotientCongruenceEquivPrimeToIdeals (m : Modulus K) :
-    idelePrimeToModulusSubgroup m ⧸
-        congruenceSubgroupInPrimeTo m ≃*
-      primeToModulusIdeals m := by
-  rw [← primeToIdealMap_ker m]
-  exact QuotientGroup.quotientKerEquivOfSurjective
-    (primeToIdealMap m) (primeToIdealMap_surjective m)
-
-/-- Principal ideles satisfying the modulus conditions, considered inside
-`I_K^(m)`. -/
-def principalSubgroupInPrimeTo (m : Modulus K) :
-    Subgroup (idelePrimeToModulusSubgroup m) :=
-  Subgroup.comap (idelePrimeToModulusSubgroup m).subtype
-    (IdeleGroup.principalSubgroup K)
-
-/-- Principal ideals generated by a totally positive element congruent to
-one modulo the finite modulus. -/
-def principalRayIdealSubgroup (m : Modulus K) :
-    Subgroup (primeToModulusIdeals m) :=
-  Subgroup.map (primeToIdealMap m)
-    (principalSubgroupInPrimeTo m)
-
-theorem mem_principalRayIdealSubgroup_iff
-    (m : Modulus K) (I : primeToModulusIdeals m) :
-    I ∈ principalRayIdealSubgroup m ↔
-      ∃ x : Kˣ,
-        ∃ _hx : IdeleGroup.principalIdele K x ∈
-          idelePrimeToModulusSubgroup m,
-        toPrincipalIdeal (𝓞 K) K x =
-          (I : FractionalIdealGroup K) := by
-  constructor
-  · rintro ⟨a, ha, hmap⟩
-    obtain ⟨x, hx⟩ := ha
-    refine ⟨x, ?_, ?_⟩
-    · rw [hx]
-      exact a.property
-    · have hval := congrArg Subtype.val hmap
-      change IdeleGroup.fractionalIdeal (a : IdeleGroup K) =
-        (I : FractionalIdealGroup K) at hval
-      rw [← IdeleGroup.fractionalIdeal_principalIdele]
-      exact (congrArg (IdeleGroup.fractionalIdeal (K := K)) hx).trans hval
-  · rintro ⟨x, hx, hideal⟩
-    let a : idelePrimeToModulusSubgroup m :=
-      ⟨IdeleGroup.principalIdele K x, hx⟩
-    have ha : a ∈ principalSubgroupInPrimeTo m := by
-      change IdeleGroup.principalIdele K x ∈
-        IdeleGroup.principalSubgroup K
-      exact ⟨x, rfl⟩
-    refine ⟨a, ha, ?_⟩
-    apply Subtype.ext
-    change IdeleGroup.fractionalIdeal
-        (IdeleGroup.principalIdele K x) =
-      (I : FractionalIdealGroup K)
-    rw [IdeleGroup.fractionalIdeal_principalIdele, hideal]
-
-/-- The ideal-theoretic ray class group `J_K^m / P_K^m`. -/
-abbrev IdealRayClassGroup (m : Modulus K) :=
-  primeToModulusIdeals m ⧸ principalRayIdealSubgroup m
-
-/-- The canonical projection from ideles prime to the modulus to the
-ideal-theoretic ray class group. -/
-def idealRayProjection (m : Modulus K) :
-    idelePrimeToModulusSubgroup m →*
-      IdealRayClassGroup m :=
-  (QuotientGroup.mk' (principalRayIdealSubgroup m)).comp
-    (primeToIdealMap m)
-
-/-- The subgroup generated by congruence ideles and principal ideles
-inside the ideles prime to a modulus. -/
-def raySubgroupInPrimeTo (m : Modulus K) :
-    Subgroup (idelePrimeToModulusSubgroup m) :=
-  congruenceSubgroupInPrimeTo m ⊔
-    principalSubgroupInPrimeTo m
-
-theorem idealRayProjection_surjective (m : Modulus K) :
-    Function.Surjective (idealRayProjection m) := by
-  intro c
-  obtain ⟨I, rfl⟩ :=
-    QuotientGroup.mk'_surjective
-      (principalRayIdealSubgroup m) c
-  obtain ⟨a, rfl⟩ := primeToIdealMap_surjective m I
-  exact ⟨a, rfl⟩
-
-theorem idealRayProjection_ker (m : Modulus K) :
-    (idealRayProjection m).ker =
-      raySubgroupInPrimeTo m := by
-  ext a
-  constructor
-  · intro ha
-    change QuotientGroup.mk'
-        (principalRayIdealSubgroup m)
-        (primeToIdealMap m a) = 1 at ha
-    rw [QuotientGroup.mk'_apply,
-      QuotientGroup.eq_one_iff] at ha
-    obtain ⟨p, hp, hpa⟩ := ha
-    let n : idelePrimeToModulusSubgroup m := a * p⁻¹
-    have hn : n ∈ congruenceSubgroupInPrimeTo m := by
-      rw [← primeToIdealMap_ker m, MonoidHom.mem_ker]
-      change primeToIdealMap m (a * p⁻¹) = 1
-      rw [map_mul, map_inv, hpa]
-      simp
-    rw [raySubgroupInPrimeTo, Subgroup.mem_sup]
-    refine ⟨n, hn, p, hp, ?_⟩
-    dsimp [n]
-    group
-  · intro ha
-    rw [raySubgroupInPrimeTo, Subgroup.mem_sup] at ha
-    obtain ⟨n, hn, p, hp, rfl⟩ := ha
-    change QuotientGroup.mk'
-        (principalRayIdealSubgroup m)
-        (primeToIdealMap m (n * p)) = 1
-    rw [map_mul]
-    have hn' : primeToIdealMap m n = 1 :=
-      MonoidHom.mem_ker.mp
-        ((primeToIdealMap_ker m).symm ▸ hn)
-    rw [hn', one_mul]
-    rw [QuotientGroup.mk'_apply,
-      QuotientGroup.eq_one_iff]
-    exact ⟨p, hp, rfl⟩
-
-/-- The quotient of ideles prime to the modulus by the full ray subgroup,
-identified with the ideal-theoretic ray class group. -/
-def quotientRaySubgroupEquivIdealRayClassGroup
-    (m : Modulus K) :
-    idelePrimeToModulusSubgroup m ⧸
-        raySubgroupInPrimeTo m ≃*
-      IdealRayClassGroup m :=
-  (QuotientGroup.quotientMulEquivOfEq
-      (idealRayProjection_ker m).symm).trans
-    (QuotientGroup.quotientKerEquivOfSurjective
-      (idealRayProjection m)
-      (idealRayProjection_surjective m))
-
-/-- The quotient equivalence induced by the ideal-ray projection evaluates on
-the class of a prime-to-modulus idele as the original projection. -/
-@[simp]
-theorem quotientRaySubgroupEquivIdealRayClassGroup_mk
-    (m : Modulus K) (a : idelePrimeToModulusSubgroup m) :
-    quotientRaySubgroupEquivIdealRayClassGroup m
-        (QuotientGroup.mk' (raySubgroupInPrimeTo m) a) =
-      idealRayProjection m a := by
-  rw [quotientRaySubgroupEquivIdealRayClassGroup,
-    MulEquiv.trans_apply, QuotientGroup.mk'_apply,
-    QuotientGroup.quotientMulEquivOfEq_mk]
-  exact QuotientGroup.kerLift_mk (idealRayProjection m) a
-
-/-! ### Simultaneous approximation at the places in a modulus -/
-
-/-- The finite primes in `m`, together with all infinite places. -/
-abbrev ApproximationPlace (m : Modulus K) :=
-  (↥m.finitePart.support) ⊕ InfinitePlace K
-
-/-- The absolute value represented by an approximation place. -/
-abbrev approximationAbsoluteValue (m : Modulus K) :
-    ApproximationPlace m → AbsoluteValue K ℝ
-  | Sum.inl v => NumberField.HeightOneSpectrum.adicAbv K v.1
-  | Sum.inr w => w.1
-
-theorem adicAbv_isNontrivial
-    (v : HeightOneSpectrum (𝓞 K)) :
-    (NumberField.HeightOneSpectrum.adicAbv K v).IsNontrivial := by
-  obtain ⟨x, hxv, hx0⟩ :=
-    Submodule.exists_mem_ne_zero_of_ne_bot v.ne_bot
-  refine ⟨algebraMap (𝓞 K) K x, ?_, ?_⟩
-  · exact (FaithfulSMul.algebraMap_eq_zero_iff (𝓞 K) K).not.mpr hx0
-  · apply ne_of_lt
-    rw [← FinitePlace.norm_embedding]
-    exact (FinitePlace.norm_lt_one_iff_mem (K := K) v x).2 hxv
-
-theorem adicAbv_not_isEquiv_of_ne
-    {v w : HeightOneSpectrum (𝓞 K)} (hvw : v ≠ w) :
-    ¬ (NumberField.HeightOneSpectrum.adicAbv K v).IsEquiv
-      (NumberField.HeightOneSpectrum.adicAbv K w) := by
-  intro h
-  have hnotle : ¬ v.asIdeal ≤ w.asIdeal := by
-    intro hvw_le
-    have htop_le : (⊤ : Ideal (𝓞 K)) ≤ w.asIdeal := by
-      rw [← (v.isCoprime_of_ne w hvw).sup_eq]
-      exact sup_le hvw_le le_rfl
-    exact w.isPrime.ne_top (top_unique htop_le)
-  obtain ⟨x, hxv, hxw⟩ := Set.not_subset.mp hnotle
-  have hvlt :
-      NumberField.HeightOneSpectrum.adicAbv K v
-        (algebraMap (𝓞 K) K x) < 1 := by
-    rw [← FinitePlace.norm_embedding]
-    exact (FinitePlace.norm_lt_one_iff_mem (K := K) v x).2 hxv
-  have hweq :
-      NumberField.HeightOneSpectrum.adicAbv K w
-        (algebraMap (𝓞 K) K x) = 1 := by
-    rw [← FinitePlace.norm_embedding]
-    exact (FinitePlace.norm_eq_one_iff_notMem (K := K) w x).2 hxw
-  exact (ne_of_lt hvlt) (h.eq_one_iff.mpr hweq)
-
-theorem adicAbv_not_isEquiv_infinitePlace
-    (v : HeightOneSpectrum (𝓞 K)) (w : InfinitePlace K) :
-    ¬ (NumberField.HeightOneSpectrum.adicAbv K v).IsEquiv w.1 := by
-  intro h
-  have hle :
-      w.1 ((2 : ℕ) : K) ≤ 1 :=
-    h.le_one_iff.mp
-      (NumberField.HeightOneSpectrum.adicAbv_natCast_le_one K v 2)
-  have hw :
-      w.1 ((2 : ℕ) : K) = (2 : ℝ) :=
-    NumberField.InfinitePlace.map_natCast w 2
-  have hfalse : (2 : ℝ) ≤ 1 := hw ▸ hle
-  norm_num at hfalse
-
-theorem approximationAbsoluteValue_isNontrivial
-    (m : Modulus K) :
-    ∀ i, (approximationAbsoluteValue m i).IsNontrivial
-  | Sum.inl v => by
-      change
-        (NumberField.HeightOneSpectrum.adicAbv K v.1).IsNontrivial
-      exact adicAbv_isNontrivial v.1
-  | Sum.inr w => by
-      change w.1.IsNontrivial
-      exact w.isNontrivial
-
-theorem approximationAbsoluteValue_pairwise
-    (m : Modulus K) :
-    Pairwise fun i j =>
-      ¬ (approximationAbsoluteValue m i).IsEquiv
-        (approximationAbsoluteValue m j) := by
-  intro i j hij
-  cases i with
-  | inl v =>
-      cases j with
-      | inl w =>
-          apply adicAbv_not_isEquiv_of_ne
-          intro hvw
-          apply hij
-          exact congrArg Sum.inl (Subtype.ext hvw)
-      | inr w =>
-          exact adicAbv_not_isEquiv_infinitePlace v.1 w
-  | inr v =>
-      cases j with
-      | inl w =>
-          exact fun h =>
-            adicAbv_not_isEquiv_infinitePlace w.1 v h.symm
-      | inr w =>
-          intro h
-          apply hij
-          congr
-          change v.1.IsEquiv w.1 at h
-          exact
-            (InfinitePlace.eq_iff_isEquiv (K := K)).mpr h
-
-/-- The corresponding product of local completions. -/
-abbrev approximationCompletion (m : Modulus K) :
-    ApproximationPlace m → Type _
-  | Sum.inl v => v.1.adicCompletion K
-  | Sum.inr w => w.Completion
-
-noncomputable instance approximationCompletionTopologicalSpace
-    (m : Modulus K) (i : ApproximationPlace m) :
-    TopologicalSpace (approximationCompletion m i) := by
-  cases i <;> simp only [approximationCompletion] <;> infer_instance
-
-/-- Coordinatewise completion of the valued copies of `K`. -/
-def approximationCompletionMap (m : Modulus K) :
-    ∀ i : ApproximationPlace m,
-      WithAbs (approximationAbsoluteValue m i) →
-        approximationCompletion m i
-  | Sum.inl v =>
-      fun x =>
-        FinitePlace.embedding v.1
-          (WithAbs.equiv
-            (NumberField.HeightOneSpectrum.adicAbv K v.1) x)
-  | Sum.inr w => fun x => (x : w.Completion)
-
-theorem denseRange_finiteApproximationCompletionMap
-    (v : HeightOneSpectrum (𝓞 K)) :
-    DenseRange
-      (fun x :
-          WithAbs (NumberField.HeightOneSpectrum.adicAbv K v) =>
-        FinitePlace.embedding v
-          (WithAbs.equiv
-            (NumberField.HeightOneSpectrum.adicAbv K v) x)) := by
-  have hrange :
-      Set.range
-          (fun x :
-              WithAbs (NumberField.HeightOneSpectrum.adicAbv K v) =>
-            FinitePlace.embedding v
-              (WithAbs.equiv
-                (NumberField.HeightOneSpectrum.adicAbv K v) x)) =
-        Set.range (algebraMap K (v.adicCompletion K)) := by
-    ext y
-    constructor
-    · rintro ⟨x, rfl⟩
-      exact
-        ⟨WithAbs.equiv
-            (NumberField.HeightOneSpectrum.adicAbv K v) x, rfl⟩
-    · rintro ⟨x, rfl⟩
-      refine
-        ⟨(WithAbs.equiv
-            (NumberField.HeightOneSpectrum.adicAbv K v)).symm x, ?_⟩
-      rfl
-  rw [DenseRange, hrange]
-  exact v.denseRange_algebraMap K
-
-theorem continuous_finiteApproximationCompletionMap
-    (v : HeightOneSpectrum (𝓞 K)) :
-    Continuous
-      (fun x :
-          WithAbs (NumberField.HeightOneSpectrum.adicAbv K v) =>
-        FinitePlace.embedding v
-          (WithAbs.equiv
-            (NumberField.HeightOneSpectrum.adicAbv K v) x)) := by
-  apply Isometry.continuous
-  apply Isometry.of_dist_eq
-  intro x y
-  rw [dist_eq_norm, dist_eq_norm, ← map_sub,
-    FinitePlace.norm_embedding]
-  rfl
-
-theorem denseRange_approximationCompletionMap
-    (m : Modulus K) :
-    ∀ i, DenseRange (approximationCompletionMap m i)
-  | Sum.inl v => denseRange_finiteApproximationCompletionMap v.1
-  | Sum.inr w =>
-      NumberField.InfinitePlace.Completion.denseRange_coe w
-
-theorem continuous_approximationCompletionMap
-    (m : Modulus K) :
-    ∀ i, Continuous (approximationCompletionMap m i)
-  | Sum.inl v => continuous_finiteApproximationCompletionMap v.1
-  | Sum.inr w =>
-      NumberField.InfinitePlace.Completion.continuous_coe w
-
-/-- The diagonal embedding into the finite product of the relevant
-completions. -/
-def approximationEmbedding (m : Modulus K) :
-    K → (i : ApproximationPlace m) → approximationCompletion m i :=
-  (Pi.map (approximationCompletionMap m)) ∘
-    algebraMap K
-      ((i : ApproximationPlace m) →
-        WithAbs (approximationAbsoluteValue m i))
-
-@[simp]
-theorem approximationEmbedding_finite
-    (m : Modulus K) (x : K) (v : ↥m.finitePart.support) :
-    approximationEmbedding m x (Sum.inl v) =
-      FinitePlace.embedding v.1 x :=
-  rfl
-
-@[simp]
-theorem approximationEmbedding_infinite
-    (m : Modulus K) (x : K) (w : InfinitePlace K) :
-    approximationEmbedding m x (Sum.inr w) =
-      (x : w.Completion) :=
-  rfl
-
-theorem denseRange_approximationEmbedding (m : Modulus K) :
-    DenseRange (approximationEmbedding m) := by
-  exact
-    (DenseRange.piMap
-      (denseRange_approximationCompletionMap m)).comp
-      (AbsoluteValue.denseRange_algebraMap_pi
-        (approximationAbsoluteValue_isNontrivial m)
-        (approximationAbsoluteValue_pairwise m))
-      (.piMap (continuous_approximationCompletionMap m))
-
-/-- The open set of field elements whose ratio with a fixed unit lies in
-a prescribed open unit set. -/
-def unitRatioSet
-    {F : Type*} [Field F] (a : Fˣ) (U : Subgroup Fˣ) :
-    Set F :=
-  Units.val '' (fun y : Fˣ => a * y⁻¹) ⁻¹' (U : Set Fˣ)
-
-/-- The unit-ratio set associated to an open set of units is open. -/
-theorem isOpen_unitRatioSet
-    {F : Type*} [Field F] [TopologicalSpace F]
-    [IsTopologicalRing F] [ContinuousInv₀ F] [T1Space F]
-    (a : Fˣ) (U : Subgroup Fˣ)
-    (hU : IsOpen (U : Set Fˣ)) :
-    IsOpen (unitRatioSet a U) := by
-  apply IsOpenUnits.isOpenEmbedding_unitsVal.isOpenMap
-  exact hU.preimage (continuous_const.mul continuous_inv)
-
-/-- The value of the distinguished unit belongs to its unit-ratio set. -/
-theorem val_mem_unitRatioSet
-    {F : Type*} [Field F] (a : Fˣ) (U : Subgroup Fˣ) :
-    (a : F) ∈ unitRatioSet a U := by
-  exact ⟨a, by simp, rfl⟩
-
-/-- The open local conditions that make `a / x` prime to `m`. -/
-def approximationTarget (m : Modulus K) (a : IdeleGroup K) :
-    ∀ i : ApproximationPlace m, Set (approximationCompletion m i)
-  | Sum.inl v =>
-      unitRatioSet (a.2 v.1)
-        (localHigherUnitGroup v.1 (m.finitePart v.1))
-  | Sum.inr w =>
-      unitRatioSet
-        (ContinuousMulEquiv.piUnits a.1 w)
-        (m.localInfiniteCongruenceSubgroup w)
-
-theorem isOpen_approximationTarget
-    (m : Modulus K) (a : IdeleGroup K) :
-    ∀ i, IsOpen (approximationTarget m a i)
-  | Sum.inl v => by
-      change IsOpen
-        (unitRatioSet (a.2 v.1)
-          (localHigherUnitGroup v.1 (m.finitePart v.1)))
-      exact isOpen_unitRatioSet _ _
-        (isOpen_localHigherUnitGroup v.1 (m.finitePart v.1))
-  | Sum.inr w => by
-      change IsOpen
-        (unitRatioSet
-          (ContinuousMulEquiv.piUnits a.1 w)
-          (m.localInfiniteCongruenceSubgroup w))
-      apply isOpen_unitRatioSet _ _
-      classical
-      by_cases hw : w.IsReal
-      · by_cases hmem : (⟨w, hw⟩ : RealPlace K) ∈ m.infinitePart
-        · rw [Modulus.localInfiniteCongruenceSubgroup,
-            dite_eq_left hw, dite_eq_left hmem]
-          exact isOpen_infinitePositiveSubgroup w
-        · rw [Modulus.localInfiniteCongruenceSubgroup,
-            dite_eq_left hw, dite_eq_right hmem]
-          exact isOpen_univ
-      · rw [Modulus.localInfiniteCongruenceSubgroup, dite_eq_right hw]
-        exact isOpen_univ
-
-/-- The given idele itself lies in the product of its approximation
-neighborhoods. -/
-def approximationTargetPoint
-    (m : Modulus K) (a : IdeleGroup K) :
-    (i : ApproximationPlace m) → approximationCompletion m i
-  | Sum.inl v => (a.2 v.1 : v.1.adicCompletion K)
-  | Sum.inr w =>
-      (ContinuousMulEquiv.piUnits a.1 w : w.Completion)
-
-theorem approximationTargetPoint_mem
-    (m : Modulus K) (a : IdeleGroup K) :
-    approximationTargetPoint m a ∈
-      Set.univ.pi (approximationTarget m a) := by
-  intro i _hi
-  cases i with
-  | inl v =>
-      exact val_mem_unitRatioSet _ _
-  | inr w =>
-      exact val_mem_unitRatioSet _ _
-
 /-- Weak approximation in the precise open local cosets required by the
 modulus. -/
 theorem exists_principal_quotient_mem_primeTo
diff --git a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/IdealApproximation.lean b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/IdealApproximation.lean
new file mode 100644
index 0000000..168aa31
--- /dev/null
+++ b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/IdealApproximation.lean
@@ -0,0 +1,322 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.AlgebraicNumberTheory.RayClass.IdealGroups
+
+
+set_option autoImplicit false
+
+
+open scoped NumberField WithZero Classical
+open NumberField IsDedekindDomain
+
+noncomputable section
+
+
+variable {K : Type*} [Field K] [NumberField K]
+
+namespace RayClass
+
+/-! ### Simultaneous approximation at the places in a modulus -/
+
+/-- The finite primes in `m`, together with all infinite places. -/
+abbrev ApproximationPlace (m : Modulus K) :=
+  (↥m.finitePart.support) ⊕ InfinitePlace K
+
+/-- The absolute value represented by an approximation place. -/
+abbrev approximationAbsoluteValue (m : Modulus K) :
+    ApproximationPlace m → AbsoluteValue K ℝ
+  | Sum.inl v => NumberField.HeightOneSpectrum.adicAbv K v.1
+  | Sum.inr w => w.1
+
+theorem adicAbv_isNontrivial
+    (v : HeightOneSpectrum (𝓞 K)) :
+    (NumberField.HeightOneSpectrum.adicAbv K v).IsNontrivial := by
+  obtain ⟨x, hxv, hx0⟩ :=
+    Submodule.exists_mem_ne_zero_of_ne_bot v.ne_bot
+  refine ⟨algebraMap (𝓞 K) K x, ?_, ?_⟩
+  · exact (FaithfulSMul.algebraMap_eq_zero_iff (𝓞 K) K).not.mpr hx0
+  · apply ne_of_lt
+    rw [← FinitePlace.norm_embedding]
+    exact (FinitePlace.norm_lt_one_iff_mem (K := K) v x).2 hxv
+
+theorem adicAbv_not_isEquiv_of_ne
+    {v w : HeightOneSpectrum (𝓞 K)} (hvw : v ≠ w) :
+    ¬ (NumberField.HeightOneSpectrum.adicAbv K v).IsEquiv
+      (NumberField.HeightOneSpectrum.adicAbv K w) := by
+  intro h
+  have hnotle : ¬ v.asIdeal ≤ w.asIdeal := by
+    intro hvw_le
+    have htop_le : (⊤ : Ideal (𝓞 K)) ≤ w.asIdeal := by
+      rw [← (v.isCoprime_of_ne w hvw).sup_eq]
+      exact sup_le hvw_le le_rfl
+    exact w.isPrime.ne_top (top_unique htop_le)
+  obtain ⟨x, hxv, hxw⟩ := Set.not_subset.mp hnotle
+  have hvlt :
+      NumberField.HeightOneSpectrum.adicAbv K v
+        (algebraMap (𝓞 K) K x) < 1 := by
+    rw [← FinitePlace.norm_embedding]
+    exact (FinitePlace.norm_lt_one_iff_mem (K := K) v x).2 hxv
+  have hweq :
+      NumberField.HeightOneSpectrum.adicAbv K w
+        (algebraMap (𝓞 K) K x) = 1 := by
+    rw [← FinitePlace.norm_embedding]
+    exact (FinitePlace.norm_eq_one_iff_notMem (K := K) w x).2 hxw
+  exact (ne_of_lt hvlt) (h.eq_one_iff.mpr hweq)
+
+theorem adicAbv_not_isEquiv_infinitePlace
+    (v : HeightOneSpectrum (𝓞 K)) (w : InfinitePlace K) :
+    ¬ (NumberField.HeightOneSpectrum.adicAbv K v).IsEquiv w.1 := by
+  intro h
+  have hle :
+      w.1 ((2 : ℕ) : K) ≤ 1 :=
+    h.le_one_iff.mp
+      (NumberField.HeightOneSpectrum.adicAbv_natCast_le_one K v 2)
+  have hw :
+      w.1 ((2 : ℕ) : K) = (2 : ℝ) :=
+    NumberField.InfinitePlace.map_natCast w 2
+  have hfalse : (2 : ℝ) ≤ 1 := hw ▸ hle
+  norm_num at hfalse
+
+theorem approximationAbsoluteValue_isNontrivial
+    (m : Modulus K) :
+    ∀ i, (approximationAbsoluteValue m i).IsNontrivial
+  | Sum.inl v => by
+      change
+        (NumberField.HeightOneSpectrum.adicAbv K v.1).IsNontrivial
+      exact adicAbv_isNontrivial v.1
+  | Sum.inr w => by
+      change w.1.IsNontrivial
+      exact w.isNontrivial
+
+theorem approximationAbsoluteValue_pairwise
+    (m : Modulus K) :
+    Pairwise fun i j =>
+      ¬ (approximationAbsoluteValue m i).IsEquiv
+        (approximationAbsoluteValue m j) := by
+  intro i j hij
+  cases i with
+  | inl v =>
+      cases j with
+      | inl w =>
+          apply adicAbv_not_isEquiv_of_ne
+          intro hvw
+          apply hij
+          exact congrArg Sum.inl (Subtype.ext hvw)
+      | inr w =>
+          exact adicAbv_not_isEquiv_infinitePlace v.1 w
+  | inr v =>
+      cases j with
+      | inl w =>
+          exact fun h =>
+            adicAbv_not_isEquiv_infinitePlace w.1 v h.symm
+      | inr w =>
+          intro h
+          apply hij
+          congr
+          change v.1.IsEquiv w.1 at h
+          exact
+            (InfinitePlace.eq_iff_isEquiv (K := K)).mpr h
+
+/-- The corresponding product of local completions. -/
+abbrev approximationCompletion (m : Modulus K) :
+    ApproximationPlace m → Type _
+  | Sum.inl v => v.1.adicCompletion K
+  | Sum.inr w => w.Completion
+
+noncomputable instance approximationCompletionTopologicalSpace
+    (m : Modulus K) (i : ApproximationPlace m) :
+    TopologicalSpace (approximationCompletion m i) := by
+  cases i <;> simp only [approximationCompletion] <;> infer_instance
+
+/-- Coordinatewise completion of the valued copies of `K`. -/
+def approximationCompletionMap (m : Modulus K) :
+    ∀ i : ApproximationPlace m,
+      WithAbs (approximationAbsoluteValue m i) →
+        approximationCompletion m i
+  | Sum.inl v =>
+      fun x =>
+        FinitePlace.embedding v.1
+          (WithAbs.equiv
+            (NumberField.HeightOneSpectrum.adicAbv K v.1) x)
+  | Sum.inr w => fun x => (x : w.Completion)
+
+theorem denseRange_finiteApproximationCompletionMap
+    (v : HeightOneSpectrum (𝓞 K)) :
+    DenseRange
+      (fun x :
+          WithAbs (NumberField.HeightOneSpectrum.adicAbv K v) =>
+        FinitePlace.embedding v
+          (WithAbs.equiv
+            (NumberField.HeightOneSpectrum.adicAbv K v) x)) := by
+  have hrange :
+      Set.range
+          (fun x :
+              WithAbs (NumberField.HeightOneSpectrum.adicAbv K v) =>
+            FinitePlace.embedding v
+              (WithAbs.equiv
+                (NumberField.HeightOneSpectrum.adicAbv K v) x)) =
+        Set.range (algebraMap K (v.adicCompletion K)) := by
+    ext y
+    constructor
+    · rintro ⟨x, rfl⟩
+      exact
+        ⟨WithAbs.equiv
+            (NumberField.HeightOneSpectrum.adicAbv K v) x, rfl⟩
+    · rintro ⟨x, rfl⟩
+      refine
+        ⟨(WithAbs.equiv
+            (NumberField.HeightOneSpectrum.adicAbv K v)).symm x, ?_⟩
+      rfl
+  rw [DenseRange, hrange]
+  exact v.denseRange_algebraMap K
+
+theorem continuous_finiteApproximationCompletionMap
+    (v : HeightOneSpectrum (𝓞 K)) :
+    Continuous
+      (fun x :
+          WithAbs (NumberField.HeightOneSpectrum.adicAbv K v) =>
+        FinitePlace.embedding v
+          (WithAbs.equiv
+            (NumberField.HeightOneSpectrum.adicAbv K v) x)) := by
+  apply Isometry.continuous
+  apply Isometry.of_dist_eq
+  intro x y
+  rw [dist_eq_norm, dist_eq_norm, ← map_sub,
+    FinitePlace.norm_embedding]
+  rfl
+
+theorem denseRange_approximationCompletionMap
+    (m : Modulus K) :
+    ∀ i, DenseRange (approximationCompletionMap m i)
+  | Sum.inl v => denseRange_finiteApproximationCompletionMap v.1
+  | Sum.inr w =>
+      NumberField.InfinitePlace.Completion.denseRange_coe w
+
+theorem continuous_approximationCompletionMap
+    (m : Modulus K) :
+    ∀ i, Continuous (approximationCompletionMap m i)
+  | Sum.inl v => continuous_finiteApproximationCompletionMap v.1
+  | Sum.inr w =>
+      NumberField.InfinitePlace.Completion.continuous_coe w
+
+/-- The diagonal embedding into the finite product of the relevant
+completions. -/
+def approximationEmbedding (m : Modulus K) :
+    K → (i : ApproximationPlace m) → approximationCompletion m i :=
+  (Pi.map (approximationCompletionMap m)) ∘
+    algebraMap K
+      ((i : ApproximationPlace m) →
+        WithAbs (approximationAbsoluteValue m i))
+
+@[simp]
+theorem approximationEmbedding_finite
+    (m : Modulus K) (x : K) (v : ↥m.finitePart.support) :
+    approximationEmbedding m x (Sum.inl v) =
+      FinitePlace.embedding v.1 x :=
+  rfl
+
+@[simp]
+theorem approximationEmbedding_infinite
+    (m : Modulus K) (x : K) (w : InfinitePlace K) :
+    approximationEmbedding m x (Sum.inr w) =
+      (x : w.Completion) :=
+  rfl
+
+theorem denseRange_approximationEmbedding (m : Modulus K) :
+    DenseRange (approximationEmbedding m) := by
+  exact
+    (DenseRange.piMap
+      (denseRange_approximationCompletionMap m)).comp
+      (AbsoluteValue.denseRange_algebraMap_pi
+        (approximationAbsoluteValue_isNontrivial m)
+        (approximationAbsoluteValue_pairwise m))
+      (.piMap (continuous_approximationCompletionMap m))
+
+/-- The open set of field elements whose ratio with a fixed unit lies in
+a prescribed open unit set. -/
+def unitRatioSet
+    {F : Type*} [Field F] (a : Fˣ) (U : Subgroup Fˣ) :
+    Set F :=
+  Units.val '' (fun y : Fˣ => a * y⁻¹) ⁻¹' (U : Set Fˣ)
+
+/-- The unit-ratio set associated to an open set of units is open. -/
+theorem isOpen_unitRatioSet
+    {F : Type*} [Field F] [TopologicalSpace F]
+    [IsTopologicalRing F] [ContinuousInv₀ F] [T1Space F]
+    (a : Fˣ) (U : Subgroup Fˣ)
+    (hU : IsOpen (U : Set Fˣ)) :
+    IsOpen (unitRatioSet a U) := by
+  apply IsOpenUnits.isOpenEmbedding_unitsVal.isOpenMap
+  exact hU.preimage (continuous_const.mul continuous_inv)
+
+/-- The value of the distinguished unit belongs to its unit-ratio set. -/
+theorem val_mem_unitRatioSet
+    {F : Type*} [Field F] (a : Fˣ) (U : Subgroup Fˣ) :
+    (a : F) ∈ unitRatioSet a U := by
+  exact ⟨a, by simp, rfl⟩
+
+/-- The open local conditions that make `a / x` prime to `m`. -/
+def approximationTarget (m : Modulus K) (a : IdeleGroup K) :
+    ∀ i : ApproximationPlace m, Set (approximationCompletion m i)
+  | Sum.inl v =>
+      unitRatioSet (a.2 v.1)
+        (localHigherUnitGroup v.1 (m.finitePart v.1))
+  | Sum.inr w =>
+      unitRatioSet
+        (ContinuousMulEquiv.piUnits a.1 w)
+        (m.localInfiniteCongruenceSubgroup w)
+
+theorem isOpen_approximationTarget
+    (m : Modulus K) (a : IdeleGroup K) :
+    ∀ i, IsOpen (approximationTarget m a i)
+  | Sum.inl v => by
+      change IsOpen
+        (unitRatioSet (a.2 v.1)
+          (localHigherUnitGroup v.1 (m.finitePart v.1)))
+      exact isOpen_unitRatioSet _ _
+        (isOpen_localHigherUnitGroup v.1 (m.finitePart v.1))
+  | Sum.inr w => by
+      change IsOpen
+        (unitRatioSet
+          (ContinuousMulEquiv.piUnits a.1 w)
+          (m.localInfiniteCongruenceSubgroup w))
+      apply isOpen_unitRatioSet _ _
+      classical
+      by_cases hw : w.IsReal
+      · by_cases hmem : (⟨w, hw⟩ : RealPlace K) ∈ m.infinitePart
+        · rw [Modulus.localInfiniteCongruenceSubgroup,
+            dite_eq_left hw, dite_eq_left hmem]
+          exact isOpen_infinitePositiveSubgroup w
+        · rw [Modulus.localInfiniteCongruenceSubgroup,
+            dite_eq_left hw, dite_eq_right hmem]
+          exact isOpen_univ
+      · rw [Modulus.localInfiniteCongruenceSubgroup, dite_eq_right hw]
+        exact isOpen_univ
+
+/-- The given idele itself lies in the product of its approximation
+neighborhoods. -/
+def approximationTargetPoint
+    (m : Modulus K) (a : IdeleGroup K) :
+    (i : ApproximationPlace m) → approximationCompletion m i
+  | Sum.inl v => (a.2 v.1 : v.1.adicCompletion K)
+  | Sum.inr w =>
+      (ContinuousMulEquiv.piUnits a.1 w : w.Completion)
+
+theorem approximationTargetPoint_mem
+    (m : Modulus K) (a : IdeleGroup K) :
+    approximationTargetPoint m a ∈
+      Set.univ.pi (approximationTarget m a) := by
+  intro i _hi
+  cases i with
+  | inl v =>
+      exact val_mem_unitRatioSet _ _
+  | inr w =>
+      exact val_mem_unitRatioSet _ _
+
+
+end RayClass
diff --git a/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/IdealGroups.lean b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/IdealGroups.lean
new file mode 100644
index 0000000..2b15f82
--- /dev/null
+++ b/Lean4/ClassFieldTheory/AlgebraicNumberTheory/RayClass/IdealGroups.lean
@@ -0,0 +1,417 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.AlgebraicNumberTheory.RayClass.Topology
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.LocallyCompact
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.PrincipalNorm
+import ValuedFieldTheory.Valuation.AbsoluteValue.Theory.AbsoluteValues
+import Mathlib.NumberTheory.NumberField.Completion.InfinitePlace
+import Mathlib.Topology.Algebra.IsOpenUnits
+import Mathlib.Topology.Algebra.Ring.Compact
+
+
+set_option autoImplicit false
+
+
+open scoped NumberField WithZero Classical
+open NumberField IsDedekindDomain
+
+noncomputable section
+
+
+variable {K : Type*} [Field K] [NumberField K]
+
+namespace RayClass
+
+/-- Fractional ideals having valuation zero at every finite prime in the
+support of the modulus. -/
+def primeToModulusIdeals (m : Modulus K) :
+    Subgroup (FractionalIdealGroup K) where
+  carrier := {I | ∀ v, v ∈ m.finitePart.support →
+    FractionalIdeal.count K v
+      (I : FractionalIdeal (nonZeroDivisors (𝓞 K)) K) = 0}
+  one_mem' v _ := FractionalIdeal.count_one K v
+  mul_mem' {I J} hI hJ v hv := by
+    rw [Units.val_mul,
+      FractionalIdeal.count_mul K v (Units.ne_zero I) (Units.ne_zero J),
+      hI v hv, hJ v hv, add_zero]
+  inv_mem' {I} hI v hv := by
+    rw [Units.val_inv_eq_inv_val, FractionalIdeal.count_inv K v,
+      hI v hv, neg_zero]
+
+@[simp]
+theorem mem_primeToModulusIdeals_iff
+    (m : Modulus K) (I : FractionalIdealGroup K) :
+    I ∈ primeToModulusIdeals m ↔
+      ∀ v, v ∈ m.finitePart.support →
+        FractionalIdeal.count K v
+          (I : FractionalIdeal (nonZeroDivisors (𝓞 K)) K) = 0 :=
+  Iff.rfl
+
+/-- A finite prime outside the support of `m`, regarded as an element of
+the group of fractional ideals prime to `m`. -/
+def primeToModulusIdeal
+    (m : Modulus K)
+    (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ m.finitePart.support) :
+    primeToModulusIdeals m :=
+  ⟨FractionalIdealGroup.prime v, by
+    intro w hw
+    have hwv : w ≠ v := by
+      intro h
+      exact hv (h ▸ hw)
+    change
+      FractionalIdeal.count K w
+          (v.asIdeal :
+            FractionalIdeal (nonZeroDivisors (𝓞 K)) K) =
+        0
+    exact
+      FractionalIdeal.count_maximal_coprime
+        K w hwv.symm⟩
+
+/-- Coercing a prime outside the modulus support recovers its prime
+fractional ideal. -/
+@[simp]
+theorem primeToModulusIdeal_coe
+    (m : Modulus K)
+    (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ m.finitePart.support) :
+    (primeToModulusIdeal m v hv :
+        FractionalIdealGroup K) =
+      FractionalIdealGroup.prime v :=
+  rfl
+
+/-- Finite ideles satisfying the higher-unit condition at every prime in
+the support of the modulus. -/
+def finitePrimeToModulusSubgroup (m : Modulus K) :
+    Subgroup (FiniteIdeleGroup K) where
+  carrier := {a | ∀ v, v ∈ m.finitePart.support →
+    a v ∈ localHigherUnitGroup v (m.finitePart v)}
+  one_mem' v _ := (localHigherUnitGroup v (m.finitePart v)).one_mem
+  mul_mem' ha hb v hv :=
+    (localHigherUnitGroup v (m.finitePart v)).mul_mem (ha v hv) (hb v hv)
+  inv_mem' ha v hv :=
+    (localHigherUnitGroup v (m.finitePart v)).inv_mem (ha v hv)
+
+/-- Ideles satisfying the infinite positivity and finite higher-unit
+conditions of a modulus. -/
+def idelePrimeToModulusSubgroup (m : Modulus K) :
+    Subgroup (IdeleGroup K) :=
+  m.infiniteCongruenceSubgroup.prod
+    (finitePrimeToModulusSubgroup m)
+
+theorem localHigherUnitGroup_le_integralUnits
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ) :
+    localHigherUnitGroup v n ≤
+      (v.adicCompletionIntegers K).units := by
+  intro x hx
+  rw [mem_localHigherUnitGroup_iff] at hx
+  obtain ⟨y, rfl, _⟩ := hx
+  exact y.property
+
+theorem fractionalIdeal_mem_primeToModulusIdeals
+    (m : Modulus K) (a : IdeleGroup K)
+    (ha : a ∈ idelePrimeToModulusSubgroup m) :
+    IdeleGroup.fractionalIdeal a ∈ primeToModulusIdeals m := by
+  intro v hv
+  change FractionalIdeal.count K v
+      (((FractionalIdealGroup.factorization (K := K))
+        (FiniteIdeleGroup.valuationVector a.2) :
+          FractionalIdealGroup K) :
+        FractionalIdeal (nonZeroDivisors (𝓞 K)) K) = 0
+  rw [FractionalIdealGroup.count_factorization,
+    FiniteIdeleGroup.valuationVector_apply]
+  apply (FiniteIdeleGroup.localOrder_eq_zero_iff v (a.2 v)).2
+  exact localHigherUnitGroup_le_integralUnits v (m.finitePart v) (ha.2 v hv)
+
+/-- The fractional-ideal map restricted to ideles prime to a modulus. -/
+def primeToIdealMap (m : Modulus K) :
+    idelePrimeToModulusSubgroup m →*
+      primeToModulusIdeals m where
+  toFun a :=
+    ⟨IdeleGroup.fractionalIdeal a,
+      fractionalIdeal_mem_primeToModulusIdeals m a a.property⟩
+  map_one' := by
+    apply Subtype.ext
+    exact map_one _
+  map_mul' a b := by
+    apply Subtype.ext
+    exact map_mul _ _ _
+
+/-- A finite idele with a prescribed valuation vector away from the
+support of a modulus and value one on its support. -/
+def valuationVectorSectionPrimeTo
+    (m : Modulus K)
+  (e : HeightOneSpectrum (𝓞 K) →₀ ℤ) :
+    FiniteIdeleGroup K :=
+  ⟨fun v =>
+      if v ∈ m.finitePart.support then 1
+      else FiniteIdeleGroup.chosenLocalOrderSection v (e v), by
+    filter_upwards
+      [m.finitePart.support.eventually_cofinite_notMem,
+        e.support.eventually_cofinite_notMem] with v hvm he
+    simp only [hvm, ↓reduceIte]
+    apply (FiniteIdeleGroup.localOrder_eq_zero_iff v _).1
+    rw [FiniteIdeleGroup.localOrder_chosenLocalOrderSection,
+      Finsupp.notMem_support_iff.mp he]⟩
+
+theorem valuationVector_valuationVectorSectionPrimeTo
+    (m : Modulus K)
+    (e : HeightOneSpectrum (𝓞 K) →₀ ℤ)
+    (he : ∀ v, v ∈ m.finitePart.support → e v = 0) :
+    FiniteIdeleGroup.valuationVector
+        (valuationVectorSectionPrimeTo m e) =
+      Multiplicative.ofAdd e := by
+  apply Multiplicative.ext
+  ext v
+  rw [FiniteIdeleGroup.valuationVector_apply]
+  by_cases hv : v ∈ m.finitePart.support
+  · change
+      (FiniteIdeleGroup.localOrder v
+        (if v ∈ m.finitePart.support then 1
+          else FiniteIdeleGroup.chosenLocalOrderSection v (e v))).toAdd =
+        e v
+    rw [ite_eq_left hv, map_one]
+    exact (he v hv).symm
+  · change
+      (FiniteIdeleGroup.localOrder v
+        (if v ∈ m.finitePart.support then 1
+          else FiniteIdeleGroup.chosenLocalOrderSection v (e v))).toAdd =
+        e v
+    rw [ite_eq_right hv,
+      FiniteIdeleGroup.localOrder_chosenLocalOrderSection]
+
+theorem primeToIdealMap_surjective (m : Modulus K) :
+    Function.Surjective (primeToIdealMap m) := by
+  intro I
+  let e : HeightOneSpectrum (𝓞 K) →₀ ℤ :=
+    FractionalIdealGroup.countVector (I : FractionalIdealGroup K)
+  have he : ∀ v, v ∈ m.finitePart.support → e v = 0 := by
+    intro v hv
+    exact I.property v hv
+  let a : IdeleGroup K :=
+    (1, valuationVectorSectionPrimeTo m e)
+  have ha : a ∈ idelePrimeToModulusSubgroup m := by
+    constructor
+    · exact m.infiniteCongruenceSubgroup.one_mem
+    · intro v hv
+      change
+        (if v ∈ m.finitePart.support then 1
+          else FiniteIdeleGroup.chosenLocalOrderSection v (e v)) ∈
+            localHigherUnitGroup v (m.finitePart v)
+      rw [ite_eq_left hv]
+      exact (localHigherUnitGroup v (m.finitePart v)).one_mem
+  refine ⟨⟨a, ha⟩, ?_⟩
+  apply Subtype.ext
+  apply FractionalIdealGroup.ext_count
+  intro v
+  change FractionalIdeal.count K v
+      (((FractionalIdealGroup.factorization (K := K))
+        (FiniteIdeleGroup.valuationVector
+          (valuationVectorSectionPrimeTo m e)) :
+          FractionalIdealGroup K) :
+        FractionalIdeal (nonZeroDivisors (𝓞 K)) K) =
+      FractionalIdeal.count K v
+        ((I : FractionalIdealGroup K) :
+          FractionalIdeal (nonZeroDivisors (𝓞 K)) K)
+  rw [valuationVector_valuationVectorSectionPrimeTo m e he,
+    FractionalIdealGroup.count_factorization]
+  exact FractionalIdealGroup.countVector_apply I v
+
+theorem ideleCongruenceSubgroup_le_primeTo
+    (m : Modulus K) :
+    m.ideleCongruenceSubgroup ≤
+      idelePrimeToModulusSubgroup m := by
+  intro a ha
+  exact ⟨ha.1, fun v _ => ha.2 v⟩
+
+/-- The congruence subgroup, viewed inside the subgroup of ideles prime
+to the modulus. -/
+def congruenceSubgroupInPrimeTo (m : Modulus K) :
+    Subgroup (idelePrimeToModulusSubgroup m) :=
+  m.ideleCongruenceSubgroup.subgroupOf
+    (idelePrimeToModulusSubgroup m)
+
+theorem primeToIdealMap_ker (m : Modulus K) :
+    (primeToIdealMap m).ker =
+      congruenceSubgroupInPrimeTo m := by
+  ext a
+  constructor
+  · intro ha
+    have hintegral :
+        (a : IdeleGroup K) ∈
+          IdeleGroup.integralAtFinitePlaces (K := K) := by
+      rw [← IdeleGroup.fractionalIdeal_ker,
+        MonoidHom.mem_ker]
+      exact congrArg Subtype.val
+        (MonoidHom.mem_ker.mp ha)
+    constructor
+    · exact a.property.1
+    · intro v
+      by_cases hv : v ∈ m.finitePart.support
+      · exact a.property.2 v hv
+      · rw [Finsupp.notMem_support_iff.mp hv,
+          localHigherUnitGroup_zero]
+        exact hintegral v
+  · intro ha
+    apply MonoidHom.mem_ker.mpr
+    apply Subtype.ext
+    change IdeleGroup.fractionalIdeal (a : IdeleGroup K) = 1
+    rw [← MonoidHom.mem_ker,
+      IdeleGroup.fractionalIdeal_ker]
+    intro v
+    exact localHigherUnitGroup_le_integralUnits v (m.finitePart v) (ha.2 v)
+
+/-- The quotient of ideles prime to a modulus by the congruence subgroup,
+identified with fractional ideals prime to the modulus. -/
+def quotientCongruenceEquivPrimeToIdeals (m : Modulus K) :
+    idelePrimeToModulusSubgroup m ⧸
+        congruenceSubgroupInPrimeTo m ≃*
+      primeToModulusIdeals m := by
+  rw [← primeToIdealMap_ker m]
+  exact QuotientGroup.quotientKerEquivOfSurjective
+    (primeToIdealMap m) (primeToIdealMap_surjective m)
+
+/-- Principal ideles satisfying the modulus conditions, considered inside
+`I_K^(m)`. -/
+def principalSubgroupInPrimeTo (m : Modulus K) :
+    Subgroup (idelePrimeToModulusSubgroup m) :=
+  Subgroup.comap (idelePrimeToModulusSubgroup m).subtype
+    (IdeleGroup.principalSubgroup K)
+
+/-- Principal ideals generated by a totally positive element congruent to
+one modulo the finite modulus. -/
+def principalRayIdealSubgroup (m : Modulus K) :
+    Subgroup (primeToModulusIdeals m) :=
+  Subgroup.map (primeToIdealMap m)
+    (principalSubgroupInPrimeTo m)
+
+theorem mem_principalRayIdealSubgroup_iff
+    (m : Modulus K) (I : primeToModulusIdeals m) :
+    I ∈ principalRayIdealSubgroup m ↔
+      ∃ x : Kˣ,
+        ∃ _hx : IdeleGroup.principalIdele K x ∈
+          idelePrimeToModulusSubgroup m,
+        toPrincipalIdeal (𝓞 K) K x =
+          (I : FractionalIdealGroup K) := by
+  constructor
+  · rintro ⟨a, ha, hmap⟩
+    obtain ⟨x, hx⟩ := ha
+    refine ⟨x, ?_, ?_⟩
+    · rw [hx]
+      exact a.property
+    · have hval := congrArg Subtype.val hmap
+      change IdeleGroup.fractionalIdeal (a : IdeleGroup K) =
+        (I : FractionalIdealGroup K) at hval
+      rw [← IdeleGroup.fractionalIdeal_principalIdele]
+      exact (congrArg (IdeleGroup.fractionalIdeal (K := K)) hx).trans hval
+  · rintro ⟨x, hx, hideal⟩
+    let a : idelePrimeToModulusSubgroup m :=
+      ⟨IdeleGroup.principalIdele K x, hx⟩
+    have ha : a ∈ principalSubgroupInPrimeTo m := by
+      change IdeleGroup.principalIdele K x ∈
+        IdeleGroup.principalSubgroup K
+      exact ⟨x, rfl⟩
+    refine ⟨a, ha, ?_⟩
+    apply Subtype.ext
+    change IdeleGroup.fractionalIdeal
+        (IdeleGroup.principalIdele K x) =
+      (I : FractionalIdealGroup K)
+    rw [IdeleGroup.fractionalIdeal_principalIdele, hideal]
+
+/-- The ideal-theoretic ray class group `J_K^m / P_K^m`. -/
+abbrev IdealRayClassGroup (m : Modulus K) :=
+  primeToModulusIdeals m ⧸ principalRayIdealSubgroup m
+
+/-- The canonical projection from ideles prime to the modulus to the
+ideal-theoretic ray class group. -/
+def idealRayProjection (m : Modulus K) :
+    idelePrimeToModulusSubgroup m →*
+      IdealRayClassGroup m :=
+  (QuotientGroup.mk' (principalRayIdealSubgroup m)).comp
+    (primeToIdealMap m)
+
+/-- The subgroup generated by congruence ideles and principal ideles
+inside the ideles prime to a modulus. -/
+def raySubgroupInPrimeTo (m : Modulus K) :
+    Subgroup (idelePrimeToModulusSubgroup m) :=
+  congruenceSubgroupInPrimeTo m ⊔
+    principalSubgroupInPrimeTo m
+
+theorem idealRayProjection_surjective (m : Modulus K) :
+    Function.Surjective (idealRayProjection m) := by
+  intro c
+  obtain ⟨I, rfl⟩ :=
+    QuotientGroup.mk'_surjective
+      (principalRayIdealSubgroup m) c
+  obtain ⟨a, rfl⟩ := primeToIdealMap_surjective m I
+  exact ⟨a, rfl⟩
+
+theorem idealRayProjection_ker (m : Modulus K) :
+    (idealRayProjection m).ker =
+      raySubgroupInPrimeTo m := by
+  ext a
+  constructor
+  · intro ha
+    change QuotientGroup.mk'
+        (principalRayIdealSubgroup m)
+        (primeToIdealMap m a) = 1 at ha
+    rw [QuotientGroup.mk'_apply,
+      QuotientGroup.eq_one_iff] at ha
+    obtain ⟨p, hp, hpa⟩ := ha
+    let n : idelePrimeToModulusSubgroup m := a * p⁻¹
+    have hn : n ∈ congruenceSubgroupInPrimeTo m := by
+      rw [← primeToIdealMap_ker m, MonoidHom.mem_ker]
+      change primeToIdealMap m (a * p⁻¹) = 1
+      rw [map_mul, map_inv, hpa]
+      simp
+    rw [raySubgroupInPrimeTo, Subgroup.mem_sup]
+    refine ⟨n, hn, p, hp, ?_⟩
+    dsimp [n]
+    group
+  · intro ha
+    rw [raySubgroupInPrimeTo, Subgroup.mem_sup] at ha
+    obtain ⟨n, hn, p, hp, rfl⟩ := ha
+    change QuotientGroup.mk'
+        (principalRayIdealSubgroup m)
+        (primeToIdealMap m (n * p)) = 1
+    rw [map_mul]
+    have hn' : primeToIdealMap m n = 1 :=
+      MonoidHom.mem_ker.mp
+        ((primeToIdealMap_ker m).symm ▸ hn)
+    rw [hn', one_mul]
+    rw [QuotientGroup.mk'_apply,
+      QuotientGroup.eq_one_iff]
+    exact ⟨p, hp, rfl⟩
+
+/-- The quotient of ideles prime to the modulus by the full ray subgroup,
+identified with the ideal-theoretic ray class group. -/
+def quotientRaySubgroupEquivIdealRayClassGroup
+    (m : Modulus K) :
+    idelePrimeToModulusSubgroup m ⧸
+        raySubgroupInPrimeTo m ≃*
+      IdealRayClassGroup m :=
+  (QuotientGroup.quotientMulEquivOfEq
+      (idealRayProjection_ker m).symm).trans
+    (QuotientGroup.quotientKerEquivOfSurjective
+      (idealRayProjection m)
+      (idealRayProjection_surjective m))
+
+/-- The quotient equivalence induced by the ideal-ray projection evaluates on
+the class of a prime-to-modulus idele as the original projection. -/
+@[simp]
+theorem quotientRaySubgroupEquivIdealRayClassGroup_mk
+    (m : Modulus K) (a : idelePrimeToModulusSubgroup m) :
+    quotientRaySubgroupEquivIdealRayClassGroup m
+        (QuotientGroup.mk' (raySubgroupInPrimeTo m) a) =
+      idealRayProjection m a := by
+  rw [quotientRaySubgroupEquivIdealRayClassGroup,
+    MulEquiv.trans_apply, QuotientGroup.mk'_apply,
+    QuotientGroup.quotientMulEquivOfEq_mk]
+  exact QuotientGroup.kerLift_mk (idealRayProjection m) a
+
+
+end RayClass
diff --git a/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassIdealModulusProjection.lean b/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassIdealModulusProjection.lean
index e37e3df..415463a 100644
--- a/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassIdealModulusProjection.lean
+++ b/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassIdealModulusProjection.lean
@@ -90,7 +90,7 @@ private theorem rayCongruent_of_le
   · intro v hv
     exact hx.2 v (hmn.2 hv)
 
-private theorem rayPrimeToIdeals_antitone
+theorem rayPrimeToIdeals_antitone
     {K : Type u} [Field K] [NumberField K]
     {m n : RayClassModulus K} (hmn : m ≤ n) :
     rayClassPrimeToIdeals n ≤ rayClassPrimeToIdeals m := by
diff --git a/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassOfFinitePrime.lean b/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassOfFinitePrime.lean
index a3e9746..d8d3db5 100644
--- a/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassOfFinitePrime.lean
+++ b/Lean4/ClassFieldTheory/Definitions/ConductorsAndRayClassFields/RayClassOfFinitePrime.lean
@@ -9,6 +9,7 @@ import ClassFieldTheory.Definitions.ConductorsAndRayClassFields.OrdinaryRayClass
 import ClassFieldTheory.Definitions.ConductorsAndRayClassFields.RayClassGroup
 import ClassFieldTheory.Definitions.FrobeniusAndHilbertClassFields.FinitePrimeFractionalIdeal
 
+
 set_option autoImplicit false
 
 /-!
@@ -24,7 +25,7 @@ namespace ClassFieldTheory
 
 universe u
 
-private lemma finitePrimeFractionalIdeal_mem_primeTo
+lemma finitePrimeFractionalIdeal_mem_primeTo
     {K : Type u} [Field K] [NumberField K]
     (m : RayClassModulus K)
     (v : HeightOneSpectrum (𝓞 K))
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocity.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocity.lean
index 053f2b7..e213e03 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocity.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocity.lean
@@ -4,158 +4,18 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.RayClassFieldRealization
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.ArithmeticNormalization
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ArithmeticRayClassFieldReciprocityNormResidue
 
 set_option autoImplicit false
-
-/-!
-# Arithmetic reciprocity for actual ray class fields
-
-This is the canonical topological isomorphism
-`Gal(Kᵐ/K) ≃ₜ* C_K/C_Kᵐ` with arithmetic Frobenius
-normalization.  It uses the actual selected ray class field, its exact
-idèle norm range, the finite Krull topology, and the native ray-class
-quotient topology.
--/
-
 open scoped Classical NumberField
-
 noncomputable section
-
 namespace GlobalClassFieldTheory
 namespace GlobalClassFields
-
 open NumberField
 open Reciprocity
-
-private theorem arithmeticRayClassIdeleClassGroupIsMulCommutative
-    {F : Type} [Field F] [NumberField F] :
-    IsMulCommutative (IdeleClassGroup F) :=
-  ⟨⟨fun a b => mul_comm a b⟩⟩
-
 attribute [local instance] arithmeticRayClassIdeleClassGroupIsMulCommutative
-
 variable {K : Type} [Field K] [NumberField K]
 
-/-- Arithmetic reciprocity followed by transport between equal norm
-quotients sends a norm-residue symbol to its represented quotient class. -/
-private theorem
-    arithmeticReciprocity_quotientMulEquivOfEq_globalNormResidue
-    {F E : Type} [Field F] [NumberField F]
-    [Field E] [NumberField E] [Algebra F E]
-    [FiniteDimensional F E] [IsAbelianGalois F E]
-    (H : Subgroup (IdeleClassGroup F))
-    (h : (_root_.ideleClassNorm F E).range = H)
-    (c : IdeleClassGroup F) :
-    QuotientGroup.quotientMulEquivOfEq h
-        (arithmeticGlobalReciprocityContinuousMulEquiv F E
-          (arithmeticGlobalNormResidueMonoidHom F E c)) =
-      QuotientGroup.mk' H c := by
-  let e := arithmeticGlobalReciprocityContinuousMulEquiv F E
-  let q :=
-    QuotientGroup.mk'
-      (_root_.ideleClassNorm F E).range c
-  have hsymm :
-      e.symm q = arithmeticGlobalNormResidueMonoidHom F E c :=
-    arithmeticGlobalReciprocityContinuousMulEquiv_symm_mk F E c
-  have he :
-      e (arithmeticGlobalNormResidueMonoidHom F E c) = q := by
-    calc
-      e (arithmeticGlobalNormResidueMonoidHom F E c) =
-          e (e.symm q) :=
-        congrArg (fun σ => e σ) hsymm.symm
-      _ = q := e.apply_symm_apply q
-  calc
-    QuotientGroup.quotientMulEquivOfEq h
-        (arithmeticGlobalReciprocityContinuousMulEquiv F E
-          (arithmeticGlobalNormResidueMonoidHom F E c)) =
-        QuotientGroup.quotientMulEquivOfEq h q :=
-      congrArg (fun x => QuotientGroup.quotientMulEquivOfEq h x) he
-    _ = QuotientGroup.mk' H c :=
-      QuotientGroup.quotientMulEquivOfEq_mk h c
-
-/-- Arithmetic global reciprocity for the actual selected ray class
-field, bundled with both native topologies. -/
-noncomputable def
-    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
-    (m : RayClass.Modulus K) :
-    Gal((rayClassField K m) / K) ≃ₜ*
-      RayClass.RayClassGroup m := by
-  letI normQuotientDiscreteTopology : DiscreteTopology
-      (IdeleClassGroup K ⧸
-        (_root_.ideleClassNorm
-          K (rayClassField K m)).range) :=
-    ideleClassNormQuotient_discreteTopology
-      K (rayClassField K m)
-  letI rayClassGroupDiscreteTopology : DiscreteTopology
-      (RayClass.RayClassGroup m) :=
-    QuotientGroup.discreteTopology
-      (RayClass.isOpen_congruenceSubgroup m)
-  let quotientTransport :
-      (IdeleClassGroup K ⧸
-          (_root_.ideleClassNorm
-            K (rayClassField K m)).range) ≃ₜ*
-        RayClass.RayClassGroup m :=
-    { QuotientGroup.quotientMulEquivOfEq
-        (rayClassField_ideleClassNorm_range_over_original
-          (K := K) m) with
-      continuous_toFun := continuous_of_discreteTopology
-      continuous_invFun := continuous_of_discreteTopology }
-  exact
-    (arithmeticGlobalReciprocityContinuousMulEquiv
-      K (rayClassField K m)).trans
-        quotientTransport
-
-/-- Pointwise evaluation of arithmetic ray-class reciprocity separates
-global reciprocity from the transport between the equal norm quotients. -/
-private theorem
-    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup_apply
-    (m : RayClass.Modulus K)
-    (σ : Gal((rayClassField K m) / K)) :
-    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
-        (K := K) m σ =
-      QuotientGroup.quotientMulEquivOfEq
-        (rayClassField_ideleClassNorm_range_over_original
-          (K := K) m)
-        (arithmeticGlobalReciprocityContinuousMulEquiv
-          K (rayClassField K m) σ) := by
-  rfl
-
-/-- Arithmetic ray-class reciprocity sends the arithmetic global
-norm-residue symbol of an idèle class to its literal ray class. -/
-@[simp]
-theorem
-    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup_globalNormResidue
-    (m : RayClass.Modulus K)
-    (c : IdeleClassGroup K) :
-    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
-        (K := K) m
-        (arithmeticGlobalNormResidueMonoidHom
-          K (rayClassField K m) c) =
-      QuotientGroup.mk'
-        (RayClass.Modulus.congruenceSubgroup m) c := by
-  calc
-    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
-        (K := K) m
-        (arithmeticGlobalNormResidueMonoidHom
-          K (rayClassField K m) c) =
-      QuotientGroup.quotientMulEquivOfEq
-          (rayClassField_ideleClassNorm_range_over_original
-            (K := K) m)
-          (arithmeticGlobalReciprocityContinuousMulEquiv
-            K (rayClassField K m)
-            (arithmeticGlobalNormResidueMonoidHom
-              K (rayClassField K m) c)) :=
-      arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup_apply
-        (K := K) m _
-    _ = QuotientGroup.mk'
-          (RayClass.Modulus.congruenceSubgroup m) c :=
-      arithmeticReciprocity_quotientMulEquivOfEq_globalNormResidue
-        (RayClass.Modulus.congruenceSubgroup m)
-        (rayClassField_ideleClassNorm_range_over_original
-          (K := K) m) c
-
 /-- Inverse arithmetic ray reciprocity sends a represented ray class
 back to the arithmetic global norm-residue symbol. -/
 @[simp]
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityBase.lean
new file mode 100644
index 0000000..6519cd0
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityBase.lean
@@ -0,0 +1,112 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.RayClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.ArithmeticNormalization
+
+
+set_option autoImplicit false
+
+/-!
+# Arithmetic reciprocity for actual ray class fields
+
+This is the canonical topological isomorphism
+`Gal(Kᵐ/K) ≃ₜ* C_K/C_Kᵐ` with arithmetic Frobenius
+normalization.  It uses the actual selected ray class field, its exact
+idèle norm range, the finite Krull topology, and the native ray-class
+quotient topology.
+-/
+
+open scoped Classical NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace GlobalClassFields
+
+open NumberField
+open Reciprocity
+
+theorem arithmeticRayClassIdeleClassGroupIsMulCommutative
+    {F : Type} [Field F] [NumberField F] :
+    IsMulCommutative (IdeleClassGroup F) :=
+  ⟨⟨fun a b => mul_comm a b⟩⟩
+
+attribute [local instance] arithmeticRayClassIdeleClassGroupIsMulCommutative
+
+variable {K : Type} [Field K] [NumberField K]
+
+/-- Arithmetic reciprocity followed by transport between equal norm
+quotients sends a norm-residue symbol to its represented quotient class. -/
+theorem
+    arithmeticReciprocity_quotientMulEquivOfEq_globalNormResidue
+    {F E : Type} [Field F] [NumberField F]
+    [Field E] [NumberField E] [Algebra F E]
+    [FiniteDimensional F E] [IsAbelianGalois F E]
+    (H : Subgroup (IdeleClassGroup F))
+    (h : (_root_.ideleClassNorm F E).range = H)
+    (c : IdeleClassGroup F) :
+    QuotientGroup.quotientMulEquivOfEq h
+        (arithmeticGlobalReciprocityContinuousMulEquiv F E
+          (arithmeticGlobalNormResidueMonoidHom F E c)) =
+      QuotientGroup.mk' H c := by
+  let e := arithmeticGlobalReciprocityContinuousMulEquiv F E
+  let q :=
+    QuotientGroup.mk'
+      (_root_.ideleClassNorm F E).range c
+  have hsymm :
+      e.symm q = arithmeticGlobalNormResidueMonoidHom F E c :=
+    arithmeticGlobalReciprocityContinuousMulEquiv_symm_mk F E c
+  have he :
+      e (arithmeticGlobalNormResidueMonoidHom F E c) = q := by
+    calc
+      e (arithmeticGlobalNormResidueMonoidHom F E c) =
+          e (e.symm q) :=
+        congrArg (fun σ => e σ) hsymm.symm
+      _ = q := e.apply_symm_apply q
+  calc
+    QuotientGroup.quotientMulEquivOfEq h
+        (arithmeticGlobalReciprocityContinuousMulEquiv F E
+          (arithmeticGlobalNormResidueMonoidHom F E c)) =
+        QuotientGroup.quotientMulEquivOfEq h q :=
+      congrArg (fun x => QuotientGroup.quotientMulEquivOfEq h x) he
+    _ = QuotientGroup.mk' H c :=
+      QuotientGroup.quotientMulEquivOfEq_mk h c
+
+/-- Arithmetic global reciprocity for the actual selected ray class
+field, bundled with both native topologies. -/
+noncomputable def
+    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
+    (m : RayClass.Modulus K) :
+    Gal((rayClassField K m) / K) ≃ₜ*
+      RayClass.RayClassGroup m := by
+  letI normQuotientDiscreteTopology : DiscreteTopology
+      (IdeleClassGroup K ⧸
+        (_root_.ideleClassNorm
+          K (rayClassField K m)).range) :=
+    ideleClassNormQuotient_discreteTopology
+      K (rayClassField K m)
+  letI rayClassGroupDiscreteTopology : DiscreteTopology
+      (RayClass.RayClassGroup m) :=
+    QuotientGroup.discreteTopology
+      (RayClass.isOpen_congruenceSubgroup m)
+  let quotientTransport :
+      (IdeleClassGroup K ⧸
+          (_root_.ideleClassNorm
+            K (rayClassField K m)).range) ≃ₜ*
+        RayClass.RayClassGroup m :=
+    { QuotientGroup.quotientMulEquivOfEq
+        (rayClassField_ideleClassNorm_range_over_original
+          (K := K) m) with
+      continuous_toFun := continuous_of_discreteTopology
+      continuous_invFun := continuous_of_discreteTopology }
+  exact
+    (arithmeticGlobalReciprocityContinuousMulEquiv
+      K (rayClassField K m)).trans
+        quotientTransport
+
+end GlobalClassFields
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityEvaluation.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityEvaluation.lean
new file mode 100644
index 0000000..41d3435
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityEvaluation.lean
@@ -0,0 +1,51 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ArithmeticRayClassFieldReciprocityBase
+
+set_option autoImplicit false
+open scoped Classical NumberField
+noncomputable section
+namespace GlobalClassFieldTheory
+namespace GlobalClassFields
+open NumberField
+open Reciprocity
+attribute [local instance] arithmeticRayClassIdeleClassGroupIsMulCommutative
+private theorem mulEquiv_toEquiv_apply
+    {A B : Type*} [Mul A] [Mul B]
+    (e : A ≃* B) (x : A) : e.toEquiv x = e x := by rfl
+
+private theorem continuousMulEquiv_toMulEquiv_apply
+    {A B : Type*} [TopologicalSpace A] [TopologicalSpace B] [Mul A] [Mul B]
+    (e : A ≃ₜ* B) (x : A) : e.toMulEquiv x = e x := by rfl
+
+variable {K : Type} [Field K] [NumberField K]
+
+
+/-- Evaluation of ray class reciprocity is global reciprocity transported
+through the equality of the ray norm subgroup and the idele norm range. -/
+theorem
+    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup_apply
+    (m : RayClass.Modulus K)
+    (σ : Gal((rayClassField K m) / K)) :
+    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
+        (K := K) m σ =
+      QuotientGroup.quotientMulEquivOfEq
+        (rayClassField_ideleClassNorm_range_over_original
+          (K := K) m)
+        (arithmeticGlobalReciprocityContinuousMulEquiv
+          K (rayClassField K m) σ) := by
+  simp only [arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup,
+    ContinuousMulEquiv.trans, MulEquiv.trans, Equiv.trans,
+    ContinuousMulEquiv.coe_mk, MulEquiv.coe_mk, Equiv.coe_fn_mk,
+    Function.comp_apply,
+    mulEquiv_toEquiv_apply]
+  have h := continuousMulEquiv_toMulEquiv_apply
+    (arithmeticGlobalReciprocityContinuousMulEquiv K (rayClassField K m)) σ
+  with_reducible exact congrArg (fun q => QuotientGroup.quotientMulEquivOfEq (rayClassField_ideleClassNorm_range_over_original (K := K) m) q) h
+
+end GlobalClassFields
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityNormResidue.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityNormResidue.lean
new file mode 100644
index 0000000..ce8c03e
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/ArithmeticRayClassFieldReciprocityNormResidue.lean
@@ -0,0 +1,54 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ArithmeticRayClassFieldReciprocityEvaluation
+
+set_option autoImplicit false
+open scoped Classical NumberField
+noncomputable section
+namespace GlobalClassFieldTheory
+namespace GlobalClassFields
+open NumberField
+open Reciprocity
+attribute [local instance] arithmeticRayClassIdeleClassGroupIsMulCommutative
+variable {K : Type} [Field K] [NumberField K]
+
+/-- Arithmetic ray-class reciprocity sends the arithmetic global
+norm-residue symbol of an idèle class to its literal ray class. -/
+@[simp]
+theorem
+    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup_globalNormResidue
+    (m : RayClass.Modulus K)
+    (c : IdeleClassGroup K) :
+    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
+        (K := K) m
+        (arithmeticGlobalNormResidueMonoidHom
+          K (rayClassField K m) c) =
+      QuotientGroup.mk'
+        (RayClass.Modulus.congruenceSubgroup m) c := by
+  calc
+    arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup
+        (K := K) m
+        (arithmeticGlobalNormResidueMonoidHom
+          K (rayClassField K m) c) =
+      QuotientGroup.quotientMulEquivOfEq
+          (rayClassField_ideleClassNorm_range_over_original
+            (K := K) m)
+          (arithmeticGlobalReciprocityContinuousMulEquiv
+            K (rayClassField K m)
+            (arithmeticGlobalNormResidueMonoidHom
+              K (rayClassField K m) c)) :=
+      arithmeticRayClassFieldGaloisContinuousMulEquivRayClassGroup_apply
+        (K := K) m _
+    _ = QuotientGroup.mk'
+          (RayClass.Modulus.congruenceSubgroup m) c :=
+      arithmeticReciprocity_quotientMulEquivOfEq_globalNormResidue
+        (RayClass.Modulus.congruenceSubgroup m)
+        (rayClassField_ideleClassNorm_range_over_original
+          (K := K) m) c
+
+end GlobalClassFields
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActual.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActual.lean
index 11beea0..7ab1c83 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActual.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActual.lean
@@ -4,106 +4,17 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldReciprocity.Transport
-import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.BigHilbertClassFieldNaturality
-import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldComparison
 
-set_option autoImplicit false
-
-/-!
-# Big Hilbert reciprocity over the realized base field
-
-This leaf specializes the shared reciprocity transport to the actual base
-field of the selected big Hilbert class field.
--/
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldReciprocity.BigActualEquiv
 
+set_option autoImplicit false
 open scoped Classical IsMulCommutative NumberField
-
 noncomputable section
-
 namespace GlobalClassFieldTheory
 namespace GlobalClassFields
-
 open Reciprocity
-
 variable {K : Type} [Field K] [NumberField K]
-
-local instance
-    bigHilbertClassFieldReciprocityIdeleClassGroupIsMulCommutative
-    {F : Type} [Field F] [NumberField F] :
-    IsMulCommutative (IdeleClassGroup F) :=
-  hilbertClassFieldReciprocityIdeleClassGroupIsMulCommutative
-
-/-- The actual norm range of the selected big Hilbert class field is
-the intrinsic big-Hilbert norm subgroup of its actual base field. -/
-theorem bigHilbertClassField_ideleClassNorm_range_eq_intrinsic :
-    (_root_.ideleClassNorm
-      (bigHilbertClassFieldBase K)
-      (bigHilbertClassField K)).range =
-      bigHilbertClassFieldNormSubgroup
-        (K := bigHilbertClassFieldBase K) := by
-  rw [bigHilbertClassField_ideleClassNorm_range]
-  exact
-    bigHilbertClassFieldNormSubgroup_map_ideleClassCongr
-      (bigHilbertClassFieldBaseEquiv (K := K))
-
-/-- Global reciprocity identifies the genuine Galois group of the
-selected big Hilbert class field with the narrow ideal class group of
-the original number field. -/
-private noncomputable def bigHilbertClassFieldReciprocityData :
-    {e : Gal((bigHilbertClassField K) /
-          (bigHilbertClassFieldBase K)) ≃*
-        RayClass.NarrowClassGroup K //
-      ∀ c : IdeleClassGroup (bigHilbertClassFieldBase K),
-        e (globalNormResidueMonoidHom
-            (bigHilbertClassFieldBase K)
-            (bigHilbertClassField K) c) =
-          bigHilbertNarrowClassGroupCongr
-            (bigHilbertClassFieldBaseEquiv (K := K)).symm
-            (bigHilbertClassFieldQuotientEquivNarrowClassGroup
-              (K := bigHilbertClassFieldBase K)
-              (QuotientGroup.mk'
-                (bigHilbertClassFieldNormSubgroup
-                  (K := bigHilbertClassFieldBase K)) c))} := by
-  let d := hilbertClassFieldGlobalReciprocityTransportData
-    (bigHilbertClassFieldNormSubgroup
-      (K := bigHilbertClassFieldBase K))
-    (bigHilbertClassField_ideleClassNorm_range_eq_intrinsic
-      (K := K))
-    ((bigHilbertClassFieldQuotientEquivNarrowClassGroup
-        (K := bigHilbertClassFieldBase K)).trans
-      (bigHilbertNarrowClassGroupCongr
-        (bigHilbertClassFieldBaseEquiv (K := K)).symm))
-  refine ⟨d.1, ?_⟩
-  intro c
-  exact d.2 c
-
-/-- The reciprocity equivalence from the actual big Hilbert Galois group
-to the narrow class group of the original number field. -/
-noncomputable def bigHilbertClassFieldGaloisEquivNarrowClassGroup :
-    Gal((bigHilbertClassField K) /
-        (bigHilbertClassFieldBase K)) ≃*
-      RayClass.NarrowClassGroup K :=
-  (bigHilbertClassFieldReciprocityData (K := K)).1
-
-/-- Under big-Hilbert reciprocity, the actual global norm-residue
-symbol is the narrow ideal class of its idèle-class representative,
-transported back to the original number field. -/
-@[simp]
-theorem bigHilbertClassFieldGaloisEquivNarrowClassGroup_globalNormResidue
-    (c : IdeleClassGroup (bigHilbertClassFieldBase K)) :
-    bigHilbertClassFieldGaloisEquivNarrowClassGroup (K := K)
-        (globalNormResidueMonoidHom
-          (bigHilbertClassFieldBase K)
-          (bigHilbertClassField K) c) =
-      bigHilbertNarrowClassGroupCongr
-        (bigHilbertClassFieldBaseEquiv (K := K)).symm
-        (bigHilbertClassFieldQuotientEquivNarrowClassGroup
-          (K := bigHilbertClassFieldBase K)
-          (QuotientGroup.mk'
-            (bigHilbertClassFieldNormSubgroup
-              (K := bigHilbertClassFieldBase K)) c)) := by
-  exact (bigHilbertClassFieldReciprocityData (K := K)).2 c
+attribute [local instance] bigHilbertClassFieldReciprocityIdeleClassGroupIsMulCommutative
 
 /-- Representative form of big-Hilbert reciprocity: the global
 norm-residue symbol of an actual idèle maps to its narrow ideal
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActualData.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActualData.lean
new file mode 100644
index 0000000..4dd276f
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActualData.lean
@@ -0,0 +1,84 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldReciprocity.Transport
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.BigHilbertClassFieldNaturality
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldComparison
+
+
+set_option autoImplicit false
+
+/-!
+# Big Hilbert reciprocity over the realized base field
+
+This leaf specializes the shared reciprocity transport to the actual base
+field of the selected big Hilbert class field.
+-/
+
+open scoped Classical IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace GlobalClassFields
+
+open Reciprocity
+
+variable {K : Type} [Field K] [NumberField K]
+
+local instance
+    bigHilbertClassFieldReciprocityIdeleClassGroupIsMulCommutative
+    {F : Type} [Field F] [NumberField F] :
+    IsMulCommutative (IdeleClassGroup F) :=
+  hilbertClassFieldReciprocityIdeleClassGroupIsMulCommutative
+
+/-- The actual norm range of the selected big Hilbert class field is
+the intrinsic big-Hilbert norm subgroup of its actual base field. -/
+theorem bigHilbertClassField_ideleClassNorm_range_eq_intrinsic :
+    (_root_.ideleClassNorm
+      (bigHilbertClassFieldBase K)
+      (bigHilbertClassField K)).range =
+      bigHilbertClassFieldNormSubgroup
+        (K := bigHilbertClassFieldBase K) := by
+  rw [bigHilbertClassField_ideleClassNorm_range]
+  exact
+    bigHilbertClassFieldNormSubgroup_map_ideleClassCongr
+      (bigHilbertClassFieldBaseEquiv (K := K))
+
+/-- Global reciprocity identifies the genuine Galois group of the
+selected big Hilbert class field with the narrow ideal class group of
+the original number field. -/
+noncomputable def bigHilbertClassFieldReciprocityData :
+    {e : Gal((bigHilbertClassField K) /
+          (bigHilbertClassFieldBase K)) ≃*
+        RayClass.NarrowClassGroup K //
+      ∀ c : IdeleClassGroup (bigHilbertClassFieldBase K),
+        e (globalNormResidueMonoidHom
+            (bigHilbertClassFieldBase K)
+            (bigHilbertClassField K) c) =
+          bigHilbertNarrowClassGroupCongr
+            (bigHilbertClassFieldBaseEquiv (K := K)).symm
+            (bigHilbertClassFieldQuotientEquivNarrowClassGroup
+              (K := bigHilbertClassFieldBase K)
+              (QuotientGroup.mk'
+                (bigHilbertClassFieldNormSubgroup
+                  (K := bigHilbertClassFieldBase K)) c))} := by
+  let d := hilbertClassFieldGlobalReciprocityTransportData
+    (bigHilbertClassFieldNormSubgroup
+      (K := bigHilbertClassFieldBase K))
+    (bigHilbertClassField_ideleClassNorm_range_eq_intrinsic
+      (K := K))
+    ((bigHilbertClassFieldQuotientEquivNarrowClassGroup
+        (K := bigHilbertClassFieldBase K)).trans
+      (bigHilbertNarrowClassGroupCongr
+        (bigHilbertClassFieldBaseEquiv (K := K)).symm))
+  refine ⟨d.1, ?_⟩
+  intro c
+  exact d.2 c
+
+
+end GlobalClassFields
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActualEquiv.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActualEquiv.lean
new file mode 100644
index 0000000..19f756c
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/GlobalClassFields/HilbertClassFieldReciprocity/BigActualEquiv.lean
@@ -0,0 +1,48 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldReciprocity.BigActualData
+
+set_option autoImplicit false
+open scoped Classical IsMulCommutative NumberField
+noncomputable section
+namespace GlobalClassFieldTheory
+namespace GlobalClassFields
+open Reciprocity
+variable {K : Type} [Field K] [NumberField K]
+attribute [local instance] bigHilbertClassFieldReciprocityIdeleClassGroupIsMulCommutative
+
+/-- The reciprocity equivalence from the actual big Hilbert Galois group
+to the narrow class group of the original number field. -/
+noncomputable def bigHilbertClassFieldGaloisEquivNarrowClassGroup :
+    Gal((bigHilbertClassField K) /
+        (bigHilbertClassFieldBase K)) ≃*
+      RayClass.NarrowClassGroup K :=
+  (bigHilbertClassFieldReciprocityData (K := K)).1
+
+/-- Under big-Hilbert reciprocity, the actual global norm-residue
+symbol is the narrow ideal class of its idèle-class representative,
+transported back to the original number field. -/
+@[simp]
+theorem bigHilbertClassFieldGaloisEquivNarrowClassGroup_globalNormResidue
+    (c : IdeleClassGroup (bigHilbertClassFieldBase K)) :
+    bigHilbertClassFieldGaloisEquivNarrowClassGroup (K := K)
+        (globalNormResidueMonoidHom
+          (bigHilbertClassFieldBase K)
+          (bigHilbertClassField K) c) =
+      bigHilbertNarrowClassGroupCongr
+        (bigHilbertClassFieldBaseEquiv (K := K)).symm
+        (bigHilbertClassFieldQuotientEquivNarrowClassGroup
+          (K := bigHilbertClassFieldBase K)
+          (QuotientGroup.mk'
+            (bigHilbertClassFieldNormSubgroup
+              (K := bigHilbertClassFieldBase K)) c)) := by
+  exact (bigHilbertClassFieldReciprocityData (K := K)).2 c
+
+
+end GlobalClassFields
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/RationalFiniteNormTransfer/Representatives.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/RationalFiniteNormTransfer/Representatives.lean
index a5d0b1a..45c9a3e 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/RationalFiniteNormTransfer/Representatives.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/RationalFiniteNormTransfer/Representatives.lean
@@ -6,6 +6,7 @@ Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 
 import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.RationalFiniteNormTransfer.FieldSpine
 
+
 set_option autoImplicit false
 
 /-!
@@ -56,7 +57,8 @@ noncomputable def rationalFiniteNormTransferAbstractRepresentative
     [hfinite : Finite
       (K.toSubgroup ⧸ extensionSubgroup K L hLK)]
     (a : KummerTheory.ambientFixedAddSubgroup
-      rationalIdeleClassRepresentation K) := by
+      rationalIdeleClassRepresentation K) :
+    rationalFiniteNormTransferExtensionIdeleClass K L hLK := by
   letI hLfinite := RationalFiniteNormTransferInternal.absoluteFinite
     K L hLK (hKfinite := hKfinite) (hfinite := hfinite)
   let E := abstractRelativeFixedField ℚ (SeparableClosure ℚ) hLK
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalization.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalization.lean
index 0ce4c1f..63f8f94 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalization.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalization.lean
@@ -4,188 +4,18 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.PrincipalIdealTheorem
-import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.PrincipalIdealTransfer
-import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerUnramified
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertPrincipalizationClassGroup
 
 set_option autoImplicit false
-
-/-!
-# Principalization in the selected small Hilbert class field
-
-The selected second small Hilbert class field is Galois over the
-canonical fixed-field copy of the original base, and its maximal
-abelian intermediate field is the selected first small Hilbert class
-field.  Witt transfer therefore places every idele class extended from
-that fixed-field copy in the norm range from the second stage.
-Functoriality of actual idele extension along the degree-one
-identification of the original field with its fixed-field copy gives
-the same range inclusion for idele classes extended from the original
-field itself.
-
-The exact second-stage norm subgroup is the intrinsic small-Hilbert
-subgroup, so the genuine map from the original field on small-Hilbert
-quotients is trivial.  Its naturality with extension of ideal classes
-then gives the class-group, integral-ideal, and fractional-ideal forms
-of principalization over the original number field.
--/
-
 open scoped Classical IsMulCommutative NumberField
-
 noncomputable section
-
 namespace GlobalClassFieldTheory
 namespace IdealClassFieldTheory
-
-open ClassFormation
-open GlobalClassFields
-open KummerTheory
-open LocalClassFieldTheory
-open Reciprocity
-
-local instance
-    smallHilbertPrincipalization_ideleClassGroupIsMulCommutative
-    {F : Type} [Field F] [NumberField F] :
-    IsMulCommutative (IdeleClassGroup F) :=
-  ⟨⟨fun a b => mul_comm a b⟩⟩
-
-local instance
-    smallHilbertPrincipalization_ideleClassSubgroupNormal
-    {F : Type} [Field F] [NumberField F]
-    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
-  N.normal_of_isMulCommutative
-
-local instance
-    smallHilbertPrincipalization_smallHilbertQuotientGroup
-    {F : Type} [Field F] [NumberField F] :
-    Group
-      (IdeleClassGroup F ⧸
-        smallHilbertClassFieldNormSubgroup (K := F)) :=
-  QuotientGroup.Quotient.group
-    (smallHilbertClassFieldNormSubgroup (K := F))
-
-local instance
-    smallHilbertPrincipalization_smallHilbertQuotientOne
-    {F : Type} [Field F] [NumberField F] :
-    One
-      (IdeleClassGroup F ⧸
-        smallHilbertClassFieldNormSubgroup (K := F)) :=
-  ⟨(smallHilbertPrincipalization_smallHilbertQuotientGroup
-    (F := F)).one⟩
-
+open ClassFormation GlobalClassFields KummerTheory LocalClassFieldTheory Reciprocity
+attribute [local instance] smallHilbertPrincipalization_ideleClassGroupIsMulCommutative
+  smallHilbertPrincipalization_ideleClassSubgroupNormal
 variable (K : Type) [Field K] [NumberField K]
 
-private theorem smallHilbertClassFieldIdeleExtensionMap_apply_eq_one
-    (q : IdeleClassGroup K ⧸
-      smallHilbertClassFieldNormSubgroup (K := K)) :
-    smallHilbertClassFieldIdeleExtensionMap
-        K (smallHilbertClassField K) q =
-      (1 : IdeleClassGroup (smallHilbertClassField K) ⧸
-        smallHilbertClassFieldNormSubgroup
-          (K := smallHilbertClassField K)) := by
-  induction q using QuotientGroup.induction_on with
-  | _ c =>
-      change
-        smallHilbertClassFieldIdeleExtensionMap
-            K (smallHilbertClassField K)
-            (QuotientGroup.mk'
-              (smallHilbertClassFieldNormSubgroup (K := K)) c) =
-          1
-      have hcontainment :=
-        smallHilbertClassField_ideleClassExtension_range_le_secondNormRange K
-      unfold smallHilbertClassFieldSecondNormRangeContainment at hcontainment
-      have hmembership :
-          ideleClassExtension K (smallHilbertClassField K) c ∈
-            smallHilbertClassFieldNormSubgroup
-              (K := smallHilbertClassField K) :=
-        hcontainment ⟨c, rfl⟩
-      calc
-        smallHilbertClassFieldIdeleExtensionMap
-              K (smallHilbertClassField K)
-              (QuotientGroup.mk'
-                (smallHilbertClassFieldNormSubgroup (K := K)) c) =
-            QuotientGroup.mk'
-              (smallHilbertClassFieldNormSubgroup
-                (K := smallHilbertClassField K))
-              (ideleClassExtension K (smallHilbertClassField K) c) :=
-          smallHilbertClassFieldIdeleExtensionMap_mk'
-            K (smallHilbertClassField K) c
-        _ = 1 :=
-          (QuotientGroup.eq_one_iff
-            (ideleClassExtension K (smallHilbertClassField K) c)).2
-              hmembership
-
-/-- The map induced by genuine idele extension from the original
-number field on the two small-Hilbert reciprocity quotients is
-trivial. -/
-theorem smallHilbertClassFieldIdeleExtensionMap_eq_one :
-    @Eq
-      ((IdeleClassGroup K ⧸
-          smallHilbertClassFieldNormSubgroup (K := K)) →*
-        (IdeleClassGroup (smallHilbertClassField K) ⧸
-          smallHilbertClassFieldNormSubgroup
-            (K := smallHilbertClassField K)))
-      (smallHilbertClassFieldIdeleExtensionMap
-        K (smallHilbertClassField K))
-      (1 :
-        (IdeleClassGroup K ⧸
-          smallHilbertClassFieldNormSubgroup (K := K)) →*
-        (IdeleClassGroup (smallHilbertClassField K) ⧸
-          smallHilbertClassFieldNormSubgroup
-            (K := smallHilbertClassField K))) := by
-  apply MonoidHom.ext
-  intro q
-  simpa only [MonoidHom.one_apply] using
-    smallHilbertClassFieldIdeleExtensionMap_apply_eq_one K q
-
-private theorem smallHilbertClassFieldClassGroupExtension_apply_eq_one
-    (c : ClassGroup (𝓞 K)) :
-    ClassGroup.extendedHom
-        (𝓞 K) (𝓞 (smallHilbertClassField K)) c =
-      (1 : ClassGroup (𝓞 (smallHilbertClassField K))) := by
-  obtain ⟨q, rfl⟩ :=
-    (smallHilbertClassFieldQuotientEquivClassGroup
-      (K := K)).surjective c
-  have hidele :=
-    smallHilbertClassFieldIdeleExtensionMap_apply_eq_one K q
-  have hnaturality :=
-    smallHilbertClassFieldIdeleExtensionMap_naturality
-      K (smallHilbertClassField K) q
-  calc
-    ClassGroup.extendedHom
-          (𝓞 K) (𝓞 (smallHilbertClassField K))
-          (smallHilbertClassFieldQuotientEquivClassGroup (K := K) q) =
-        smallHilbertClassFieldQuotientEquivClassGroup
-          (K := smallHilbertClassField K)
-          (smallHilbertClassFieldIdeleExtensionMap
-            K (smallHilbertClassField K) q) :=
-      hnaturality.symm
-    _ = smallHilbertClassFieldQuotientEquivClassGroup
-          (K := smallHilbertClassField K) 1 :=
-      congrArg
-        (smallHilbertClassFieldQuotientEquivClassGroup
-          (K := smallHilbertClassField K)) hidele
-    _ = 1 :=
-      (smallHilbertClassFieldQuotientEquivClassGroup
-        (K := smallHilbertClassField K)).map_one
-
-/-- Extension of ideal classes from a number field to its selected
-small Hilbert class field is the trivial homomorphism.  This follows
-directly from the naturality equality identifying actual idele
-extension with actual extension of ideal classes. -/
-theorem smallHilbertClassFieldClassGroupExtension_eq_one :
-    @Eq
-      (ClassGroup (𝓞 K) →*
-        ClassGroup (𝓞 (smallHilbertClassField K)))
-      (ClassGroup.extendedHom
-        (𝓞 K) (𝓞 (smallHilbertClassField K)))
-      (1 : ClassGroup (𝓞 K) →*
-        ClassGroup (𝓞 (smallHilbertClassField K))) := by
-  apply MonoidHom.ext
-  intro c
-  simpa only [MonoidHom.one_apply] using
-    smallHilbertClassFieldClassGroupExtension_apply_eq_one K c
-
 /-- Every ideal of a number field becomes principal after extension
 to the selected small Hilbert class field. -/
 theorem allIdealsBecomePrincipalInSmallHilbertClassField :
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationBase.lean
new file mode 100644
index 0000000..ae72e33
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationBase.lean
@@ -0,0 +1,91 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.PrincipalIdealTheorem
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.PrincipalIdealTransfer
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerUnramified
+
+
+set_option autoImplicit false
+
+/-!
+# Principalization in the selected small Hilbert class field
+
+The selected second small Hilbert class field is Galois over the
+canonical fixed-field copy of the original base, and its maximal
+abelian intermediate field is the selected first small Hilbert class
+field.  Witt transfer therefore places every idele class extended from
+that fixed-field copy in the norm range from the second stage.
+Functoriality of actual idele extension along the degree-one
+identification of the original field with its fixed-field copy gives
+the same range inclusion for idele classes extended from the original
+field itself.
+
+The exact second-stage norm subgroup is the intrinsic small-Hilbert
+subgroup, so the genuine map from the original field on small-Hilbert
+quotients is trivial.  Its naturality with extension of ideal classes
+then gives the class-group, integral-ideal, and fractional-ideal forms
+of principalization over the original number field.
+-/
+
+open scoped Classical IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace IdealClassFieldTheory
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open LocalClassFieldTheory
+open Reciprocity
+
+local instance
+    smallHilbertPrincipalization_ideleClassGroupIsMulCommutative
+    {F : Type} [Field F] [NumberField F] :
+    IsMulCommutative (IdeleClassGroup F) :=
+  ⟨⟨fun a b => mul_comm a b⟩⟩
+
+local instance
+    smallHilbertPrincipalization_ideleClassSubgroupNormal
+    {F : Type} [Field F] [NumberField F]
+    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
+  N.normal_of_isMulCommutative
+
+/-- A generic field-extension form keeps the class-field construction opaque. -/
+theorem smallHilbertExtension_trivial_of_range
+    (K L : Type) [Field K] [NumberField K] [Field L] [NumberField L]
+    [Algebra K L] [FiniteDimensional K L]
+    (h : (ideleClassExtension K L).range ≤
+      smallHilbertClassFieldNormSubgroup (K := L))
+    (q : IdeleClassGroup K ⧸ smallHilbertClassFieldNormSubgroup (K := K)) :
+    smallHilbertClassFieldIdeleExtensionMap K L q = 1 := by
+  refine QuotientGroup.induction_on q ?_
+  intro c
+  exact (smallHilbertClassFieldIdeleExtensionMap_mk' K L c).trans
+    ((QuotientGroup.eq_one_iff (ideleClassExtension K L c)).mpr (h ⟨c, rfl⟩))
+
+variable (K : Type) [Field K] [NumberField K]
+
+/-- Extension to the small Hilbert class field sends each reciprocity
+quotient class to the identity class. -/
+theorem smallHilbertClassFieldIdeleExtensionMap_apply_eq_one
+    (q : IdeleClassGroup K ⧸
+      smallHilbertClassFieldNormSubgroup (K := K)) :
+    smallHilbertClassFieldIdeleExtensionMap
+        K (smallHilbertClassField K) q =
+      (1 : IdeleClassGroup (smallHilbertClassField K) ⧸
+        smallHilbertClassFieldNormSubgroup
+          (K := smallHilbertClassField K)) := by
+  have hcontainment :=
+    smallHilbertClassField_ideleClassExtension_range_le_secondNormRange K
+  unfold smallHilbertClassFieldSecondNormRangeContainment at hcontainment
+  exact smallHilbertExtension_trivial_of_range K (smallHilbertClassField K)
+    hcontainment q
+
+end IdealClassFieldTheory
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationClassGroup.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationClassGroup.lean
new file mode 100644
index 0000000..7889495
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationClassGroup.lean
@@ -0,0 +1,68 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertPrincipalizationIdele
+
+set_option autoImplicit false
+open scoped Classical IsMulCommutative NumberField
+noncomputable section
+namespace GlobalClassFieldTheory
+namespace IdealClassFieldTheory
+open ClassFormation GlobalClassFields KummerTheory LocalClassFieldTheory Reciprocity
+attribute [local instance] smallHilbertPrincipalization_ideleClassGroupIsMulCommutative
+  smallHilbertPrincipalization_ideleClassSubgroupNormal
+variable (K : Type) [Field K] [NumberField K]
+
+private theorem smallHilbertClassFieldClassGroupExtension_apply_eq_one
+    (c : ClassGroup (𝓞 K)) :
+    ClassGroup.extendedHom
+        (𝓞 K) (𝓞 (smallHilbertClassField K)) c =
+      (1 : ClassGroup (𝓞 (smallHilbertClassField K))) := by
+  obtain ⟨q, rfl⟩ :=
+    (smallHilbertClassFieldQuotientEquivClassGroup
+      (K := K)).surjective c
+  have hidele :=
+    smallHilbertClassFieldIdeleExtensionMap_apply_eq_one K q
+  have hnaturality :=
+    smallHilbertClassFieldIdeleExtensionMap_naturality
+      K (smallHilbertClassField K) q
+  calc
+    ClassGroup.extendedHom
+          (𝓞 K) (𝓞 (smallHilbertClassField K))
+          (smallHilbertClassFieldQuotientEquivClassGroup (K := K) q) =
+        smallHilbertClassFieldQuotientEquivClassGroup
+          (K := smallHilbertClassField K)
+          (smallHilbertClassFieldIdeleExtensionMap
+            K (smallHilbertClassField K) q) :=
+      hnaturality.symm
+    _ = smallHilbertClassFieldQuotientEquivClassGroup
+          (K := smallHilbertClassField K) 1 :=
+      congrArg
+        (smallHilbertClassFieldQuotientEquivClassGroup
+          (K := smallHilbertClassField K)) hidele
+    _ = 1 :=
+      (smallHilbertClassFieldQuotientEquivClassGroup
+        (K := smallHilbertClassField K)).map_one
+
+/-- Extension of ideal classes from a number field to its selected
+small Hilbert class field is the trivial homomorphism.  This follows
+directly from the naturality equality identifying actual idele
+extension with actual extension of ideal classes. -/
+theorem smallHilbertClassFieldClassGroupExtension_eq_one :
+    @Eq
+      (ClassGroup (𝓞 K) →*
+        ClassGroup (𝓞 (smallHilbertClassField K)))
+      (ClassGroup.extendedHom
+        (𝓞 K) (𝓞 (smallHilbertClassField K)))
+      (1 : ClassGroup (𝓞 K) →*
+        ClassGroup (𝓞 (smallHilbertClassField K))) := by
+  apply MonoidHom.ext
+  intro c
+  simpa only [MonoidHom.one_apply] using
+    smallHilbertClassFieldClassGroupExtension_apply_eq_one K c
+
+end IdealClassFieldTheory
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationIdele.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationIdele.lean
new file mode 100644
index 0000000..e11bcfb
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertPrincipalizationIdele.lean
@@ -0,0 +1,43 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertPrincipalizationBase
+
+set_option autoImplicit false
+open scoped Classical IsMulCommutative NumberField
+noncomputable section
+namespace GlobalClassFieldTheory
+namespace IdealClassFieldTheory
+open ClassFormation GlobalClassFields KummerTheory LocalClassFieldTheory Reciprocity
+attribute [local instance] smallHilbertPrincipalization_ideleClassGroupIsMulCommutative
+  smallHilbertPrincipalization_ideleClassSubgroupNormal
+variable (K : Type) [Field K] [NumberField K]
+
+/-- The map induced by genuine idele extension from the original
+number field on the two small-Hilbert reciprocity quotients is
+trivial. -/
+theorem smallHilbertClassFieldIdeleExtensionMap_eq_one :
+    @Eq
+      ((IdeleClassGroup K ⧸
+          smallHilbertClassFieldNormSubgroup (K := K)) →*
+        (IdeleClassGroup (smallHilbertClassField K) ⧸
+          smallHilbertClassFieldNormSubgroup
+            (K := smallHilbertClassField K)))
+      (smallHilbertClassFieldIdeleExtensionMap
+        K (smallHilbertClassField K))
+      (1 :
+        (IdeleClassGroup K ⧸
+          smallHilbertClassFieldNormSubgroup (K := K)) →*
+        (IdeleClassGroup (smallHilbertClassField K) ⧸
+          smallHilbertClassFieldNormSubgroup
+            (K := smallHilbertClassField K))) := by
+  apply MonoidHom.ext
+  intro q
+  simpa only [MonoidHom.one_apply] using
+    smallHilbertClassFieldIdeleExtensionMap_apply_eq_one K q
+
+end IdealClassFieldTheory
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealization.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealization.lean
index 3edfd94..f01c3d8 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealization.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealization.lean
@@ -4,28 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldRealization
-import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerConjugation
-import Mathlib.Data.Rat.Cast.Defs
 
-set_option autoImplicit false
-
-/-!
-# Actual realization of the two-stage small Hilbert tower
-
-The first small Hilbert class field is the actual finite abelian
-subextension selected in `HilbertClassFieldRealization`.  Over its actual
-fixed field, the closed finite-index small-Hilbert norm subgroup has a
-finite Galois norm neighbourhood.  We embed that neighbourhood in the
-rational separable closure compatibly with the already chosen first
-stage.  Finite abelian classification then selects the second small
-Hilbert class field over the literal first-stage subgroup.
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerRealizationSecond
 
-The compatibility of the embedding is essential: an unrelated chosen
-copy of the middle number field would produce a class field over a
-conjugate closed subgroup rather than over the first-stage subgroup
-itself.
--/
+set_option autoImplicit false
 
 open scoped Classical NumberField
 
@@ -43,33 +25,10 @@ open LocalClassFieldTheory
 open RamificationTheory
 open Reciprocity
 
-/-- The ordinary idèle-class operations used by the two-stage transport,
-fixed at the canonical principal-subgroup quotient. -/
-@[instance_reducible]
-private noncomputable def smallHilbertTowerIdeleClassCommGroup
-    (F : Type) [Field F] [NumberField F] :
-    CommGroup (IdeleClassGroup F) :=
-  QuotientGroup.Quotient.commGroup (IdeleGroup.principalSubgroup F)
 
 attribute [local instance] smallHilbertTowerIdeleClassCommGroup
-
-private theorem addSubgroup_comap_symm_eq_map
-    {A B : Type*} [AddGroup A] [AddGroup B]
-    (H : AddSubgroup A) (e : A ≃+ B) :
-    H.comap e.symm.toAddMonoidHom =
-      H.map e.toAddMonoidHom := by
-  exact (AddSubgroup.map_equiv_eq_comap_symm e H).symm
-
-private noncomputable abbrev
-    closedFiniteIndexNormAmbientCanonicalBaseAlgebra
-    (F : Type) [Field F] [NumberField F]
-    (H : Subgroup (IdeleClassGroup F))
-    (hclosed : IsClosed (H : Set (IdeleClassGroup F)))
-    [H.FiniteIndex] :
-    Algebra F
-      (closedFiniteIndexClassFieldNormAmbient
-        (K := F) H hclosed) :=
-  inferInstance
+  smallHilbertTowerNormAmbientAlgebra
+  smallHilbertTowerNormAmbientIsGalois
 
 section RationalFixedField
 
@@ -78,758 +37,9 @@ variable
       (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
     (L : FiniteAbelianSubextension K.field)
 
-private noncomputable abbrev smallHilbertTowerMiddleFiniteAbstractField :
-    FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-  { field := L.field
-    finite := by
-      let : Finite
-          ((baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-            extensionSubgroup
-              (baseField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-              K.field (le_baseField K.field)) :=
-        K.finite
-      let : Finite
-          (K.field.toSubgroup ⧸
-            extensionSubgroup K.field L.field L.below) :=
-        L.finite
-      exact
-        FiniteGaloisSubextension.finite_extension_trans
-          L.below (le_baseField K.field) }
-
-local notation "E" =>
-  abstractFixedField ℚ (SeparableClosure ℚ) L.field
-
-local notation "N" =>
-  smallHilbertClassFieldNormAmbient E
-
-private noncomputable instance
-    smallHilbertTowerMiddleAbstractQuotientFinite :
-    Finite
-      ((baseField
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-        extensionSubgroup
-          (baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-          L.field (le_baseField L.field)) :=
-  (smallHilbertTowerMiddleFiniteAbstractField K L).finite
-
-private noncomputable instance
-    smallHilbertTowerMiddleFiniteDimensional :
-    FiniteDimensional ℚ E :=
-  abstractFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ) L.field inferInstance
-
-private noncomputable instance
-    smallHilbertTowerMiddleNumberField :
-    NumberField E :=
-  NumberField.of_module_finite ℚ E
-
-/-- The canonical small-Hilbert subgroup over the literal middle field.
-This typed endpoint avoids repeatedly reducing the finite-abstract-field
-package merely to recover its `field = L.field` projection. -/
-noncomputable def smallHilbertTowerMiddleNormSubgroup :
-    AddSubgroup
-      (ambientFixedAddSubgroup
-        rationalIdeleClassRepresentation L.field) :=
-  (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup.comap
-    (rationalAbstractFixedFieldIdeleClassEquivFixed
-      L.field
-      (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom
-
-/-- The typed `comap` endpoint is the canonical transported `map` endpoint.
-This uses only the generic additive equivalence law. -/
-theorem smallHilbertTowerMiddleNormSubgroup_eq_map :
-    smallHilbertTowerMiddleNormSubgroup K L =
-      (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup.map
-        (rationalAbstractFixedFieldIdeleClassEquivFixed
-          L.field
-          (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).toAddMonoidHom := by
-  exact addSubgroup_comap_symm_eq_map
-    (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup
-    (rationalAbstractFixedFieldIdeleClassEquivFixed L.field
-      (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L))
-
-@[reducible]
-private noncomputable def
-    smallHilbertTowerNormAmbientAlgebra :
-    Algebra E N :=
-  closedFiniteIndexNormAmbientCanonicalBaseAlgebra E
-    (smallHilbertClassFieldNormSubgroup (K := E))
-    (smallHilbertClassFieldNormSubgroup_isClosed (K := E))
-
-attribute [local instance] smallHilbertTowerNormAmbientAlgebra
-
-@[reducible]
-private noncomputable def
-    smallHilbertTowerNormAmbientSMul :
-    SMul E N :=
-  Algebra.toSMul
-    (self := smallHilbertTowerNormAmbientAlgebra K L)
-
-@[reducible]
-private noncomputable def
-    smallHilbertTowerNormAmbientModule :
-    Module E N :=
-  @Algebra.toModule E N _ _
-    (smallHilbertTowerNormAmbientAlgebra K L)
-
-private theorem
-    smallHilbertTowerNormAmbientScalarTower :
-    @IsScalarTower ℚ E N
-      (Algebra.toSMul (R := ℚ) (A := E))
-      (smallHilbertTowerNormAmbientSMul K L)
-      (Algebra.toSMul (R := ℚ) (A := N)) := by
-  exact IsScalarTower.of_algebraMap_eq'
-    (R := ℚ) (S := E) (A := N)
-    (RingHom.ext_rat (algebraMap ℚ N)
-      ((algebraMap E N).comp (algebraMap ℚ E)))
-
-private noncomputable def
-    smallHilbertTowerNormAmbientAlgHom :
-    E →ₐ[ℚ] N :=
-  { toRingHom := algebraMap E N
-    commutes' := fun r =>
-      (RingHom.congr_fun
-        (RingHom.ext_rat
-          ((algebraMap E N).comp (algebraMap ℚ E))
-          (algebraMap ℚ N)) r) }
-
-private theorem smallHilbertTowerNormAmbientAlgHom_apply
-    (x : E) :
-    smallHilbertTowerNormAmbientAlgHom K L x =
-      algebraMap E N x := by
-  rfl
-
-private theorem
-    smallHilbertTowerNormAmbientIsGalois :
-    IsGalois E N := by
-  unfold smallHilbertClassFieldNormAmbient
-    closedFiniteIndexClassFieldNormAmbient
-  exact
-    closedFiniteIndexNormAmbientIsGalois
-      (smallHilbertClassFieldNormSubgroup (K := E))
-      (smallHilbertClassFieldNormSubgroup_isClosed (K := E))
-
-attribute [local instance] smallHilbertTowerNormAmbientIsGalois
-
-private noncomputable def
-    smallHilbertNormNeighborhoodForwardAlignment :
-    SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ := by
-  let j₀ : N →ₐ[ℚ] SeparableClosure ℚ :=
-    numberFieldSeparableClosureEmbedding N
-  let i₀ : E →ₐ[ℚ] SeparableClosure ℚ :=
-    j₀.comp (smallHilbertTowerNormAmbientAlgHom K L)
-  exact
-    AlgEquiv.ofBijective
-      (i₀.liftNormal (SeparableClosure ℚ))
-      (AlgHom.normal_bijective
-        ℚ (SeparableClosure ℚ) (SeparableClosure ℚ) _)
-
-/-- The separable-closure automorphism which aligns an arbitrary chosen
-embedding of the norm-neighbourhood field with the already embedded
-middle field. -/
-private noncomputable def
-    smallHilbertNormNeighborhoodAlignment :
-    SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ :=
-  (smallHilbertNormNeighborhoodForwardAlignment K L).symm
-
-@[simp]
-private theorem smallHilbertNormNeighborhoodForwardAlignment_apply
-    (x : E) :
-    smallHilbertNormNeighborhoodForwardAlignment K L
-        (x : SeparableClosure ℚ) =
-      (numberFieldSeparableClosureEmbedding N)
-        (algebraMap E N x) := by
-  let j₀ : N →ₐ[ℚ] SeparableClosure ℚ :=
-    numberFieldSeparableClosureEmbedding N
-  let i₀ : E →ₐ[ℚ] SeparableClosure ℚ :=
-    j₀.comp (smallHilbertTowerNormAmbientAlgHom K L)
-  dsimp only [smallHilbertNormNeighborhoodForwardAlignment,
-    AlgEquiv.ofBijective_apply]
-  calc
-    _ = i₀ x := by
-      simpa only [IntermediateField.algebraMap_apply,
-        Algebra.algebraMap_self, RingHom.id_apply] using
-        i₀.liftNormal_commutes (SeparableClosure ℚ) x
-    _ = j₀ (algebraMap E N x) := by
-      change
-        j₀ (smallHilbertTowerNormAmbientAlgHom K L x) =
-          j₀ (algebraMap E N x)
-      exact congrArg j₀
-        (smallHilbertTowerNormAmbientAlgHom_apply K L x)
-
-/-- A controlled embedding of the concrete finite Galois norm
-neighbourhood.  Its restriction to the middle field is the literal
-inclusion of that fixed field in `SeparableClosure ℚ`. -/
-private noncomputable def
-    smallHilbertNormNeighborhoodEmbedding :
-    N →ₐ[ℚ] SeparableClosure ℚ :=
-  (smallHilbertNormNeighborhoodAlignment K L).toAlgHom.comp
-    (numberFieldSeparableClosureEmbedding N)
-
-@[simp]
-private theorem smallHilbertNormNeighborhoodEmbedding_algebraMap
-    (x : E) :
-    smallHilbertNormNeighborhoodEmbedding K L
-        (algebraMap E N x) =
-      (x : SeparableClosure ℚ) := by
-  change
-    (smallHilbertNormNeighborhoodForwardAlignment K L).symm
-        ((numberFieldSeparableClosureEmbedding N)
-          (algebraMap E N x)) =
-      (x : SeparableClosure ℚ)
-  rw [← smallHilbertNormNeighborhoodForwardAlignment_apply K L x]
-  exact
-    (smallHilbertNormNeighborhoodForwardAlignment K L).symm_apply_apply _
-
-private abbrev smallHilbertNormNeighborhoodEmbeddedBase :
-    ClosedSubgroup
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-  closedFixingSubgroup ℚ (SeparableClosure ℚ)
-    (AlgHom.fieldRange
-      ((smallHilbertNormNeighborhoodEmbedding K L).comp
-        (smallHilbertTowerNormAmbientAlgHom K L)))
-
-private abbrev smallHilbertNormNeighborhoodEmbeddedField :
-    ClosedSubgroup
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-  closedFixingSubgroup ℚ (SeparableClosure ℚ)
-    (AlgHom.fieldRange
-      (smallHilbertNormNeighborhoodEmbedding K L))
-
-private theorem smallHilbertNormNeighborhoodEmbeddedField_le_base :
-    (smallHilbertNormNeighborhoodEmbeddedField K L).toSubgroup ≤
-      (smallHilbertNormNeighborhoodEmbeddedBase K L).toSubgroup := by
-  change
-    (AlgHom.fieldRange
-        (smallHilbertNormNeighborhoodEmbedding K L)).fixingSubgroup ≤
-      (AlgHom.fieldRange
-        ((smallHilbertNormNeighborhoodEmbedding K L).comp
-          (smallHilbertTowerNormAmbientAlgHom K L))).fixingSubgroup
-  apply
-    (AlgHom.fieldRange
-      ((smallHilbertNormNeighborhoodEmbedding K L).comp
-        (smallHilbertTowerNormAmbientAlgHom K L))).fixingSubgroup_le
-  exact
-    AlgHom.range_comp_le_range
-      (smallHilbertTowerNormAmbientAlgHom K L)
-      (smallHilbertNormNeighborhoodEmbedding K L)
-
-private theorem smallHilbertNormNeighborhoodEmbeddedBase_eq :
-    smallHilbertNormNeighborhoodEmbeddedBase K L = L.field := by
-  have hi :
-      (smallHilbertNormNeighborhoodEmbedding K L).comp
-          (smallHilbertTowerNormAmbientAlgHom K L) =
-        (abstractFixedField ℚ (SeparableClosure ℚ) L.field).val := by
-    apply AlgHom.ext
-    intro x
-    change smallHilbertNormNeighborhoodEmbedding K L
-      (smallHilbertTowerNormAmbientAlgHom K L x) = (x : SeparableClosure ℚ)
-    rw [smallHilbertTowerNormAmbientAlgHom_apply K L x]
-    exact smallHilbertNormNeighborhoodEmbedding_algebraMap K L x
-  change
-    closedFixingSubgroup ℚ (SeparableClosure ℚ)
-        (AlgHom.fieldRange
-          ((smallHilbertNormNeighborhoodEmbedding K L).comp
-            (smallHilbertTowerNormAmbientAlgHom K L))) =
-      L.field
-  rw [hi, IntermediateField.fieldRange_val]
-  exact
-    closedFixingSubgroup_abstractFixedField_eq
-      ℚ (SeparableClosure ℚ) L.field
-
-private noncomputable def
-    smallHilbertNormNeighborhoodSeparableClosureEquiv :
-    let j := smallHilbertNormNeighborhoodEmbedding K L
-    let i := j.comp (smallHilbertTowerNormAmbientAlgHom K L)
-    let : Algebra E (SeparableClosure ℚ) :=
-      i.toRingHom.toAlgebra
-    SeparableClosure E ≃ₐ[E] SeparableClosure ℚ := by
-  intro j i alg
-  let : @IsScalarTower ℚ E (SeparableClosure ℚ)
-      (Algebra.toSMul (R := ℚ) (A := E))
-      alg.toSMul
-      (Algebra.toSMul (R := ℚ) (A := SeparableClosure ℚ)) :=
-    IsScalarTower.of_algebraMap_eq' i.comp_algebraMap.symm
-  let : IsSepClosure E (SeparableClosure ℚ) :=
-    ⟨IsSepClosure.sep_closed ℚ,
-      Algebra.isSeparable_tower_top_of_isSeparable
-        ℚ E (SeparableClosure ℚ)⟩
-  exact
-    IsSepClosure.equiv E
-      (SeparableClosure E) (SeparableClosure ℚ)
-
-private noncomputable def
-    smallHilbertFiniteGaloisNormNeighborhoodRaw :
-    FiniteGaloisSubextension
-      (smallHilbertNormNeighborhoodEmbeddedBase K L) := by
-  let : @IsScalarTower ℚ E N
-      (Algebra.toSMul (R := ℚ) (A := E))
-      (smallHilbertTowerNormAmbientSMul K L)
-      (Algebra.toSMul (R := ℚ) (A := N)) :=
-    smallHilbertTowerNormAmbientScalarTower K L
-  let j : N →ₐ[ℚ] SeparableClosure ℚ :=
-    smallHilbertNormNeighborhoodEmbedding K L
-  let i : E →ₐ[ℚ] SeparableClosure ℚ :=
-    j.comp (IsScalarTower.toAlgHom ℚ E N)
-  let B : ClosedSubgroup (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-    closedFixingSubgroup ℚ (SeparableClosure ℚ) i.fieldRange
-  let T : ClosedSubgroup (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-    smallHilbertNormNeighborhoodEmbeddedField K L
-  have hTB : T.toSubgroup ≤ B.toSubgroup := by
-    change j.fieldRange.fixingSubgroup ≤ i.fieldRange.fixingSubgroup
-    apply i.fieldRange.fixingSubgroup_le
-    exact AlgHom.range_comp_le_range (IsScalarTower.toAlgHom ℚ E N) j
-  let raw : FiniteGaloisSubextension B := {
-    field := T
-    below := hTB
-    normal := ambientEmbeddedExtensionSubgroup_normal ℚ E N j
-      (smallHilbertNormNeighborhoodSeparableClosureEquiv K L)
-    finite := ambientEmbeddedExtensionQuotient_finite ℚ E N j
-      (smallHilbertNormNeighborhoodSeparableClosureEquiv K L) }
-  have hi : IsScalarTower.toAlgHom ℚ E N =
-      smallHilbertTowerNormAmbientAlgHom K L := by
-    apply AlgHom.ext
-    intro x
-    exact (IsScalarTower.toAlgHom_apply ℚ E N x).trans
-      (smallHilbertTowerNormAmbientAlgHom_apply K L x).symm
-  have hB : B = smallHilbertNormNeighborhoodEmbeddedBase K L :=
-    congrArg
-      (fun f : E →ₐ[ℚ] N =>
-        closedFixingSubgroup ℚ (SeparableClosure ℚ) (j.comp f).fieldRange) hi
-  exact hB ▸ raw
-
-private noncomputable def rebaseFiniteGaloisSubextension
-    {G : Type} [Group G] [TopologicalSpace G]
-    {B B' : ClosedSubgroup G} (h : B = B')
-    (P : FiniteGaloisSubextension B) :
-    FiniteGaloisSubextension B' :=
-  h ▸ P
-
-@[simp]
-private theorem rebaseFiniteGaloisSubextension_field
-    {G : Type} [Group G] [TopologicalSpace G]
-    {B B' : ClosedSubgroup G} (h : B = B')
-    (P : FiniteGaloisSubextension B) :
-    (rebaseFiniteGaloisSubextension h P).field = P.field := by
-  cases h
-  rfl
-
-private theorem
-    rebaseRationalFiniteGaloisSubextension_fixedField_eq
-    {B B' : ClosedSubgroup
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
-    (h : B = B') (P : FiniteGaloisSubextension B) :
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (rebaseFiniteGaloisSubextension h P).below).restrictScalars ℚ =
-        (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-          P.below).restrictScalars ℚ := by
-  cases h
-  rfl
-
-/-- An actual finite Galois norm neighbourhood over the literal
-first-stage subgroup.  It is produced by the finite-index Kummer
-construction and the controlled embedding above. -/
-noncomputable def smallHilbertFiniteGaloisNormNeighborhood :
-    FiniteGaloisSubextension L.field :=
-  rebaseFiniteGaloisSubextension
-    (smallHilbertNormNeighborhoodEmbeddedBase_eq K L)
-    (smallHilbertFiniteGaloisNormNeighborhoodRaw K L)
-
-/-- The abstract norm subgroup of the chosen neighbourhood, pinned to the
-literal middle-field carrier. -/
-noncomputable def smallHilbertFiniteGaloisNormNeighborhoodNormSubgroup :
-    AddSubgroup
-      (ambientFixedAddSubgroup
-        rationalIdeleClassRepresentation L.field) :=
-  (smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
-    rationalIdeleClassRepresentation
-
-private noncomputable abbrev
-    smallHilbertFiniteGaloisNormNeighborhoodTopField : Type :=
-  abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-    (smallHilbertFiniteGaloisNormNeighborhood K L).below
-
-local notation "E₂" =>
-  smallHilbertFiniteGaloisNormNeighborhoodTopField K L
-
-private noncomputable instance
-    smallHilbertFiniteGaloisNormNeighborhoodQuotientFinite :
-    Finite
-      (L.field.toSubgroup ⧸
-        extensionSubgroup L.field
-          (smallHilbertFiniteGaloisNormNeighborhood K L).field
-          (smallHilbertFiniteGaloisNormNeighborhood K L).below) :=
-  (smallHilbertFiniteGaloisNormNeighborhood K L).finite
-
-private noncomputable instance
-    smallHilbertFiniteGaloisNormNeighborhoodFiniteDimensional :
-    FiniteDimensional E E₂ :=
-  abstractRelativeFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ) L.field
-    (smallHilbertFiniteGaloisNormNeighborhood K L).field
-    (smallHilbertFiniteGaloisNormNeighborhood K L).below
-    inferInstance inferInstance
-
-private noncomputable instance
-    smallHilbertFiniteGaloisNormNeighborhoodScalarTower :
-    IsScalarTower ℚ E E₂ :=
-  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
-
-private noncomputable instance
-    smallHilbertFiniteGaloisNormNeighborhoodAbsoluteFiniteDimensional :
-    FiniteDimensional ℚ E₂ :=
-  FiniteDimensional.trans ℚ E E₂
-
-private noncomputable instance
-    smallHilbertFiniteGaloisNormNeighborhoodNumberField :
-    NumberField E₂ :=
-  NumberField.of_module_finite ℚ E₂
-
-private noncomputable instance
-    smallHilbertFiniteGaloisNormNeighborhoodIsGalois :
-    IsGalois E E₂ :=
-  abstractRelativeFixedField_isGalois
-    ℚ (SeparableClosure ℚ) L.field
-    (smallHilbertFiniteGaloisNormNeighborhood K L).field
-    (smallHilbertFiniteGaloisNormNeighborhood K L).below
-    (smallHilbertFiniteGaloisNormNeighborhood K L).normal
-
-private theorem
-    smallHilbertFiniteGaloisNormNeighborhood_fixedField_eq_range :
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (smallHilbertFiniteGaloisNormNeighborhood K L).below).restrictScalars ℚ =
-        AlgHom.fieldRange
-          (smallHilbertNormNeighborhoodEmbedding K L) := by
-  rw [show
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (smallHilbertFiniteGaloisNormNeighborhood K L).below).restrictScalars ℚ =
-        (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-          (smallHilbertFiniteGaloisNormNeighborhoodRaw K L).below).restrictScalars ℚ
-    from
-      rebaseRationalFiniteGaloisSubextension_fixedField_eq
-        (smallHilbertNormNeighborhoodEmbeddedBase_eq K L)
-        (smallHilbertFiniteGaloisNormNeighborhoodRaw K L)]
-  exact
-    InfiniteGalois.fixedField_fixingSubgroup
-      (AlgHom.fieldRange
-        (smallHilbertNormNeighborhoodEmbedding K L))
-
-private noncomputable def
-    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv :
-    N ≃ₐ[ℚ]
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (smallHilbertFiniteGaloisNormNeighborhood K L).below).restrictScalars ℚ :=
-  (smallHilbertNormNeighborhoodEmbedding K L).equivFieldRange.trans
-    (IntermediateField.equivOfEq
-      (smallHilbertFiniteGaloisNormNeighborhood_fixedField_eq_range
-        K L).symm)
-
-@[simp]
-private theorem
-    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv_algebraMap
-    (x : E) :
-    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv K L
-        (algebraMap E N x) =
-      algebraMap E
-        E₂ x := by
-  apply Subtype.ext
-  change
-    smallHilbertNormNeighborhoodEmbedding K L
-        (algebraMap E N x) =
-      (x : SeparableClosure ℚ)
-  exact smallHilbertNormNeighborhoodEmbedding_algebraMap K L x
-
-private noncomputable def
-    smallHilbertFiniteGaloisNormNeighborhoodRelativeTopEquiv :
-    N ≃ₐ[E] E₂ := {
-  smallHilbertFiniteGaloisNormNeighborhoodTopEquiv K L with
-  commutes' := fun x =>
-    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv_algebraMap
-      K L x }
-
-private theorem
-    smallHilbertFiniteGaloisNormNeighborhood_ordinaryNorm_le :
-    (_root_.ideleClassNorm E N).range ≤
-      smallHilbertClassFieldNormSubgroup (K := E) := by
-  simpa only [smallHilbertClassFieldNormAmbient] using
-    (closedFiniteIndexClassFieldNormAmbient_normRange_le
-      (K := E) (smallHilbertClassFieldNormSubgroup (K := E))
-      (smallHilbertClassFieldNormSubgroup_isClosed (K := E)))
-
-private theorem
-    smallHilbertFiniteGaloisNormNeighborhood_ordinaryNormRange_eq :
-    (_root_.ideleClassNorm E N).range =
-      (_root_.ideleClassNorm E E₂).range := by
-  simpa only [ordinaryIdeleClassNorm_range_eq_relative] using
-    (ideleClassNorm_range_algEquiv
-      (K := E)
-      (smallHilbertFiniteGaloisNormNeighborhoodRelativeTopEquiv
-        K L)).symm
-
-private theorem
-    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq :
-    ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
-        rationalIdeleClassRepresentation).map
-          (rationalAbstractFixedFieldIdeleClassEquivFixed
-            L.field
-            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
-      (_root_.ideleClassNorm E E₂).range.toAddSubgroup := by
-  change
-    (finiteNormSubgroup rationalIdeleClassRepresentation
-      L.field
-      (smallHilbertFiniteGaloisNormNeighborhood K L).field
-      (smallHilbertFiniteGaloisNormNeighborhood K L).below).map
-        (rationalAbstractFixedFieldIdeleClassEquivFixed
-          L.field
-          (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
-      (_root_.ideleClassNorm E E₂).range.toAddSubgroup
-  exact
-    (map_rationalFiniteNormSubgroup_eq_ordinaryIdeleClassNormRange_concrete
-      L.field
-      (smallHilbertFiniteGaloisNormNeighborhood K L).field
-      (smallHilbertFiniteGaloisNormNeighborhood K L).below
-      (smallHilbertFiniteGaloisNormNeighborhood K L).normal)
-
-/-- The abstract norm map lands directly in the ordinary norm range of the
-chosen neighbourhood.  Composing the two named subgroup equalities here
-keeps downstream membership proofs pointwise. -/
-private theorem
-    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq_ordinaryNormRange :
-    ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
-        rationalIdeleClassRepresentation).map
-          (rationalAbstractFixedFieldIdeleClassEquivFixed
-            L.field
-            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
-      (_root_.ideleClassNorm E N).range.toAddSubgroup :=
-  (smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq K L).trans
-    (congrArg Subgroup.toAddSubgroup
-      (smallHilbertFiniteGaloisNormNeighborhood_ordinaryNormRange_eq K L).symm)
-
-/-- Pointwise form of the combined norm-range equality. -/
-private theorem
-    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_mem_ordinaryNormRange
-    (a : Additive (IdeleClassGroup E))
-    (ha :
-      a ∈ ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
-        rationalIdeleClassRepresentation).map
-          (rationalAbstractFixedFieldIdeleClassEquivFixed
-            L.field
-            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom) :
-    a ∈ (_root_.ideleClassNorm E N).range.toAddSubgroup :=
-  (le_of_eq
-    (smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq_ordinaryNormRange
-      K L)) ha
-
-/-- The actual finite Galois norm neighbourhood has abstract norm
-subgroup contained in the canonical small-Hilbert subgroup of the
-middle fixed field.  This is the source-producing norm-topology input;
-no norm-openness premise is assumed. -/
-theorem smallHilbertFiniteGaloisNormNeighborhood_normSubgroup_le :
-    ∀ a : ambientFixedAddSubgroup
-        rationalIdeleClassRepresentation L.field,
-      a ∈ smallHilbertFiniteGaloisNormNeighborhoodNormSubgroup K L →
-        a ∈ smallHilbertTowerMiddleNormSubgroup K L := by
-  intro a ha
-  change
-    a ∈ (smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
-      rationalIdeleClassRepresentation at ha
-  have haMap :
-      (rationalAbstractFixedFieldIdeleClassEquivFixed
-          L.field
-          (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm a ∈
-        ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
-          rationalIdeleClassRepresentation).map
-            (rationalAbstractFixedFieldIdeleClassEquivFixed
-              L.field
-              (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom :=
-    ⟨a, ha, rfl⟩
-  have haOrdinary :
-      (rationalAbstractFixedFieldIdeleClassEquivFixed L.field
-        (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm a ∈
-        (_root_.ideleClassNorm E N).range.toAddSubgroup :=
-    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_mem_ordinaryNormRange
-      K L _ haMap
-  exact smallHilbertFiniteGaloisNormNeighborhood_ordinaryNorm_le K L haOrdinary
-
-/-- The canonical small-Hilbert subgroup of the actual middle fixed
-field is open in the genuine norm topology. -/
-theorem smallHilbertNormSubgroupInRationalClassFormation_isNormOpen :
-    IsNormOpen rationalIdeleClassRepresentation L.field
-      (smallHilbertTowerMiddleNormSubgroup K L :
-        Set
-          (ambientFixedAddSubgroup
-            rationalIdeleClassRepresentation L.field)) := by
-  rw [normTopology_addSubgroup_isOpen_iff]
-  refine
-    ⟨smallHilbertFiniteGaloisNormNeighborhood K L, ?_⟩
-  change
-    smallHilbertFiniteGaloisNormNeighborhoodNormSubgroup K L ≤
-      smallHilbertTowerMiddleNormSubgroup K L
-  exact smallHilbertFiniteGaloisNormNeighborhood_normSubgroup_le K L
-
-/-- The second small Hilbert class field as an actual finite abelian
-subextension of the literal first-stage field. -/
-noncomputable def secondSmallHilbertClassFieldSubextension :
-    FiniteAbelianSubextension L.field := by
-  let H : FiniteAbelianSubextension.NormOpenAddSubgroup
-      rationalIdeleClassRepresentation L.field :=
-    ⟨smallHilbertTowerMiddleNormSubgroup K L,
-      smallHilbertNormSubgroupInRationalClassFormation_isNormOpen K L⟩
-  exact
-    Classical.choose
-      (FiniteAbelianSubextension.normSubgroupMap_surjective
-        rationalCyclotomicIdeleClassValuationData
-        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-        (smallHilbertTowerMiddleFiniteAbstractField K L) H)
-
-/-- The second-stage extension realizes exactly the canonical
-small-Hilbert norm subgroup of the actual middle field. -/
-@[simp]
-theorem secondSmallHilbertClassFieldSubextension_normSubgroup :
-    (secondSmallHilbertClassFieldSubextension K L).normSubgroup
-        rationalIdeleClassRepresentation =
-      smallHilbertTowerMiddleNormSubgroup K L := by
-  let H : FiniteAbelianSubextension.NormOpenAddSubgroup
-      rationalIdeleClassRepresentation L.field :=
-    ⟨smallHilbertTowerMiddleNormSubgroup K L,
-      smallHilbertNormSubgroupInRationalClassFormation_isNormOpen K L⟩
-  have h :=
-    Classical.choose_spec
-      (FiniteAbelianSubextension.normSubgroupMap_surjective
-        rationalCyclotomicIdeleClassValuationData
-        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-        (smallHilbertTowerMiddleFiniteAbstractField K L) H)
-  exact congrArg Subtype.val h
-
-private noncomputable abbrev secondSmallHilbertClassFieldTopField : Type :=
-  abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-    (secondSmallHilbertClassFieldSubextension K L).below
-
-local notation "T₂" => secondSmallHilbertClassFieldTopField K L
-
-private noncomputable instance
-    secondSmallHilbertClassFieldSubextensionQuotientFinite :
-    Finite
-      (L.field.toSubgroup ⧸
-        extensionSubgroup L.field
-          (secondSmallHilbertClassFieldSubextension K L).field
-          (secondSmallHilbertClassFieldSubextension K L).below) :=
-  (secondSmallHilbertClassFieldSubextension K L).finite
-
-private noncomputable instance
-    secondSmallHilbertClassFieldTopFiniteDimensional :
-    FiniteDimensional E T₂ :=
-  abstractRelativeFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ) L.field
-    (secondSmallHilbertClassFieldSubextension K L).field
-    (secondSmallHilbertClassFieldSubextension K L).below
-    inferInstance inferInstance
-
-private noncomputable instance
-    secondSmallHilbertClassFieldTopScalarTower :
-    IsScalarTower ℚ E T₂ :=
-  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
-
-private noncomputable instance
-    secondSmallHilbertClassFieldTopAbsoluteFiniteDimensional :
-    FiniteDimensional ℚ T₂ :=
-  FiniteDimensional.trans ℚ E T₂
-
-private noncomputable instance
-    secondSmallHilbertClassFieldTopNumberField :
-    NumberField T₂ :=
-  NumberField.of_module_finite ℚ T₂
-
-private noncomputable instance
-    secondSmallHilbertClassFieldTopIsGalois :
-    IsGalois E T₂ :=
-  abstractRelativeFixedField_isGalois
-    ℚ (SeparableClosure ℚ) L.field
-    (secondSmallHilbertClassFieldSubextension K L).field
-    (secondSmallHilbertClassFieldSubextension K L).below
-    (secondSmallHilbertClassFieldSubextension K L).normal
-
-private theorem
-    secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_actualNormRange :
-    ((secondSmallHilbertClassFieldSubextension K L).normSubgroup
-        rationalIdeleClassRepresentation).map
-          (rationalAbstractFixedFieldIdeleClassEquivFixed
-            L.field
-            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
-      (_root_.ideleClassNorm E T₂).range.toAddSubgroup := by
-  change
-    (finiteNormSubgroup rationalIdeleClassRepresentation
-        L.field
-        (secondSmallHilbertClassFieldSubextension K L).field
-        (secondSmallHilbertClassFieldSubextension K L).below).map
-          (rationalAbstractFixedFieldIdeleClassEquivFixed
-            L.field
-            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
-      (_root_.ideleClassNorm E T₂).range.toAddSubgroup
-  exact
-    map_rationalFiniteNormSubgroup_eq_ordinaryIdeleClassNormRange_concrete
-      L.field
-      (secondSmallHilbertClassFieldSubextension K L).field
-      (secondSmallHilbertClassFieldSubextension K L).below
-      (secondSmallHilbertClassFieldSubextension K L).normal
-
-private theorem
-    secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_smallHilbertNormSubgroup :
-    ((secondSmallHilbertClassFieldSubextension K L).normSubgroup
-        rationalIdeleClassRepresentation).map
-          (rationalAbstractFixedFieldIdeleClassEquivFixed
-            L.field
-            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
-      (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup := by
-  let e : Additive (IdeleClassGroup E) ≃+
-      ambientFixedAddSubgroup rationalIdeleClassRepresentation L.field :=
-    rationalAbstractFixedFieldIdeleClassEquivFixed L.field
-      (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)
-  let H : AddSubgroup (Additive (IdeleClassGroup E)) :=
-    (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup
-  let back : AddSubgroup
-      (ambientFixedAddSubgroup rationalIdeleClassRepresentation L.field) →
-      AddSubgroup (Additive (IdeleClassGroup E)) :=
-    fun n => n.map e.symm.toAddMonoidHom
-  have hNorm :
-      back ((secondSmallHilbertClassFieldSubextension K L).normSubgroup
-        rationalIdeleClassRepresentation) =
-      back (smallHilbertTowerMiddleNormSubgroup K L) :=
-    congrArg back (secondSmallHilbertClassFieldSubextension_normSubgroup K L)
-  have hCancel : back (smallHilbertTowerMiddleNormSubgroup K L) = H :=
-    AddSubgroup.map_comap_eq_self_of_surjective e.symm.surjective H
-  exact hNorm.trans hCancel
-
-/-- The actual second small Hilbert class field has exactly the intrinsic
-small-Hilbert norm range over the literal middle fixed field. -/
-@[simp]
-theorem secondSmallHilbertClassFieldSubextension_ideleClassNorm_range :
-    (_root_.ideleClassNorm E T₂).range =
-      smallHilbertClassFieldNormSubgroup (K := E) := by
-  apply Subgroup.toAddSubgroup.injective
-  exact
-    (secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_actualNormRange
-      K L).symm.trans
-      (secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_smallHilbertNormSubgroup
-        K L)
 
-/-- Compatibility of the typed middle endpoint with the canonical endpoint
-used by the conjugation API. -/
-theorem smallHilbertTowerMiddleNormSubgroup_eq_conjugationEndpoint :
-    smallHilbertTowerMiddleNormSubgroup K L =
-      smallHilbertNormSubgroupInRationalClassFormation
-        (L.toFiniteGaloisExtension.toFiniteAbstractFieldExtension).field := by
-  have hField :
-      (L.toFiniteGaloisExtension.toFiniteAbstractFieldExtension).field.field =
-        L.field := by
-    rfl
-  unfold smallHilbertNormSubgroupInRationalClassFormation
-  cases hField
-  exact smallHilbertTowerMiddleNormSubgroup_eq_map K L
+local notation "E" => abstractFixedField ℚ (SeparableClosure ℚ) L.field
+local notation "N" => smallHilbertClassFieldNormAmbient E
 
 end RationalFixedField
 
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationBase.lean
new file mode 100644
index 0000000..31a4787
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationBase.lean
@@ -0,0 +1,278 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.HilbertClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerConjugation
+import Mathlib.Data.Rat.Cast.Defs
+
+
+set_option autoImplicit false
+
+/-!
+# Actual realization of the two-stage small Hilbert tower
+
+The first small Hilbert class field is the actual finite abelian
+subextension selected in `HilbertClassFieldRealization`.  Over its actual
+fixed field, the closed finite-index small-Hilbert norm subgroup has a
+finite Galois norm neighbourhood.  We embed that neighbourhood in the
+rational separable closure compatibly with the already chosen first
+stage.  Finite abelian classification then selects the second small
+Hilbert class field over the literal first-stage subgroup.
+
+The compatibility of the embedding is essential: an unrelated chosen
+copy of the middle number field would produce a class field over a
+conjugate closed subgroup rather than over the first-stage subgroup
+itself.
+-/
+
+open scoped Classical NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace IdealClassFieldTheory
+
+open AlgebraicNumberTheory
+open ClassFormation
+open CyclicCohomology
+open GlobalClassFields
+open KummerTheory
+open LocalClassFieldTheory
+open RamificationTheory
+open Reciprocity
+
+/-- The ordinary idèle-class operations used by the two-stage transport,
+fixed at the canonical principal-subgroup quotient. -/
+@[instance_reducible]
+noncomputable def smallHilbertTowerIdeleClassCommGroup
+    (F : Type) [Field F] [NumberField F] :
+    CommGroup (IdeleClassGroup F) :=
+  QuotientGroup.Quotient.commGroup (IdeleGroup.principalSubgroup F)
+
+attribute [local instance] smallHilbertTowerIdeleClassCommGroup
+
+private theorem addSubgroup_comap_symm_eq_map
+    {A B : Type*} [AddGroup A] [AddGroup B]
+    (H : AddSubgroup A) (e : A ≃+ B) :
+    H.comap e.symm.toAddMonoidHom =
+      H.map e.toAddMonoidHom := by
+  exact (AddSubgroup.map_equiv_eq_comap_symm e H).symm
+
+private noncomputable abbrev
+    closedFiniteIndexNormAmbientCanonicalBaseAlgebra
+    (F : Type) [Field F] [NumberField F]
+    (H : Subgroup (IdeleClassGroup F))
+    (hclosed : IsClosed (H : Set (IdeleClassGroup F)))
+    [H.FiniteIndex] :
+    Algebra F
+      (closedFiniteIndexClassFieldNormAmbient
+        (K := F) H hclosed) :=
+  inferInstance
+
+section RationalFixedField
+
+variable
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (L : FiniteAbelianSubextension K.field)
+
+/-- The finite abstract field attached to the intermediate field of the
+finite abelian extension, with finiteness obtained by transitivity. -/
+noncomputable abbrev smallHilbertTowerMiddleFiniteAbstractField :
+    FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+  { field := L.field
+    finite := by
+      let : Finite
+          ((baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+            extensionSubgroup
+              (baseField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+              K.field (le_baseField K.field)) :=
+        K.finite
+      let : Finite
+          (K.field.toSubgroup ⧸
+            extensionSubgroup K.field L.field L.below) :=
+        L.finite
+      exact
+        FiniteGaloisSubextension.finite_extension_trans
+          L.below (le_baseField K.field) }
+
+local notation "E" =>
+  abstractFixedField ℚ (SeparableClosure ℚ) L.field
+
+local notation "N" =>
+  smallHilbertClassFieldNormAmbient E
+
+noncomputable instance
+    smallHilbertTowerMiddleAbstractQuotientFinite :
+    Finite
+      ((baseField
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+        extensionSubgroup
+          (baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+          L.field (le_baseField L.field)) :=
+  (smallHilbertTowerMiddleFiniteAbstractField K L).finite
+
+private noncomputable instance
+    smallHilbertTowerMiddleFiniteDimensional :
+    FiniteDimensional ℚ E :=
+  abstractFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ) L.field inferInstance
+
+private noncomputable instance
+    smallHilbertTowerMiddleNumberField :
+    NumberField E :=
+  NumberField.of_module_finite ℚ E
+
+
+/-- The small Hilbert norm subgroup of the intermediate number field,
+transported to its ambient fixed idele-class representation. -/
+noncomputable def smallHilbertTowerMiddleNormSubgroup :
+    AddSubgroup
+      (ambientFixedAddSubgroup
+        rationalIdeleClassRepresentation L.field) :=
+  (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup.comap
+    (rationalAbstractFixedFieldIdeleClassEquivFixed
+      L.field
+      (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom
+
+/-- The typed `comap` endpoint is the canonical transported `map` endpoint.
+This uses only the generic additive equivalence law. -/
+theorem smallHilbertTowerMiddleNormSubgroup_eq_map :
+    smallHilbertTowerMiddleNormSubgroup K L =
+      (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup.map
+        (rationalAbstractFixedFieldIdeleClassEquivFixed
+          L.field
+          (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).toAddMonoidHom := by
+  exact addSubgroup_comap_symm_eq_map
+    (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup
+    (rationalAbstractFixedFieldIdeleClassEquivFixed L.field
+      (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L))
+
+/-- The algebra structure of the intermediate field on its small Hilbert
+class-field norm ambient. -/
+@[reducible]
+noncomputable def
+    smallHilbertTowerNormAmbientAlgebra :
+    Algebra E N :=
+  closedFiniteIndexNormAmbientCanonicalBaseAlgebra E
+    (smallHilbertClassFieldNormSubgroup (K := E))
+    (smallHilbertClassFieldNormSubgroup_isClosed (K := E))
+
+attribute [local instance] smallHilbertTowerNormAmbientAlgebra
+
+/-- Scalar multiplication of the intermediate field on its small Hilbert
+class-field norm ambient. -/
+@[reducible]
+noncomputable def
+    smallHilbertTowerNormAmbientSMul :
+    SMul E N :=
+  Algebra.toSMul
+    (self := smallHilbertTowerNormAmbientAlgebra K L)
+
+@[reducible]
+private noncomputable def
+    smallHilbertTowerNormAmbientModule :
+    Module E N :=
+  @Algebra.toModule E N _ _
+    (smallHilbertTowerNormAmbientAlgebra K L)
+
+theorem
+    smallHilbertTowerNormAmbientScalarTower :
+    @IsScalarTower ℚ E N
+      (Algebra.toSMul (R := ℚ) (A := E))
+      (smallHilbertTowerNormAmbientSMul K L)
+      (Algebra.toSMul (R := ℚ) (A := N)) := by
+  exact IsScalarTower.of_algebraMap_eq'
+    (R := ℚ) (S := E) (A := N)
+    (RingHom.ext_rat (algebraMap ℚ N)
+      ((algebraMap E N).comp (algebraMap ℚ E)))
+
+/-- The rational algebra embedding of the intermediate field into its
+small Hilbert class-field norm ambient. -/
+noncomputable def
+    smallHilbertTowerNormAmbientAlgHom :
+    E →ₐ[ℚ] N :=
+  { toRingHom := algebraMap E N
+    commutes' := fun r =>
+      (RingHom.congr_fun
+        (RingHom.ext_rat
+          ((algebraMap E N).comp (algebraMap ℚ E))
+          (algebraMap ℚ N)) r) }
+
+theorem smallHilbertTowerNormAmbientAlgHom_apply
+    (x : E) :
+    smallHilbertTowerNormAmbientAlgHom K L x =
+      algebraMap E N x := by
+  rfl
+
+theorem
+    smallHilbertTowerNormAmbientIsGalois :
+    IsGalois E N := by
+  unfold smallHilbertClassFieldNormAmbient
+    closedFiniteIndexClassFieldNormAmbient
+  exact
+    closedFiniteIndexNormAmbientIsGalois
+      (smallHilbertClassFieldNormSubgroup (K := E))
+      (smallHilbertClassFieldNormSubgroup_isClosed (K := E))
+
+attribute [local instance] smallHilbertTowerNormAmbientIsGalois
+
+/-- The separable-closure automorphism extending the induced embedding
+of the intermediate field into the norm-neighborhood field. -/
+noncomputable def
+    smallHilbertNormNeighborhoodForwardAlignment :
+    SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ := by
+  let j₀ : N →ₐ[ℚ] SeparableClosure ℚ :=
+    numberFieldSeparableClosureEmbedding N
+  let i₀ : E →ₐ[ℚ] SeparableClosure ℚ :=
+    j₀.comp (smallHilbertTowerNormAmbientAlgHom K L)
+  exact
+    AlgEquiv.ofBijective
+      (i₀.liftNormal (SeparableClosure ℚ))
+      (AlgHom.normal_bijective
+        ℚ (SeparableClosure ℚ) (SeparableClosure ℚ) _)
+
+/-- The separable-closure automorphism which aligns an arbitrary chosen
+embedding of the norm-neighbourhood field with the already embedded
+middle field. -/
+noncomputable def
+    smallHilbertNormNeighborhoodAlignment :
+    SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ :=
+  (smallHilbertNormNeighborhoodForwardAlignment K L).symm
+
+@[simp]
+theorem smallHilbertNormNeighborhoodForwardAlignment_apply
+    (x : E) :
+    smallHilbertNormNeighborhoodForwardAlignment K L
+        (x : SeparableClosure ℚ) =
+      (numberFieldSeparableClosureEmbedding N)
+        (algebraMap E N x) := by
+  let j₀ : N →ₐ[ℚ] SeparableClosure ℚ :=
+    numberFieldSeparableClosureEmbedding N
+  let i₀ : E →ₐ[ℚ] SeparableClosure ℚ :=
+    j₀.comp (smallHilbertTowerNormAmbientAlgHom K L)
+  dsimp only [smallHilbertNormNeighborhoodForwardAlignment,
+    AlgEquiv.ofBijective_apply]
+  calc
+    _ = i₀ x := by
+      simpa only [IntermediateField.algebraMap_apply,
+        Algebra.algebraMap_self, RingHom.id_apply] using
+        i₀.liftNormal_commutes (SeparableClosure ℚ) x
+    _ = j₀ (algebraMap E N x) := by
+      change
+        j₀ (smallHilbertTowerNormAmbientAlgHom K L x) =
+          j₀ (algebraMap E N x)
+      exact congrArg j₀
+        (smallHilbertTowerNormAmbientAlgHom_apply K L x)
+
+
+end RationalFixedField
+end IdealClassFieldTheory
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationEmbedding.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationEmbedding.lean
new file mode 100644
index 0000000..b76d2bd
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationEmbedding.lean
@@ -0,0 +1,190 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerRealizationBase
+
+set_option autoImplicit false
+
+open scoped Classical NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace IdealClassFieldTheory
+
+open AlgebraicNumberTheory
+open ClassFormation
+open CyclicCohomology
+open GlobalClassFields
+open KummerTheory
+open LocalClassFieldTheory
+open RamificationTheory
+open Reciprocity
+
+
+attribute [local instance] smallHilbertTowerIdeleClassCommGroup
+  smallHilbertTowerNormAmbientAlgebra
+  smallHilbertTowerNormAmbientIsGalois
+
+section RationalFixedField
+
+variable
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (L : FiniteAbelianSubextension K.field)
+
+
+local notation "E" => abstractFixedField ℚ (SeparableClosure ℚ) L.field
+local notation "N" => smallHilbertClassFieldNormAmbient E
+
+/-- A controlled embedding of the concrete finite Galois norm
+neighbourhood.  Its restriction to the middle field is the literal
+inclusion of that fixed field in `SeparableClosure ℚ`. -/
+noncomputable def
+    smallHilbertNormNeighborhoodEmbedding :
+    N →ₐ[ℚ] SeparableClosure ℚ :=
+  (smallHilbertNormNeighborhoodAlignment K L).toAlgHom.comp
+    (numberFieldSeparableClosureEmbedding N)
+
+@[simp]
+theorem smallHilbertNormNeighborhoodEmbedding_algebraMap
+    (x : E) :
+    smallHilbertNormNeighborhoodEmbedding K L
+        (algebraMap E N x) =
+      (x : SeparableClosure ℚ) := by
+  change
+    (smallHilbertNormNeighborhoodForwardAlignment K L).symm
+        ((numberFieldSeparableClosureEmbedding N)
+          (algebraMap E N x)) =
+      (x : SeparableClosure ℚ)
+  rw [← smallHilbertNormNeighborhoodForwardAlignment_apply K L x]
+  exact
+    (smallHilbertNormNeighborhoodForwardAlignment K L).symm_apply_apply _
+
+private abbrev smallHilbertNormNeighborhoodEmbeddedBase :
+    ClosedSubgroup
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+  closedFixingSubgroup ℚ (SeparableClosure ℚ)
+    (AlgHom.fieldRange
+      ((smallHilbertNormNeighborhoodEmbedding K L).comp
+        (smallHilbertTowerNormAmbientAlgHom K L)))
+
+private abbrev smallHilbertNormNeighborhoodEmbeddedField :
+    ClosedSubgroup
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+  closedFixingSubgroup ℚ (SeparableClosure ℚ)
+    (AlgHom.fieldRange
+      (smallHilbertNormNeighborhoodEmbedding K L))
+
+private theorem smallHilbertNormNeighborhoodEmbeddedField_le_base :
+    (smallHilbertNormNeighborhoodEmbeddedField K L).toSubgroup ≤
+      (smallHilbertNormNeighborhoodEmbeddedBase K L).toSubgroup := by
+  change
+    (AlgHom.fieldRange
+        (smallHilbertNormNeighborhoodEmbedding K L)).fixingSubgroup ≤
+      (AlgHom.fieldRange
+        ((smallHilbertNormNeighborhoodEmbedding K L).comp
+          (smallHilbertTowerNormAmbientAlgHom K L))).fixingSubgroup
+  apply
+    (AlgHom.fieldRange
+      ((smallHilbertNormNeighborhoodEmbedding K L).comp
+        (smallHilbertTowerNormAmbientAlgHom K L))).fixingSubgroup_le
+  exact
+    AlgHom.range_comp_le_range
+      (smallHilbertTowerNormAmbientAlgHom K L)
+      (smallHilbertNormNeighborhoodEmbedding K L)
+
+theorem smallHilbertNormNeighborhoodEmbeddedBase_eq :
+    smallHilbertNormNeighborhoodEmbeddedBase K L = L.field := by
+  have hi :
+      (smallHilbertNormNeighborhoodEmbedding K L).comp
+          (smallHilbertTowerNormAmbientAlgHom K L) =
+        (abstractFixedField ℚ (SeparableClosure ℚ) L.field).val := by
+    apply AlgHom.ext
+    intro x
+    change smallHilbertNormNeighborhoodEmbedding K L
+      (smallHilbertTowerNormAmbientAlgHom K L x) = (x : SeparableClosure ℚ)
+    rw [smallHilbertTowerNormAmbientAlgHom_apply K L x]
+    exact smallHilbertNormNeighborhoodEmbedding_algebraMap K L x
+  change
+    closedFixingSubgroup ℚ (SeparableClosure ℚ)
+        (AlgHom.fieldRange
+          ((smallHilbertNormNeighborhoodEmbedding K L).comp
+            (smallHilbertTowerNormAmbientAlgHom K L))) =
+      L.field
+  rw [hi, IntermediateField.fieldRange_val]
+  exact
+    closedFixingSubgroup_abstractFixedField_eq
+      ℚ (SeparableClosure ℚ) L.field
+
+private noncomputable def
+    smallHilbertNormNeighborhoodSeparableClosureEquiv :
+    let j := smallHilbertNormNeighborhoodEmbedding K L
+    let i := j.comp (smallHilbertTowerNormAmbientAlgHom K L)
+    let : Algebra E (SeparableClosure ℚ) :=
+      i.toRingHom.toAlgebra
+    SeparableClosure E ≃ₐ[E] SeparableClosure ℚ := by
+  intro j i alg
+  let : @IsScalarTower ℚ E (SeparableClosure ℚ)
+      (Algebra.toSMul (R := ℚ) (A := E))
+      alg.toSMul
+      (Algebra.toSMul (R := ℚ) (A := SeparableClosure ℚ)) :=
+    IsScalarTower.of_algebraMap_eq' i.comp_algebraMap.symm
+  let : IsSepClosure E (SeparableClosure ℚ) :=
+    ⟨IsSepClosure.sep_closed ℚ,
+      Algebra.isSeparable_tower_top_of_isSeparable
+        ℚ E (SeparableClosure ℚ)⟩
+  exact
+    IsSepClosure.equiv E
+      (SeparableClosure E) (SeparableClosure ℚ)
+
+/-- The finite Galois extension given by the embedded small Hilbert
+norm-neighborhood field over the embedded intermediate field. -/
+noncomputable def
+    smallHilbertFiniteGaloisNormNeighborhoodRaw :
+    FiniteGaloisSubextension
+      (smallHilbertNormNeighborhoodEmbeddedBase K L) := by
+  let : @IsScalarTower ℚ E N
+      (Algebra.toSMul (R := ℚ) (A := E))
+      (smallHilbertTowerNormAmbientSMul K L)
+      (Algebra.toSMul (R := ℚ) (A := N)) :=
+    smallHilbertTowerNormAmbientScalarTower K L
+  let j : N →ₐ[ℚ] SeparableClosure ℚ :=
+    smallHilbertNormNeighborhoodEmbedding K L
+  let i : E →ₐ[ℚ] SeparableClosure ℚ :=
+    j.comp (IsScalarTower.toAlgHom ℚ E N)
+  let B : ClosedSubgroup (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+    closedFixingSubgroup ℚ (SeparableClosure ℚ) i.fieldRange
+  let T : ClosedSubgroup (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+    smallHilbertNormNeighborhoodEmbeddedField K L
+  have hTB : T.toSubgroup ≤ B.toSubgroup := by
+    change j.fieldRange.fixingSubgroup ≤ i.fieldRange.fixingSubgroup
+    apply i.fieldRange.fixingSubgroup_le
+    exact AlgHom.range_comp_le_range (IsScalarTower.toAlgHom ℚ E N) j
+  let raw : FiniteGaloisSubextension B := {
+    field := T
+    below := hTB
+    normal := ambientEmbeddedExtensionSubgroup_normal ℚ E N j
+      (smallHilbertNormNeighborhoodSeparableClosureEquiv K L)
+    finite := ambientEmbeddedExtensionQuotient_finite ℚ E N j
+      (smallHilbertNormNeighborhoodSeparableClosureEquiv K L) }
+  have hi : IsScalarTower.toAlgHom ℚ E N =
+      smallHilbertTowerNormAmbientAlgHom K L := by
+    apply AlgHom.ext
+    intro x
+    exact (IsScalarTower.toAlgHom_apply ℚ E N x).trans
+      (smallHilbertTowerNormAmbientAlgHom_apply K L x).symm
+  have hB : B = smallHilbertNormNeighborhoodEmbeddedBase K L :=
+    congrArg
+      (fun f : E →ₐ[ℚ] N =>
+        closedFixingSubgroup ℚ (SeparableClosure ℚ) (j.comp f).fieldRange) hi
+  exact hB ▸ raw
+
+
+end RationalFixedField
+end IdealClassFieldTheory
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationNorm.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationNorm.lean
new file mode 100644
index 0000000..7c2a69a
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationNorm.lean
@@ -0,0 +1,272 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerRealizationEmbedding
+
+set_option autoImplicit false
+
+open scoped Classical NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace IdealClassFieldTheory
+
+open AlgebraicNumberTheory
+open ClassFormation
+open CyclicCohomology
+open GlobalClassFields
+open KummerTheory
+open LocalClassFieldTheory
+open RamificationTheory
+open Reciprocity
+
+
+attribute [local instance] smallHilbertTowerIdeleClassCommGroup
+  smallHilbertTowerNormAmbientAlgebra
+  smallHilbertTowerNormAmbientIsGalois
+
+section RationalFixedField
+
+variable
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (L : FiniteAbelianSubextension K.field)
+
+
+local notation "E" => abstractFixedField ℚ (SeparableClosure ℚ) L.field
+local notation "N" => smallHilbertClassFieldNormAmbient E
+
+private noncomputable def rebaseFiniteGaloisSubextension
+    {G : Type} [Group G] [TopologicalSpace G]
+    {B B' : ClosedSubgroup G} (h : B = B')
+    (P : FiniteGaloisSubextension B) :
+    FiniteGaloisSubextension B' :=
+  h ▸ P
+
+@[simp]
+private theorem rebaseFiniteGaloisSubextension_field
+    {G : Type} [Group G] [TopologicalSpace G]
+    {B B' : ClosedSubgroup G} (h : B = B')
+    (P : FiniteGaloisSubextension B) :
+    (rebaseFiniteGaloisSubextension h P).field = P.field := by
+  cases h
+  rfl
+
+private theorem
+    rebaseRationalFiniteGaloisSubextension_fixedField_eq
+    {B B' : ClosedSubgroup
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
+    (h : B = B') (P : FiniteGaloisSubextension B) :
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (rebaseFiniteGaloisSubextension h P).below).restrictScalars ℚ =
+        (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+          P.below).restrictScalars ℚ := by
+  cases h
+  rfl
+
+/-- An actual finite Galois norm neighbourhood over the literal
+first-stage subgroup.  It is produced by the finite-index Kummer
+construction and the controlled embedding above. -/
+noncomputable def smallHilbertFiniteGaloisNormNeighborhood :
+    FiniteGaloisSubextension L.field :=
+  rebaseFiniteGaloisSubextension
+    (smallHilbertNormNeighborhoodEmbeddedBase_eq K L)
+    (smallHilbertFiniteGaloisNormNeighborhoodRaw K L)
+
+
+/-- The norm subgroup of the small Hilbert norm-neighborhood extension
+in the ambient fixed idele-class representation. -/
+noncomputable def smallHilbertFiniteGaloisNormNeighborhoodNormSubgroup :
+    AddSubgroup
+      (ambientFixedAddSubgroup
+        rationalIdeleClassRepresentation L.field) :=
+  (smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
+    rationalIdeleClassRepresentation
+
+/-- The relative fixed field realizing the top field of the small
+Hilbert norm-neighborhood extension. -/
+noncomputable abbrev
+    smallHilbertFiniteGaloisNormNeighborhoodTopField : Type :=
+  abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+    (smallHilbertFiniteGaloisNormNeighborhood K L).below
+
+local notation "E₂" =>
+  smallHilbertFiniteGaloisNormNeighborhoodTopField K L
+
+private noncomputable instance
+    smallHilbertFiniteGaloisNormNeighborhoodQuotientFinite :
+    Finite
+      (L.field.toSubgroup ⧸
+        extensionSubgroup L.field
+          (smallHilbertFiniteGaloisNormNeighborhood K L).field
+          (smallHilbertFiniteGaloisNormNeighborhood K L).below) :=
+  (smallHilbertFiniteGaloisNormNeighborhood K L).finite
+
+private noncomputable instance
+    smallHilbertFiniteGaloisNormNeighborhoodFiniteDimensional :
+    FiniteDimensional E E₂ :=
+  abstractRelativeFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ) L.field
+    (smallHilbertFiniteGaloisNormNeighborhood K L).field
+    (smallHilbertFiniteGaloisNormNeighborhood K L).below
+    inferInstance inferInstance
+
+private noncomputable instance
+    smallHilbertFiniteGaloisNormNeighborhoodScalarTower :
+    IsScalarTower ℚ E E₂ :=
+  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
+
+private noncomputable instance
+    smallHilbertFiniteGaloisNormNeighborhoodAbsoluteFiniteDimensional :
+    FiniteDimensional ℚ E₂ :=
+  FiniteDimensional.trans ℚ E E₂
+
+private noncomputable instance
+    smallHilbertFiniteGaloisNormNeighborhoodNumberField :
+    NumberField E₂ :=
+  NumberField.of_module_finite ℚ E₂
+
+private noncomputable instance
+    smallHilbertFiniteGaloisNormNeighborhoodIsGalois :
+    IsGalois E E₂ :=
+  abstractRelativeFixedField_isGalois
+    ℚ (SeparableClosure ℚ) L.field
+    (smallHilbertFiniteGaloisNormNeighborhood K L).field
+    (smallHilbertFiniteGaloisNormNeighborhood K L).below
+    (smallHilbertFiniteGaloisNormNeighborhood K L).normal
+
+private theorem
+    smallHilbertFiniteGaloisNormNeighborhood_fixedField_eq_range :
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (smallHilbertFiniteGaloisNormNeighborhood K L).below).restrictScalars ℚ =
+        AlgHom.fieldRange
+          (smallHilbertNormNeighborhoodEmbedding K L) := by
+  rw [show
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (smallHilbertFiniteGaloisNormNeighborhood K L).below).restrictScalars ℚ =
+        (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+          (smallHilbertFiniteGaloisNormNeighborhoodRaw K L).below).restrictScalars ℚ
+    from
+      rebaseRationalFiniteGaloisSubextension_fixedField_eq
+        (smallHilbertNormNeighborhoodEmbeddedBase_eq K L)
+        (smallHilbertFiniteGaloisNormNeighborhoodRaw K L)]
+  exact
+    InfiniteGalois.fixedField_fixingSubgroup
+      (AlgHom.fieldRange
+        (smallHilbertNormNeighborhoodEmbedding K L))
+
+private noncomputable def
+    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv :
+    N ≃ₐ[ℚ]
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (smallHilbertFiniteGaloisNormNeighborhood K L).below).restrictScalars ℚ :=
+  (smallHilbertNormNeighborhoodEmbedding K L).equivFieldRange.trans
+    (IntermediateField.equivOfEq
+      (smallHilbertFiniteGaloisNormNeighborhood_fixedField_eq_range
+        K L).symm)
+
+@[simp]
+private theorem
+    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv_algebraMap
+    (x : E) :
+    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv K L
+        (algebraMap E N x) =
+      algebraMap E
+        E₂ x := by
+  apply Subtype.ext
+  change
+    smallHilbertNormNeighborhoodEmbedding K L
+        (algebraMap E N x) =
+      (x : SeparableClosure ℚ)
+  exact smallHilbertNormNeighborhoodEmbedding_algebraMap K L x
+
+private noncomputable def
+    smallHilbertFiniteGaloisNormNeighborhoodRelativeTopEquiv :
+    N ≃ₐ[E] E₂ := {
+  smallHilbertFiniteGaloisNormNeighborhoodTopEquiv K L with
+  commutes' := fun x =>
+    smallHilbertFiniteGaloisNormNeighborhoodTopEquiv_algebraMap
+      K L x }
+
+theorem
+    smallHilbertFiniteGaloisNormNeighborhood_ordinaryNorm_le :
+    (_root_.ideleClassNorm E N).range ≤
+      smallHilbertClassFieldNormSubgroup (K := E) := by
+  simpa only [smallHilbertClassFieldNormAmbient] using
+    (closedFiniteIndexClassFieldNormAmbient_normRange_le
+      (K := E) (smallHilbertClassFieldNormSubgroup (K := E))
+      (smallHilbertClassFieldNormSubgroup_isClosed (K := E)))
+
+private theorem
+    smallHilbertFiniteGaloisNormNeighborhood_ordinaryNormRange_eq :
+    (_root_.ideleClassNorm E N).range =
+      (_root_.ideleClassNorm E E₂).range := by
+  simpa only [ordinaryIdeleClassNorm_range_eq_relative] using
+    (ideleClassNorm_range_algEquiv
+      (K := E)
+      (smallHilbertFiniteGaloisNormNeighborhoodRelativeTopEquiv
+        K L)).symm
+
+private theorem
+    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq :
+    ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
+        rationalIdeleClassRepresentation).map
+          (rationalAbstractFixedFieldIdeleClassEquivFixed
+            L.field
+            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
+      (_root_.ideleClassNorm E E₂).range.toAddSubgroup := by
+  change
+    (finiteNormSubgroup rationalIdeleClassRepresentation
+      L.field
+      (smallHilbertFiniteGaloisNormNeighborhood K L).field
+      (smallHilbertFiniteGaloisNormNeighborhood K L).below).map
+        (rationalAbstractFixedFieldIdeleClassEquivFixed
+          L.field
+          (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
+      (_root_.ideleClassNorm E E₂).range.toAddSubgroup
+  exact
+    (map_rationalFiniteNormSubgroup_eq_ordinaryIdeleClassNormRange_concrete
+      L.field
+      (smallHilbertFiniteGaloisNormNeighborhood K L).field
+      (smallHilbertFiniteGaloisNormNeighborhood K L).below
+      (smallHilbertFiniteGaloisNormNeighborhood K L).normal)
+
+/-- The abstract norm map lands directly in the ordinary norm range of the
+chosen neighbourhood.  Composing the two named subgroup equalities here
+keeps downstream membership proofs pointwise. -/
+private theorem
+    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq_ordinaryNormRange :
+    ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
+        rationalIdeleClassRepresentation).map
+          (rationalAbstractFixedFieldIdeleClassEquivFixed
+            L.field
+            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
+      (_root_.ideleClassNorm E N).range.toAddSubgroup :=
+  (smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq K L).trans
+    (congrArg Subgroup.toAddSubgroup
+      (smallHilbertFiniteGaloisNormNeighborhood_ordinaryNormRange_eq K L).symm)
+
+/-- Pointwise form of the combined norm-range equality. -/
+theorem
+    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_mem_ordinaryNormRange
+    (a : Additive (IdeleClassGroup E))
+    (ha :
+      a ∈ ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
+        rationalIdeleClassRepresentation).map
+          (rationalAbstractFixedFieldIdeleClassEquivFixed
+            L.field
+            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom) :
+    a ∈ (_root_.ideleClassNorm E N).range.toAddSubgroup :=
+  (le_of_eq
+    (smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_eq_ordinaryNormRange
+      K L)) ha
+
+
+end RationalFixedField
+end IdealClassFieldTheory
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationSecond.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationSecond.lean
new file mode 100644
index 0000000..768d9b0
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/IdealClassFieldTheory/SmallHilbertTowerRealizationSecond.lean
@@ -0,0 +1,256 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+
+import ClassFieldTheory.GlobalClassFieldTheory.IdealClassFieldTheory.SmallHilbertTowerRealizationNorm
+
+set_option autoImplicit false
+
+open scoped Classical NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace IdealClassFieldTheory
+
+open AlgebraicNumberTheory
+open ClassFormation
+open CyclicCohomology
+open GlobalClassFields
+open KummerTheory
+open LocalClassFieldTheory
+open RamificationTheory
+open Reciprocity
+
+
+attribute [local instance] smallHilbertTowerIdeleClassCommGroup
+  smallHilbertTowerNormAmbientAlgebra
+  smallHilbertTowerNormAmbientIsGalois
+
+section RationalFixedField
+
+variable
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (L : FiniteAbelianSubextension K.field)
+
+
+local notation "E" => abstractFixedField ℚ (SeparableClosure ℚ) L.field
+local notation "N" => smallHilbertClassFieldNormAmbient E
+
+local notation "E₂" => smallHilbertFiniteGaloisNormNeighborhoodTopField K L
+
+/-- The actual finite Galois norm neighbourhood has abstract norm
+subgroup contained in the canonical small-Hilbert subgroup of the
+middle fixed field.  This is the source-producing norm-topology input;
+no norm-openness premise is assumed. -/
+theorem smallHilbertFiniteGaloisNormNeighborhood_normSubgroup_le :
+    ∀ a : ambientFixedAddSubgroup
+        rationalIdeleClassRepresentation L.field,
+      a ∈ smallHilbertFiniteGaloisNormNeighborhoodNormSubgroup K L →
+        a ∈ smallHilbertTowerMiddleNormSubgroup K L := by
+  intro a ha
+  change
+    a ∈ (smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
+      rationalIdeleClassRepresentation at ha
+  have haMap :
+      (rationalAbstractFixedFieldIdeleClassEquivFixed
+          L.field
+          (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm a ∈
+        ((smallHilbertFiniteGaloisNormNeighborhood K L).normSubgroup
+          rationalIdeleClassRepresentation).map
+            (rationalAbstractFixedFieldIdeleClassEquivFixed
+              L.field
+              (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom :=
+    ⟨a, ha, rfl⟩
+  have haOrdinary :
+      (rationalAbstractFixedFieldIdeleClassEquivFixed L.field
+        (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm a ∈
+        (_root_.ideleClassNorm E N).range.toAddSubgroup :=
+    smallHilbertFiniteGaloisNormNeighborhood_abstractNormMap_mem_ordinaryNormRange
+      K L _ haMap
+  exact smallHilbertFiniteGaloisNormNeighborhood_ordinaryNorm_le K L haOrdinary
+
+/-- The canonical small-Hilbert subgroup of the actual middle fixed
+field is open in the genuine norm topology. -/
+theorem smallHilbertNormSubgroupInRationalClassFormation_isNormOpen :
+    IsNormOpen rationalIdeleClassRepresentation L.field
+      (smallHilbertTowerMiddleNormSubgroup K L :
+        Set
+          (ambientFixedAddSubgroup
+            rationalIdeleClassRepresentation L.field)) := by
+  rw [normTopology_addSubgroup_isOpen_iff]
+  refine
+    ⟨smallHilbertFiniteGaloisNormNeighborhood K L, ?_⟩
+  change
+    smallHilbertFiniteGaloisNormNeighborhoodNormSubgroup K L ≤
+      smallHilbertTowerMiddleNormSubgroup K L
+  exact smallHilbertFiniteGaloisNormNeighborhood_normSubgroup_le K L
+
+/-- The second small Hilbert class field as an actual finite abelian
+subextension of the literal first-stage field. -/
+noncomputable def secondSmallHilbertClassFieldSubextension :
+    FiniteAbelianSubextension L.field := by
+  let H : FiniteAbelianSubextension.NormOpenAddSubgroup
+      rationalIdeleClassRepresentation L.field :=
+    ⟨smallHilbertTowerMiddleNormSubgroup K L,
+      smallHilbertNormSubgroupInRationalClassFormation_isNormOpen K L⟩
+  exact
+    Classical.choose
+      (FiniteAbelianSubextension.normSubgroupMap_surjective
+        rationalCyclotomicIdeleClassValuationData
+        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+        (smallHilbertTowerMiddleFiniteAbstractField K L) H)
+
+/-- The second-stage extension realizes exactly the canonical
+small-Hilbert norm subgroup of the actual middle field. -/
+@[simp]
+theorem secondSmallHilbertClassFieldSubextension_normSubgroup :
+    (secondSmallHilbertClassFieldSubextension K L).normSubgroup
+        rationalIdeleClassRepresentation =
+      smallHilbertTowerMiddleNormSubgroup K L := by
+  let H : FiniteAbelianSubextension.NormOpenAddSubgroup
+      rationalIdeleClassRepresentation L.field :=
+    ⟨smallHilbertTowerMiddleNormSubgroup K L,
+      smallHilbertNormSubgroupInRationalClassFormation_isNormOpen K L⟩
+  have h :=
+    Classical.choose_spec
+      (FiniteAbelianSubextension.normSubgroupMap_surjective
+        rationalCyclotomicIdeleClassValuationData
+        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+        (smallHilbertTowerMiddleFiniteAbstractField K L) H)
+  exact congrArg Subtype.val h
+
+private noncomputable abbrev secondSmallHilbertClassFieldTopField : Type :=
+  abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+    (secondSmallHilbertClassFieldSubextension K L).below
+
+local notation "T₂" => secondSmallHilbertClassFieldTopField K L
+
+private noncomputable instance
+    secondSmallHilbertClassFieldSubextensionQuotientFinite :
+    Finite
+      (L.field.toSubgroup ⧸
+        extensionSubgroup L.field
+          (secondSmallHilbertClassFieldSubextension K L).field
+          (secondSmallHilbertClassFieldSubextension K L).below) :=
+  (secondSmallHilbertClassFieldSubextension K L).finite
+
+private noncomputable instance
+    secondSmallHilbertClassFieldTopFiniteDimensional :
+    FiniteDimensional E T₂ :=
+  abstractRelativeFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ) L.field
+    (secondSmallHilbertClassFieldSubextension K L).field
+    (secondSmallHilbertClassFieldSubextension K L).below
+    inferInstance inferInstance
+
+private noncomputable instance
+    secondSmallHilbertClassFieldTopScalarTower :
+    IsScalarTower ℚ E T₂ :=
+  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
+
+private noncomputable instance
+    secondSmallHilbertClassFieldTopAbsoluteFiniteDimensional :
+    FiniteDimensional ℚ T₂ :=
+  FiniteDimensional.trans ℚ E T₂
+
+private noncomputable instance
+    secondSmallHilbertClassFieldTopNumberField :
+    NumberField T₂ :=
+  NumberField.of_module_finite ℚ T₂
+
+private noncomputable instance
+    secondSmallHilbertClassFieldTopIsGalois :
+    IsGalois E T₂ :=
+  abstractRelativeFixedField_isGalois
+    ℚ (SeparableClosure ℚ) L.field
+    (secondSmallHilbertClassFieldSubextension K L).field
+    (secondSmallHilbertClassFieldSubextension K L).below
+    (secondSmallHilbertClassFieldSubextension K L).normal
+
+private theorem
+    secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_actualNormRange :
+    ((secondSmallHilbertClassFieldSubextension K L).normSubgroup
+        rationalIdeleClassRepresentation).map
+          (rationalAbstractFixedFieldIdeleClassEquivFixed
+            L.field
+            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
+      (_root_.ideleClassNorm E T₂).range.toAddSubgroup := by
+  change
+    (finiteNormSubgroup rationalIdeleClassRepresentation
+        L.field
+        (secondSmallHilbertClassFieldSubextension K L).field
+        (secondSmallHilbertClassFieldSubextension K L).below).map
+          (rationalAbstractFixedFieldIdeleClassEquivFixed
+            L.field
+            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
+      (_root_.ideleClassNorm E T₂).range.toAddSubgroup
+  exact
+    map_rationalFiniteNormSubgroup_eq_ordinaryIdeleClassNormRange_concrete
+      L.field
+      (secondSmallHilbertClassFieldSubextension K L).field
+      (secondSmallHilbertClassFieldSubextension K L).below
+      (secondSmallHilbertClassFieldSubextension K L).normal
+
+private theorem
+    secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_smallHilbertNormSubgroup :
+    ((secondSmallHilbertClassFieldSubextension K L).normSubgroup
+        rationalIdeleClassRepresentation).map
+          (rationalAbstractFixedFieldIdeleClassEquivFixed
+            L.field
+            (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)).symm.toAddMonoidHom =
+      (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup := by
+  let e : Additive (IdeleClassGroup E) ≃+
+      ambientFixedAddSubgroup rationalIdeleClassRepresentation L.field :=
+    rationalAbstractFixedFieldIdeleClassEquivFixed L.field
+      (hfinite := smallHilbertTowerMiddleAbstractQuotientFinite K L)
+  let H : AddSubgroup (Additive (IdeleClassGroup E)) :=
+    (smallHilbertClassFieldNormSubgroup (K := E)).toAddSubgroup
+  let back : AddSubgroup
+      (ambientFixedAddSubgroup rationalIdeleClassRepresentation L.field) →
+      AddSubgroup (Additive (IdeleClassGroup E)) :=
+    fun n => n.map e.symm.toAddMonoidHom
+  have hNorm :
+      back ((secondSmallHilbertClassFieldSubextension K L).normSubgroup
+        rationalIdeleClassRepresentation) =
+      back (smallHilbertTowerMiddleNormSubgroup K L) :=
+    congrArg back (secondSmallHilbertClassFieldSubextension_normSubgroup K L)
+  have hCancel : back (smallHilbertTowerMiddleNormSubgroup K L) = H :=
+    AddSubgroup.map_comap_eq_self_of_surjective e.symm.surjective H
+  exact hNorm.trans hCancel
+
+/-- The actual second small Hilbert class field has exactly the intrinsic
+small-Hilbert norm range over the literal middle fixed field. -/
+@[simp]
+theorem secondSmallHilbertClassFieldSubextension_ideleClassNorm_range :
+    (_root_.ideleClassNorm E T₂).range =
+      smallHilbertClassFieldNormSubgroup (K := E) := by
+  apply Subgroup.toAddSubgroup.injective
+  exact
+    (secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_actualNormRange
+      K L).symm.trans
+      (secondSmallHilbertClassFieldSubextension_abstractNormMap_eq_smallHilbertNormSubgroup
+        K L)
+
+/-- Compatibility of the typed middle endpoint with the canonical endpoint
+used by the conjugation API. -/
+theorem smallHilbertTowerMiddleNormSubgroup_eq_conjugationEndpoint :
+    smallHilbertTowerMiddleNormSubgroup K L =
+      smallHilbertNormSubgroupInRationalClassFormation
+        (L.toFiniteGaloisExtension.toFiniteAbstractFieldExtension).field := by
+  have hField :
+      (L.toFiniteGaloisExtension.toFiniteAbstractFieldExtension).field.field =
+        L.field := by
+    rfl
+  unfold smallHilbertNormSubgroupInRationalClassFormation
+  cases hField
+  exact smallHilbertTowerMiddleNormSubgroup_eq_map K L
+
+
+end RationalFixedField
+end IdealClassFieldTheory
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidue.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidue.lean
index 8984e0a..3f1f8ad 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidue.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidue.lean
@@ -4,24 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidue
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.AbstractFixedFieldGlobalNormResidueFixedFields
 
-set_option autoImplicit false
-
-/-!
-# Global norm residue on actual fixed fields
 
-An abstract finite abelian subextension of the rational absolute Galois
-group determines an actual finite abelian extension between its two
-fixed number fields.  This file transports the abstract norm-residue
-symbol directly to the ordinary idele-class norm quotient of those
-fixed fields.
+set_option autoImplicit false
 
-Keeping this construction in one ambient separable closure is essential
-for the norm--restriction diagrams: no independently chosen embedding of
-either field is introduced.
--/
 
 open scoped IsMulCommutative NumberField
 open NumberField
@@ -38,1460 +25,26 @@ open AlgebraicNumberTheory
 open LocalClassFieldTheory
 open RamificationTheory
 
-/-- The algebra structure on the rational separable closure induced by a
-specified rational field embedding.  It is deliberately not an instance:
-different embeddings of the same field need not induce definitionally equal
-algebra structures. -/
-@[reducible]
-noncomputable def rationalEmbeddingSeparableClosureAlgebra
-    {F : Type} [Field F] [Algebra ℚ F]
-    (i : F →ₐ[ℚ] SeparableClosure ℚ) :
-    Algebra F (SeparableClosure ℚ) :=
-  i.toRingHom.toAlgebra
-
-/-- Two rational embeddings of the same number field into the fixed
-rational separable closure differ by an automorphism of that
-separable closure. -/
-theorem exists_numberFieldEmbeddingComparisonAutomorphism
-    {F : Type} [Field F] [NumberField F]
-    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
-    ∃ σ : SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ,
-      ∀ x : F, σ (i x) = j x := by
-  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
-    rationalEmbeddingSeparableClosureAlgebra i
-  let hScalarTower : IsScalarTower ℚ F (SeparableClosure ℚ) :=
-    IsScalarTower.of_algebraMap_eq' i.comp_algebraMap.symm
-  let hSeparable : Algebra.IsSeparable F (SeparableClosure ℚ) :=
-    Algebra.isSeparable_tower_top_of_isSeparable
-      ℚ F (SeparableClosure ℚ)
-  obtain ⟨φ, hφ⟩ :=
-    (IsSepClosed.surjective_domRestrict_of_isSeparable
-      (K := ℚ) (L := F)
-      (M := SeparableClosure ℚ)
-      (E := SeparableClosure ℚ)) j
-  let σ : SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ :=
-    AlgEquiv.ofBijective φ
-      (Normal.toIsAlgebraic.algHom_bijective₂
-        φ (AlgHom.id ℚ (SeparableClosure ℚ))).1
-  refine ⟨σ, ?_⟩
-  intro x
-  have hx :=
-    congrArg (fun ψ : F →ₐ[ℚ] SeparableClosure ℚ => ψ x) hφ
-  exact hx
-
-/-- The canonical comparison automorphism between two rational
-embeddings of one number field into the fixed separable closure. -/
-noncomputable def numberFieldEmbeddingComparisonAutomorphism
-    {F : Type} [Field F] [NumberField F]
-    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
-    SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ :=
-  Classical.choose
-    (exists_numberFieldEmbeddingComparisonAutomorphism i j)
-
-/-- The comparison automorphism carries the first embedded copy of the
-number field to the second one pointwise. -/
-@[simp]
-theorem numberFieldEmbeddingComparisonAutomorphism_apply
-    {F : Type} [Field F] [NumberField F]
-    (i j : F →ₐ[ℚ] SeparableClosure ℚ)
-    (x : F) :
-    numberFieldEmbeddingComparisonAutomorphism i j (i x) =
-      j x :=
-  Classical.choose_spec
-    (exists_numberFieldEmbeddingComparisonAutomorphism i j) x
-
-/-- Conjugating the fixing subgroup of one embedded copy of a number
-field by the comparison automorphism gives the fixing subgroup of the
-other embedded copy. -/
-theorem conjugateClosedFixingSubgroup_embeddingRange
-    {F : Type} [Field F] [NumberField F]
-    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
-    conjugateClosedSubgroup
-        (RamificationTheory.closedFixingSubgroup
-          ℚ (SeparableClosure ℚ) i.fieldRange)
-        (numberFieldEmbeddingComparisonAutomorphism j i) =
-      RamificationTheory.closedFixingSubgroup
-        ℚ (SeparableClosure ℚ) j.fieldRange := by
-  let s :=
-    numberFieldEmbeddingComparisonAutomorphism j i
-  ext τ
-  change
-    τ ∈ conjugateClosedSubgroup
-        (closedFixingSubgroup ℚ (SeparableClosure ℚ) i.fieldRange) s ↔
-      τ ∈ closedFixingSubgroup ℚ (SeparableClosure ℚ) j.fieldRange
-  rw [conjugateClosedSubgroup_mem]
-  change
-    s * τ * s⁻¹ ∈ i.fieldRange.fixingSubgroup ↔
-      τ ∈ j.fieldRange.fixingSubgroup
-  rw [IntermediateField.mem_fixingSubgroup_iff,
-    IntermediateField.mem_fixingSubgroup_iff]
-  constructor
-  · intro h x hx
-    rcases hx with ⟨y, rfl⟩
-    have hi := h (i y) ⟨y, rfl⟩
-    have hs :
-        s (j y) = i y :=
-      numberFieldEmbeddingComparisonAutomorphism_apply j i y
-    change s (τ (s.symm (i y))) = i y at hi
-    have hpre : s.symm (i y) = j y := by
-      rw [← hs, s.symm_apply_apply]
-    rw [hpre, ← hs] at hi
-    exact s.injective hi
-  · intro h x hx
-    rcases hx with ⟨y, rfl⟩
-    have hj := h (j y) ⟨y, rfl⟩
-    have hs :
-        s (j y) = i y :=
-      numberFieldEmbeddingComparisonAutomorphism_apply j i y
-    change s (τ (s.symm (i y))) = i y
-    have hpre : s.symm (i y) = j y := by
-      rw [← hs, s.symm_apply_apply]
-    rw [hpre, hj, hs]
-
-section EmbeddedNumberFieldRealization
-
-local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
-    {F : Type} [Field F] [NumberField F]
-    : IsMulCommutative (IdeleClassGroup F) :=
-  ⟨⟨fun a b => mul_comm a b⟩⟩
-
-local instance numberFieldEmbeddedIdeleClassSubgroupNormal
-    {F : Type} [Field F] [NumberField F]
-    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
-  N.normal_of_isMulCommutative
-
-variable
-    (K L : Type) [Field K] [NumberField K]
-    [Field L] [NumberField L] [Algebra K L]
-
-/-- The lower embedding obtained by restricting an explicitly supplied
-embedding of the top field into the rational separable closure. -/
-noncomputable def numberFieldEmbeddedLowerEmbedding
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    K →ₐ[ℚ] SeparableClosure ℚ :=
-  j.comp (IsScalarTower.toAlgHom ℚ K L)
-
-/-- The exact algebra structure on the rational separable closure induced by
-the lower embedding of an explicitly embedded number-field tower.  Keeping
-this as a reducible definition lets every use of the associated separable-
-closure equivalence share one definitionally identical algebra structure. -/
-@[reducible]
-noncomputable def numberFieldEmbeddedSeparableClosureAlgebra
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Algebra K (SeparableClosure ℚ) :=
-  rationalEmbeddingSeparableClosureAlgebra
-    (numberFieldEmbeddedLowerEmbedding K L j)
-
-/-- The fixing subgroup of the explicitly embedded lower field. -/
-abbrev numberFieldEmbeddedBaseSubgroup
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    ClosedSubgroup
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-  closedFixingSubgroup ℚ (SeparableClosure ℚ)
-    (numberFieldEmbeddedLowerEmbedding K L j).fieldRange
-
-/-- The fixing subgroup of the explicitly embedded top field. -/
-abbrev numberFieldEmbeddedTopSubgroup
-    (_K L : Type) [Field L] [NumberField L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    ClosedSubgroup
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-  closedFixingSubgroup ℚ (SeparableClosure ℚ) j.fieldRange
-
-/-- The top fixing subgroup lies in the lower fixing subgroup. -/
-theorem numberFieldEmbeddedTopSubgroup_le_baseSubgroup
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    (numberFieldEmbeddedTopSubgroup K L j).toSubgroup ≤
-      (numberFieldEmbeddedBaseSubgroup K L j).toSubgroup := by
-  change
-    j.fieldRange.fixingSubgroup ≤
-      (numberFieldEmbeddedLowerEmbedding K L j).fieldRange.fixingSubgroup
-  apply
-    (numberFieldEmbeddedLowerEmbedding K L j).fieldRange.fixingSubgroup_le
-  intro x hx
-  rcases hx with ⟨y, rfl⟩
-  exact ⟨algebraMap K L y, rfl⟩
-
-/-- The separable closure of the actual lower field, identified with
-the rational separable closure carrying the algebra structure induced
-by an explicit compatible embedding. -/
-noncomputable def numberFieldEmbeddedSeparableClosureEquiv
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    letI : Algebra K (SeparableClosure ℚ) :=
-      numberFieldEmbeddedSeparableClosureAlgebra K L j
-    SeparableClosure K ≃ₐ[K] SeparableClosure ℚ := by
-  letI : Algebra K (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra K L j
-  letI hScalarTower : IsScalarTower ℚ K (SeparableClosure ℚ) :=
-    IsScalarTower.of_algebraMap_eq'
-      (numberFieldEmbeddedLowerEmbedding K L j).comp_algebraMap.symm
-  letI hseparable : Algebra.IsSeparable K (SeparableClosure ℚ) :=
-    Algebra.isSeparable_tower_top_of_isSeparable
-      ℚ K (SeparableClosure ℚ)
-  letI hSepClosure : IsSepClosure K (SeparableClosure ℚ) :=
-    ⟨IsSepClosure.sep_closed ℚ, hseparable⟩
-  exact
-    IsSepClosure.equiv K
-      (SeparableClosure K) (SeparableClosure ℚ)
-
-/-- The relative subgroup arising from an explicit compatible
-number-field embedding is normal. -/
-theorem numberFieldEmbeddedExtensionSubgroup_normal
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    (CyclicCohomology.extensionSubgroup
-      (numberFieldEmbeddedBaseSubgroup K L j)
-      (numberFieldEmbeddedTopSubgroup K L j)
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal := by
-  let i := numberFieldEmbeddedLowerEmbedding K L j
-  let hAlgebra : Algebra K (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra K L j
-  let e := numberFieldEmbeddedSeparableClosureEquiv K L j
-  change
-    (CyclicCohomology.extensionSubgroup
-      (closedFixingSubgroup ℚ (SeparableClosure ℚ)
-        (AlgHom.fieldRange i))
-      (closedFixingSubgroup ℚ (SeparableClosure ℚ)
-        (AlgHom.fieldRange j)) _).Normal
-  exact ambientEmbeddedExtensionSubgroup_normal ℚ K L j e
-
-/-- The normality witness for an explicitly embedded tower, registered at
-the precise subgroup used by the downstream quotient constructions. -/
-noncomputable local instance
-    numberFieldEmbeddedExtensionSubgroupNormal
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    (CyclicCohomology.extensionSubgroup
-      (numberFieldEmbeddedBaseSubgroup K L j)
-      (numberFieldEmbeddedTopSubgroup K L j)
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
-  numberFieldEmbeddedExtensionSubgroup_normal K L j
-
-/-- The relative quotient arising from an explicit compatible
-number-field embedding is finite. -/
-theorem numberFieldEmbeddedExtensionQuotient_finite
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Finite
-      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (numberFieldEmbeddedBaseSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) := by
-  let i := numberFieldEmbeddedLowerEmbedding K L j
-  let hAlgebra : Algebra K (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra K L j
-  let e := numberFieldEmbeddedSeparableClosureEquiv K L j
-  change
-    Finite
-      ((closedFixingSubgroup ℚ (SeparableClosure ℚ)
-          (AlgHom.fieldRange i)).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (closedFixingSubgroup ℚ (SeparableClosure ℚ)
-            (AlgHom.fieldRange i))
-          (closedFixingSubgroup ℚ (SeparableClosure ℚ)
-            (AlgHom.fieldRange j)) _)
-  exact ambientEmbeddedExtensionQuotient_finite ℚ K L j e
-
-/-- The relative-index witness for an explicitly embedded tower, registered
-at the exact quotient consumed by `FiniteNormQuotient`. -/
-noncomputable local instance
-    numberFieldEmbeddedExtensionQuotientFinite
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Finite
-      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (numberFieldEmbeddedBaseSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
-  numberFieldEmbeddedExtensionQuotient_finite K L j
-
-/-- The finite abstract field determined by the lower member of an
-explicitly embedded number-field tower. -/
-noncomputable abbrev numberFieldEmbeddedFiniteAbstractField
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) where
-  field := numberFieldEmbeddedBaseSubgroup K L j
-  finite := by
-    simpa only [numberFieldEmbeddedBaseSubgroup] using
-      (ambientEmbeddedAbsoluteQuotientFinite
-        ℚ K (numberFieldEmbeddedLowerEmbedding K L j))
-
-/-- The absolute-index witness for the lower member of an explicitly embedded
-tower, registered at its specialized quotient type. -/
-noncomputable local instance
-    numberFieldEmbeddedAbsoluteQuotientFinite
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Finite
-      ((baseField
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-          (numberFieldEmbeddedBaseSubgroup K L j)
-          (le_baseField
-            (numberFieldEmbeddedBaseSubgroup K L j))) :=
-  (numberFieldEmbeddedFiniteAbstractField K L j).finite
-
-/-- The finite Galois subextension determined by an explicitly embedded
-number-field tower. -/
-noncomputable abbrev numberFieldEmbeddedFiniteGaloisSubextension
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteGaloisSubextension
-      (numberFieldEmbeddedBaseSubgroup K L j) where
-  field := numberFieldEmbeddedTopSubgroup K L j
-  below := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
-  normal := numberFieldEmbeddedExtensionSubgroup_normal K L j
-  finite := numberFieldEmbeddedExtensionQuotient_finite K L j
-
-/-- Shared finite-dimensional data for the fixed field of the lower subgroup
-in an explicitly embedded number-field tower. -/
-noncomputable local instance
-    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteDimensional ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L j)) :=
-  abstractFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ)
-    (numberFieldEmbeddedBaseSubgroup K L j)
-    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
-
-/-- Shared relative finite-dimensional data for the two fixed fields of an
-explicitly embedded number-field tower. -/
-noncomputable local instance
-    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteDimensional
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L j))
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
-  abstractRelativeFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ)
-    (numberFieldEmbeddedBaseSubgroup K L j)
-    (numberFieldEmbeddedTopSubgroup K L j)
-    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
-    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
-    (numberFieldEmbeddedExtensionQuotientFinite K L j)
-
-local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    IsScalarTower ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L j))
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
-  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
-
-noncomputable local instance
-    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteDimensional ℚ
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
-  FiniteDimensional.trans ℚ
-    (abstractFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedBaseSubgroup K L j))
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
-
-noncomputable local instance
-    numberFieldEmbeddedAbstractFixedFieldNumberField
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    NumberField
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L j)) :=
-  NumberField.of_module_finite ℚ
-    (abstractFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedBaseSubgroup K L j))
-
-noncomputable local instance
-    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    NumberField
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
-  NumberField.of_module_finite ℚ
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
-
-noncomputable local instance
-    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteDimensional ℚ
-      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
-  change FiniteDimensional ℚ
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
-  infer_instance
-
-noncomputable local instance
-    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    NumberField
-      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
-  NumberField.of_module_finite ℚ _
-
-noncomputable local instance
-    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Algebra
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L j))
-      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
-  (IntermediateField.inclusion
-    (abstractFixedField_le ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
-
-noncomputable local instance
-    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    IsGalois
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L j))
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
-  abstractRelativeFixedField_isGalois
-    ℚ (SeparableClosure ℚ)
-    (numberFieldEmbeddedBaseSubgroup K L j)
-    (numberFieldEmbeddedTopSubgroup K L j)
-    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
-    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
-
-/-- The quotient of the two explicitly embedded fixing subgroups is
-the actual relative Galois group. -/
-noncomputable def
-    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient ≃*
-      Gal(L / K) := by
-  let i := numberFieldEmbeddedLowerEmbedding K L j
-  letI hAlgebra : Algebra K (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra K L j
-  let e := numberFieldEmbeddedSeparableClosureEquiv K L j
-  let H₀ :=
-    closedFixingSubgroup ℚ (SeparableClosure ℚ)
-      (AlgHom.fieldRange i)
-  let J₀ :=
-    closedFixingSubgroup ℚ (SeparableClosure ℚ)
-      (AlgHom.fieldRange j)
-  let hJH : J₀.toSubgroup ≤ H₀.toSubgroup := by
-    change j.fieldRange.fixingSubgroup ≤ i.fieldRange.fixingSubgroup
-    apply i.fieldRange.fixingSubgroup_le
-    intro x hx
-    rcases hx with ⟨y, rfl⟩
-    exact ⟨algebraMap K L y, rfl⟩
-  letI : (CyclicCohomology.extensionSubgroup H₀ J₀ hJH).Normal :=
-    ambientEmbeddedExtensionSubgroup_normal ℚ K L j e
-  change
-    (H₀.toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup H₀ J₀ hJH) ≃*
-      Gal(L / K)
-  exact ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ K L j e
-
-/-- The original lower field is canonically equivalent to the fixed
-field of its explicitly embedded fixing subgroup. -/
-noncomputable def numberFieldEmbeddedAbstractBaseFieldEquiv
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    K ≃ₐ[ℚ]
-      abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L j) :=
-  (numberFieldEmbeddedLowerEmbedding K L j).equivFieldRange.trans
-    (IntermediateField.equivOfEq
-      (InfiniteGalois.fixedField_fixingSubgroup
-        (numberFieldEmbeddedLowerEmbedding K L j).fieldRange).symm)
-
-/-- The original top field is canonically equivalent to the relative
-fixed field of its explicitly embedded fixing subgroup. -/
-noncomputable def numberFieldEmbeddedAbstractTopFieldEquiv
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    L ≃ₐ[ℚ]
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup
-          K L j)).restrictScalars ℚ :=
-  j.equivFieldRange.trans
-    (IntermediateField.equivOfEq
-      (InfiniteGalois.fixedField_fixingSubgroup j.fieldRange).symm)
-
-/-- The two explicit fixed-field equivalences commute with the tower
-algebra maps. -/
-@[simp]
-theorem numberFieldEmbeddedAbstractFieldEquiv_algebraMap
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (x : K) :
-    numberFieldEmbeddedAbstractTopFieldEquiv K L j
-        (algebraMap K L x) =
-      algebraMap
-        (abstractFixedField ℚ (SeparableClosure ℚ)
-          (numberFieldEmbeddedBaseSubgroup K L j))
-        (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
-        (numberFieldEmbeddedAbstractBaseFieldEquiv K L j x) := by
-  apply Subtype.ext
-  rfl
-
-/-- The ordinary idele class group of the explicitly embedded lower
-field, transported to the fixed part of the rational absolute
-idele-class representation. -/
-noncomputable def numberFieldEmbeddedIdeleClassEquivAmbientFixed
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Additive (IdeleClassGroup K) ≃+
-      ambientFixedAddSubgroup
-        rationalIdeleClassRepresentation
-        (numberFieldEmbeddedBaseSubgroup K L j) := by
-  let H := numberFieldEmbeddedBaseSubgroup K L j
-  exact
-    (MulEquiv.toAdditive
-      (ideleClassCongr
-        (numberFieldEmbeddedAbstractBaseFieldEquiv K L j))).trans
-      (rationalAbstractFixedFieldIdeleClassEquivFixed H)
-
-/-- The abstract finite norm quotient of an explicitly embedded tower
-is its genuine ordinary idele-class norm quotient. -/
-noncomputable def
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteNormQuotient rationalIdeleClassRepresentation
-        (numberFieldEmbeddedBaseSubgroup K L j)
-        (numberFieldEmbeddedTopSubgroup K L j)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) ≃+
-      Additive
-        (IdeleClassGroup K ⧸
-          (_root_.ideleClassNorm K L).range) := by
-  let hnormal :=
-    numberFieldEmbeddedExtensionSubgroupNormal K L j
-  let H := numberFieldEmbeddedBaseSubgroup K L j
-  let J := numberFieldEmbeddedTopSubgroup K L j
-  let hJH :=
-    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
-  let fixedFieldEquiv :=
-    rationalFiniteNormQuotientEquivIdeleClassNormQuotient
-      H J hJH hnormal
-  let actualFieldEquiv :=
-    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
-      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
-  exact
-    fixedFieldEquiv.trans
-      (MulEquiv.toAdditive actualFieldEquiv.symm)
-
-private theorem numberFieldEmbeddedOrdinaryNormQuotientCongr_symm_mk
-    {K₀ L₀ K₁ L₁ : Type}
-    [Field K₀] [NumberField K₀]
-    [Field L₀] [NumberField L₀] [Algebra K₀ L₀]
-    [Field K₁] [NumberField K₁]
-    [Field L₁] [NumberField L₁] [Algebra K₁ L₁]
-    (eK : K₀ ≃ₐ[ℚ] K₁)
-    (eL : L₀ ≃ₐ[ℚ] L₁)
-    (h : ∀ x : K₀,
-      eL (algebraMap K₀ L₀ x) =
-        algebraMap K₁ L₁ (eK x))
-    (c : IdeleClassGroup K₁) :
-    (ordinaryIdeleClassNormQuotientCongrOfAlgEquiv eK eL h).symm
-        (QuotientGroup.mk'
-          (_root_.ideleClassNorm K₁ L₁).range c) =
-      QuotientGroup.mk'
-        (_root_.ideleClassNorm K₀ L₀).range
-        ((ideleClassCongr eK).symm c) := by
-  let e := ordinaryIdeleClassNormQuotientCongrOfAlgEquiv eK eL h
-  apply e.injective
-  rw [e.apply_symm_apply,
-    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv_mk,
-    MulEquiv.apply_symm_apply]
-
-private noncomputable def numberFieldEmbeddedFiniteNormClassPublicValue
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    Additive
-      (IdeleClassGroup K ⧸
-        (_root_.ideleClassNorm K L).range) :=
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
-        K L j
-        (finiteNormClass rationalIdeleClassRepresentation
-          (numberFieldEmbeddedBaseSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
-          a)
-
-private noncomputable def numberFieldEmbeddedFiniteNormClassExpectedValue
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    Additive
-      (IdeleClassGroup K ⧸
-        (_root_.ideleClassNorm K L).range) :=
-  Additive.ofMul
-    (QuotientGroup.mk'
-      (_root_.ideleClassNorm K L).range
-      (Additive.toMul
-        ((numberFieldEmbeddedIdeleClassEquivAmbientFixed K L j).symm a)))
-
-private noncomputable def
-    numberFieldEmbeddedFiniteNormClassDirectComparisonValue
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    Additive
-      (IdeleClassGroup K ⧸
-        (_root_.ideleClassNorm K L).range) := by
-  let hnormal :=
-    numberFieldEmbeddedExtensionSubgroupNormal K L j
-  let H := numberFieldEmbeddedBaseSubgroup K L j
-  let J := numberFieldEmbeddedTopSubgroup K L j
-  let hJH :=
-    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
-  let fixedFieldEquiv :=
-    rationalFiniteNormQuotientEquivIdeleClassNormQuotient
-      H J hJH hnormal
-  let actualFieldEquiv :=
-    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
-      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
-  exact
-    MulEquiv.toAdditive actualFieldEquiv.symm
-      (fixedFieldEquiv
-        (finiteNormClass rationalIdeleClassRepresentation
-          H J hJH a))
-
-private theorem
-    numberFieldEmbeddedFiniteNormClassPublicValue_eq_directComparison
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    numberFieldEmbeddedFiniteNormClassPublicValue K L j a =
-      numberFieldEmbeddedFiniteNormClassDirectComparisonValue K L j a := by
-  unfold numberFieldEmbeddedFiniteNormClassPublicValue
-  unfold numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
-  unfold numberFieldEmbeddedFiniteNormClassDirectComparisonValue
-  rfl
-
-private noncomputable def numberFieldEmbeddedActualNormClassRepresentativeValue
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    Additive
-      (IdeleClassGroup K ⧸
-        (_root_.ideleClassNorm K L).range) := by
-  let H := numberFieldEmbeddedBaseSubgroup K L j
-  let hJH := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
-  let F := abstractFixedField ℚ (SeparableClosure ℚ) H
-  let E := abstractRelativeFixedField ℚ (SeparableClosure ℚ) hJH
-  let actualFieldEquiv :=
-    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
-      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
-  exact
-    Additive.ofMul
-      (actualFieldEquiv.symm
-        (QuotientGroup.mk'
-          (_root_.ideleClassNorm F E).range
-          (Additive.toMul
-            ((rationalAbstractFixedFieldIdeleClassEquivFixed H).symm a))))
-
-private theorem
-    numberFieldEmbeddedFiniteNormClassDirectComparison_eq_actualValue
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    numberFieldEmbeddedFiniteNormClassDirectComparisonValue K L j a =
-      numberFieldEmbeddedActualNormClassRepresentativeValue K L j a := by
-  let hnormal := numberFieldEmbeddedExtensionSubgroupNormal K L j
-  let H := numberFieldEmbeddedBaseSubgroup K L j
-  let J := numberFieldEmbeddedTopSubgroup K L j
-  let hJH := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
-  let actualFieldEquiv :=
-    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
-      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
-  have hfixed :=
-    rationalFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass
-      H J hJH hnormal a
-  unfold numberFieldEmbeddedFiniteNormClassDirectComparisonValue
-  unfold numberFieldEmbeddedActualNormClassRepresentativeValue
-  exact congrArg (MulEquiv.toAdditive actualFieldEquiv.symm) hfixed
-
-private theorem numberFieldEmbeddedActualNormClassRepresentativeValue_eq_expected
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    numberFieldEmbeddedActualNormClassRepresentativeValue K L j a =
-      numberFieldEmbeddedFiniteNormClassExpectedValue K L j a := by
-  let H := numberFieldEmbeddedBaseSubgroup K L j
-  let hJH := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
-  let F := abstractFixedField ℚ (SeparableClosure ℚ) H
-  let E := abstractRelativeFixedField ℚ (SeparableClosure ℚ) hJH
-  let actualFieldEquiv :=
-    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
-      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
-      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
-  let c : IdeleClassGroup F :=
-    Additive.toMul
-      ((rationalAbstractFixedFieldIdeleClassEquivFixed H).symm a)
-  unfold numberFieldEmbeddedActualNormClassRepresentativeValue
-  unfold numberFieldEmbeddedFiniteNormClassExpectedValue
-  change
-    Additive.ofMul
-        (actualFieldEquiv.symm
-          (QuotientGroup.mk'
-            (_root_.ideleClassNorm F E).range c)) =
-      Additive.ofMul
-        (QuotientGroup.mk'
-          (_root_.ideleClassNorm K L).range
-          ((ideleClassCongr
-            (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)).symm c))
-  exact
-    congrArg Additive.ofMul
-      (numberFieldEmbeddedOrdinaryNormQuotientCongr_symm_mk
-        (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
-        (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
-        (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j) c)
-
-/-- On a finite norm-class representative, the explicit fixed-field
-comparison is the genuine ordinary idele-class quotient. -/
-@[simp]
-theorem
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)) :
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient K L j
-        (finiteNormClass rationalIdeleClassRepresentation
-          (numberFieldEmbeddedBaseSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) a) =
-      Additive.ofMul
-        (QuotientGroup.mk'
-          (_root_.ideleClassNorm K L).range
-          (Additive.toMul
-            ((numberFieldEmbeddedIdeleClassEquivAmbientFixed K L j).symm a))) := by
-  change
-    numberFieldEmbeddedFiniteNormClassPublicValue K L j a =
-      numberFieldEmbeddedFiniteNormClassExpectedValue K L j a
-  exact
-    (numberFieldEmbeddedFiniteNormClassPublicValue_eq_directComparison
-      K L j a).trans
-      ((numberFieldEmbeddedFiniteNormClassDirectComparison_eq_actualValue
-        K L j a).trans
-        (numberFieldEmbeddedActualNormClassRepresentativeValue_eq_expected
-          K L j a))
-
-/-- On an ordinary idele class, the explicit fixed-part realization
-followed by the abstract finite norm-class map is the genuine quotient
-class modulo the ordinary idele-class norm. -/
-@[simp]
-theorem
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
-    [FiniteDimensional K L] [IsGalois K L]
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (c : IdeleClassGroup K) :
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient K L j
-        (finiteNormClass rationalIdeleClassRepresentation
-          (numberFieldEmbeddedBaseSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
-          (numberFieldEmbeddedIdeleClassEquivAmbientFixed
-            K L j (Additive.ofMul c))) =
-      Additive.ofMul
-        (QuotientGroup.mk'
-          (_root_.ideleClassNorm K L).range c) := by
-  simpa only [AddEquiv.symm_apply_apply, toMul_ofMul] using
-    (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass
-      K L j
-      (numberFieldEmbeddedIdeleClassEquivAmbientFixed
-        K L j (Additive.ofMul c)))
-
-variable [FiniteDimensional K L] [IsAbelianGalois K L]
-
-/-- The abelianized quotient of the explicitly embedded tower is the
-actual abelian Galois group. -/
-noncomputable def
-    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Additive
-      (Abelianization
-        (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient) ≃+
-      Additive Gal(L / K) :=
-  MulEquiv.toAdditive
-    ((MulEquiv.abelianizationCongr
-      (numberFieldEmbeddedExtensionQuotientEquivGaloisGroup K L j)).trans
-        (Abelianization.equivOfComm :
-          Gal(L / K) ≃*
-            Abelianization Gal(L / K)).symm)
-
-/-- The actual global norm-residue equivalence constructed from an
-explicit compatible embedding of a finite abelian number-field
-extension into the rational separable closure. -/
-noncomputable def globalNormResidueEquivOfEmbedding
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    Additive
-        (IdeleClassGroup K ⧸
-          (_root_.ideleClassNorm K L).range) ≃+
-      Additive Gal(L / K) := by
-  let eNorm :
-      FiniteNormQuotient rationalIdeleClassRepresentation
-          (numberFieldEmbeddedBaseSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup K L j)
-          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) ≃+
-        Additive
-          (Abelianization
-            (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient) :=
-    rationalCyclotomicDegreeData.normResidueSymbol
-      rationalIdeleClassRepresentation
-      rationalCyclotomicIdeleClassValuationData
-      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-      (numberFieldEmbeddedFiniteAbstractField K L j)
-      (numberFieldEmbeddedFiniteGaloisSubextension K L j)
-  exact
-    (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
-        K L j).symm.trans
-      (eNorm.trans
-        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-          K L j))
-
-/-- The explicit-embedding norm-residue equivalence on a finite norm class. -/
-theorem globalNormResidueEquivOfEmbedding_finiteNormClass
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (x : FiniteNormQuotient rationalIdeleClassRepresentation
-      (numberFieldEmbeddedBaseSubgroup K L j)
-      (numberFieldEmbeddedTopSubgroup K L j)
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :
-    globalNormResidueEquivOfEmbedding K L j
-        (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
-          K L j x) =
-      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup K L j
-        (rationalCyclotomicDegreeData.normResidueSymbol
-          rationalIdeleClassRepresentation
-          rationalCyclotomicIdeleClassValuationData
-          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-          (numberFieldEmbeddedFiniteAbstractField K L j)
-          (numberFieldEmbeddedFiniteGaloisSubextension K L j) x) := by
-  simp only [globalNormResidueEquivOfEmbedding, AddEquiv.trans_apply,
-    AddEquiv.symm_apply_apply]
-
-/-- The global norm-residue homomorphism obtained from an explicit
-compatible embedding. -/
-noncomputable def globalNormResidueMonoidHomOfEmbedding
-    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
-    IdeleClassGroup K →* Gal(L / K) := by
-  let e :
-      (IdeleClassGroup K ⧸
-          (_root_.ideleClassNorm K L).range) ≃*
-        Gal(L / K) :=
-    AddEquiv.toMultiplicative
-      (globalNormResidueEquivOfEmbedding K L j)
-  exact
-    e.toMonoidHom.comp
-      (QuotientGroup.mk'
-        (_root_.ideleClassNorm K L).range)
-
-/-- Evaluation of the explicit-embedding global norm-residue map is
-the abstract norm-residue symbol evaluated on the corresponding genuine
-fixed-part finite norm class. -/
-@[simp]
-theorem globalNormResidueMonoidHomOfEmbedding_apply
-    (j : L →ₐ[ℚ] SeparableClosure ℚ)
-    (c : IdeleClassGroup K) :
-    globalNormResidueMonoidHomOfEmbedding K L j c =
-      Additive.toMul
-        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-          K L j
-          (rationalCyclotomicDegreeData.normResidueSymbol
-            rationalIdeleClassRepresentation
-            rationalCyclotomicIdeleClassValuationData
-            rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-            (numberFieldEmbeddedFiniteAbstractField K L j)
-            (numberFieldEmbeddedFiniteGaloisSubextension K L j)
-            (finiteNormClass rationalIdeleClassRepresentation
-              (numberFieldEmbeddedBaseSubgroup K L j)
-              (numberFieldEmbeddedTopSubgroup K L j)
-              (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
-              (numberFieldEmbeddedIdeleClassEquivAmbientFixed
-                K L j (Additive.ofMul c))))) := by
-  have hclass :=
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
-      K L j c
-  change
-    Additive.toMul
-      (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-        K L j
-        (rationalCyclotomicDegreeData.normResidueSymbol
-          rationalIdeleClassRepresentation
-          rationalCyclotomicIdeleClassValuationData
-          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-          (numberFieldEmbeddedFiniteAbstractField K L j)
-          (numberFieldEmbeddedFiniteGaloisSubextension K L j)
-          ((numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
-            K L j).symm
-            (Additive.ofMul
-              (QuotientGroup.mk'
-                (_root_.ideleClassNorm K L).range c))))) =
-      _
-  rw [← hclass,
-    (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
-      K L j).symm_apply_apply]
-
-omit [FiniteDimensional K L] [IsAbelianGalois K L] in
-/-- The ambient-fixed idèle-class transport for the standard embedding is
-the same map as the transport for an explicitly supplied embedding.  This
-comparison is kept at the transport boundary, before forming norm quotients
-or applying reciprocity. -/
-theorem numberFieldTowerIdeleClassEquivAmbientFixed_eq_embedded_standard :
-    numberFieldTowerIdeleClassEquivAmbientFixed K L =
-      numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
-        (numberFieldSeparableClosureEmbedding L) := by
-  let j := numberFieldSeparableClosureEmbedding L
-  have hBase :
-      numberFieldTowerAbstractBaseFieldEquiv K L =
-        numberFieldEmbeddedAbstractBaseFieldEquiv K L j := by
-    rfl
-  unfold numberFieldTowerIdeleClassEquivAmbientFixed
-    numberFieldEmbeddedIdeleClassEquivAmbientFixed
-  rw [hBase]
-  dsimp only
-  congr 1
-
-/- At the chosen embedding, both constructions use the same fixed tower and
-abstract reciprocity data.  We compare their values on the particular fixed
-idele class needed below; the two implementations of the finite norm-quotient
-equivalence are deliberately not compared as dependent structures. -/
-private theorem numberFieldTowerNormResidueValue_eq_embedded_standard
-    (c : IdeleClassGroup K) :
-    numberFieldTowerAbelianizedExtensionQuotientEquivGaloisGroup K L
-        (rationalCyclotomicDegreeData.normResidueSymbol
-          rationalIdeleClassRepresentation
-          rationalCyclotomicIdeleClassValuationData
-          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-          (numberFieldTowerReciprocityFiniteAbstractField K L)
-          (numberFieldTowerFiniteGaloisSubextension K L)
-          (finiteNormClass rationalIdeleClassRepresentation
-            (numberFieldTowerBaseSubgroup K L)
-            (numberFieldTowerTopSubgroup L)
-            (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
-            (numberFieldTowerIdeleClassEquivAmbientFixed K L (Additive.ofMul c)))) =
-      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-        K L (numberFieldSeparableClosureEmbedding L)
-        (rationalCyclotomicDegreeData.normResidueSymbol
-          rationalIdeleClassRepresentation
-          rationalCyclotomicIdeleClassValuationData
-          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-          (numberFieldEmbeddedFiniteAbstractField K L
-            (numberFieldSeparableClosureEmbedding L))
-          (numberFieldEmbeddedFiniteGaloisSubextension K L
-            (numberFieldSeparableClosureEmbedding L))
-          (finiteNormClass rationalIdeleClassRepresentation
-            (numberFieldEmbeddedBaseSubgroup K L
-              (numberFieldSeparableClosureEmbedding L))
-            (numberFieldEmbeddedTopSubgroup K L
-              (numberFieldSeparableClosureEmbedding L))
-            (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L
-              (numberFieldSeparableClosureEmbedding L))
-            (numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
-              (numberFieldSeparableClosureEmbedding L) (Additive.ofMul c)))) := by
-  have hIdeleClassEquiv :
-      numberFieldTowerIdeleClassEquivAmbientFixed K L =
-        numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
-          (numberFieldSeparableClosureEmbedding L) := by
-    exact numberFieldTowerIdeleClassEquivAmbientFixed_eq_embedded_standard K L
-  have hFiniteAbstractField :
-      numberFieldTowerReciprocityFiniteAbstractField K L =
-        numberFieldEmbeddedFiniteAbstractField K L
-          (numberFieldSeparableClosureEmbedding L) := by
-    rfl
-  have hSubextension :
-      numberFieldTowerFiniteGaloisSubextension K L =
-        numberFieldEmbeddedFiniteGaloisSubextension K L
-          (numberFieldSeparableClosureEmbedding L) := by
-    rfl
-  have hGaloisComparison :
-      numberFieldTowerAbelianizedExtensionQuotientEquivGaloisGroup K L =
-        numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-          K L (numberFieldSeparableClosureEmbedding L) := by
-    rfl
-  simp only [← hGaloisComparison]
-  cases hFiniteAbstractField
-  cases hSubextension
-  rw [← hIdeleClassEquiv]
-  rfl
-
-/-- At the standard embedding, the two global norm-residue equivalences
-agree on the actual norm quotient.  The comparison is extensional: it uses
-surjectivity of the quotient map and the established evaluation formulas,
-not definitional equality of the two quotient constructions. -/
-theorem globalNormResidueEquiv_eq_ofEmbedding_standard :
-    globalNormResidueEquiv K L =
-      globalNormResidueEquivOfEmbedding K L
-        (numberFieldSeparableClosureEmbedding L) := by
-  apply AddEquiv.ext
-  intro q
-  obtain ⟨c, hc⟩ :=
-    QuotientGroup.mk'_surjective
-      (_root_.ideleClassNorm K L).range (Additive.toMul q)
-  have hq :
-      Additive.ofMul
-          (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c) = q :=
-    Additive.toMul.injective hc
-  rw [← hq]
-  have hTower :=
-    numberFieldTowerFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
-      K L c
-  have hEmbedded :=
-    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
-      K L (numberFieldSeparableClosureEmbedding L) c
-  conv_lhs =>
-    rw [← hTower, globalNormResidueEquiv_finiteNormClass]
-  conv_rhs =>
-    rw [← hEmbedded, globalNormResidueEquivOfEmbedding_finiteNormClass]
-  exact numberFieldTowerNormResidueValue_eq_embedded_standard K L c
-
-/-- The existing global norm-residue map is the explicit-embedding
-construction for the standard chosen embedding of the top field. -/
-theorem globalNormResidueMonoidHom_eq_ofEmbedding_standard :
-    globalNormResidueMonoidHom K L =
-      globalNormResidueMonoidHomOfEmbedding K L
-        (numberFieldSeparableClosureEmbedding L) := by
-  apply MonoidHom.ext
-  intro c
-  apply Additive.toMul.injective
-  change
-    globalNormResidueEquiv K L
-        (Additive.ofMul
-          (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c)) =
-      globalNormResidueEquivOfEmbedding K L
-        (numberFieldSeparableClosureEmbedding L)
-        (Additive.ofMul
-          (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c))
-  exact congrArg
-    (fun e :
-      Additive
-          (IdeleClassGroup K ⧸
-            (_root_.ideleClassNorm K L).range) ≃+
-        Additive (Gal(L / K)) =>
-      e (Additive.ofMul
-        (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c)))
-    (globalNormResidueEquiv_eq_ofEmbedding_standard K L)
-
-end EmbeddedNumberFieldRealization
-
-section EmbeddedNumberFieldRestriction
-
-variable
-    (K K' L L' : Type)
-    [Field K] [NumberField K]
-    [Field K'] [NumberField K']
-    [Field L] [NumberField L]
-    [Field L'] [NumberField L']
-    [Algebra K K'] [Algebra K L] [Algebra K L']
-    [Algebra K' L'] [Algebra L L']
-    [IsScalarTower K K' L'] [IsScalarTower K L L']
-
-/-- A compatible common embedding reverses the inclusion of the two base
-fields into an inclusion of their fixing subgroups. -/
-theorem numberFieldEmbeddedBaseSubgroup_le_of_tower
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
-      j.comp (IsScalarTower.toAlgHom ℚ L L')
-    (numberFieldEmbeddedBaseSubgroup K' L' j).toSubgroup ≤
-      (numberFieldEmbeddedBaseSubgroup K L jLower).toSubgroup := by
-  dsimp only
-  change
-    (numberFieldEmbeddedLowerEmbedding K' L' j).fieldRange.fixingSubgroup ≤
-      (numberFieldEmbeddedLowerEmbedding K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L'))).fieldRange.fixingSubgroup
-  apply
-    (numberFieldEmbeddedLowerEmbedding K L
-      (j.comp (IsScalarTower.toAlgHom ℚ L L'))).fieldRange.fixingSubgroup_le
-  intro x hx
-  rcases hx with ⟨y, rfl⟩
-  refine ⟨algebraMap K K' y, ?_⟩
-  change
-    j (algebraMap K' L' (algebraMap K K' y)) =
-      j (algebraMap L L' (algebraMap K L y))
-  rw [← IsScalarTower.algebraMap_apply K K' L',
-    ← IsScalarTower.algebraMap_apply K L L']
-
-omit [Field K] [NumberField K]
-    [Field K'] [NumberField K']
-    [Algebra K K'] [Algebra K L] [Algebra K L'] [Algebra K' L']
-    [IsScalarTower K K' L'] [IsScalarTower K L L'] in
-/-- A compatible common embedding reverses the inclusion of the two top
-fields into an inclusion of their fixing subgroups. -/
-theorem numberFieldEmbeddedTopSubgroup_le_of_tower
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
-      j.comp (IsScalarTower.toAlgHom ℚ L L')
-    (numberFieldEmbeddedTopSubgroup K' L' j).toSubgroup ≤
-      (numberFieldEmbeddedTopSubgroup K L jLower).toSubgroup := by
-  dsimp only
-  change
-    j.fieldRange.fixingSubgroup ≤
-      (j.comp
-        (IsScalarTower.toAlgHom ℚ L L')).fieldRange.fixingSubgroup
-  apply
-    (j.comp
-      (IsScalarTower.toAlgHom ℚ L L')).fieldRange.fixingSubgroup_le
-  intro x hx
-  rcases hx with ⟨y, rfl⟩
-  exact ⟨algebraMap L L' y, rfl⟩
-
-end EmbeddedNumberFieldRestriction
-
 variable
     (K : FiniteAbstractField
       (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
     (L : FiniteAbelianSubextension K.field)
 
-local instance abstractFixedFieldBaseQuotientFinite :
-    Finite
-      ((baseField
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-          K.field (le_baseField K.field)) :=
-  K.finite
-
-local instance abstractFixedFieldRelativeQuotientFinite :
-    Finite
-      (K.field.toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
-  L.finite
-
-local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
-    IsMulCommutative L.extensionQuotient :=
-  L.commutative
 
-noncomputable local instance abstractFixedFieldFiniteDimensional :
-    FiniteDimensional ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
-  abstractFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ) K.field K.finite
-
-noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
-    FiniteDimensional
-      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below) :=
-  abstractRelativeFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ)
-    K.field L.field L.below K.finite L.finite
-
-local instance abstractFixedFieldRelativeScalarTower :
-    IsScalarTower ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below) :=
-  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
-
-noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
-    FiniteDimensional ℚ
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below) :=
-  FiniteDimensional.trans ℚ
-    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
-    (abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) L.below)
-
-noncomputable local instance abstractFixedFieldNumberField :
-    NumberField
-      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
-  NumberField.of_module_finite ℚ
-    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
-
-/-- The lower fixed idèle-class group is commutative.  Naming the mixin
-before the public quotient declarations avoids delayed normality synthesis
-inside their definition bodies. -/
-local instance
-    abstractFixedFieldIdeleClassGroupIsMulCommutative :
-    IsMulCommutative
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
-  ⟨⟨fun a b => mul_comm a b⟩⟩
-
-noncomputable local instance abstractRelativeFixedFieldNumberField :
-    NumberField
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below) :=
-  NumberField.of_module_finite ℚ
-    (abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) L.below)
-
-/-- Use the same explicit Galois witness as the fixed-field quotient
-comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
-but much larger dependent instance path. -/
-noncomputable local instance
-    abstractRelativeFixedFieldIsGalois :
-    IsGalois
-      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below) :=
-  abstractRelativeFixedField_isGalois
-    ℚ (SeparableClosure ℚ)
-    K.field L.field L.below L.normal
-
-noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
-    IsAbelianGalois
-      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below) :=
-  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
-
-/-- Use one opaque normality witness for the actual fixed-field norm range.
-This keeps every occurrence of its quotient group on the same instance path. -/
-local instance
-    abstractFixedFieldIdeleClassNormRangeNormal :
-    ((_root_.ideleClassNorm
-      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
-  infer_instance
-
-/-- The abelianized abstract extension quotient is the actual Galois
-group of the corresponding pair of fixed fields. -/
-noncomputable def
-    abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) K.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below
-    Additive
-        (Abelianization
-          (FiniteGaloisSubextension.extensionQuotient
-            L.toFiniteGaloisExtension)) ≃+
-      Additive (Gal(E / F)) := by
-  dsimp only
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) K.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) L.below
-  let e :
-      L.extensionQuotient ≃*
-        Gal(E / F) :=
-    L.extensionQuotientMulEquiv.trans
-      (abstractExtensionQuotientEquivGaloisGroup
-        ℚ (SeparableClosure ℚ)
-        K.field L.field L.below L.normal)
-  exact
-    MulEquiv.toAdditive
-      ((Abelianization.equivOfComm :
-          L.extensionQuotient ≃*
-            Abelianization L.extensionQuotient).symm.trans e)
-
-/-- The actual fixed-field global norm-residue equivalence
-
-`C_F / N_{E/F} C_E ≃ Gal(E/F)`
-
-attached to an abstract finite abelian subextension in the rational
-absolute class formation. -/
-noncomputable def abstractFixedFieldGlobalNormResidueEquiv :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) K.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below
-    Additive
-        (IdeleClassGroup F ⧸
-          (_root_.ideleClassNorm F E).range) ≃+
-      Additive (Gal(E / F)) := by
-  dsimp only
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) K.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) L.below
-  let eNorm :
-      FiniteNormQuotient rationalIdeleClassRepresentation
-          K.field L.field L.below ≃+
-        Additive
-          (Abelianization L.toFiniteGaloisExtension.extensionQuotient) :=
-    rationalCyclotomicDegreeData.normResidueSymbol
-      rationalIdeleClassRepresentation
-      rationalCyclotomicIdeleClassValuationData
-      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-      K L.toFiniteGaloisExtension
-  exact
-    (rationalFiniteNormQuotientEquivIdeleClassNormQuotient
-        K.field L.field L.below L.normal).symm.trans
-      (eNorm.trans
-        (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
-          K L))
-
-/-- The abstract finite norm-residue equivalence with its dependent source
-instance fixed to the public finite norm quotient. -/
-private noncomputable def abstractFixedFieldFiniteNormResidueGaloisEquiv :
-    FiniteNormQuotient rationalIdeleClassRepresentation
-        K.field L.field L.below ≃+
-      Additive
-        (Gal(
-          (abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) L.below) /
-          (abstractFixedField ℚ (SeparableClosure ℚ) K.field))) := by
-  letI : AddCommGroup
-      (FiniteNormQuotient rationalIdeleClassRepresentation
-        K.field L.field L.below) :=
-    finiteNormQuotientAddCommGroup rationalIdeleClassRepresentation
-      K.field L.field L.below
-  exact
-    @AddEquiv.trans
-      (FiniteNormQuotient rationalIdeleClassRepresentation
-        K.field L.field L.below)
-      (Additive
-        (Abelianization
-          L.toFiniteGaloisExtension.extensionQuotient))
-      (Additive
-        (Gal(
-          (abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) L.below) /
-          (abstractFixedField ℚ (SeparableClosure ℚ) K.field))))
-      inferInstance inferInstance inferInstance
-      (rationalCyclotomicDegreeData.normResidueSymbol
-        rationalIdeleClassRepresentation
-        rationalCyclotomicIdeleClassValuationData
-        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-        K L.toFiniteGaloisExtension)
-      (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
-        K L)
-
-/-- The abstract norm-residue symbol on the fixed part of the rational
-absolute idele-class representation, with its value transported to the
-actual Galois group of the two fixed fields.  This is the form consumed
-directly by the abstract norm--restriction naturality theorem. -/
-noncomputable def ambientFixedGlobalNormResidueAddMonoidHom :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) K.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below
-    ambientFixedAddSubgroup
-        rationalIdeleClassRepresentation K.field →+
-      Additive (Gal(E / F)) := by
-  dsimp only
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) K.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) L.below
-  exact
-    (abstractFixedFieldFiniteNormResidueGaloisEquiv K L).toAddMonoidHom.comp
-      (finiteNormClassHom rationalIdeleClassRepresentation
-        K.field L.field L.below)
-
-/-- The ordinary idele class group of the lower fixed field, transported
-to the fixed part of the rational absolute idele-class representation. -/
-private noncomputable def abstractFixedFieldIdeleClassToAmbientFixedMonoidHom :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) K.field
-    IdeleClassGroup F →*
-      Multiplicative
-        (ambientFixedAddSubgroup
-          rationalIdeleClassRepresentation K.field) :=
-  (rationalAbstractFixedFieldIdeleClassEquivFixed
-    K.field).toAddMonoidHom.toMultiplicativeRight
-
-/-- The actual norm-residue homomorphism on the ordinary idele class
-group of the lower fixed field, constructed without choosing a second
-field embedding. -/
-noncomputable def abstractFixedFieldGlobalNormResidueMonoidHom :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) K.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) L.below
-    IdeleClassGroup F →* Gal(E / F) :=
-  (ambientFixedGlobalNormResidueAddMonoidHom K L).toMultiplicative.comp
-    (abstractFixedFieldIdeleClassToAmbientFixedMonoidHom K)
-
-/-- Pointwise form of the ambient fixed-part norm-residue homomorphism. -/
-private theorem ambientFixedGlobalNormResidueAddMonoidHom_apply
-    (a :
-      ambientFixedAddSubgroup
-        rationalIdeleClassRepresentation K.field) :
-    ambientFixedGlobalNormResidueAddMonoidHom K L a =
-      abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
-        K L
-        (rationalCyclotomicDegreeData.normResidueSymbol
-          rationalIdeleClassRepresentation
-          rationalCyclotomicIdeleClassValuationData
-          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-          K L.toFiniteGaloisExtension
-          (finiteNormClass rationalIdeleClassRepresentation
-            K.field L.field L.below a)) := by
-  change
-    abstractFixedFieldFiniteNormResidueGaloisEquiv K L
-        (finiteNormClass rationalIdeleClassRepresentation
-          K.field L.field L.below a) = _
-  rfl
-
-/-- Pointwise form of the transported fixed-field norm-residue homomorphism. -/
-private theorem abstractFixedFieldGlobalNormResidueMonoidHom_apply :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) K.field
-    ∀ c : IdeleClassGroup F,
-      abstractFixedFieldGlobalNormResidueMonoidHom K L c =
-        Additive.toMul
-          (ambientFixedGlobalNormResidueAddMonoidHom K L
-            ((rationalAbstractFixedFieldIdeleClassEquivFixed K.field)
-              (Additive.ofMul c))) := by
-  dsimp only
-  intro c
-  rfl
-
-/-- Transporting an ordinary fixed-field idele class to the ambient
-fixed part and applying the abstract norm-residue map gives exactly the
-actual fixed-field norm-residue value. -/
-@[simp]
-theorem abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply
-    (a :
-      ambientFixedAddSubgroup
-        rationalIdeleClassRepresentation K.field) :
-    abstractFixedFieldGlobalNormResidueMonoidHom K L
-        (Additive.toMul
-          ((rationalAbstractFixedFieldIdeleClassEquivFixed
-            K.field).symm a)) =
-      Additive.toMul
-        (ambientFixedGlobalNormResidueAddMonoidHom K L a) := by
-  rw [abstractFixedFieldGlobalNormResidueMonoidHom_apply]
-  change
-    Additive.toMul
-        (ambientFixedGlobalNormResidueAddMonoidHom K L
-          ((rationalAbstractFixedFieldIdeleClassEquivFixed K.field)
-            ((rationalAbstractFixedFieldIdeleClassEquivFixed
-              K.field).symm a))) =
-      Additive.toMul
-        (ambientFixedGlobalNormResidueAddMonoidHom K L a)
-  rw [(rationalAbstractFixedFieldIdeleClassEquivFixed
-    K.field).apply_symm_apply]
+attribute [local instance]
+  abstractFixedFieldBaseQuotientFinite
+  abstractFixedFieldRelativeQuotientFinite
+  abstractFixedFieldRelativeQuotientIsMulCommutative
+  abstractFixedFieldFiniteDimensional
+  abstractRelativeFixedFieldFiniteDimensional
+  abstractFixedFieldRelativeScalarTower
+  abstractRelativeFixedFieldAbsoluteFiniteDimensional
+  abstractFixedFieldNumberField
+  abstractFixedFieldIdeleClassGroupIsMulCommutative
+  abstractRelativeFixedFieldNumberField
+  abstractRelativeFixedFieldIsGalois
+  abstractRelativeFixedFieldIsAbelianGalois
+  abstractFixedFieldIdeleClassNormRangeNormal
 
 /-- The ambient fixed-part reciprocity value vanishes precisely when its
 finite norm class vanishes. -/
@@ -1644,8 +197,7 @@ theorem abstractFixedFieldGlobalNormResidueMonoidHom_surjective :
   rw [abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply, ha]
   rfl
 
-/-- The kernel of the fixed-field global norm-residue homomorphism is
-the genuine ordinary idele-class norm range. -/
+
 @[simp]
 theorem abstractFixedFieldGlobalNormResidueMonoidHom_ker :
     (abstractFixedFieldGlobalNormResidueMonoidHom K L).ker =
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueEmbeddedReciprocity.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueEmbeddedReciprocity.lean
new file mode 100644
index 0000000..c07e4ec
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueEmbeddedReciprocity.lean
@@ -0,0 +1,386 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.AbstractFixedFieldGlobalNormResidueNormClasses
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+section EmbeddedNumberFieldRealization
+
+variable
+    (K L : Type) [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+
+
+attribute [local instance]
+  numberFieldEmbeddedIdeleClassGroupIsMulCommutative
+  numberFieldEmbeddedIdeleClassSubgroupNormal
+  numberFieldEmbeddedExtensionSubgroupNormal
+  numberFieldEmbeddedExtensionQuotientFinite
+  numberFieldEmbeddedAbsoluteQuotientFinite
+  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
+  numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
+  numberFieldEmbeddedAbstractFixedFieldScalarTower
+  numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
+  numberFieldEmbeddedAbstractFixedFieldNumberField
+  numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
+  numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
+  numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
+  numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
+  numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
+
+variable [FiniteDimensional K L] [IsAbelianGalois K L]
+
+/-- The abelianized quotient of the explicitly embedded tower is the
+actual abelian Galois group. -/
+noncomputable def
+    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Additive
+      (Abelianization
+        (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient) ≃+
+      Additive Gal(L / K) :=
+  MulEquiv.toAdditive
+    ((MulEquiv.abelianizationCongr
+      (numberFieldEmbeddedExtensionQuotientEquivGaloisGroup K L j)).trans
+        (Abelianization.equivOfComm :
+          Gal(L / K) ≃*
+            Abelianization Gal(L / K)).symm)
+
+/-- The actual global norm-residue equivalence constructed from an
+explicit compatible embedding of a finite abelian number-field
+extension into the rational separable closure. -/
+noncomputable def globalNormResidueEquivOfEmbedding
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Additive
+        (IdeleClassGroup K ⧸
+          (_root_.ideleClassNorm K L).range) ≃+
+      Additive Gal(L / K) := by
+  let eNorm :
+      FiniteNormQuotient rationalIdeleClassRepresentation
+          (numberFieldEmbeddedBaseSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) ≃+
+        Additive
+          (Abelianization
+            (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient) :=
+    rationalCyclotomicDegreeData.normResidueSymbol
+      rationalIdeleClassRepresentation
+      rationalCyclotomicIdeleClassValuationData
+      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+      (numberFieldEmbeddedFiniteAbstractField K L j)
+      (numberFieldEmbeddedFiniteGaloisSubextension K L j)
+  exact
+    (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
+        K L j).symm.trans
+      (eNorm.trans
+        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+          K L j))
+
+/-- The explicit-embedding norm-residue equivalence on a finite norm class. -/
+theorem globalNormResidueEquivOfEmbedding_finiteNormClass
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (x : FiniteNormQuotient rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)
+      (numberFieldEmbeddedTopSubgroup K L j)
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :
+    globalNormResidueEquivOfEmbedding K L j
+        (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
+          K L j x) =
+      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup K L j
+        (rationalCyclotomicDegreeData.normResidueSymbol
+          rationalIdeleClassRepresentation
+          rationalCyclotomicIdeleClassValuationData
+          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+          (numberFieldEmbeddedFiniteAbstractField K L j)
+          (numberFieldEmbeddedFiniteGaloisSubextension K L j) x) := by
+  simp only [globalNormResidueEquivOfEmbedding, AddEquiv.trans_apply,
+    AddEquiv.symm_apply_apply]
+
+/-- The global norm-residue homomorphism obtained from an explicit
+compatible embedding. -/
+noncomputable def globalNormResidueMonoidHomOfEmbedding
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    IdeleClassGroup K →* Gal(L / K) := by
+  let e :
+      (IdeleClassGroup K ⧸
+          (_root_.ideleClassNorm K L).range) ≃*
+        Gal(L / K) :=
+    AddEquiv.toMultiplicative
+      (globalNormResidueEquivOfEmbedding K L j)
+  exact
+    e.toMonoidHom.comp
+      (QuotientGroup.mk'
+        (_root_.ideleClassNorm K L).range)
+
+/-- Evaluation of the explicit-embedding global norm-residue map is
+the abstract norm-residue symbol evaluated on the corresponding genuine
+fixed-part finite norm class. -/
+@[simp]
+theorem globalNormResidueMonoidHomOfEmbedding_apply
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (c : IdeleClassGroup K) :
+    globalNormResidueMonoidHomOfEmbedding K L j c =
+      Additive.toMul
+        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+          K L j
+          (rationalCyclotomicDegreeData.normResidueSymbol
+            rationalIdeleClassRepresentation
+            rationalCyclotomicIdeleClassValuationData
+            rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+            (numberFieldEmbeddedFiniteAbstractField K L j)
+            (numberFieldEmbeddedFiniteGaloisSubextension K L j)
+            (finiteNormClass rationalIdeleClassRepresentation
+              (numberFieldEmbeddedBaseSubgroup K L j)
+              (numberFieldEmbeddedTopSubgroup K L j)
+              (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
+              (numberFieldEmbeddedIdeleClassEquivAmbientFixed
+                K L j (Additive.ofMul c))))) := by
+  have hclass :=
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
+      K L j c
+  change
+    Additive.toMul
+      (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+        K L j
+        (rationalCyclotomicDegreeData.normResidueSymbol
+          rationalIdeleClassRepresentation
+          rationalCyclotomicIdeleClassValuationData
+          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+          (numberFieldEmbeddedFiniteAbstractField K L j)
+          (numberFieldEmbeddedFiniteGaloisSubextension K L j)
+          ((numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
+            K L j).symm
+            (Additive.ofMul
+              (QuotientGroup.mk'
+                (_root_.ideleClassNorm K L).range c))))) =
+      _
+  rw [← hclass,
+    (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
+      K L j).symm_apply_apply]
+
+omit [FiniteDimensional K L] [IsAbelianGalois K L] in
+
+
+theorem numberFieldTowerIdeleClassEquivAmbientFixed_eq_embedded_standard :
+    numberFieldTowerIdeleClassEquivAmbientFixed K L =
+      numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
+        (numberFieldSeparableClosureEmbedding L) := by
+  let j := numberFieldSeparableClosureEmbedding L
+  have hBase :
+      numberFieldTowerAbstractBaseFieldEquiv K L =
+        numberFieldEmbeddedAbstractBaseFieldEquiv K L j := by
+    rfl
+  unfold numberFieldTowerIdeleClassEquivAmbientFixed
+    numberFieldEmbeddedIdeleClassEquivAmbientFixed
+  rw [hBase]
+  dsimp only
+  congr 1
+
+/- At the chosen embedding, both constructions use the same fixed tower and
+abstract reciprocity data.  We compare their values on the particular fixed
+idele class needed below; the two implementations of the finite norm-quotient
+equivalence are deliberately not compared as dependent structures. -/
+private theorem numberFieldTowerNormResidueValue_eq_embedded_standard
+    (c : IdeleClassGroup K) :
+    numberFieldTowerAbelianizedExtensionQuotientEquivGaloisGroup K L
+        (rationalCyclotomicDegreeData.normResidueSymbol
+          rationalIdeleClassRepresentation
+          rationalCyclotomicIdeleClassValuationData
+          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+          (numberFieldTowerReciprocityFiniteAbstractField K L)
+          (numberFieldTowerFiniteGaloisSubextension K L)
+          (finiteNormClass rationalIdeleClassRepresentation
+            (numberFieldTowerBaseSubgroup K L)
+            (numberFieldTowerTopSubgroup L)
+            (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
+            (numberFieldTowerIdeleClassEquivAmbientFixed K L (Additive.ofMul c)))) =
+      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+        K L (numberFieldSeparableClosureEmbedding L)
+        (rationalCyclotomicDegreeData.normResidueSymbol
+          rationalIdeleClassRepresentation
+          rationalCyclotomicIdeleClassValuationData
+          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+          (numberFieldEmbeddedFiniteAbstractField K L
+            (numberFieldSeparableClosureEmbedding L))
+          (numberFieldEmbeddedFiniteGaloisSubextension K L
+            (numberFieldSeparableClosureEmbedding L))
+          (finiteNormClass rationalIdeleClassRepresentation
+            (numberFieldEmbeddedBaseSubgroup K L
+              (numberFieldSeparableClosureEmbedding L))
+            (numberFieldEmbeddedTopSubgroup K L
+              (numberFieldSeparableClosureEmbedding L))
+            (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L
+              (numberFieldSeparableClosureEmbedding L))
+            (numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
+              (numberFieldSeparableClosureEmbedding L) (Additive.ofMul c)))) := by
+  have hIdeleClassEquiv :
+      numberFieldTowerIdeleClassEquivAmbientFixed K L =
+        numberFieldEmbeddedIdeleClassEquivAmbientFixed K L
+          (numberFieldSeparableClosureEmbedding L) := by
+    exact numberFieldTowerIdeleClassEquivAmbientFixed_eq_embedded_standard K L
+  have hFiniteAbstractField :
+      numberFieldTowerReciprocityFiniteAbstractField K L =
+        numberFieldEmbeddedFiniteAbstractField K L
+          (numberFieldSeparableClosureEmbedding L) := by
+    rfl
+  have hSubextension :
+      numberFieldTowerFiniteGaloisSubextension K L =
+        numberFieldEmbeddedFiniteGaloisSubextension K L
+          (numberFieldSeparableClosureEmbedding L) := by
+    rfl
+  have hGaloisComparison :
+      numberFieldTowerAbelianizedExtensionQuotientEquivGaloisGroup K L =
+        numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+          K L (numberFieldSeparableClosureEmbedding L) := by
+    rfl
+  simp only [← hGaloisComparison]
+  cases hFiniteAbstractField
+  cases hSubextension
+  rw [← hIdeleClassEquiv]
+  rfl
+
+/-- At the standard embedding, the two global norm-residue equivalences
+agree on the actual norm quotient.  The comparison is extensional: it uses
+surjectivity of the quotient map and the established evaluation formulas,
+not definitional equality of the two quotient constructions. -/
+theorem globalNormResidueEquiv_eq_ofEmbedding_standard :
+    globalNormResidueEquiv K L =
+      globalNormResidueEquivOfEmbedding K L
+        (numberFieldSeparableClosureEmbedding L) := by
+  apply AddEquiv.ext
+  intro q
+  obtain ⟨c, hc⟩ :=
+    QuotientGroup.mk'_surjective
+      (_root_.ideleClassNorm K L).range (Additive.toMul q)
+  have hq :
+      Additive.ofMul
+          (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c) = q :=
+    Additive.toMul.injective hc
+  rw [← hq]
+  have hTower :=
+    numberFieldTowerFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
+      K L c
+  have hEmbedded :=
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
+      K L (numberFieldSeparableClosureEmbedding L) c
+  conv_lhs =>
+    rw [← hTower, globalNormResidueEquiv_finiteNormClass]
+  conv_rhs =>
+    rw [← hEmbedded, globalNormResidueEquivOfEmbedding_finiteNormClass]
+  exact numberFieldTowerNormResidueValue_eq_embedded_standard K L c
+
+/-- The existing global norm-residue map is the explicit-embedding
+construction for the standard chosen embedding of the top field. -/
+theorem globalNormResidueMonoidHom_eq_ofEmbedding_standard :
+    globalNormResidueMonoidHom K L =
+      globalNormResidueMonoidHomOfEmbedding K L
+        (numberFieldSeparableClosureEmbedding L) := by
+  apply MonoidHom.ext
+  intro c
+  apply Additive.toMul.injective
+  change
+    globalNormResidueEquiv K L
+        (Additive.ofMul
+          (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c)) =
+      globalNormResidueEquivOfEmbedding K L
+        (numberFieldSeparableClosureEmbedding L)
+        (Additive.ofMul
+          (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c))
+  exact congrArg
+    (fun e :
+      Additive
+          (IdeleClassGroup K ⧸
+            (_root_.ideleClassNorm K L).range) ≃+
+        Additive (Gal(L / K)) =>
+      e (Additive.ofMul
+        (QuotientGroup.mk' (_root_.ideleClassNorm K L).range c)))
+    (globalNormResidueEquiv_eq_ofEmbedding_standard K L)
+
+end EmbeddedNumberFieldRealization
+
+section EmbeddedNumberFieldRestriction
+
+variable
+    (K K' L L' : Type)
+    [Field K] [NumberField K]
+    [Field K'] [NumberField K']
+    [Field L] [NumberField L]
+    [Field L'] [NumberField L']
+    [Algebra K K'] [Algebra K L] [Algebra K L']
+    [Algebra K' L'] [Algebra L L']
+    [IsScalarTower K K' L'] [IsScalarTower K L L']
+
+/-- A compatible common embedding reverses the inclusion of the two base
+fields into an inclusion of their fixing subgroups. -/
+theorem numberFieldEmbeddedBaseSubgroup_le_of_tower
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
+      j.comp (IsScalarTower.toAlgHom ℚ L L')
+    (numberFieldEmbeddedBaseSubgroup K' L' j).toSubgroup ≤
+      (numberFieldEmbeddedBaseSubgroup K L jLower).toSubgroup := by
+  dsimp only
+  change
+    (numberFieldEmbeddedLowerEmbedding K' L' j).fieldRange.fixingSubgroup ≤
+      (numberFieldEmbeddedLowerEmbedding K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L'))).fieldRange.fixingSubgroup
+  apply
+    (numberFieldEmbeddedLowerEmbedding K L
+      (j.comp (IsScalarTower.toAlgHom ℚ L L'))).fieldRange.fixingSubgroup_le
+  intro x hx
+  rcases hx with ⟨y, rfl⟩
+  refine ⟨algebraMap K K' y, ?_⟩
+  change
+    j (algebraMap K' L' (algebraMap K K' y)) =
+      j (algebraMap L L' (algebraMap K L y))
+  rw [← IsScalarTower.algebraMap_apply K K' L',
+    ← IsScalarTower.algebraMap_apply K L L']
+
+omit [Field K] [NumberField K]
+    [Field K'] [NumberField K']
+    [Algebra K K'] [Algebra K L] [Algebra K L'] [Algebra K' L']
+    [IsScalarTower K K' L'] [IsScalarTower K L L'] in
+/-- A compatible common embedding reverses the inclusion of the two top
+fields into an inclusion of their fixing subgroups. -/
+theorem numberFieldEmbeddedTopSubgroup_le_of_tower
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
+      j.comp (IsScalarTower.toAlgHom ℚ L L')
+    (numberFieldEmbeddedTopSubgroup K' L' j).toSubgroup ≤
+      (numberFieldEmbeddedTopSubgroup K L jLower).toSubgroup := by
+  dsimp only
+  change
+    j.fieldRange.fixingSubgroup ≤
+      (j.comp
+        (IsScalarTower.toAlgHom ℚ L L')).fieldRange.fixingSubgroup
+  apply
+    (j.comp
+      (IsScalarTower.toAlgHom ℚ L L')).fieldRange.fixingSubgroup_le
+  intro x hx
+  rcases hx with ⟨y, rfl⟩
+  exact ⟨algebraMap L L' y, rfl⟩
+
+end EmbeddedNumberFieldRestriction
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueEmbeddings.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueEmbeddings.lean
new file mode 100644
index 0000000..2451e8f
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueEmbeddings.lean
@@ -0,0 +1,586 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidue
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+/-- The algebra structure on the rational separable closure induced by a
+specified rational field embedding.  It is deliberately not an instance:
+different embeddings of the same field need not induce definitionally equal
+algebra structures. -/
+@[reducible]
+noncomputable def rationalEmbeddingSeparableClosureAlgebra
+    {F : Type} [Field F] [Algebra ℚ F]
+    (i : F →ₐ[ℚ] SeparableClosure ℚ) :
+    Algebra F (SeparableClosure ℚ) :=
+  i.toRingHom.toAlgebra
+
+/-- Two rational embeddings of the same number field into the fixed
+rational separable closure differ by an automorphism of that
+separable closure. -/
+theorem exists_numberFieldEmbeddingComparisonAutomorphism
+    {F : Type} [Field F] [NumberField F]
+    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
+    ∃ σ : SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ,
+      ∀ x : F, σ (i x) = j x := by
+  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
+    rationalEmbeddingSeparableClosureAlgebra i
+  let hScalarTower : IsScalarTower ℚ F (SeparableClosure ℚ) :=
+    IsScalarTower.of_algebraMap_eq' i.comp_algebraMap.symm
+  let hSeparable : Algebra.IsSeparable F (SeparableClosure ℚ) :=
+    Algebra.isSeparable_tower_top_of_isSeparable
+      ℚ F (SeparableClosure ℚ)
+  obtain ⟨φ, hφ⟩ :=
+    (IsSepClosed.surjective_domRestrict_of_isSeparable
+      (K := ℚ) (L := F)
+      (M := SeparableClosure ℚ)
+      (E := SeparableClosure ℚ)) j
+  let σ : SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ :=
+    AlgEquiv.ofBijective φ
+      (Normal.toIsAlgebraic.algHom_bijective₂
+        φ (AlgHom.id ℚ (SeparableClosure ℚ))).1
+  refine ⟨σ, ?_⟩
+  intro x
+  have hx :=
+    congrArg (fun ψ : F →ₐ[ℚ] SeparableClosure ℚ => ψ x) hφ
+  exact hx
+
+/-- The canonical comparison automorphism between two rational
+embeddings of one number field into the fixed separable closure. -/
+noncomputable def numberFieldEmbeddingComparisonAutomorphism
+    {F : Type} [Field F] [NumberField F]
+    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
+    SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ :=
+  Classical.choose
+    (exists_numberFieldEmbeddingComparisonAutomorphism i j)
+
+/-- The comparison automorphism carries the first embedded copy of the
+number field to the second one pointwise. -/
+@[simp]
+theorem numberFieldEmbeddingComparisonAutomorphism_apply
+    {F : Type} [Field F] [NumberField F]
+    (i j : F →ₐ[ℚ] SeparableClosure ℚ)
+    (x : F) :
+    numberFieldEmbeddingComparisonAutomorphism i j (i x) =
+      j x :=
+  Classical.choose_spec
+    (exists_numberFieldEmbeddingComparisonAutomorphism i j) x
+
+/-- Conjugating the fixing subgroup of one embedded copy of a number
+field by the comparison automorphism gives the fixing subgroup of the
+other embedded copy. -/
+theorem conjugateClosedFixingSubgroup_embeddingRange
+    {F : Type} [Field F] [NumberField F]
+    (i j : F →ₐ[ℚ] SeparableClosure ℚ) :
+    conjugateClosedSubgroup
+        (RamificationTheory.closedFixingSubgroup
+          ℚ (SeparableClosure ℚ) i.fieldRange)
+        (numberFieldEmbeddingComparisonAutomorphism j i) =
+      RamificationTheory.closedFixingSubgroup
+        ℚ (SeparableClosure ℚ) j.fieldRange := by
+  let s :=
+    numberFieldEmbeddingComparisonAutomorphism j i
+  ext τ
+  change
+    τ ∈ conjugateClosedSubgroup
+        (closedFixingSubgroup ℚ (SeparableClosure ℚ) i.fieldRange) s ↔
+      τ ∈ closedFixingSubgroup ℚ (SeparableClosure ℚ) j.fieldRange
+  rw [conjugateClosedSubgroup_mem]
+  change
+    s * τ * s⁻¹ ∈ i.fieldRange.fixingSubgroup ↔
+      τ ∈ j.fieldRange.fixingSubgroup
+  rw [IntermediateField.mem_fixingSubgroup_iff,
+    IntermediateField.mem_fixingSubgroup_iff]
+  constructor
+  · intro h x hx
+    rcases hx with ⟨y, rfl⟩
+    have hi := h (i y) ⟨y, rfl⟩
+    have hs :
+        s (j y) = i y :=
+      numberFieldEmbeddingComparisonAutomorphism_apply j i y
+    change s (τ (s.symm (i y))) = i y at hi
+    have hpre : s.symm (i y) = j y := by
+      rw [← hs, s.symm_apply_apply]
+    rw [hpre, ← hs] at hi
+    exact s.injective hi
+  · intro h x hx
+    rcases hx with ⟨y, rfl⟩
+    have hj := h (j y) ⟨y, rfl⟩
+    have hs :
+        s (j y) = i y :=
+      numberFieldEmbeddingComparisonAutomorphism_apply j i y
+    change s (τ (s.symm (i y))) = i y
+    have hpre : s.symm (i y) = j y := by
+      rw [← hs, s.symm_apply_apply]
+    rw [hpre, hj, hs]
+
+section EmbeddedNumberFieldRealization
+
+local instance numberFieldEmbeddedIdeleClassGroupIsMulCommutative
+    {F : Type} [Field F] [NumberField F]
+    : IsMulCommutative (IdeleClassGroup F) :=
+  ⟨⟨fun a b => mul_comm a b⟩⟩
+
+local instance numberFieldEmbeddedIdeleClassSubgroupNormal
+    {F : Type} [Field F] [NumberField F]
+    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
+  N.normal_of_isMulCommutative
+
+variable
+    (K L : Type) [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+
+/-- The lower embedding obtained by restricting an explicitly supplied
+embedding of the top field into the rational separable closure. -/
+noncomputable def numberFieldEmbeddedLowerEmbedding
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    K →ₐ[ℚ] SeparableClosure ℚ :=
+  j.comp (IsScalarTower.toAlgHom ℚ K L)
+
+/-- The exact algebra structure on the rational separable closure induced by
+the lower embedding of an explicitly embedded number-field tower.  Keeping
+this as a reducible definition lets every use of the associated separable-
+closure equivalence share one definitionally identical algebra structure. -/
+@[reducible]
+noncomputable def numberFieldEmbeddedSeparableClosureAlgebra
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Algebra K (SeparableClosure ℚ) :=
+  rationalEmbeddingSeparableClosureAlgebra
+    (numberFieldEmbeddedLowerEmbedding K L j)
+
+/-- The fixing subgroup of the explicitly embedded lower field. -/
+abbrev numberFieldEmbeddedBaseSubgroup
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    ClosedSubgroup
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+  closedFixingSubgroup ℚ (SeparableClosure ℚ)
+    (numberFieldEmbeddedLowerEmbedding K L j).fieldRange
+
+/-- The fixing subgroup of the explicitly embedded top field. -/
+abbrev numberFieldEmbeddedTopSubgroup
+    (_K L : Type) [Field L] [NumberField L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    ClosedSubgroup
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+  closedFixingSubgroup ℚ (SeparableClosure ℚ) j.fieldRange
+
+/-- The top fixing subgroup lies in the lower fixing subgroup. -/
+theorem numberFieldEmbeddedTopSubgroup_le_baseSubgroup
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    (numberFieldEmbeddedTopSubgroup K L j).toSubgroup ≤
+      (numberFieldEmbeddedBaseSubgroup K L j).toSubgroup := by
+  change
+    j.fieldRange.fixingSubgroup ≤
+      (numberFieldEmbeddedLowerEmbedding K L j).fieldRange.fixingSubgroup
+  apply
+    (numberFieldEmbeddedLowerEmbedding K L j).fieldRange.fixingSubgroup_le
+  intro x hx
+  rcases hx with ⟨y, rfl⟩
+  exact ⟨algebraMap K L y, rfl⟩
+
+/-- The separable closure of the actual lower field, identified with
+the rational separable closure carrying the algebra structure induced
+by an explicit compatible embedding. -/
+noncomputable def numberFieldEmbeddedSeparableClosureEquiv
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    letI : Algebra K (SeparableClosure ℚ) :=
+      numberFieldEmbeddedSeparableClosureAlgebra K L j
+    SeparableClosure K ≃ₐ[K] SeparableClosure ℚ := by
+  letI : Algebra K (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra K L j
+  letI hScalarTower : IsScalarTower ℚ K (SeparableClosure ℚ) :=
+    IsScalarTower.of_algebraMap_eq'
+      (numberFieldEmbeddedLowerEmbedding K L j).comp_algebraMap.symm
+  letI hseparable : Algebra.IsSeparable K (SeparableClosure ℚ) :=
+    Algebra.isSeparable_tower_top_of_isSeparable
+      ℚ K (SeparableClosure ℚ)
+  letI hSepClosure : IsSepClosure K (SeparableClosure ℚ) :=
+    ⟨IsSepClosure.sep_closed ℚ, hseparable⟩
+  exact
+    IsSepClosure.equiv K
+      (SeparableClosure K) (SeparableClosure ℚ)
+
+/-- The relative subgroup arising from an explicit compatible
+number-field embedding is normal. -/
+theorem numberFieldEmbeddedExtensionSubgroup_normal
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    (CyclicCohomology.extensionSubgroup
+      (numberFieldEmbeddedBaseSubgroup K L j)
+      (numberFieldEmbeddedTopSubgroup K L j)
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal := by
+  let i := numberFieldEmbeddedLowerEmbedding K L j
+  let hAlgebra : Algebra K (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra K L j
+  let e := numberFieldEmbeddedSeparableClosureEquiv K L j
+  change
+    (CyclicCohomology.extensionSubgroup
+      (closedFixingSubgroup ℚ (SeparableClosure ℚ)
+        (AlgHom.fieldRange i))
+      (closedFixingSubgroup ℚ (SeparableClosure ℚ)
+        (AlgHom.fieldRange j)) _).Normal
+  exact ambientEmbeddedExtensionSubgroup_normal ℚ K L j e
+
+/-- The normality witness for an explicitly embedded tower, registered at
+the precise subgroup used by the downstream quotient constructions. -/
+noncomputable local instance
+    numberFieldEmbeddedExtensionSubgroupNormal
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    (CyclicCohomology.extensionSubgroup
+      (numberFieldEmbeddedBaseSubgroup K L j)
+      (numberFieldEmbeddedTopSubgroup K L j)
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).Normal :=
+  numberFieldEmbeddedExtensionSubgroup_normal K L j
+
+/-- The relative quotient arising from an explicit compatible
+number-field embedding is finite. -/
+theorem numberFieldEmbeddedExtensionQuotient_finite
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Finite
+      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (numberFieldEmbeddedBaseSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) := by
+  let i := numberFieldEmbeddedLowerEmbedding K L j
+  let hAlgebra : Algebra K (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra K L j
+  let e := numberFieldEmbeddedSeparableClosureEquiv K L j
+  change
+    Finite
+      ((closedFixingSubgroup ℚ (SeparableClosure ℚ)
+          (AlgHom.fieldRange i)).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (closedFixingSubgroup ℚ (SeparableClosure ℚ)
+            (AlgHom.fieldRange i))
+          (closedFixingSubgroup ℚ (SeparableClosure ℚ)
+            (AlgHom.fieldRange j)) _)
+  exact ambientEmbeddedExtensionQuotient_finite ℚ K L j e
+
+/-- The relative-index witness for an explicitly embedded tower, registered
+at the exact quotient consumed by `FiniteNormQuotient`. -/
+noncomputable local instance
+    numberFieldEmbeddedExtensionQuotientFinite
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Finite
+      ((numberFieldEmbeddedBaseSubgroup K L j).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (numberFieldEmbeddedBaseSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
+  numberFieldEmbeddedExtensionQuotient_finite K L j
+
+/-- The finite abstract field determined by the lower member of an
+explicitly embedded number-field tower. -/
+noncomputable abbrev numberFieldEmbeddedFiniteAbstractField
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) where
+  field := numberFieldEmbeddedBaseSubgroup K L j
+  finite := by
+    simpa only [numberFieldEmbeddedBaseSubgroup] using
+      (ambientEmbeddedAbsoluteQuotientFinite
+        ℚ K (numberFieldEmbeddedLowerEmbedding K L j))
+
+/-- The absolute-index witness for the lower member of an explicitly embedded
+tower, registered at its specialized quotient type. -/
+noncomputable local instance
+    numberFieldEmbeddedAbsoluteQuotientFinite
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Finite
+      ((baseField
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+          (numberFieldEmbeddedBaseSubgroup K L j)
+          (le_baseField
+            (numberFieldEmbeddedBaseSubgroup K L j))) :=
+  (numberFieldEmbeddedFiniteAbstractField K L j).finite
+
+/-- The finite Galois subextension determined by an explicitly embedded
+number-field tower. -/
+noncomputable abbrev numberFieldEmbeddedFiniteGaloisSubextension
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteGaloisSubextension
+      (numberFieldEmbeddedBaseSubgroup K L j) where
+  field := numberFieldEmbeddedTopSubgroup K L j
+  below := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
+  normal := numberFieldEmbeddedExtensionSubgroup_normal K L j
+  finite := numberFieldEmbeddedExtensionQuotient_finite K L j
+
+/-- Shared finite-dimensional data for the fixed field of the lower subgroup
+in an explicitly embedded number-field tower. -/
+noncomputable local instance
+    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteDimensional ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L j)) :=
+  abstractFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ)
+    (numberFieldEmbeddedBaseSubgroup K L j)
+    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
+
+/-- Shared relative finite-dimensional data for the two fixed fields of an
+explicitly embedded number-field tower. -/
+noncomputable local instance
+    numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteDimensional
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L j))
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
+  abstractRelativeFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ)
+    (numberFieldEmbeddedBaseSubgroup K L j)
+    (numberFieldEmbeddedTopSubgroup K L j)
+    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
+    (numberFieldEmbeddedAbsoluteQuotientFinite K L j)
+    (numberFieldEmbeddedExtensionQuotientFinite K L j)
+
+local instance numberFieldEmbeddedAbstractFixedFieldScalarTower
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    IsScalarTower ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L j))
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
+  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
+
+noncomputable local instance
+    numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteDimensional ℚ
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
+  FiniteDimensional.trans ℚ
+    (abstractFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedBaseSubgroup K L j))
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
+
+noncomputable local instance
+    numberFieldEmbeddedAbstractFixedFieldNumberField
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    NumberField
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L j)) :=
+  NumberField.of_module_finite ℚ
+    (abstractFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedBaseSubgroup K L j))
+
+noncomputable local instance
+    numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    NumberField
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
+  NumberField.of_module_finite ℚ
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
+
+noncomputable local instance
+    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteDimensional ℚ
+      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) := by
+  change FiniteDimensional ℚ
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
+  infer_instance
+
+noncomputable local instance
+    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    NumberField
+      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
+  NumberField.of_module_finite ℚ _
+
+noncomputable local instance
+    numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Algebra
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L j))
+      ((abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)).restrictScalars ℚ) :=
+  (IntermediateField.inclusion
+    (abstractFixedField_le ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))).toRingHom.toAlgebra
+
+noncomputable local instance
+    numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    IsGalois
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L j))
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)) :=
+  abstractRelativeFixedField_isGalois
+    ℚ (SeparableClosure ℚ)
+    (numberFieldEmbeddedBaseSubgroup K L j)
+    (numberFieldEmbeddedTopSubgroup K L j)
+    (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
+    (numberFieldEmbeddedExtensionSubgroupNormal K L j)
+
+/-- The quotient of the two explicitly embedded fixing subgroups is
+the actual relative Galois group. -/
+noncomputable def
+    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    (numberFieldEmbeddedFiniteGaloisSubextension K L j).extensionQuotient ≃*
+      Gal(L / K) := by
+  let i := numberFieldEmbeddedLowerEmbedding K L j
+  letI hAlgebra : Algebra K (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra K L j
+  let e := numberFieldEmbeddedSeparableClosureEquiv K L j
+  let H₀ :=
+    closedFixingSubgroup ℚ (SeparableClosure ℚ)
+      (AlgHom.fieldRange i)
+  let J₀ :=
+    closedFixingSubgroup ℚ (SeparableClosure ℚ)
+      (AlgHom.fieldRange j)
+  let hJH : J₀.toSubgroup ≤ H₀.toSubgroup := by
+    change j.fieldRange.fixingSubgroup ≤ i.fieldRange.fixingSubgroup
+    apply i.fieldRange.fixingSubgroup_le
+    intro x hx
+    rcases hx with ⟨y, rfl⟩
+    exact ⟨algebraMap K L y, rfl⟩
+  letI : (CyclicCohomology.extensionSubgroup H₀ J₀ hJH).Normal :=
+    ambientEmbeddedExtensionSubgroup_normal ℚ K L j e
+  change
+    (H₀.toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup H₀ J₀ hJH) ≃*
+      Gal(L / K)
+  exact ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ K L j e
+
+/-- The original lower field is canonically equivalent to the fixed
+field of its explicitly embedded fixing subgroup. -/
+noncomputable def numberFieldEmbeddedAbstractBaseFieldEquiv
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    K ≃ₐ[ℚ]
+      abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L j) :=
+  (numberFieldEmbeddedLowerEmbedding K L j).equivFieldRange.trans
+    (IntermediateField.equivOfEq
+      (InfiniteGalois.fixedField_fixingSubgroup
+        (numberFieldEmbeddedLowerEmbedding K L j).fieldRange).symm)
+
+/-- The original top field is canonically equivalent to the relative
+fixed field of its explicitly embedded fixing subgroup. -/
+noncomputable def numberFieldEmbeddedAbstractTopFieldEquiv
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    L ≃ₐ[ℚ]
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup
+          K L j)).restrictScalars ℚ :=
+  j.equivFieldRange.trans
+    (IntermediateField.equivOfEq
+      (InfiniteGalois.fixedField_fixingSubgroup j.fieldRange).symm)
+
+/-- The two explicit fixed-field equivalences commute with the tower
+algebra maps. -/
+@[simp]
+theorem numberFieldEmbeddedAbstractFieldEquiv_algebraMap
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (x : K) :
+    numberFieldEmbeddedAbstractTopFieldEquiv K L j
+        (algebraMap K L x) =
+      algebraMap
+        (abstractFixedField ℚ (SeparableClosure ℚ)
+          (numberFieldEmbeddedBaseSubgroup K L j))
+        (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j))
+        (numberFieldEmbeddedAbstractBaseFieldEquiv K L j x) := by
+  apply Subtype.ext
+  rfl
+
+/-- The ordinary idele class group of the explicitly embedded lower
+field, transported to the fixed part of the rational absolute
+idele-class representation. -/
+noncomputable def numberFieldEmbeddedIdeleClassEquivAmbientFixed
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    Additive (IdeleClassGroup K) ≃+
+      ambientFixedAddSubgroup
+        rationalIdeleClassRepresentation
+        (numberFieldEmbeddedBaseSubgroup K L j) := by
+  let H := numberFieldEmbeddedBaseSubgroup K L j
+  exact
+    (MulEquiv.toAdditive
+      (ideleClassCongr
+        (numberFieldEmbeddedAbstractBaseFieldEquiv K L j))).trans
+      (rationalAbstractFixedFieldIdeleClassEquivFixed H)
+
+/-- The abstract finite norm quotient of an explicitly embedded tower
+is its genuine ordinary idele-class norm quotient. -/
+noncomputable def
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteNormQuotient rationalIdeleClassRepresentation
+        (numberFieldEmbeddedBaseSubgroup K L j)
+        (numberFieldEmbeddedTopSubgroup K L j)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) ≃+
+      Additive
+        (IdeleClassGroup K ⧸
+          (_root_.ideleClassNorm K L).range) := by
+  let hnormal :=
+    numberFieldEmbeddedExtensionSubgroupNormal K L j
+  let H := numberFieldEmbeddedBaseSubgroup K L j
+  let J := numberFieldEmbeddedTopSubgroup K L j
+  let hJH :=
+    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
+  let fixedFieldEquiv :=
+    rationalFiniteNormQuotientEquivIdeleClassNormQuotient
+      H J hJH hnormal
+  let actualFieldEquiv :=
+    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
+      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
+  exact
+    fixedFieldEquiv.trans
+      (MulEquiv.toAdditive actualFieldEquiv.symm)
+
+
+end EmbeddedNumberFieldRealization
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueFixedFields.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueFixedFields.lean
new file mode 100644
index 0000000..7520ff3
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueFixedFields.lean
@@ -0,0 +1,359 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.AbstractFixedFieldGlobalNormResidueEmbeddedReciprocity
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+
+variable
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (L : FiniteAbelianSubextension K.field)
+
+local instance abstractFixedFieldBaseQuotientFinite :
+    Finite
+      ((baseField
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+          K.field (le_baseField K.field)) :=
+  K.finite
+
+local instance abstractFixedFieldRelativeQuotientFinite :
+    Finite
+      (K.field.toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup K.field L.field L.below) :=
+  L.finite
+
+local instance abstractFixedFieldRelativeQuotientIsMulCommutative :
+    IsMulCommutative L.extensionQuotient :=
+  L.commutative
+
+noncomputable local instance abstractFixedFieldFiniteDimensional :
+    FiniteDimensional ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
+  abstractFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ) K.field K.finite
+
+noncomputable local instance abstractRelativeFixedFieldFiniteDimensional :
+    FiniteDimensional
+      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below) :=
+  abstractRelativeFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ)
+    K.field L.field L.below K.finite L.finite
+
+local instance abstractFixedFieldRelativeScalarTower :
+    IsScalarTower ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below) :=
+  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
+
+noncomputable local instance abstractRelativeFixedFieldAbsoluteFiniteDimensional :
+    FiniteDimensional ℚ
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below) :=
+  FiniteDimensional.trans ℚ
+    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
+    (abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) L.below)
+
+noncomputable local instance abstractFixedFieldNumberField :
+    NumberField
+      (abstractFixedField ℚ (SeparableClosure ℚ) K.field) :=
+  NumberField.of_module_finite ℚ
+    (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
+
+/-- The lower fixed idèle-class group is commutative.  Naming the mixin
+before the public quotient declarations avoids delayed normality synthesis
+inside their definition bodies. -/
+local instance
+    abstractFixedFieldIdeleClassGroupIsMulCommutative :
+    IsMulCommutative
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) K.field)) :=
+  ⟨⟨fun a b => mul_comm a b⟩⟩
+
+noncomputable local instance abstractRelativeFixedFieldNumberField :
+    NumberField
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below) :=
+  NumberField.of_module_finite ℚ
+    (abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) L.below)
+
+/-- Use the same explicit Galois witness as the fixed-field quotient
+comparison.  Deriving it through `IsAbelianGalois` produces an equivalent
+but much larger dependent instance path. -/
+noncomputable local instance
+    abstractRelativeFixedFieldIsGalois :
+    IsGalois
+      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below) :=
+  abstractRelativeFixedField_isGalois
+    ℚ (SeparableClosure ℚ)
+    K.field L.field L.below L.normal
+
+noncomputable local instance abstractRelativeFixedFieldIsAbelianGalois :
+    IsAbelianGalois
+      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below) :=
+  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois L
+
+/-- Use one opaque normality witness for the actual fixed-field norm range.
+This keeps every occurrence of its quotient group on the same instance path. -/
+local instance
+    abstractFixedFieldIdeleClassNormRangeNormal :
+    ((_root_.ideleClassNorm
+      (abstractFixedField ℚ (SeparableClosure ℚ) K.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below)).range).Normal := by
+  infer_instance
+
+/-- The abelianized abstract extension quotient is the actual Galois
+group of the corresponding pair of fixed fields. -/
+noncomputable def
+    abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) K.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below
+    Additive
+        (Abelianization
+          (FiniteGaloisSubextension.extensionQuotient
+            L.toFiniteGaloisExtension)) ≃+
+      Additive (Gal(E / F)) := by
+  dsimp only
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) K.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) L.below
+  let e :
+      L.extensionQuotient ≃*
+        Gal(E / F) :=
+    L.extensionQuotientMulEquiv.trans
+      (abstractExtensionQuotientEquivGaloisGroup
+        ℚ (SeparableClosure ℚ)
+        K.field L.field L.below L.normal)
+  exact
+    MulEquiv.toAdditive
+      ((Abelianization.equivOfComm :
+          L.extensionQuotient ≃*
+            Abelianization L.extensionQuotient).symm.trans e)
+
+/-- The actual fixed-field global norm-residue equivalence
+
+`C_F / N_{E/F} C_E ≃ Gal(E/F)`
+
+attached to an abstract finite abelian subextension in the rational
+absolute class formation. -/
+noncomputable def abstractFixedFieldGlobalNormResidueEquiv :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) K.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below
+    Additive
+        (IdeleClassGroup F ⧸
+          (_root_.ideleClassNorm F E).range) ≃+
+      Additive (Gal(E / F)) := by
+  dsimp only
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) K.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) L.below
+  let eNorm :
+      FiniteNormQuotient rationalIdeleClassRepresentation
+          K.field L.field L.below ≃+
+        Additive
+          (Abelianization L.toFiniteGaloisExtension.extensionQuotient) :=
+    rationalCyclotomicDegreeData.normResidueSymbol
+      rationalIdeleClassRepresentation
+      rationalCyclotomicIdeleClassValuationData
+      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+      K L.toFiniteGaloisExtension
+  exact
+    (rationalFiniteNormQuotientEquivIdeleClassNormQuotient
+        K.field L.field L.below L.normal).symm.trans
+      (eNorm.trans
+        (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
+          K L))
+
+/-- The abstract finite norm-residue equivalence with its dependent source
+instance fixed to the public finite norm quotient. -/
+noncomputable def abstractFixedFieldFiniteNormResidueGaloisEquiv :
+    FiniteNormQuotient rationalIdeleClassRepresentation
+        K.field L.field L.below ≃+
+      Additive
+        (Gal(
+          (abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) L.below) /
+          (abstractFixedField ℚ (SeparableClosure ℚ) K.field))) := by
+  letI : AddCommGroup
+      (FiniteNormQuotient rationalIdeleClassRepresentation
+        K.field L.field L.below) :=
+    finiteNormQuotientAddCommGroup rationalIdeleClassRepresentation
+      K.field L.field L.below
+  exact
+    @AddEquiv.trans
+      (FiniteNormQuotient rationalIdeleClassRepresentation
+        K.field L.field L.below)
+      (Additive
+        (Abelianization
+          L.toFiniteGaloisExtension.extensionQuotient))
+      (Additive
+        (Gal(
+          (abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) L.below) /
+          (abstractFixedField ℚ (SeparableClosure ℚ) K.field))))
+      inferInstance inferInstance inferInstance
+      (rationalCyclotomicDegreeData.normResidueSymbol
+        rationalIdeleClassRepresentation
+        rationalCyclotomicIdeleClassValuationData
+        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+        K L.toFiniteGaloisExtension)
+      (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
+        K L)
+
+/-- The abstract norm-residue symbol on the fixed part of the rational
+absolute idele-class representation, with its value transported to the
+actual Galois group of the two fixed fields.  This is the form consumed
+directly by the abstract norm--restriction naturality theorem. -/
+noncomputable def ambientFixedGlobalNormResidueAddMonoidHom :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) K.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below
+    ambientFixedAddSubgroup
+        rationalIdeleClassRepresentation K.field →+
+      Additive (Gal(E / F)) := by
+  dsimp only
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) K.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) L.below
+  exact
+    (abstractFixedFieldFiniteNormResidueGaloisEquiv K L).toAddMonoidHom.comp
+      (finiteNormClassHom rationalIdeleClassRepresentation
+        K.field L.field L.below)
+
+/-- The ordinary idele class group of the lower fixed field, transported
+to the fixed part of the rational absolute idele-class representation. -/
+private noncomputable def abstractFixedFieldIdeleClassToAmbientFixedMonoidHom :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) K.field
+    IdeleClassGroup F →*
+      Multiplicative
+        (ambientFixedAddSubgroup
+          rationalIdeleClassRepresentation K.field) :=
+  (rationalAbstractFixedFieldIdeleClassEquivFixed
+    K.field).toAddMonoidHom.toMultiplicativeRight
+
+/-- The actual norm-residue homomorphism on the ordinary idele class
+group of the lower fixed field, constructed without choosing a second
+field embedding. -/
+noncomputable def abstractFixedFieldGlobalNormResidueMonoidHom :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) K.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) L.below
+    IdeleClassGroup F →* Gal(E / F) :=
+  (ambientFixedGlobalNormResidueAddMonoidHom K L).toMultiplicative.comp
+    (abstractFixedFieldIdeleClassToAmbientFixedMonoidHom K)
+
+/-- Pointwise form of the ambient fixed-part norm-residue homomorphism. -/
+theorem ambientFixedGlobalNormResidueAddMonoidHom_apply
+    (a :
+      ambientFixedAddSubgroup
+        rationalIdeleClassRepresentation K.field) :
+    ambientFixedGlobalNormResidueAddMonoidHom K L a =
+      abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
+        K L
+        (rationalCyclotomicDegreeData.normResidueSymbol
+          rationalIdeleClassRepresentation
+          rationalCyclotomicIdeleClassValuationData
+          rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+          K L.toFiniteGaloisExtension
+          (finiteNormClass rationalIdeleClassRepresentation
+            K.field L.field L.below a)) := by
+  change
+    abstractFixedFieldFiniteNormResidueGaloisEquiv K L
+        (finiteNormClass rationalIdeleClassRepresentation
+          K.field L.field L.below a) = _
+  rfl
+
+/-- Pointwise form of the transported fixed-field norm-residue homomorphism. -/
+private theorem abstractFixedFieldGlobalNormResidueMonoidHom_apply :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) K.field
+    ∀ c : IdeleClassGroup F,
+      abstractFixedFieldGlobalNormResidueMonoidHom K L c =
+        Additive.toMul
+          (ambientFixedGlobalNormResidueAddMonoidHom K L
+            ((rationalAbstractFixedFieldIdeleClassEquivFixed K.field)
+              (Additive.ofMul c))) := by
+  dsimp only
+  intro c
+  rfl
+
+/-- Transporting an ordinary fixed-field idele class to the ambient
+fixed part and applying the abstract norm-residue map gives exactly the
+actual fixed-field norm-residue value. -/
+@[simp]
+theorem abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply
+    (a :
+      ambientFixedAddSubgroup
+        rationalIdeleClassRepresentation K.field) :
+    abstractFixedFieldGlobalNormResidueMonoidHom K L
+        (Additive.toMul
+          ((rationalAbstractFixedFieldIdeleClassEquivFixed
+            K.field).symm a)) =
+      Additive.toMul
+        (ambientFixedGlobalNormResidueAddMonoidHom K L a) := by
+  rw [abstractFixedFieldGlobalNormResidueMonoidHom_apply]
+  change
+    Additive.toMul
+        (ambientFixedGlobalNormResidueAddMonoidHom K L
+          ((rationalAbstractFixedFieldIdeleClassEquivFixed K.field)
+            ((rationalAbstractFixedFieldIdeleClassEquivFixed
+              K.field).symm a))) =
+      Additive.toMul
+        (ambientFixedGlobalNormResidueAddMonoidHom K L a)
+  rw [(rationalAbstractFixedFieldIdeleClassEquivFixed
+    K.field).apply_symm_apply]
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueNormClasses.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueNormClasses.lean
new file mode 100644
index 0000000..9a8db28
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/AbstractFixedFieldGlobalNormResidueNormClasses.lean
@@ -0,0 +1,294 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.AbstractFixedFieldGlobalNormResidueEmbeddings
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+section EmbeddedNumberFieldRealization
+
+variable
+    (K L : Type) [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+
+
+attribute [local instance]
+  numberFieldEmbeddedIdeleClassGroupIsMulCommutative
+  numberFieldEmbeddedIdeleClassSubgroupNormal
+  numberFieldEmbeddedExtensionSubgroupNormal
+  numberFieldEmbeddedExtensionQuotientFinite
+  numberFieldEmbeddedAbsoluteQuotientFinite
+  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional
+  numberFieldEmbeddedAbstractRelativeFixedFieldFiniteDimensional
+  numberFieldEmbeddedAbstractFixedFieldScalarTower
+  numberFieldEmbeddedAbstractRelativeFixedFieldAbsoluteFiniteDimensional
+  numberFieldEmbeddedAbstractFixedFieldNumberField
+  numberFieldEmbeddedAbstractRelativeFixedFieldNumberField
+  numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsFiniteDimensional
+  numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsNumberField
+  numberFieldEmbeddedAbstractRelativeFixedFieldRestrictScalarsAlgebra
+  numberFieldEmbeddedAbstractRelativeFixedFieldIsGalois
+
+private theorem numberFieldEmbeddedOrdinaryNormQuotientCongr_symm_mk
+    {K₀ L₀ K₁ L₁ : Type}
+    [Field K₀] [NumberField K₀]
+    [Field L₀] [NumberField L₀] [Algebra K₀ L₀]
+    [Field K₁] [NumberField K₁]
+    [Field L₁] [NumberField L₁] [Algebra K₁ L₁]
+    (eK : K₀ ≃ₐ[ℚ] K₁)
+    (eL : L₀ ≃ₐ[ℚ] L₁)
+    (h : ∀ x : K₀,
+      eL (algebraMap K₀ L₀ x) =
+        algebraMap K₁ L₁ (eK x))
+    (c : IdeleClassGroup K₁) :
+    (ordinaryIdeleClassNormQuotientCongrOfAlgEquiv eK eL h).symm
+        (QuotientGroup.mk'
+          (_root_.ideleClassNorm K₁ L₁).range c) =
+      QuotientGroup.mk'
+        (_root_.ideleClassNorm K₀ L₀).range
+        ((ideleClassCongr eK).symm c) := by
+  let e := ordinaryIdeleClassNormQuotientCongrOfAlgEquiv eK eL h
+  apply e.injective
+  rw [e.apply_symm_apply,
+    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv_mk,
+    MulEquiv.apply_symm_apply]
+
+private noncomputable def numberFieldEmbeddedFiniteNormClassPublicValue
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    Additive
+      (IdeleClassGroup K ⧸
+        (_root_.ideleClassNorm K L).range) :=
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
+        K L j
+        (finiteNormClass rationalIdeleClassRepresentation
+          (numberFieldEmbeddedBaseSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
+          a)
+
+private noncomputable def numberFieldEmbeddedFiniteNormClassExpectedValue
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    Additive
+      (IdeleClassGroup K ⧸
+        (_root_.ideleClassNorm K L).range) :=
+  Additive.ofMul
+    (QuotientGroup.mk'
+      (_root_.ideleClassNorm K L).range
+      (Additive.toMul
+        ((numberFieldEmbeddedIdeleClassEquivAmbientFixed K L j).symm a)))
+
+private noncomputable def
+    numberFieldEmbeddedFiniteNormClassDirectComparisonValue
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    Additive
+      (IdeleClassGroup K ⧸
+        (_root_.ideleClassNorm K L).range) := by
+  let hnormal :=
+    numberFieldEmbeddedExtensionSubgroupNormal K L j
+  let H := numberFieldEmbeddedBaseSubgroup K L j
+  let J := numberFieldEmbeddedTopSubgroup K L j
+  let hJH :=
+    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
+  let fixedFieldEquiv :=
+    rationalFiniteNormQuotientEquivIdeleClassNormQuotient
+      H J hJH hnormal
+  let actualFieldEquiv :=
+    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
+      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
+  exact
+    MulEquiv.toAdditive actualFieldEquiv.symm
+      (fixedFieldEquiv
+        (finiteNormClass rationalIdeleClassRepresentation
+          H J hJH a))
+
+private theorem
+    numberFieldEmbeddedFiniteNormClassPublicValue_eq_directComparison
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    numberFieldEmbeddedFiniteNormClassPublicValue K L j a =
+      numberFieldEmbeddedFiniteNormClassDirectComparisonValue K L j a := by
+  unfold numberFieldEmbeddedFiniteNormClassPublicValue
+  unfold numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient
+  unfold numberFieldEmbeddedFiniteNormClassDirectComparisonValue
+  rfl
+
+private noncomputable def numberFieldEmbeddedActualNormClassRepresentativeValue
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    Additive
+      (IdeleClassGroup K ⧸
+        (_root_.ideleClassNorm K L).range) := by
+  let H := numberFieldEmbeddedBaseSubgroup K L j
+  let hJH := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
+  let F := abstractFixedField ℚ (SeparableClosure ℚ) H
+  let E := abstractRelativeFixedField ℚ (SeparableClosure ℚ) hJH
+  let actualFieldEquiv :=
+    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
+      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
+  exact
+    Additive.ofMul
+      (actualFieldEquiv.symm
+        (QuotientGroup.mk'
+          (_root_.ideleClassNorm F E).range
+          (Additive.toMul
+            ((rationalAbstractFixedFieldIdeleClassEquivFixed H).symm a))))
+
+private theorem
+    numberFieldEmbeddedFiniteNormClassDirectComparison_eq_actualValue
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    numberFieldEmbeddedFiniteNormClassDirectComparisonValue K L j a =
+      numberFieldEmbeddedActualNormClassRepresentativeValue K L j a := by
+  let hnormal := numberFieldEmbeddedExtensionSubgroupNormal K L j
+  let H := numberFieldEmbeddedBaseSubgroup K L j
+  let J := numberFieldEmbeddedTopSubgroup K L j
+  let hJH := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
+  let actualFieldEquiv :=
+    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
+      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
+  have hfixed :=
+    rationalFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass
+      H J hJH hnormal a
+  unfold numberFieldEmbeddedFiniteNormClassDirectComparisonValue
+  unfold numberFieldEmbeddedActualNormClassRepresentativeValue
+  exact congrArg (MulEquiv.toAdditive actualFieldEquiv.symm) hfixed
+
+private theorem numberFieldEmbeddedActualNormClassRepresentativeValue_eq_expected
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    numberFieldEmbeddedActualNormClassRepresentativeValue K L j a =
+      numberFieldEmbeddedFiniteNormClassExpectedValue K L j a := by
+  let H := numberFieldEmbeddedBaseSubgroup K L j
+  let hJH := numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j
+  let F := abstractFixedField ℚ (SeparableClosure ℚ) H
+  let E := abstractRelativeFixedField ℚ (SeparableClosure ℚ) hJH
+  let actualFieldEquiv :=
+    ordinaryIdeleClassNormQuotientCongrOfAlgEquiv
+      (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
+      (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j)
+  let c : IdeleClassGroup F :=
+    Additive.toMul
+      ((rationalAbstractFixedFieldIdeleClassEquivFixed H).symm a)
+  unfold numberFieldEmbeddedActualNormClassRepresentativeValue
+  unfold numberFieldEmbeddedFiniteNormClassExpectedValue
+  change
+    Additive.ofMul
+        (actualFieldEquiv.symm
+          (QuotientGroup.mk'
+            (_root_.ideleClassNorm F E).range c)) =
+      Additive.ofMul
+        (QuotientGroup.mk'
+          (_root_.ideleClassNorm K L).range
+          ((ideleClassCongr
+            (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)).symm c))
+  exact
+    congrArg Additive.ofMul
+      (numberFieldEmbeddedOrdinaryNormQuotientCongr_symm_mk
+        (numberFieldEmbeddedAbstractBaseFieldEquiv K L j)
+        (numberFieldEmbeddedAbstractTopFieldEquiv K L j)
+        (numberFieldEmbeddedAbstractFieldEquiv_algebraMap K L j) c)
+
+/-- On a finite norm-class representative, the explicit fixed-field
+comparison is the genuine ordinary idele-class quotient. -/
+@[simp]
+theorem
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (a : ambientFixedAddSubgroup rationalIdeleClassRepresentation
+      (numberFieldEmbeddedBaseSubgroup K L j)) :
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient K L j
+        (finiteNormClass rationalIdeleClassRepresentation
+          (numberFieldEmbeddedBaseSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j) a) =
+      Additive.ofMul
+        (QuotientGroup.mk'
+          (_root_.ideleClassNorm K L).range
+          (Additive.toMul
+            ((numberFieldEmbeddedIdeleClassEquivAmbientFixed K L j).symm a))) := by
+  change
+    numberFieldEmbeddedFiniteNormClassPublicValue K L j a =
+      numberFieldEmbeddedFiniteNormClassExpectedValue K L j a
+  exact
+    (numberFieldEmbeddedFiniteNormClassPublicValue_eq_directComparison
+      K L j a).trans
+      ((numberFieldEmbeddedFiniteNormClassDirectComparison_eq_actualValue
+        K L j a).trans
+        (numberFieldEmbeddedActualNormClassRepresentativeValue_eq_expected
+          K L j a))
+
+/-- On an ordinary idele class, the explicit fixed-part realization
+followed by the abstract finite norm-class map is the genuine quotient
+class modulo the ordinary idele-class norm. -/
+@[simp]
+theorem
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_ideleClass
+    [FiniteDimensional K L] [IsGalois K L]
+    (j : L →ₐ[ℚ] SeparableClosure ℚ)
+    (c : IdeleClassGroup K) :
+    numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient K L j
+        (finiteNormClass rationalIdeleClassRepresentation
+          (numberFieldEmbeddedBaseSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup K L j)
+          (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L j)
+          (numberFieldEmbeddedIdeleClassEquivAmbientFixed
+            K L j (Additive.ofMul c))) =
+      Additive.ofMul
+        (QuotientGroup.mk'
+          (_root_.ideleClassNorm K L).range c) := by
+  simpa only [AddEquiv.symm_apply_apply, toMul_ofMul] using
+    (numberFieldEmbeddedFiniteNormQuotientEquivIdeleClassNormQuotient_finiteNormClass
+      K L j
+      (numberFieldEmbeddedIdeleClassEquivAmbientFixed
+        K L j (Additive.ofMul c)))
+
+
+end EmbeddedNumberFieldRealization
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtin.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtin.lean
index d3e6e51..75c1574 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtin.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtin.lean
@@ -4,26 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicIdeleClassValuation
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.InfiniteGlobalArtin
-import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.MaximalUnramifiedReciprocity
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicAbstractFixedFieldArtinComparison
 
-set_option autoImplicit false
-
-/-!
-# Cyclotomic Artin coordinates over abstract fixed fields
 
-The cyclotomic degree datum on the rational absolute Galois group has
-an actual maximal-unramified field over every finite abstract fixed
-field.  This file identifies its genuine Galois group with
-`Multiplicative ZHat`, using the normalized degree map, and supplies
-the abelian Galois structure needed by the actual infinite global
-Artin homomorphism.
+set_option autoImplicit false
 
-These constructions are the source side of the comparison between
-abstract finite reciprocity and the chosen local-factor product.  No
-reciprocity comparison is assumed in their definitions.
--/
 
 open scoped IsMulCommutative NumberField
 
@@ -35,2336 +20,11 @@ namespace Reciprocity
 open ClassFormation
 open KummerTheory
 
-/-- Keep this module on the rational algebra structures used by the
-cyclotomic fixed-field API.  Generic intermediate-field instances are
-propositionally equal here but not definitionally interchangeable. -/
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
-    Algebra ℚ (SeparableClosure ℚ) :=
-  DivisionRing.toRatAlgebra
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
-    Algebra ℚ rationalCyclotomicZHatField :=
-  DivisionRing.toRatAlgebra
-
-/-- Cyclotomic field inertia is contained in the original abstract
-field subgroup. -/
-theorem rationalCyclotomicFieldInertia_le
-    (H : ClosedSubgroup
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    (rationalCyclotomicDegreeData.fieldInertia H).toSubgroup ≤
-      H.toSubgroup := by
-  intro σ hσ
-  exact hσ.1
-
-/-- Viewing cyclotomic field inertia inside its ambient field subgroup
-gives exactly the kernel of normalized degree. -/
-theorem extensionSubgroup_rationalCyclotomicFieldInertia
-    (H : ClosedSubgroup
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    CyclicCohomology.extensionSubgroup H
-        (rationalCyclotomicDegreeData.fieldInertia H)
-        (rationalCyclotomicFieldInertia_le H) =
-      rationalCyclotomicDegreeData.fieldInertiaWithin H := by
-  ext σ
-  rw [
-    mem_extensionSubgroup_iff,
-    rationalCyclotomicDegreeData.mem_fieldInertiaWithin_iff,
-    rationalCyclotomicDegreeData.mem_fieldInertia_iff]
-  exact and_iff_right σ.2
-
-/-- The actual Galois group of the cyclotomic maximal-unramified
-extension of an abstract fixed field, in its normalized `ZHat`
-coordinate. -/
-noncomputable def abstractFixedFieldCyclotomicGalEquivZHat
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) hI /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field) ≃*
-      Multiplicative ZHat := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let hnormal :
-      (CyclicCohomology.extensionSubgroup H.field
-        (rationalCyclotomicDegreeData.fieldInertia H.field)
-        hI).Normal := by
-    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-    infer_instance
-  let qField :
-      H.field.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H.field
-            (rationalCyclotomicDegreeData.fieldInertia H.field)
-            hI ≃*
-        Gal(
-          LocalClassFieldTheory.abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) hI /
-          LocalClassFieldTheory.abstractFixedField
-            ℚ (SeparableClosure ℚ) H.field) :=
-    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-      ℚ (SeparableClosure ℚ) H.field
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-      hI hnormal
-  let qInertia :
-      H.field.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H.field
-            (rationalCyclotomicDegreeData.fieldInertia H.field)
-            hI ≃*
-        H.field.toSubgroup ⧸
-          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
-    QuotientGroup.quotientMulEquivOfEq
-      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
-  exact
-    qField.symm.trans
-      (qInertia.trans
-        (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
-          (H.toFiniteResidueAbstractField
-            rationalCyclotomicDegreeData)))
-
-/-- On an absolute-Galois representative fixing the lower field, the
-actual cyclotomic Galois coordinate is its normalized degree. -/
-@[simp]
-theorem abstractFixedFieldCyclotomicGalEquivZHat_extensionClass
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (σ : H.field.toSubgroup) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    let hnormal :
-        (CyclicCohomology.extensionSubgroup H.field
-          (rationalCyclotomicDegreeData.fieldInertia H.field)
-          hI).Normal := by
-      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-      infer_instance
-    abstractFixedFieldCyclotomicGalEquivZHat H
-        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-          ℚ (SeparableClosure ℚ) H.field
-          (rationalCyclotomicDegreeData.fieldInertia H.field)
-          hI hnormal
-          (QuotientGroup.mk σ)) =
-      rationalCyclotomicDegreeData.normalizedDegree
-        (H.toFiniteResidueAbstractField
-          rationalCyclotomicDegreeData) σ := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let hnormal :
-      (CyclicCohomology.extensionSubgroup H.field
-        (rationalCyclotomicDegreeData.fieldInertia H.field)
-        hI).Normal := by
-    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-    infer_instance
-  let qField :=
-    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-      ℚ (SeparableClosure ℚ) H.field
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-      hI hnormal
-  change
-    (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
-      (H.toFiniteResidueAbstractField
-        rationalCyclotomicDegreeData))
-      (QuotientGroup.quotientMulEquivOfEq
-        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
-        (qField.symm (qField (QuotientGroup.mk σ)))) =
-      rationalCyclotomicDegreeData.normalizedDegree
-        (H.toFiniteResidueAbstractField
-          rationalCyclotomicDegreeData) σ
-  rw [qField.symm_apply_apply]
-  rfl
-
-/-- Quotient-level evaluation of the actual cyclotomic Galois
-coordinate. -/
-@[simp]
-theorem abstractFixedFieldCyclotomicGalEquivZHat_quotientClass
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (q :
-      H.field.toSubgroup ⧸
-        rationalCyclotomicDegreeData.fieldInertiaWithin H.field) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    let hnormal :
-        (CyclicCohomology.extensionSubgroup H.field
-          (rationalCyclotomicDegreeData.fieldInertia H.field)
-          hI).Normal := by
-      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-      infer_instance
-    let qInertia :
-        H.field.toSubgroup ⧸
-            CyclicCohomology.extensionSubgroup H.field
-              (rationalCyclotomicDegreeData.fieldInertia H.field)
-              hI ≃*
-          H.field.toSubgroup ⧸
-            rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
-      QuotientGroup.quotientMulEquivOfEq
-        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
-    abstractFixedFieldCyclotomicGalEquivZHat H
-        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-          ℚ (SeparableClosure ℚ) H.field
-          (rationalCyclotomicDegreeData.fieldInertia H.field)
-          hI hnormal
-          (qInertia.symm q)) =
-      rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
-        (H.toFiniteResidueAbstractField
-          rationalCyclotomicDegreeData) q := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let hnormal :
-      (CyclicCohomology.extensionSubgroup H.field
-        (rationalCyclotomicDegreeData.fieldInertia H.field)
-        hI).Normal := by
-    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-    infer_instance
-  let qField :=
-    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-      ℚ (SeparableClosure ℚ) H.field
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-      hI hnormal
-  let qInertia :
-      H.field.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H.field
-            (rationalCyclotomicDegreeData.fieldInertia H.field)
-            hI ≃*
-        H.field.toSubgroup ⧸
-          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
-    QuotientGroup.quotientMulEquivOfEq
-      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
-  change
-    (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
-      (H.toFiniteResidueAbstractField
-        rationalCyclotomicDegreeData))
-      (qInertia
-        (qField.symm
-          (qField (qInertia.symm q)))) =
-      rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
-        (H.toFiniteResidueAbstractField
-          rationalCyclotomicDegreeData) q
-  rw [qField.symm_apply_apply, qInertia.apply_symm_apply]
-
-/-- The extension fixed by cyclotomic field inertia is an actual
-abelian Galois extension of the lower abstract fixed field. -/
-theorem abstractFixedFieldCyclotomic_isAbelianGalois
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    IsAbelianGalois
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let hnormal :
-      (CyclicCohomology.extensionSubgroup H.field
-        (rationalCyclotomicDegreeData.fieldInertia H.field)
-        hI).Normal := by
-    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-    infer_instance
-  let : IsGalois
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) :=
-    LocalClassFieldTheory.abstractRelativeFixedField_isGalois
-      ℚ (SeparableClosure ℚ) H.field
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-      hI hnormal
-  let e :=
-    abstractFixedFieldCyclotomicGalEquivZHat H
-  exact
-    { is_comm.comm := by
-        intro σ τ
-        apply e.injective
-        simpa only [map_mul] using mul_comm (e σ) (e τ) }
-
-/-- The rational cyclotomic `ZHat`-field embedded in the actual
-maximal-unramified compositum of an abstract fixed field. -/
-noncomputable def
-    rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    rationalCyclotomicZHatField →ₐ[ℚ]
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let J :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ)
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-  have hTJ :
-      rationalCyclotomicZHatField ≤ J := by
-    change
-      rationalCyclotomicZHatField ≤
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicDegreeData.fieldInertia H.field)
-    rw [
-      rationalCyclotomicDegreeData_fixedField_fieldInertia]
-    exact le_sup_right
-  exact IntermediateField.inclusion hTJ
-
-noncomputable instance
-    abstractFixedFieldCyclotomicCompositum_algebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    Algebra rationalCyclotomicZHatField
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) :=
-  ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-    H).toRingHom).toAlgebra
-
-instance abstractFixedFieldCyclotomicCompositum_scalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    IsScalarTower ℚ rationalCyclotomicZHatField
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) :=
-  IsScalarTower.of_algebraMap_eq'
-    ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-      H).comp_algebraMap).symm
-
-instance abstractFixedFieldCyclotomicCompositum_baseScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    IsScalarTower ℚ
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  apply IsScalarTower.of_algebraMap_eq
-  intro x
-  apply Subtype.ext
-  rfl
-
-/-- Restriction from the actual maximal-unramified compositum of an
-abstract fixed field to the rational cyclotomic factor. -/
-noncomputable def abstractFixedFieldCyclotomicRestriction
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) hI /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field) →*
-      Gal(rationalCyclotomicZHatField / ℚ) := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  exact
-    IntermediateField.restrictRestrictAlgEquivMapHom
-      ℚ rationalCyclotomicZHatField
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI)
-
-/-- On a quotient representative, cyclotomic restriction of the
-actual relative automorphism is ordinary restriction of the same
-ambient absolute-Galois automorphism. -/
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
-    Normal ℚ rationalCyclotomicZHatField :=
-  rationalCyclotomicZHatField_isNormal
-
-@[simp]
-theorem abstractFixedFieldCyclotomicRestriction_extensionClass
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (σ : H.field.toSubgroup) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    let hnormal :
-        (CyclicCohomology.extensionSubgroup H.field
-          (rationalCyclotomicDegreeData.fieldInertia H.field)
-          hI).Normal := by
-      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-      infer_instance
-    abstractFixedFieldCyclotomicRestriction H
-        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-          ℚ (SeparableClosure ℚ) H.field
-          (rationalCyclotomicDegreeData.fieldInertia H.field)
-          hI hnormal
-          (QuotientGroup.mk σ)) =
-      AlgEquiv.restrictNormalHom
-        rationalCyclotomicZHatField σ.1 := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let hnormal :
-      (CyclicCohomology.extensionSubgroup H.field
-        (rationalCyclotomicDegreeData.fieldInertia H.field)
-        hI).Normal := by
-    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-    infer_instance
-  let τ :=
-    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-      ℚ (SeparableClosure ℚ) H.field
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-      hI hnormal
-      (QuotientGroup.mk'
-        (CyclicCohomology.extensionSubgroup H.field
-          (rationalCyclotomicDegreeData.fieldInertia H.field) hI)
-        σ)
-  apply AlgEquiv.ext
-  intro x
-  apply Subtype.ext
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) hI
-  have hrestrict :
-      algebraMap rationalCyclotomicZHatField U
-          ((abstractFixedFieldCyclotomicRestriction H τ) x) =
-        τ (algebraMap rationalCyclotomicZHatField U x) := by
-    change
-      algebraMap rationalCyclotomicZHatField U
-          ((AlgEquiv.restrictNormal
-            (MulSemiringAction.toAlgEquiv ℚ U τ)
-            rationalCyclotomicZHatField) x) =
-        (MulSemiringAction.toAlgEquiv ℚ U τ)
-          (algebraMap rationalCyclotomicZHatField U x)
-    exact
-      AlgEquiv.restrictNormal_commutes
-        (MulSemiringAction.toAlgEquiv ℚ U τ)
-        rationalCyclotomicZHatField x
-  have halgebraMap_eq_embedding
-      (z : rationalCyclotomicZHatField) :
-      algebraMap rationalCyclotomicZHatField U z =
-        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-          H z : U) := by
-    rfl
-  have hembedding_coe
-      (z : rationalCyclotomicZHatField) :
-      (((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-          H z : U) : SeparableClosure ℚ)) =
-        (z : SeparableClosure ℚ) := by
-    rfl
-  have halgebraMap_coe
-      (z : rationalCyclotomicZHatField) :
-      ((algebraMap rationalCyclotomicZHatField U z : U) :
-          SeparableClosure ℚ) =
-        (z : SeparableClosure ℚ) := by
-    rw [halgebraMap_eq_embedding]
-    exact hembedding_coe z
-  have hambient :=
-    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup_mk_apply_val
-        ℚ (SeparableClosure ℚ) H.field
-        (rationalCyclotomicDegreeData.fieldInertia H.field)
-        hI hnormal σ
-        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-          H x)
-  calc
-    (((abstractFixedFieldCyclotomicRestriction H τ) x :
-        rationalCyclotomicZHatField) :
-        SeparableClosure ℚ) =
-        ((τ
-          (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-            H x) :
-          LocalClassFieldTheory.abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) hI) :
-          SeparableClosure ℚ) := by
-      calc
-        (((abstractFixedFieldCyclotomicRestriction H τ) x :
-            rationalCyclotomicZHatField) : SeparableClosure ℚ) =
-            ((algebraMap rationalCyclotomicZHatField U
-              ((abstractFixedFieldCyclotomicRestriction H τ) x) : U) :
-              SeparableClosure ℚ) :=
-          (halgebraMap_coe
-            ((abstractFixedFieldCyclotomicRestriction H τ) x)).symm
-        _ = ((τ (algebraMap rationalCyclotomicZHatField U x) : U) :
-            SeparableClosure ℚ) :=
-          congrArg (fun y : U => (y : SeparableClosure ℚ)) hrestrict
-        _ = ((τ
-            (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-              H x) : U) : SeparableClosure ℚ) := by
-          rw [halgebraMap_eq_embedding]
-    _ = σ.1 (x : SeparableClosure ℚ) := by
-      calc
-        ((τ
-            (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-              H x) : U) : SeparableClosure ℚ) =
-            σ.1
-              ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-                H x : U) : SeparableClosure ℚ) := by
-          simpa only [τ, U] using hambient.symm
-        _ = σ.1 (x : SeparableClosure ℚ) := by
-          rw [hembedding_coe]
-    _ =
-        (((AlgEquiv.restrictNormalHom
-          rationalCyclotomicZHatField σ.1) x :
-          rationalCyclotomicZHatField) :
-          SeparableClosure ℚ) := by
-      exact
-        (AlgEquiv.restrictNormal_commutes
-          σ.1 rationalCyclotomicZHatField x).symm
-
-/-- Raw cyclotomic restriction is residue-degree multiplication of
-the normalized actual Galois coordinate. -/
-theorem
-    abstractFixedFieldCyclotomicRestriction_coordinate
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (τ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field)) :
-    Multiplicative.toAdd
-        (rationalCyclotomicZHatFieldGalEquivZHat
-          (abstractFixedFieldCyclotomicRestriction H τ)) =
-      (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
-        Multiplicative.toAdd
-          (abstractFixedFieldCyclotomicGalEquivZHat H τ) := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let hnormal :
-      (CyclicCohomology.extensionSubgroup H.field
-        (rationalCyclotomicDegreeData.fieldInertia H.field)
-        hI).Normal := by
-    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
-    infer_instance
-  let qField :=
-    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-      ℚ (SeparableClosure ℚ) H.field
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-      hI hnormal
-  obtain ⟨q, rfl⟩ := qField.surjective τ
-  refine Quotient.inductionOn' q ?_
-  intro σ
-  rw [
-    abstractFixedFieldCyclotomicRestriction_extensionClass,
-    abstractFixedFieldCyclotomicGalEquivZHat_extensionClass]
-  exact
-    (rationalCyclotomicDegreeData.residueDegree_nsmul_normalizedDegree
-        (H.toFiniteResidueAbstractField
-          rationalCyclotomicDegreeData) σ).symm
-
-/-- The canonical compositum of an abstract fixed field with a finite
-layer of the rational cyclotomic `ZHat`-extension. -/
-def abstractFixedFieldCyclotomicFiniteCompositum
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IntermediateField ℚ (SeparableClosure ℚ) :=
-  LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field ⊔
-    IntermediateField.lift E.toIntermediateField
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteCompositum_finiteDimensional
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    FiniteDimensional ℚ
-      (abstractFixedFieldCyclotomicFiniteCompositum H E) := by
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field
-  let : FiniteDimensional ℚ F :=
-    LocalClassFieldTheory.abstractFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ) H.field H.finite
-  let : FiniteDimensional ℚ
-      (IntermediateField.lift E.toIntermediateField) :=
-    ((IntermediateField.liftAlgEquiv
-      E.toIntermediateField).toLinearEquiv).finiteDimensional
-  exact IntermediateField.finiteDimensional_sup
-    F (IntermediateField.lift E.toIntermediateField)
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteCompositum_numberField
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    NumberField
-      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
-  NumberField.of_module_finite ℚ
-    (abstractFixedFieldCyclotomicFiniteCompositum H E)
-
-/-- The lower abstract fixed field embedded into its finite
-cyclotomic compositum. -/
-noncomputable def
-    abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field →ₐ[ℚ]
-      abstractFixedFieldCyclotomicFiniteCompositum H E :=
-  IntermediateField.inclusion
-    (show
-      LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field ≤
-        abstractFixedFieldCyclotomicFiniteCompositum H E from
-      le_sup_left)
-
-/-- The fixed field attached to a finite abstract field is a number
-field.  Keeping this as the single file-local instance makes it
-available while later theorem binders are elaborated. -/
-noncomputable local instance
-    abstractFixedFieldCyclotomic_numberField
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    NumberField
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field) := by
-  let : FiniteDimensional ℚ
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field) :=
-    LocalClassFieldTheory.abstractFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ) H.field H.finite
-  exact
-    NumberField.of_module_finite ℚ
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-
-/-- A finite rational cyclotomic layer embedded into its compositum
-with the abstract fixed field. -/
-noncomputable def
-    abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    E →ₐ[ℚ]
-      abstractFixedFieldCyclotomicFiniteCompositum H E :=
-  (IntermediateField.inclusion
-      (show
-        IntermediateField.lift E.toIntermediateField ≤
-          abstractFixedFieldCyclotomicFiniteCompositum H E from
-        le_sup_right)).comp
-    (IntermediateField.liftAlgEquiv E.toIntermediateField).toAlgHom
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteCompositum_baseAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
-  RingHom.toAlgebra
-    (AlgHom.toRingHom
-      (abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding H E))
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteCompositum_layerAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra E
-      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
-  RingHom.toAlgebra
-    (AlgHom.toRingHom
-      (abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding H E))
-
-/-- The finite-layer action induced by its explicit embedding into the
-finite compositum. -/
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteCompositum_layerSMul
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    SMul E
-      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
-  Algebra.toSMul
-    (self :=
-      abstractFixedFieldCyclotomicFiniteCompositum_layerAlgebra H E)
-
-instance
-    abstractFixedFieldCyclotomicFiniteCompositum_baseScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower ℚ
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
-  IsScalarTower.of_algebraMap_eq'
-    (AlgHom.comp_algebraMap
-      (abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding H E)).symm
-
-instance
-    abstractFixedFieldCyclotomicFiniteCompositum_layerScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower ℚ E
-      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
-  IsScalarTower.of_algebraMap_eq'
-    (AlgHom.comp_algebraMap
-      (abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding H E)).symm
-
-/-- Inclusion of the finite cyclotomic compositum into the actual
-maximal-unramified compositum. -/
-noncomputable def
-    abstractFixedFieldCyclotomicFiniteCompositumInclusion
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    abstractFixedFieldCyclotomicFiniteCompositum H E →ₐ[ℚ]
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let J :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ)
-      (rationalCyclotomicDegreeData.fieldInertia H.field)
-  have hle :
-      abstractFixedFieldCyclotomicFiniteCompositum H E ≤ J := by
-    dsimp only [J]
-    rw [
-      rationalCyclotomicDegreeData_fixedField_fieldInertia H.field]
-    exact
-      sup_le_sup le_rfl
-        (IntermediateField.lift_le E.toIntermediateField)
-  exact IntermediateField.inclusion hle
-
-/-- The same finite-compositum inclusion over the lower abstract fixed
-field. -/
-noncomputable def
-    abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    abstractFixedFieldCyclotomicFiniteCompositum H E →ₐ[
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field]
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI := by
-  let f :=
-    abstractFixedFieldCyclotomicFiniteCompositumInclusion H E
-  exact
-    { f.toRingHom with
-      commutes' := by
-        intro x
-        rfl }
-
-/-- The finite cyclotomic compositum as an intermediate field of the
-actual maximal-unramified extension. -/
-noncomputable def abstractFixedFieldCyclotomicFiniteLayer
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    IntermediateField
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) :=
-  (abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase H E).fieldRange
-
-/-- The base algebra on the finite field range, obtained from the
-explicit base embedding followed by the field-range equivalence. -/
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  RingHom.toAlgebra
-    ((AlgHom.toRingHom
-      (AlgEquiv.toAlgHom
-        (AlgHom.equivFieldRange
-          (abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
-            H E)))).comp
-      (AlgHom.toRingHom
-        (abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding H E)))
-
-/-- The scalar action belonging to the canonical base algebra on the
-finite field range. -/
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_baseSMul
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    SMul
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  Algebra.toSMul
-    (self := abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra H E)
-
-/-- The module structure belonging to the canonical base algebra on
-the finite field range. -/
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_baseModule
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Module
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  @Algebra.toModule
-    (LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field)
-    (abstractFixedFieldCyclotomicFiniteLayer H E)
-    _ _
-    (abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra H E)
-
-/-- The field-range equivalence rebuilt over the explicit base
-algebras.  Its underlying ring equivalence is the canonical one. -/
-noncomputable def
-    abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    abstractFixedFieldCyclotomicFiniteCompositum H E ≃ₐ[
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field]
-      abstractFixedFieldCyclotomicFiniteLayer H E :=
-  AlgEquiv.ofRingEquiv
-    (f :=
-      (AlgHom.equivFieldRange
-        (abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
-          H E)).toRingEquiv)
-    (fun _ => rfl)
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    FiniteDimensional
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  (abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
-    H E).toLinearEquiv.finiteDimensional
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_numberField
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    NumberField
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  NumberField.of_module_finite
-    (LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field)
-    (abstractFixedFieldCyclotomicFiniteLayer H E)
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsAbelianGalois
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E) := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let : IsAbelianGalois
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) :=
-    abstractFixedFieldCyclotomic_isAbelianGalois H
-  exact
-    IsAbelianGalois.of_algHom
-      ((abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
-          H E).comp
-        (AlgEquiv.toAlgHom
-          (AlgEquiv.symm
-            (abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
-              H E))))
-
-instance
-    abstractFixedFieldCyclotomicFiniteLayer_baseScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower ℚ
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E) := by
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  apply IsScalarTower.of_algebraMap_eq
-  intro x
-  apply Subtype.ext
-  change
-    algebraMap ℚ
-        (LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) hI) x =
-      algebraMap
-        (LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field)
-        (LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) hI)
-        (algebraMap ℚ
-          (LocalClassFieldTheory.abstractFixedField
-            ℚ (SeparableClosure ℚ) H.field) x)
-  exact
-    IsScalarTower.algebraMap_apply
-      ℚ
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) x
-
-/-- The finite rational layer embedded into its corresponding
-intermediate field over the abstract fixed field. -/
-noncomputable def
-    abstractFixedFieldCyclotomicFiniteLayerEmbedding
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    E →ₐ[ℚ]
-      abstractFixedFieldCyclotomicFiniteLayer H E := by
-  exact
-    (AlgEquiv.toAlgHom
-      (AlgEquiv.restrictScalars ℚ
-        (abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
-          H E))).comp
-      (abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding H E)
-
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra E
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  RingHom.toAlgebra
-    (AlgHom.toRingHom
-      (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E))
-
-/-- The finite-layer action on its actual image in the relative fixed
-field. -/
-noncomputable instance
-    abstractFixedFieldCyclotomicFiniteLayer_layerSMul
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    SMul E
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  Algebra.toSMul
-    (self := abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E)
-
-instance
-    abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower ℚ E
-      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
-  IsScalarTower.of_algebraMap_eq'
-    (AlgHom.comp_algebraMap
-      (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E)).symm
-
-/-- The finite cyclotomic layer as an object of the finite-Galois
-inverse system of the actual maximal-unramified extension. -/
-@[reducible]
-noncomputable def
-    abstractFixedFieldCyclotomicFiniteGaloisLayer
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    FiniteGaloisIntermediateField
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (rationalCyclotomicFieldInertia_le H.field)) where
-  toIntermediateField :=
-    abstractFixedFieldCyclotomicFiniteLayer H E
-  finiteDimensional :=
-    abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional H E
-  isGalois :=
-    (abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E).toIsGalois
-
-/-- The explicit base algebra on a finite cyclotomic layer is the canonical
-intermediate-field inclusion used by the finite Galois inverse system. -/
-theorem abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra_eq_algebra'
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra H E =
-      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra' := by
-  apply Algebra.algebra_ext
-  intro x
-  apply Subtype.ext
-  rfl
-
-/-- The finite Galois layer uses its canonical inclusion into the full
-relative fixed field for the upper scalar action. -/
-instance
-    abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).toIntermediateField
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (rationalCyclotomicFieldInertia_le H.field)) := by
-  let i :
-      (abstractFixedFieldCyclotomicFiniteGaloisLayer
-          H E).toIntermediateField →ₐ[
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field]
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) :=
-    { (abstractFixedFieldCyclotomicFiniteGaloisLayer
-          H E).toIntermediateField.val.toRingHom with
-      commutes' := by
-        intro x
-        rfl }
-  exact
-    IsScalarTower.of_algebraMap_eq'
-      (AlgHom.comp_algebraMap i).symm
-
-/-- The canonical inclusion of the finite cyclotomic layer into the
-full abstract-fixed-field compositum. -/
-private noncomputable def
-    abstractFixedFieldCyclotomicFiniteLayerInclusion
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    abstractFixedFieldCyclotomicFiniteLayer H E →ₐ[
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field]
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (rationalCyclotomicFieldInertia_le H.field) :=
-  IntermediateField.val
-    (abstractFixedFieldCyclotomicFiniteLayer H E)
-
-/-- The two embeddings of a finite rational cyclotomic layer into the
-full abstract-fixed-field compositum agree. -/
-private theorem
-    abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    (z : E) :
-    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E z) =
-      rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
-        (z : rationalCyclotomicZHatField) := by
-  exact Subtype.ext rfl
-
-/-- Restriction to `E` commutes pointwise with the restriction from the
-full rational cyclotomic tower. -/
-private theorem
-    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    [Normal ℚ E]
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field))
-    (z : E) :
-    ((AlgEquiv.restrictNormalHom E
-      (abstractFixedFieldCyclotomicRestriction H σ)) z :
-        rationalCyclotomicZHatField) =
-      (abstractFixedFieldCyclotomicRestriction H σ)
-        (z : rationalCyclotomicZHatField) := by
-  exact
-    AlgEquiv.restrictNormal_commutes
-      (abstractFixedFieldCyclotomicRestriction H σ) E z
-
-/-- The raw cyclotomic restriction commutes with the canonical embedding
-of the full rational cyclotomic tower. -/
-private theorem
-    abstractFixedFieldCyclotomicRestriction_embedding_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field))
-    (z : rationalCyclotomicZHatField) :
-    rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
-        (abstractFixedFieldCyclotomicRestriction H σ z) =
-      σ
-        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-          H z) := by
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ)
-      (rationalCyclotomicFieldInertia_le H.field)
-  exact
-    AlgEquiv.restrictNormal_commutes
-      (MulSemiringAction.toAlgEquiv ℚ U σ)
-      rationalCyclotomicZHatField z
-
-/-- Restriction to the finite compositum layer commutes with its
-canonical inclusion into the full compositum. -/
-private theorem
-    restrictNormalHom_abstractFixedFieldCyclotomicFiniteLayer_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    [Normal
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E)]
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field))
-    (z : abstractFixedFieldCyclotomicFiniteLayer H E) :
-    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-        (AlgEquiv.restrictNormalHom
-          (abstractFixedFieldCyclotomicFiniteLayer H E) σ z) =
-      σ (abstractFixedFieldCyclotomicFiniteLayerInclusion H E z) := by
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ)
-      (rationalCyclotomicFieldInertia_le H.field)
-  let P : IntermediateField F U :=
-    abstractFixedFieldCyclotomicFiniteLayer H E
-  exact
-    AlgEquiv.restrictNormal_commutes
-      (MulSemiringAction.toAlgEquiv F U σ) P z
-
-/-- Restricting the finite-compositum action further to `E` commutes
-with the explicit embedding of `E` into that finite layer. -/
-private theorem
-    abstractFixedFieldCyclotomicFiniteLayerEmbedding_restrict_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    [Normal ℚ E]
-    [Normal
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E)]
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field))
-    (z : E) :
-    abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
-        (IntermediateField.restrictRestrictAlgEquivMapHom
-          ℚ E
-          (LocalClassFieldTheory.abstractFixedField
-            ℚ (SeparableClosure ℚ) H.field)
-          (abstractFixedFieldCyclotomicFiniteLayer H E)
-          (AlgEquiv.restrictNormalHom
-            (abstractFixedFieldCyclotomicFiniteLayer H E) σ) z) =
-      (AlgEquiv.restrictNormalHom
-        (abstractFixedFieldCyclotomicFiniteLayer H E) σ)
-        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E z) := by
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ)
-      (rationalCyclotomicFieldInertia_le H.field)
-  let P : IntermediateField F U :=
-    abstractFixedFieldCyclotomicFiniteLayer H E
-  exact
-    AlgEquiv.restrictNormal_commutes
-      (MulSemiringAction.toAlgEquiv ℚ P
-        (AlgEquiv.restrictNormalHom P σ)) E z
-
-/-- The left finite-level restriction, after both canonical embeddings into
-the full compositum, is the action of `σ` on the cyclotomic embedding. -/
-private theorem
-    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_left_embedded
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    [Normal ℚ E]
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field))
-    (x : E) :
-    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
-          (AlgEquiv.restrictNormalHom E
-            (abstractFixedFieldCyclotomicRestriction H σ) x)) =
-      σ
-        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
-          (x : rationalCyclotomicZHatField)) := by
-  calc
-    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-          (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
-            (AlgEquiv.restrictNormalHom E
-              (abstractFixedFieldCyclotomicRestriction H σ) x)) =
-        rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
-          (((AlgEquiv.restrictNormalHom E
-            (abstractFixedFieldCyclotomicRestriction H σ) x) :
-              rationalCyclotomicZHatField)) :=
-      abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion H E _
-    _ = rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
-          (abstractFixedFieldCyclotomicRestriction H σ
-            (x : rationalCyclotomicZHatField)) :=
-      congrArg
-        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H)
-        (restrictNormalHom_abstractFixedFieldCyclotomicRestriction_apply
-          H E σ x)
-    _ = σ
-          (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-            H (x : rationalCyclotomicZHatField)) :=
-      abstractFixedFieldCyclotomicRestriction_embedding_apply
-        H σ (x : rationalCyclotomicZHatField)
-
-/-- The right finite-level restriction, after both canonical embeddings into
-the full compositum, is the same action of `σ`. -/
-private theorem
-    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_right_embedded
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    [Normal ℚ E]
-    [Normal
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E)]
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field))
-    (x : E) :
-    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
-          (IntermediateField.restrictRestrictAlgEquivMapHom
-            ℚ E
-            (LocalClassFieldTheory.abstractFixedField
-              ℚ (SeparableClosure ℚ) H.field)
-            (abstractFixedFieldCyclotomicFiniteLayer H E)
-            (AlgEquiv.restrictNormalHom
-              (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x)) =
-      σ
-        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
-          (x : rationalCyclotomicZHatField)) := by
-  calc
-    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-          (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
-            (IntermediateField.restrictRestrictAlgEquivMapHom
-              ℚ E
-              (LocalClassFieldTheory.abstractFixedField
-                ℚ (SeparableClosure ℚ) H.field)
-              (abstractFixedFieldCyclotomicFiniteLayer H E)
-              (AlgEquiv.restrictNormalHom
-                (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x)) =
-        abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-          ((AlgEquiv.restrictNormalHom
-            (abstractFixedFieldCyclotomicFiniteLayer H E) σ)
-            (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E x)) :=
-      congrArg
-        (abstractFixedFieldCyclotomicFiniteLayerInclusion H E)
-        (abstractFixedFieldCyclotomicFiniteLayerEmbedding_restrict_apply
-          H E σ x)
-    _ = σ
-          (abstractFixedFieldCyclotomicFiniteLayerInclusion H E
-            (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E x)) :=
-      restrictNormalHom_abstractFixedFieldCyclotomicFiniteLayer_apply
-        H E σ (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E x)
-    _ = σ
-          (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
-            H (x : rationalCyclotomicZHatField)) :=
-      congrArg σ
-        (abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion H E x)
-
-/-- Pointwise form of finite-layer restriction compatibility. -/
-private theorem
-    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    [Normal ℚ E]
-    [Normal
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (abstractFixedFieldCyclotomicFiniteLayer H E)]
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field))
-    (x : E) :
-    AlgEquiv.restrictNormalHom E
-        (abstractFixedFieldCyclotomicRestriction H σ) x =
-      IntermediateField.restrictRestrictAlgEquivMapHom
-          ℚ E
-          (LocalClassFieldTheory.abstractFixedField
-            ℚ (SeparableClosure ℚ) H.field)
-          (abstractFixedFieldCyclotomicFiniteLayer H E)
-        (AlgEquiv.restrictNormalHom
-          (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x := by
-  apply
-    (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E).injective
-  apply
-    (abstractFixedFieldCyclotomicFiniteLayerInclusion H E).injective
-  exact
-    (restrictNormalHom_abstractFixedFieldCyclotomicRestriction_left_embedded
-      H E σ x).trans
-      (restrictNormalHom_abstractFixedFieldCyclotomicRestriction_right_embedded
-        H E σ x).symm
-
-/-- Restricting through a finite layer commutes with restriction from
-the full abstract-fixed-field compositum. -/
-theorem
-    restrictNormalHom_abstractFixedFieldCyclotomicRestriction
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    (σ :
-      Gal(
-        LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ)
-          (rationalCyclotomicFieldInertia_le H.field) /
-        LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field)) :
-    letI : Normal ℚ E := E.isGalois.to_normal
-    AlgEquiv.restrictNormalHom E
-        (abstractFixedFieldCyclotomicRestriction H σ) =
-      IntermediateField.restrictRestrictAlgEquivMapHom
-          ℚ E
-          (LocalClassFieldTheory.abstractFixedField
-            ℚ (SeparableClosure ℚ) H.field)
-          (abstractFixedFieldCyclotomicFiniteLayer H E)
-        (AlgEquiv.restrictNormalHom
-          (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-          σ) := by
-  let : Normal ℚ E := E.isGalois.to_normal
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ)
-      (rationalCyclotomicFieldInertia_le H.field)
-  let T := rationalCyclotomicZHatField
-  let P : IntermediateField F U :=
-    abstractFixedFieldCyclotomicFiniteLayer H E
-  let : Algebra T U :=
-    abstractFixedFieldCyclotomicCompositum_algebra H
-  let : IsScalarTower ℚ T U :=
-    abstractFixedFieldCyclotomicCompositum_scalarTower H
-  let : Algebra E P :=
-    abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
-  let : IsScalarTower ℚ E P :=
-    abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
-  let : IsAbelianGalois F P := by
-    change IsAbelianGalois F
-      (abstractFixedFieldCyclotomicFiniteLayer H E)
-    exact
-      abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E
-  let : Normal F P := IsGalois.to_normal
-  apply AlgEquiv.ext
-  intro x
-  exact
-    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise
-      H E σ x
-
-section FiniteCoordinateHelpers
-
-/-- Opaque three-step equality composition used to keep large dependent
-finite-level coordinates out of endpoint proof normalization. -/
-private theorem cyclotomicAbstractFixedFieldArtin_eqTransThree
-    {α : Type} {a b c d : α}
-    (hab : a = b) (hbc : b = c) (hcd : c = d) :
-    a = d :=
-  hab.trans (hbc.trans hcd)
-
-private abbrev cyclotomicAbstractFixedFieldArtinCoordinateBase
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :=
-  LocalClassFieldTheory.abstractFixedField
-    ℚ (SeparableClosure ℚ) H.field
-
-private abbrev cyclotomicAbstractFixedFieldArtinCoordinateRelative
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :=
-  LocalClassFieldTheory.abstractRelativeFixedField
-    ℚ (SeparableClosure ℚ)
-    (rationalCyclotomicFieldInertia_le H.field)
-
-private abbrev cyclotomicAbstractFixedFieldArtinCoordinateLayer
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IntermediateField
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
-  abstractFixedFieldCyclotomicFiniteLayer H E
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    Algebra ℚ
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    FiniteDimensional ℚ
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
-  LocalClassFieldTheory.abstractFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ) H.field H.finite
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    NumberField
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
-  NumberField.of_module_finite ℚ
-    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    Algebra
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (SeparableClosure ℚ) :=
-  IntermediateField.toAlgebra
-    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    Algebra
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
-
-local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    IsScalarTower ℚ
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
-  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
-    IsAbelianGalois
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
-  abstractFixedFieldCyclotomic_isAbelianGalois H
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra ℚ E :=
-  E.toIntermediateField.algebra'
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    NumberField E :=
-  NumberField.of_module_finite ℚ E
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsAbelianGalois ℚ E :=
-  IsAbelianGalois.of_algHom E.toIntermediateField.val
-
-local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateNormal
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Normal ℚ E :=
-  E.isGalois.to_normal
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
-  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra ℚ
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
-  DivisionRing.toRatAlgebra
-
-local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower ℚ
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
-  let hI := rationalCyclotomicFieldInertia_le H.field
-  apply IsScalarTower.of_algebraMap_eq
-  intro x
-  apply Subtype.ext
-  change
-    algebraMap ℚ
-        (LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) hI) x =
-      algebraMap
-        (LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field)
-        (LocalClassFieldTheory.abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) hI)
-        (algebraMap ℚ
-          (LocalClassFieldTheory.abstractFixedField
-            ℚ (SeparableClosure ℚ) H.field) x)
-  exact
-    IsScalarTower.algebraMap_apply
-      ℚ
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI) x
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    NumberField
-      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
-  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra E
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
-  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    SMul E
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
-  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
-
-local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower ℚ E
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
-  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Algebra
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
-  IntermediateField.toAlgebra
-    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
-
-local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsScalarTower
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
-  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
-
-noncomputable local instance
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    IsAbelianGalois
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
-  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E
-
-/-- The full abstract Artin symbol whose finite coordinates are compared
-below.  Naming this endpoint keeps its relative fixed-field data opaque. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinAbstractEndpoint
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
-    Gal(rationalCyclotomicZHatField / ℚ) :=
-  abstractFixedFieldCyclotomicRestriction H
-    (infiniteGlobalArtinMonoidHom
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) a)
-
-/-- The rational norm Artin symbol serving as the other full endpoint. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinRationalEndpoint
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
-    Gal(rationalCyclotomicZHatField / ℚ) :=
-  rationalCyclotomicZHatGlobalArtin
-    (IdeleGroup.norm ℚ
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) a)
-
-/-- The finite restriction map packaged together with its pointwise Artin
-naturality law.  The map is inferred from the generic hom-level theorem, so
-no concrete instance tower is compared after the opaque boundary. -/
-private noncomputable def cyclotomicAbstractFixedFieldArtinCoordinateMapData
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    {f : Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
-        cyclotomicAbstractFixedFieldArtinCoordinateBase H) →*
-        Gal(E / ℚ) //
-      f.comp
-          (@globalArtinMonoidHomOfNumberField
-            (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-            (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-            (inferInstance : Field
-              (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-            (cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField H)
-            (inferInstance : Field
-              (abstractFixedFieldCyclotomicFiniteGaloisLayer H E))
-            (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
-            (cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois H E)
-            (cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField H E)) =
-        (globalArtinMonoidHom (K := ℚ) (L := E)).comp
-          (IdeleGroup.norm ℚ
-            (cyclotomicAbstractFixedFieldArtinCoordinateBase H))} := by
-  have hnat :=
-    @globalArtinMonoidHomOfNumberField_norm_restriction
-      ℚ E
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-      inferInstance inferInstance
-      inferInstance
-      (cyclotomicAbstractFixedFieldArtinCoordinateNumberField E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateAlgebra E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois E)
-      inferInstance
-      (cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField H)
-      inferInstance
-      (cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField H E)
-  exact
-    ⟨(@AlgEquiv.restrictNormalHom
-          ℚ inferInstance
-          (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-          inferInstance
-          (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
-          E inferInstance
-          (cyclotomicAbstractFixedFieldArtinCoordinateAlgebra E)
-          (cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra H E)
-          (cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower H E)
-          (cyclotomicAbstractFixedFieldArtinCoordinateNormal E)).comp
-        (@AlgEquiv.restrictScalarsHom
-          ℚ
-          (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-          (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-          inferInstance inferInstance inferInstance
-          (cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra H)
-          (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
-          (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
-          (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower H E)),
-      hnat⟩
-
-/-- The fixed restriction map from the relative finite layer to one rational
-cyclotomic coordinate. -/
-private noncomputable def cyclotomicAbstractFixedFieldArtinCoordinateMap
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
-        cyclotomicAbstractFixedFieldArtinCoordinateBase H) →*
-      Gal(E / ℚ) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateMapData H E).1
-
-/-- Naturality of the named coordinate map, kept at the hom level so later
-pointwise rewrites match the opaque map without unfolding its data package. -/
-private theorem cyclotomicAbstractFixedFieldArtinCoordinateMap_naturality
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    (cyclotomicAbstractFixedFieldArtinCoordinateMap H E).comp
-        (globalArtinMonoidHom
-          (K := cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-          (L := abstractFixedFieldCyclotomicFiniteGaloisLayer H E)) =
-      (globalArtinMonoidHom (K := ℚ) (L := E)).comp
-        (IdeleGroup.norm ℚ
-          (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateMapData H E).2
-
-/-- Pointwise identification of the named coordinate map with the concrete
-two-stage restriction used by the abstract fixed-field comparison. -/
-private theorem cyclotomicAbstractFixedFieldArtinCoordinateMap_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField)
-    (σ : Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
-      cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
-    cyclotomicAbstractFixedFieldArtinCoordinateMap H E σ =
-      @IntermediateField.restrictRestrictAlgEquivMapHom
-        ℚ E
-        (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-        (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-        inferInstance inferInstance inferInstance inferInstance
-        (cyclotomicAbstractFixedFieldArtinCoordinateAlgebra E)
-        (cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra H)
-        (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
-        (cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra H E)
-        (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
-        (cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower H E)
-        (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower H E)
-        (cyclotomicAbstractFixedFieldArtinCoordinateNormal E) σ :=
-  rfl
-
-/-- The relative projection, common rational finite value, and rational
-infinite projection, with both comparison steps packaged by the generic
-provider before this concrete tower becomes opaque. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinCoordinateBridgeData
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :=
-    @compRestrictNormalHomInfiniteGlobalArtinRationalCyclotomicDataOfNumberField
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)
-      (inferInstance : Field
-        (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-      (cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField H)
-      (inferInstance : Field
-        (cyclotomicAbstractFixedFieldArtinCoordinateRelative H))
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra H)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois H)
-      a
-      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField H E)
-      E
-      (cyclotomicAbstractFixedFieldArtinCoordinateNumberField E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateMap H E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateMap_naturality H E)
-
-/-- The abstract endpoint after projection to one finite coordinate. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinAbstractCoordinate
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Gal(E / ℚ) :=
-  AlgEquiv.restrictNormalHom E
-    (cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a)
-
-/-- The relative infinite Artin symbol, restricted to a finite layer and
-then mapped to the corresponding rational coordinate. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Gal(E / ℚ) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.1
-
-/-- The finite relative Artin symbol mapped to one rational coordinate. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinFiniteCoordinate
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Gal(E / ℚ) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.2.1
-
-/-- The finite rational Artin coordinate of the idele norm. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinRationalCoordinate
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Gal(E / ℚ) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.2.1
-
-/-- Naturality of the finite global Artin map at the concrete cyclotomic
-coordinate. -/
-private theorem
-    cyclotomicAbstractFixedFieldArtinCoordinateNaturality
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E =
-      cyclotomicAbstractFixedFieldArtinRationalCoordinate H a E := by
-  rfl
-
-/-- The rational endpoint after projection to one finite coordinate. -/
-private noncomputable def
-    cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    Gal(E / ℚ) :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.2.2
-
-/-- The abstract restriction map projected to the concrete finite layer. -/
-private theorem
-    cyclotomicAbstractFixedFieldArtinCoordinateRestriction
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    cyclotomicAbstractFixedFieldArtinAbstractCoordinate H a E =
-      cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E := by
-  calc
-    cyclotomicAbstractFixedFieldArtinAbstractCoordinate H a E =
-        AlgEquiv.restrictNormalHom E
-          (cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a) := rfl
-    _ = cyclotomicAbstractFixedFieldArtinCoordinateMap H E
-          (AlgEquiv.restrictNormalHom
-            (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
-            (infiniteGlobalArtinMonoidHom
-              (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-              (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) a)) :=
-      (restrictNormalHom_abstractFixedFieldCyclotomicRestriction
-          H E
-            (infiniteGlobalArtinMonoidHom
-              (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
-              (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) a)).trans
-        (cyclotomicAbstractFixedFieldArtinCoordinateMap_apply H E _).symm
-    _ = cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E := rfl
-
-/-- Restricting the infinite relative Artin symbol supplies exactly the
-finite Artin coordinate. -/
-private theorem
-    cyclotomicAbstractFixedFieldArtinCoordinateLayerProjection
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E =
-      cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E :=
-  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).2.1
-
-/-- The rational cyclotomic Artin map projected to the same finite
-coordinate. -/
-private theorem
-    cyclotomicAbstractFixedFieldArtinCoordinateRationalProjection
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate H a E =
-      cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E := by
-  exact
-    (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).2.2.symm
-
-/-- Equality of the two full endpoints at one opaque finite coordinate. -/
-private theorem
-    cyclotomicAbstractFixedFieldArtinFiniteCoordinateComparison
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a : IdeleGroup
-      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
-    (E :
-      FiniteGaloisIntermediateField
-        ℚ rationalCyclotomicZHatField) :
-    AlgEquiv.restrictNormalHom E
-        (cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a) =
-      AlgEquiv.restrictNormalHom E
-        (cyclotomicAbstractFixedFieldArtinRationalEndpoint H a) := by
-  change
-    cyclotomicAbstractFixedFieldArtinAbstractCoordinate H a E =
-      cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate H a E
-  exact
-    cyclotomicAbstractFixedFieldArtin_eqTransThree
-      (cyclotomicAbstractFixedFieldArtinCoordinateRestriction H a E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateLayerProjection H a E)
-      (cyclotomicAbstractFixedFieldArtinCoordinateRationalProjection
-        H a E).symm
-
-/-- The finite-coordinate comparison assembled in the rational cyclotomic
-inverse limit. -/
-private theorem
-    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a :
-      IdeleGroup
-        (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
-    cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a =
-      cyclotomicAbstractFixedFieldArtinRationalEndpoint H a := by
-  apply
-    (InfiniteGalois.continuousMulEquivToLimit
-      ℚ rationalCyclotomicZHatField).injective
-  apply Subtype.ext
-  funext Eop
-  exact
-    cyclotomicAbstractFixedFieldArtinFiniteCoordinateComparison
-      H a Eop.unop
-
-end FiniteCoordinateHelpers
-
-/-- The actual infinite global Artin map on an abstract fixed field
-restricts to the rational cyclotomic Artin map of the ordinary idele
-norm. -/
-@[simp]
-theorem
-    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a :
-      IdeleGroup
-        (LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field)) :
-    let F :=
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field
-    letI : FiniteDimensional ℚ F :=
-      LocalClassFieldTheory.abstractFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) H.field H.finite
-    letI : NumberField F :=
-      NumberField.of_module_finite ℚ F
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    let U :=
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI
-    letI : IsAbelianGalois F U :=
-      abstractFixedFieldCyclotomic_isAbelianGalois H
-    abstractFixedFieldCyclotomicRestriction H
-        (infiniteGlobalArtinMonoidHom F U a) =
-      rationalCyclotomicZHatGlobalArtin
-        (IdeleGroup.norm ℚ F a) := by
-  exact
-    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit
-      H a
-
-/-- In the normalized actual Galois coordinate, the infinite global
-Artin symbol is exactly the normalized cyclotomic idele value. -/
-theorem
-    abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a :
-      IdeleGroup
-        (LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field)) :
-    let F :=
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field
-    letI : FiniteDimensional ℚ F :=
-      LocalClassFieldTheory.abstractFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) H.field H.finite
-    letI : NumberField F :=
-      NumberField.of_module_finite ℚ F
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    let U :=
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI
-    letI : IsAbelianGalois F U :=
-      abstractFixedFieldCyclotomic_isAbelianGalois H
-    Multiplicative.toAdd
-        (abstractFixedFieldCyclotomicGalEquivZHat H
-          (infiniteGlobalArtinMonoidHom F U a)) =
-      normalizedCyclotomicZHatIdeleValue F
-        (Additive.ofMul a) := by
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field
-  let : FiniteDimensional ℚ F :=
-    LocalClassFieldTheory.abstractFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ) H.field H.finite
-  let : NumberField F :=
-    NumberField.of_module_finite ℚ F
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) hI
-  let : IsAbelianGalois F U :=
-    abstractFixedFieldCyclotomic_isAbelianGalois H
-  apply
-    zHatMulNat_injective
-      (H.residueDegree rationalCyclotomicDegreeData).pos
-  calc
-    (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
-          Multiplicative.toAdd
-            (abstractFixedFieldCyclotomicGalEquivZHat H
-              (infiniteGlobalArtinMonoidHom F U a)) =
-        Multiplicative.toAdd
-          (rationalCyclotomicZHatFieldGalEquivZHat
-            (abstractFixedFieldCyclotomicRestriction H
-              (infiniteGlobalArtinMonoidHom F U a))) := by
-      exact
-        (abstractFixedFieldCyclotomicRestriction_coordinate H
-          (infiniteGlobalArtinMonoidHom F U a)).symm
-    _ =
-        Multiplicative.toAdd
-          (rationalCyclotomicZHatFieldGalEquivZHat
-            (rationalCyclotomicZHatGlobalArtin
-              (IdeleGroup.norm ℚ F a))) := by
-      rw [
-        abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom]
-    _ =
-        cyclotomicZHatNormComposite F
-          (Additive.ofMul a) := by
-      rfl
-    _ =
-        cyclotomicZHatIntersectionDegree F •
-          normalizedCyclotomicZHatIdeleValue F
-            (Additive.ofMul a) := by
-      exact
-        (cyclotomicZHatIntersectionDegree_nsmul_normalizedIdeleValue
-          F (Additive.ofMul a)).symm
-    _ =
-        (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
-          normalizedCyclotomicZHatIdeleValue F
-            (Additive.ofMul a) := by
-      rw [
-        cyclotomicZHatIntersectionDegree_abstractFixedField_eq_residueDegree
-          H]
-
-/-- Representative form of the actual cyclotomic Artin-coordinate
-identity after descent of the normalized value to the idele class
-group. -/
-theorem
-    abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom_mk
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (a :
-      IdeleGroup
-        (LocalClassFieldTheory.abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field)) :
-    let F :=
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field
-    letI : FiniteDimensional ℚ F :=
-      LocalClassFieldTheory.abstractFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) H.field H.finite
-    letI : NumberField F :=
-      NumberField.of_module_finite ℚ F
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    let U :=
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI
-    letI : IsAbelianGalois F U :=
-      abstractFixedFieldCyclotomic_isAbelianGalois H
-    Multiplicative.toAdd
-        (abstractFixedFieldCyclotomicGalEquivZHat H
-          (infiniteGlobalArtinMonoidHom F U a)) =
-      normalizedCyclotomicZHatIdeleClassValueContinuous F
-        (Additive.ofMul
-          (QuotientGroup.mk'
-            (IdeleGroup.principalSubgroup F) a)) := by
-  have hSeparableClosureAlgebra :
-      cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra =
-        rationalSeparableClosureAlgebra :=
-    Subsingleton.elim _ _
-  cases hSeparableClosureAlgebra
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) hI
-  let : IsAbelianGalois F U :=
-    abstractFixedFieldCyclotomic_isAbelianGalois H
-  calc
-    Multiplicative.toAdd
-        (abstractFixedFieldCyclotomicGalEquivZHat H
-          (infiniteGlobalArtinMonoidHom F U a)) =
-        normalizedCyclotomicZHatIdeleValue F
-          (Additive.ofMul a) :=
-      abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
-        H a
-    _ =
-        normalizedCyclotomicZHatIdeleClassValueContinuous F
-          (Additive.ofMul
-            (QuotientGroup.mk'
-              (IdeleGroup.principalSubgroup F) a)) :=
-      (normalizedCyclotomicZHatIdeleClassValueContinuous_mk
-        (K := F) a).symm
-
-/-- The genuine infinite global Artin symbol of the cyclotomic
-maximal-unramified extension kills every principal idele of the
-abstract fixed field. -/
-@[simp]
-theorem
-    infiniteGlobalArtinMonoidHom_abstractFixedFieldCyclotomic_principalIdele
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (x :
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field)ˣ) :
-    let F :=
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) H.field
-    letI : FiniteDimensional ℚ F :=
-      LocalClassFieldTheory.abstractFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) H.field H.finite
-    letI : NumberField F :=
-      NumberField.of_module_finite ℚ F
-    let hI :=
-      rationalCyclotomicFieldInertia_le H.field
-    let U :=
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) hI
-    letI : IsAbelianGalois F U :=
-      abstractFixedFieldCyclotomic_isAbelianGalois H
-    infiniteGlobalArtinMonoidHom F U
-        (IdeleGroup.principalIdele F x) =
-      1 := by
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H.field
-  let : FiniteDimensional ℚ F :=
-    LocalClassFieldTheory.abstractFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ) H.field H.finite
-  let : NumberField F :=
-    NumberField.of_module_finite ℚ F
-  let hI :=
-    rationalCyclotomicFieldInertia_le H.field
-  let U :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) hI
-  let : IsAbelianGalois F U :=
-    abstractFixedFieldCyclotomic_isAbelianGalois H
-  have hcoord :
-      Multiplicative.toAdd
-        (abstractFixedFieldCyclotomicGalEquivZHat H
-          (infiniteGlobalArtinMonoidHom F U
-            (IdeleGroup.principalIdele F x))) = 0 :=
-    (abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
-      H (IdeleGroup.principalIdele F x)).trans
-      (normalizedCyclotomicZHatIdeleValue_principalIdele_eq_zero F x)
-  apply (abstractFixedFieldCyclotomicGalEquivZHat H).injective
-  rw [map_one]
-  apply Multiplicative.ext
-  exact hcoord.trans toAdd_one.symm
+attribute [local instance]
+  cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal
+  abstractFixedFieldCyclotomic_numberField
 
 /-- The genuine cyclotomic maximal-unramified Artin map descended to
 the idele class group of an abstract fixed field. -/
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinBase.lean
new file mode 100644
index 0000000..dc971fc
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinBase.lean
@@ -0,0 +1,545 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicIdeleClassValuation
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.InfiniteGlobalArtin
+import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.MaximalUnramifiedReciprocity
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open KummerTheory
+
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra :
+    Algebra ℚ (SeparableClosure ℚ) :=
+  DivisionRing.toRatAlgebra
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra :
+    Algebra ℚ rationalCyclotomicZHatField :=
+  DivisionRing.toRatAlgebra
+
+/-- Cyclotomic field inertia is contained in the original abstract
+field subgroup. -/
+theorem rationalCyclotomicFieldInertia_le
+    (H : ClosedSubgroup
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    (rationalCyclotomicDegreeData.fieldInertia H).toSubgroup ≤
+      H.toSubgroup := by
+  intro σ hσ
+  exact hσ.1
+
+
+theorem extensionSubgroup_rationalCyclotomicFieldInertia
+    (H : ClosedSubgroup
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    CyclicCohomology.extensionSubgroup H
+        (rationalCyclotomicDegreeData.fieldInertia H)
+        (rationalCyclotomicFieldInertia_le H) =
+      rationalCyclotomicDegreeData.fieldInertiaWithin H := by
+  ext σ
+  rw [
+    mem_extensionSubgroup_iff,
+    rationalCyclotomicDegreeData.mem_fieldInertiaWithin_iff,
+    rationalCyclotomicDegreeData.mem_fieldInertia_iff]
+  exact and_iff_right σ.2
+
+/-- The actual Galois group of the cyclotomic maximal-unramified
+extension of an abstract fixed field, in its normalized `ZHat`
+coordinate. -/
+noncomputable def abstractFixedFieldCyclotomicGalEquivZHat
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) hI /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field) ≃*
+      Multiplicative ZHat := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let hnormal :
+      (CyclicCohomology.extensionSubgroup H.field
+        (rationalCyclotomicDegreeData.fieldInertia H.field)
+        hI).Normal := by
+    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+    infer_instance
+  let qField :
+      H.field.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H.field
+            (rationalCyclotomicDegreeData.fieldInertia H.field)
+            hI ≃*
+        Gal(
+          LocalClassFieldTheory.abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) hI /
+          LocalClassFieldTheory.abstractFixedField
+            ℚ (SeparableClosure ℚ) H.field) :=
+    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+      ℚ (SeparableClosure ℚ) H.field
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+      hI hnormal
+  let qInertia :
+      H.field.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H.field
+            (rationalCyclotomicDegreeData.fieldInertia H.field)
+            hI ≃*
+        H.field.toSubgroup ⧸
+          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
+    QuotientGroup.quotientMulEquivOfEq
+      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
+  exact
+    qField.symm.trans
+      (qInertia.trans
+        (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
+          (H.toFiniteResidueAbstractField
+            rationalCyclotomicDegreeData)))
+
+/-- On an absolute-Galois representative fixing the lower field, the
+actual cyclotomic Galois coordinate is its normalized degree. -/
+@[simp]
+theorem abstractFixedFieldCyclotomicGalEquivZHat_extensionClass
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (σ : H.field.toSubgroup) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    let hnormal :
+        (CyclicCohomology.extensionSubgroup H.field
+          (rationalCyclotomicDegreeData.fieldInertia H.field)
+          hI).Normal := by
+      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+      infer_instance
+    abstractFixedFieldCyclotomicGalEquivZHat H
+        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+          ℚ (SeparableClosure ℚ) H.field
+          (rationalCyclotomicDegreeData.fieldInertia H.field)
+          hI hnormal
+          (QuotientGroup.mk σ)) =
+      rationalCyclotomicDegreeData.normalizedDegree
+        (H.toFiniteResidueAbstractField
+          rationalCyclotomicDegreeData) σ := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let hnormal :
+      (CyclicCohomology.extensionSubgroup H.field
+        (rationalCyclotomicDegreeData.fieldInertia H.field)
+        hI).Normal := by
+    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+    infer_instance
+  let qField :=
+    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+      ℚ (SeparableClosure ℚ) H.field
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+      hI hnormal
+  change
+    (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
+      (H.toFiniteResidueAbstractField
+        rationalCyclotomicDegreeData))
+      (QuotientGroup.quotientMulEquivOfEq
+        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
+        (qField.symm (qField (QuotientGroup.mk σ)))) =
+      rationalCyclotomicDegreeData.normalizedDegree
+        (H.toFiniteResidueAbstractField
+          rationalCyclotomicDegreeData) σ
+  rw [qField.symm_apply_apply]
+  rfl
+
+/-- Quotient-level evaluation of the actual cyclotomic Galois
+coordinate. -/
+@[simp]
+theorem abstractFixedFieldCyclotomicGalEquivZHat_quotientClass
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (q :
+      H.field.toSubgroup ⧸
+        rationalCyclotomicDegreeData.fieldInertiaWithin H.field) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    let hnormal :
+        (CyclicCohomology.extensionSubgroup H.field
+          (rationalCyclotomicDegreeData.fieldInertia H.field)
+          hI).Normal := by
+      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+      infer_instance
+    let qInertia :
+        H.field.toSubgroup ⧸
+            CyclicCohomology.extensionSubgroup H.field
+              (rationalCyclotomicDegreeData.fieldInertia H.field)
+              hI ≃*
+          H.field.toSubgroup ⧸
+            rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
+      QuotientGroup.quotientMulEquivOfEq
+        (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
+    abstractFixedFieldCyclotomicGalEquivZHat H
+        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+          ℚ (SeparableClosure ℚ) H.field
+          (rationalCyclotomicDegreeData.fieldInertia H.field)
+          hI hnormal
+          (qInertia.symm q)) =
+      rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
+        (H.toFiniteResidueAbstractField
+          rationalCyclotomicDegreeData) q := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let hnormal :
+      (CyclicCohomology.extensionSubgroup H.field
+        (rationalCyclotomicDegreeData.fieldInertia H.field)
+        hI).Normal := by
+    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+    infer_instance
+  let qField :=
+    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+      ℚ (SeparableClosure ℚ) H.field
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+      hI hnormal
+  let qInertia :
+      H.field.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H.field
+            (rationalCyclotomicDegreeData.fieldInertia H.field)
+            hI ≃*
+        H.field.toSubgroup ⧸
+          rationalCyclotomicDegreeData.fieldInertiaWithin H.field :=
+    QuotientGroup.quotientMulEquivOfEq
+      (extensionSubgroup_rationalCyclotomicFieldInertia H.field)
+  change
+    (rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
+      (H.toFiniteResidueAbstractField
+        rationalCyclotomicDegreeData))
+      (qInertia
+        (qField.symm
+          (qField (qInertia.symm q)))) =
+      rationalCyclotomicDegreeData.maximalUnramifiedDegreeEquiv
+        (H.toFiniteResidueAbstractField
+          rationalCyclotomicDegreeData) q
+  rw [qField.symm_apply_apply, qInertia.apply_symm_apply]
+
+/-- The extension fixed by cyclotomic field inertia is an actual
+abelian Galois extension of the lower abstract fixed field. -/
+theorem abstractFixedFieldCyclotomic_isAbelianGalois
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    IsAbelianGalois
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let hnormal :
+      (CyclicCohomology.extensionSubgroup H.field
+        (rationalCyclotomicDegreeData.fieldInertia H.field)
+        hI).Normal := by
+    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+    infer_instance
+  let : IsGalois
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) :=
+    LocalClassFieldTheory.abstractRelativeFixedField_isGalois
+      ℚ (SeparableClosure ℚ) H.field
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+      hI hnormal
+  let e :=
+    abstractFixedFieldCyclotomicGalEquivZHat H
+  exact
+    { is_comm.comm := by
+        intro σ τ
+        apply e.injective
+        simpa only [map_mul] using mul_comm (e σ) (e τ) }
+
+/-- The rational cyclotomic `ZHat`-field embedded in the actual
+maximal-unramified compositum of an abstract fixed field. -/
+noncomputable def
+    rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    rationalCyclotomicZHatField →ₐ[ℚ]
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let J :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ)
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+  have hTJ :
+      rationalCyclotomicZHatField ≤ J := by
+    change
+      rationalCyclotomicZHatField ≤
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicDegreeData.fieldInertia H.field)
+    rw [
+      rationalCyclotomicDegreeData_fixedField_fieldInertia]
+    exact le_sup_right
+  exact IntermediateField.inclusion hTJ
+
+noncomputable instance
+    abstractFixedFieldCyclotomicCompositum_algebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    Algebra rationalCyclotomicZHatField
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) :=
+  ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+    H).toRingHom).toAlgebra
+
+instance abstractFixedFieldCyclotomicCompositum_scalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    IsScalarTower ℚ rationalCyclotomicZHatField
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) :=
+  IsScalarTower.of_algebraMap_eq'
+    ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+      H).comp_algebraMap).symm
+
+instance abstractFixedFieldCyclotomicCompositum_baseScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    IsScalarTower ℚ
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  apply IsScalarTower.of_algebraMap_eq
+  intro x
+  apply Subtype.ext
+  rfl
+
+/-- Restriction from the actual maximal-unramified compositum of an
+abstract fixed field to the rational cyclotomic factor. -/
+noncomputable def abstractFixedFieldCyclotomicRestriction
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) hI /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field) →*
+      Gal(rationalCyclotomicZHatField / ℚ) := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  exact
+    IntermediateField.restrictRestrictAlgEquivMapHom
+      ℚ rationalCyclotomicZHatField
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI)
+
+/-- On a quotient representative, cyclotomic restriction of the
+actual relative automorphism is ordinary restriction of the same
+ambient absolute-Galois automorphism. -/
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal :
+    Normal ℚ rationalCyclotomicZHatField :=
+  rationalCyclotomicZHatField_isNormal
+
+@[simp]
+theorem abstractFixedFieldCyclotomicRestriction_extensionClass
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (σ : H.field.toSubgroup) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    let hnormal :
+        (CyclicCohomology.extensionSubgroup H.field
+          (rationalCyclotomicDegreeData.fieldInertia H.field)
+          hI).Normal := by
+      rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+      infer_instance
+    abstractFixedFieldCyclotomicRestriction H
+        (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+          ℚ (SeparableClosure ℚ) H.field
+          (rationalCyclotomicDegreeData.fieldInertia H.field)
+          hI hnormal
+          (QuotientGroup.mk σ)) =
+      AlgEquiv.restrictNormalHom
+        rationalCyclotomicZHatField σ.1 := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let hnormal :
+      (CyclicCohomology.extensionSubgroup H.field
+        (rationalCyclotomicDegreeData.fieldInertia H.field)
+        hI).Normal := by
+    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+    infer_instance
+  let τ :=
+    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+      ℚ (SeparableClosure ℚ) H.field
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+      hI hnormal
+      (QuotientGroup.mk'
+        (CyclicCohomology.extensionSubgroup H.field
+          (rationalCyclotomicDegreeData.fieldInertia H.field) hI)
+        σ)
+  apply AlgEquiv.ext
+  intro x
+  apply Subtype.ext
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) hI
+  have hrestrict :
+      algebraMap rationalCyclotomicZHatField U
+          ((abstractFixedFieldCyclotomicRestriction H τ) x) =
+        τ (algebraMap rationalCyclotomicZHatField U x) := by
+    change
+      algebraMap rationalCyclotomicZHatField U
+          ((AlgEquiv.restrictNormal
+            (MulSemiringAction.toAlgEquiv ℚ U τ)
+            rationalCyclotomicZHatField) x) =
+        (MulSemiringAction.toAlgEquiv ℚ U τ)
+          (algebraMap rationalCyclotomicZHatField U x)
+    exact
+      AlgEquiv.restrictNormal_commutes
+        (MulSemiringAction.toAlgEquiv ℚ U τ)
+        rationalCyclotomicZHatField x
+  have halgebraMap_eq_embedding
+      (z : rationalCyclotomicZHatField) :
+      algebraMap rationalCyclotomicZHatField U z =
+        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+          H z : U) := by
+    rfl
+  have hembedding_coe
+      (z : rationalCyclotomicZHatField) :
+      (((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+          H z : U) : SeparableClosure ℚ)) =
+        (z : SeparableClosure ℚ) := by
+    rfl
+  have halgebraMap_coe
+      (z : rationalCyclotomicZHatField) :
+      ((algebraMap rationalCyclotomicZHatField U z : U) :
+          SeparableClosure ℚ) =
+        (z : SeparableClosure ℚ) := by
+    rw [halgebraMap_eq_embedding]
+    exact hembedding_coe z
+  have hambient :=
+    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup_mk_apply_val
+        ℚ (SeparableClosure ℚ) H.field
+        (rationalCyclotomicDegreeData.fieldInertia H.field)
+        hI hnormal σ
+        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+          H x)
+  calc
+    (((abstractFixedFieldCyclotomicRestriction H τ) x :
+        rationalCyclotomicZHatField) :
+        SeparableClosure ℚ) =
+        ((τ
+          (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+            H x) :
+          LocalClassFieldTheory.abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) hI) :
+          SeparableClosure ℚ) := by
+      calc
+        (((abstractFixedFieldCyclotomicRestriction H τ) x :
+            rationalCyclotomicZHatField) : SeparableClosure ℚ) =
+            ((algebraMap rationalCyclotomicZHatField U
+              ((abstractFixedFieldCyclotomicRestriction H τ) x) : U) :
+              SeparableClosure ℚ) :=
+          (halgebraMap_coe
+            ((abstractFixedFieldCyclotomicRestriction H τ) x)).symm
+        _ = ((τ (algebraMap rationalCyclotomicZHatField U x) : U) :
+            SeparableClosure ℚ) :=
+          congrArg (fun y : U => (y : SeparableClosure ℚ)) hrestrict
+        _ = ((τ
+            (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+              H x) : U) : SeparableClosure ℚ) := by
+          rw [halgebraMap_eq_embedding]
+    _ = σ.1 (x : SeparableClosure ℚ) := by
+      calc
+        ((τ
+            (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+              H x) : U) : SeparableClosure ℚ) =
+            σ.1
+              ((rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+                H x : U) : SeparableClosure ℚ) := by
+          simpa only [τ, U] using hambient.symm
+        _ = σ.1 (x : SeparableClosure ℚ) := by
+          rw [hembedding_coe]
+    _ =
+        (((AlgEquiv.restrictNormalHom
+          rationalCyclotomicZHatField σ.1) x :
+          rationalCyclotomicZHatField) :
+          SeparableClosure ℚ) := by
+      exact
+        (AlgEquiv.restrictNormal_commutes
+          σ.1 rationalCyclotomicZHatField x).symm
+
+/-- Raw cyclotomic restriction is residue-degree multiplication of
+the normalized actual Galois coordinate. -/
+theorem
+    abstractFixedFieldCyclotomicRestriction_coordinate
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (τ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field)) :
+    Multiplicative.toAdd
+        (rationalCyclotomicZHatFieldGalEquivZHat
+          (abstractFixedFieldCyclotomicRestriction H τ)) =
+      (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
+        Multiplicative.toAdd
+          (abstractFixedFieldCyclotomicGalEquivZHat H τ) := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let hnormal :
+      (CyclicCohomology.extensionSubgroup H.field
+        (rationalCyclotomicDegreeData.fieldInertia H.field)
+        hI).Normal := by
+    rw [extensionSubgroup_rationalCyclotomicFieldInertia]
+    infer_instance
+  let qField :=
+    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+      ℚ (SeparableClosure ℚ) H.field
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+      hI hnormal
+  obtain ⟨q, rfl⟩ := qField.surjective τ
+  refine Quotient.inductionOn' q ?_
+  intro σ
+  rw [
+    abstractFixedFieldCyclotomicRestriction_extensionClass,
+    abstractFixedFieldCyclotomicGalEquivZHat_extensionClass]
+  exact
+    (rationalCyclotomicDegreeData.residueDegree_nsmul_normalizedDegree
+        (H.toFiniteResidueAbstractField
+          rationalCyclotomicDegreeData) σ).symm
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinComparison.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinComparison.lean
new file mode 100644
index 0000000..0788ef1
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinComparison.lean
@@ -0,0 +1,273 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicAbstractFixedFieldArtinCoordinates
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open KummerTheory
+
+attribute [local instance]
+  cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal
+  abstractFixedFieldCyclotomic_numberField
+
+/-- The actual infinite global Artin map on an abstract fixed field
+restricts to the rational cyclotomic Artin map of the ordinary idele
+norm. -/
+@[simp]
+theorem
+    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a :
+      IdeleGroup
+        (LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field)) :
+    let F :=
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field
+    letI : FiniteDimensional ℚ F :=
+      LocalClassFieldTheory.abstractFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) H.field H.finite
+    letI : NumberField F :=
+      NumberField.of_module_finite ℚ F
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    let U :=
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI
+    letI : IsAbelianGalois F U :=
+      abstractFixedFieldCyclotomic_isAbelianGalois H
+    abstractFixedFieldCyclotomicRestriction H
+        (infiniteGlobalArtinMonoidHom F U a) =
+      rationalCyclotomicZHatGlobalArtin
+        (IdeleGroup.norm ℚ F a) := by
+  exact
+    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit
+      H a
+
+/-- In the normalized actual Galois coordinate, the infinite global
+Artin symbol is exactly the normalized cyclotomic idele value. -/
+theorem
+    abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a :
+      IdeleGroup
+        (LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field)) :
+    let F :=
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field
+    letI : FiniteDimensional ℚ F :=
+      LocalClassFieldTheory.abstractFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) H.field H.finite
+    letI : NumberField F :=
+      NumberField.of_module_finite ℚ F
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    let U :=
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI
+    letI : IsAbelianGalois F U :=
+      abstractFixedFieldCyclotomic_isAbelianGalois H
+    Multiplicative.toAdd
+        (abstractFixedFieldCyclotomicGalEquivZHat H
+          (infiniteGlobalArtinMonoidHom F U a)) =
+      normalizedCyclotomicZHatIdeleValue F
+        (Additive.ofMul a) := by
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field
+  let : FiniteDimensional ℚ F :=
+    LocalClassFieldTheory.abstractFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ) H.field H.finite
+  let : NumberField F :=
+    NumberField.of_module_finite ℚ F
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) hI
+  let : IsAbelianGalois F U :=
+    abstractFixedFieldCyclotomic_isAbelianGalois H
+  apply
+    zHatMulNat_injective
+      (H.residueDegree rationalCyclotomicDegreeData).pos
+  calc
+    (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
+          Multiplicative.toAdd
+            (abstractFixedFieldCyclotomicGalEquivZHat H
+              (infiniteGlobalArtinMonoidHom F U a)) =
+        Multiplicative.toAdd
+          (rationalCyclotomicZHatFieldGalEquivZHat
+            (abstractFixedFieldCyclotomicRestriction H
+              (infiniteGlobalArtinMonoidHom F U a))) := by
+      exact
+        (abstractFixedFieldCyclotomicRestriction_coordinate H
+          (infiniteGlobalArtinMonoidHom F U a)).symm
+    _ =
+        Multiplicative.toAdd
+          (rationalCyclotomicZHatFieldGalEquivZHat
+            (rationalCyclotomicZHatGlobalArtin
+              (IdeleGroup.norm ℚ F a))) := by
+      rw [
+        abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtinMonoidHom]
+    _ =
+        cyclotomicZHatNormComposite F
+          (Additive.ofMul a) := by
+      rfl
+    _ =
+        cyclotomicZHatIntersectionDegree F •
+          normalizedCyclotomicZHatIdeleValue F
+            (Additive.ofMul a) := by
+      exact
+        (cyclotomicZHatIntersectionDegree_nsmul_normalizedIdeleValue
+          F (Additive.ofMul a)).symm
+    _ =
+        (H.residueDegree rationalCyclotomicDegreeData : ℕ) •
+          normalizedCyclotomicZHatIdeleValue F
+            (Additive.ofMul a) := by
+      rw [
+        cyclotomicZHatIntersectionDegree_abstractFixedField_eq_residueDegree
+          H]
+
+/-- Representative form of the actual cyclotomic Artin-coordinate
+identity after descent of the normalized value to the idele class
+group. -/
+theorem
+    abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom_mk
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a :
+      IdeleGroup
+        (LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field)) :
+    let F :=
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field
+    letI : FiniteDimensional ℚ F :=
+      LocalClassFieldTheory.abstractFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) H.field H.finite
+    letI : NumberField F :=
+      NumberField.of_module_finite ℚ F
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    let U :=
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI
+    letI : IsAbelianGalois F U :=
+      abstractFixedFieldCyclotomic_isAbelianGalois H
+    Multiplicative.toAdd
+        (abstractFixedFieldCyclotomicGalEquivZHat H
+          (infiniteGlobalArtinMonoidHom F U a)) =
+      normalizedCyclotomicZHatIdeleClassValueContinuous F
+        (Additive.ofMul
+          (QuotientGroup.mk'
+            (IdeleGroup.principalSubgroup F) a)) := by
+  have hSeparableClosureAlgebra :
+      cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra =
+        rationalSeparableClosureAlgebra :=
+    Subsingleton.elim _ _
+  cases hSeparableClosureAlgebra
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) hI
+  let : IsAbelianGalois F U :=
+    abstractFixedFieldCyclotomic_isAbelianGalois H
+  calc
+    Multiplicative.toAdd
+        (abstractFixedFieldCyclotomicGalEquivZHat H
+          (infiniteGlobalArtinMonoidHom F U a)) =
+        normalizedCyclotomicZHatIdeleValue F
+          (Additive.ofMul a) :=
+      abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
+        H a
+    _ =
+        normalizedCyclotomicZHatIdeleClassValueContinuous F
+          (Additive.ofMul
+            (QuotientGroup.mk'
+              (IdeleGroup.principalSubgroup F) a)) :=
+      (normalizedCyclotomicZHatIdeleClassValueContinuous_mk
+        (K := F) a).symm
+
+/-- The genuine infinite global Artin symbol of the cyclotomic
+maximal-unramified extension kills every principal idele of the
+abstract fixed field. -/
+@[simp]
+theorem
+    infiniteGlobalArtinMonoidHom_abstractFixedFieldCyclotomic_principalIdele
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (x :
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)ˣ) :
+    let F :=
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field
+    letI : FiniteDimensional ℚ F :=
+      LocalClassFieldTheory.abstractFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) H.field H.finite
+    letI : NumberField F :=
+      NumberField.of_module_finite ℚ F
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    let U :=
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI
+    letI : IsAbelianGalois F U :=
+      abstractFixedFieldCyclotomic_isAbelianGalois H
+    infiniteGlobalArtinMonoidHom F U
+        (IdeleGroup.principalIdele F x) =
+      1 := by
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field
+  let : FiniteDimensional ℚ F :=
+    LocalClassFieldTheory.abstractFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ) H.field H.finite
+  let : NumberField F :=
+    NumberField.of_module_finite ℚ F
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) hI
+  let : IsAbelianGalois F U :=
+    abstractFixedFieldCyclotomic_isAbelianGalois H
+  have hcoord :
+      Multiplicative.toAdd
+        (abstractFixedFieldCyclotomicGalEquivZHat H
+          (infiniteGlobalArtinMonoidHom F U
+            (IdeleGroup.principalIdele F x))) = 0 :=
+    (abstractFixedFieldCyclotomicGalEquivZHat_infiniteGlobalArtinMonoidHom
+      H (IdeleGroup.principalIdele F x)).trans
+      (normalizedCyclotomicZHatIdeleValue_principalIdele_eq_zero F x)
+  apply (abstractFixedFieldCyclotomicGalEquivZHat H).injective
+  rw [map_one]
+  apply Multiplicative.ext
+  exact hcoord.trans toAdd_one.symm
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinCompositum.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinCompositum.lean
new file mode 100644
index 0000000..1295a7e
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinCompositum.lean
@@ -0,0 +1,257 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicAbstractFixedFieldArtinBase
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open KummerTheory
+
+attribute [local instance]
+  cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal
+
+/-- The canonical compositum of an abstract fixed field with a finite
+layer of the rational cyclotomic `ZHat`-extension. -/
+def abstractFixedFieldCyclotomicFiniteCompositum
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IntermediateField ℚ (SeparableClosure ℚ) :=
+  LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field ⊔
+    IntermediateField.lift E.toIntermediateField
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteCompositum_finiteDimensional
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    FiniteDimensional ℚ
+      (abstractFixedFieldCyclotomicFiniteCompositum H E) := by
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field
+  let : FiniteDimensional ℚ F :=
+    LocalClassFieldTheory.abstractFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ) H.field H.finite
+  let : FiniteDimensional ℚ
+      (IntermediateField.lift E.toIntermediateField) :=
+    ((IntermediateField.liftAlgEquiv
+      E.toIntermediateField).toLinearEquiv).finiteDimensional
+  exact IntermediateField.finiteDimensional_sup
+    F (IntermediateField.lift E.toIntermediateField)
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteCompositum_numberField
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    NumberField
+      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
+  NumberField.of_module_finite ℚ
+    (abstractFixedFieldCyclotomicFiniteCompositum H E)
+
+/-- The lower abstract fixed field embedded into its finite
+cyclotomic compositum. -/
+noncomputable def
+    abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field →ₐ[ℚ]
+      abstractFixedFieldCyclotomicFiniteCompositum H E :=
+  IntermediateField.inclusion
+    (show
+      LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field ≤
+        abstractFixedFieldCyclotomicFiniteCompositum H E from
+      le_sup_left)
+
+
+noncomputable local instance
+    abstractFixedFieldCyclotomic_numberField
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    NumberField
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field) := by
+  let : FiniteDimensional ℚ
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field) :=
+    LocalClassFieldTheory.abstractFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ) H.field H.finite
+  exact
+    NumberField.of_module_finite ℚ
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+
+/-- A finite rational cyclotomic layer embedded into its compositum
+with the abstract fixed field. -/
+noncomputable def
+    abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    E →ₐ[ℚ]
+      abstractFixedFieldCyclotomicFiniteCompositum H E :=
+  (IntermediateField.inclusion
+      (show
+        IntermediateField.lift E.toIntermediateField ≤
+          abstractFixedFieldCyclotomicFiniteCompositum H E from
+        le_sup_right)).comp
+    (IntermediateField.liftAlgEquiv E.toIntermediateField).toAlgHom
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteCompositum_baseAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
+  RingHom.toAlgebra
+    (AlgHom.toRingHom
+      (abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding H E))
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteCompositum_layerAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra E
+      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
+  RingHom.toAlgebra
+    (AlgHom.toRingHom
+      (abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding H E))
+
+/-- The finite-layer action induced by its explicit embedding into the
+finite compositum. -/
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteCompositum_layerSMul
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    SMul E
+      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
+  Algebra.toSMul
+    (self :=
+      abstractFixedFieldCyclotomicFiniteCompositum_layerAlgebra H E)
+
+instance
+    abstractFixedFieldCyclotomicFiniteCompositum_baseScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower ℚ
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
+  IsScalarTower.of_algebraMap_eq'
+    (AlgHom.comp_algebraMap
+      (abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding H E)).symm
+
+instance
+    abstractFixedFieldCyclotomicFiniteCompositum_layerScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower ℚ E
+      (abstractFixedFieldCyclotomicFiniteCompositum H E) :=
+  IsScalarTower.of_algebraMap_eq'
+    (AlgHom.comp_algebraMap
+      (abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding H E)).symm
+
+/-- Inclusion of the finite cyclotomic compositum into the actual
+maximal-unramified compositum. -/
+noncomputable def
+    abstractFixedFieldCyclotomicFiniteCompositumInclusion
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    abstractFixedFieldCyclotomicFiniteCompositum H E →ₐ[ℚ]
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let J :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ)
+      (rationalCyclotomicDegreeData.fieldInertia H.field)
+  have hle :
+      abstractFixedFieldCyclotomicFiniteCompositum H E ≤ J := by
+    dsimp only [J]
+    rw [
+      rationalCyclotomicDegreeData_fixedField_fieldInertia H.field]
+    exact
+      sup_le_sup le_rfl
+        (IntermediateField.lift_le E.toIntermediateField)
+  exact IntermediateField.inclusion hle
+
+/-- The same finite-compositum inclusion over the lower abstract fixed
+field. -/
+noncomputable def
+    abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    abstractFixedFieldCyclotomicFiniteCompositum H E →ₐ[
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field]
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI := by
+  let f :=
+    abstractFixedFieldCyclotomicFiniteCompositumInclusion H E
+  exact
+    { f.toRingHom with
+      commutes' := by
+        intro x
+        rfl }
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinCoordinates.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinCoordinates.lean
new file mode 100644
index 0000000..64ed13e
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinCoordinates.lean
@@ -0,0 +1,668 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicAbstractFixedFieldArtinFiniteRestriction
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open KummerTheory
+
+attribute [local instance]
+  cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal
+  abstractFixedFieldCyclotomic_numberField
+
+section FiniteCoordinateHelpers
+
+/-- Opaque three-step equality composition used to keep large dependent
+finite-level coordinates out of endpoint proof normalization. -/
+private theorem cyclotomicAbstractFixedFieldArtin_eqTransThree
+    {α : Type} {a b c d : α}
+    (hab : a = b) (hbc : b = c) (hcd : c = d) :
+    a = d :=
+  hab.trans (hbc.trans hcd)
+
+private abbrev cyclotomicAbstractFixedFieldArtinCoordinateBase
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :=
+  LocalClassFieldTheory.abstractFixedField
+    ℚ (SeparableClosure ℚ) H.field
+
+private abbrev cyclotomicAbstractFixedFieldArtinCoordinateRelative
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :=
+  LocalClassFieldTheory.abstractRelativeFixedField
+    ℚ (SeparableClosure ℚ)
+    (rationalCyclotomicFieldInertia_le H.field)
+
+private abbrev cyclotomicAbstractFixedFieldArtinCoordinateLayer
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IntermediateField
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
+  abstractFixedFieldCyclotomicFiniteLayer H E
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    Algebra ℚ
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateBase H).algebra'
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateBaseFiniteDimensional
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    FiniteDimensional ℚ
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
+  LocalClassFieldTheory.abstractFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ) H.field H.finite
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    NumberField
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) :=
+  NumberField.of_module_finite ℚ
+    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateBaseSeparableAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    Algebra
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (SeparableClosure ℚ) :=
+  IntermediateField.toAlgebra
+    (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    Algebra
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateRelative H).algebra'
+
+local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateRelativeScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    IsScalarTower ℚ
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
+  abstractFixedFieldCyclotomicCompositum_baseScalarTower H
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)) :
+    IsAbelianGalois
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
+  abstractFixedFieldCyclotomic_isAbelianGalois H
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateAlgebra
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra ℚ E :=
+  E.toIntermediateField.algebra'
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateNumberField
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    NumberField E :=
+  NumberField.of_module_finite ℚ E
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsAbelianGalois ℚ E :=
+  IsAbelianGalois.of_algHom E.toIntermediateField.val
+
+local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateNormal
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Normal ℚ E :=
+  E.isGalois.to_normal
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
+  (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra'
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra ℚ
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
+  DivisionRing.toRatAlgebra
+
+local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower ℚ
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) := by
+  let hI := rationalCyclotomicFieldInertia_le H.field
+  apply IsScalarTower.of_algebraMap_eq
+  intro x
+  apply Subtype.ext
+  change
+    algebraMap ℚ
+        (LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) hI) x =
+      algebraMap
+        (LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field)
+        (LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) hI)
+        (algebraMap ℚ
+          (LocalClassFieldTheory.abstractFixedField
+            ℚ (SeparableClosure ℚ) H.field) x)
+  exact
+    IsScalarTower.algebraMap_apply
+      ℚ
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) x
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    NumberField
+      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E) :=
+  abstractFixedFieldCyclotomicFiniteLayer_numberField H E
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra E
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
+  abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerSMul
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    SMul E
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
+  abstractFixedFieldCyclotomicFiniteLayer_layerSMul H E
+
+local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower ℚ E
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
+  abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
+  IntermediateField.toAlgebra
+    (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
+
+local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerRelativeScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) :=
+  abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower H E
+
+noncomputable local instance
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsAbelianGalois
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayer H E) :=
+  abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E
+
+/-- The full abstract Artin symbol whose finite coordinates are compared
+below.  Naming this endpoint keeps its relative fixed-field data opaque. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinAbstractEndpoint
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
+    Gal(rationalCyclotomicZHatField / ℚ) :=
+  abstractFixedFieldCyclotomicRestriction H
+    (infiniteGlobalArtinMonoidHom
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) a)
+
+/-- The rational norm Artin symbol serving as the other full endpoint. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinRationalEndpoint
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
+    Gal(rationalCyclotomicZHatField / ℚ) :=
+  rationalCyclotomicZHatGlobalArtin
+    (IdeleGroup.norm ℚ
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H) a)
+
+
+private noncomputable def cyclotomicAbstractFixedFieldArtinCoordinateMapData
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    {f : Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
+        cyclotomicAbstractFixedFieldArtinCoordinateBase H) →*
+        Gal(E / ℚ) //
+      f.comp
+          (@globalArtinMonoidHomOfNumberField
+            (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+            (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+            (inferInstance : Field
+              (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+            (cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField H)
+            (inferInstance : Field
+              (abstractFixedFieldCyclotomicFiniteGaloisLayer H E))
+            (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
+            (cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois H E)
+            (cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField H E)) =
+        (globalArtinMonoidHom (K := ℚ) (L := E)).comp
+          (IdeleGroup.norm ℚ
+            (cyclotomicAbstractFixedFieldArtinCoordinateBase H))} := by
+  have hnat :=
+    @globalArtinMonoidHomOfNumberField_norm_restriction
+      ℚ E
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+      inferInstance inferInstance
+      inferInstance
+      (cyclotomicAbstractFixedFieldArtinCoordinateNumberField E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateAlgebra E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois E)
+      inferInstance
+      (cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField H)
+      inferInstance
+      (cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerIsAbelianGalois H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField H E)
+  exact
+    ⟨(@AlgEquiv.restrictNormalHom
+          ℚ inferInstance
+          (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+          inferInstance
+          (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
+          E inferInstance
+          (cyclotomicAbstractFixedFieldArtinCoordinateAlgebra E)
+          (cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra H E)
+          (cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower H E)
+          (cyclotomicAbstractFixedFieldArtinCoordinateNormal E)).comp
+        (@AlgEquiv.restrictScalarsHom
+          ℚ
+          (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+          (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+          inferInstance inferInstance inferInstance
+          (cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra H)
+          (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
+          (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
+          (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower H E)),
+      hnat⟩
+
+/-- The fixed restriction map from the relative finite layer to one rational
+cyclotomic coordinate. -/
+private noncomputable def cyclotomicAbstractFixedFieldArtinCoordinateMap
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
+        cyclotomicAbstractFixedFieldArtinCoordinateBase H) →*
+      Gal(E / ℚ) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateMapData H E).1
+
+
+private theorem cyclotomicAbstractFixedFieldArtinCoordinateMap_naturality
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    (cyclotomicAbstractFixedFieldArtinCoordinateMap H E).comp
+        (globalArtinMonoidHom
+          (K := cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+          (L := abstractFixedFieldCyclotomicFiniteGaloisLayer H E)) =
+      (globalArtinMonoidHom (K := ℚ) (L := E)).comp
+        (IdeleGroup.norm ℚ
+          (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateMapData H E).2
+
+/-- Pointwise identification of the named coordinate map with the concrete
+two-stage restriction used by the abstract fixed-field comparison. -/
+private theorem cyclotomicAbstractFixedFieldArtinCoordinateMap_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    (σ : Gal(abstractFixedFieldCyclotomicFiniteGaloisLayer H E /
+      cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
+    cyclotomicAbstractFixedFieldArtinCoordinateMap H E σ =
+      @IntermediateField.restrictRestrictAlgEquivMapHom
+        ℚ E
+        (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+        (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+        inferInstance inferInstance inferInstance inferInstance
+        (cyclotomicAbstractFixedFieldArtinCoordinateAlgebra E)
+        (cyclotomicAbstractFixedFieldArtinCoordinateBaseAlgebra H)
+        (cyclotomicAbstractFixedFieldArtinCoordinateLayerRatAlgebra H E)
+        (cyclotomicAbstractFixedFieldArtinCoordinateLayerAlgebra H E)
+        (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseAlgebra H E)
+        (cyclotomicAbstractFixedFieldArtinCoordinateLayerScalarTower H E)
+        (cyclotomicAbstractFixedFieldArtinCoordinateLayerBaseScalarTower H E)
+        (cyclotomicAbstractFixedFieldArtinCoordinateNormal E) σ :=
+  rfl
+
+/-- The relative projection, common rational finite value, and rational
+infinite projection, with both comparison steps packaged by the generic
+provider before this concrete tower becomes opaque. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinCoordinateBridgeData
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :=
+    @compRestrictNormalHomInfiniteGlobalArtinRationalCyclotomicDataOfNumberField
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelative H)
+      (inferInstance : Field
+        (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+      (cyclotomicAbstractFixedFieldArtinCoordinateBaseNumberField H)
+      (inferInstance : Field
+        (cyclotomicAbstractFixedFieldArtinCoordinateRelative H))
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelativeAlgebra H)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRelativeIsAbelianGalois H)
+      a
+      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerNumberField H E)
+      E
+      (cyclotomicAbstractFixedFieldArtinCoordinateNumberField E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateIsAbelianGalois E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateMap H E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateMap_naturality H E)
+
+/-- The abstract endpoint after projection to one finite coordinate. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinAbstractCoordinate
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Gal(E / ℚ) :=
+  AlgEquiv.restrictNormalHom E
+    (cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a)
+
+/-- The relative infinite Artin symbol, restricted to a finite layer and
+then mapped to the corresponding rational coordinate. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Gal(E / ℚ) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.1
+
+/-- The finite relative Artin symbol mapped to one rational coordinate. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinFiniteCoordinate
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Gal(E / ℚ) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.2.1
+
+/-- The finite rational Artin coordinate of the idele norm. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinRationalCoordinate
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Gal(E / ℚ) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.2.1
+
+/-- Naturality of the finite global Artin map at the concrete cyclotomic
+coordinate. -/
+private theorem
+    cyclotomicAbstractFixedFieldArtinCoordinateNaturality
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E =
+      cyclotomicAbstractFixedFieldArtinRationalCoordinate H a E := by
+  rfl
+
+/-- The rational endpoint after projection to one finite coordinate. -/
+private noncomputable def
+    cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Gal(E / ℚ) :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).1.2.2
+
+/-- The abstract restriction map projected to the concrete finite layer. -/
+private theorem
+    cyclotomicAbstractFixedFieldArtinCoordinateRestriction
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    cyclotomicAbstractFixedFieldArtinAbstractCoordinate H a E =
+      cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E := by
+  calc
+    cyclotomicAbstractFixedFieldArtinAbstractCoordinate H a E =
+        AlgEquiv.restrictNormalHom E
+          (cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a) := rfl
+    _ = cyclotomicAbstractFixedFieldArtinCoordinateMap H E
+          (AlgEquiv.restrictNormalHom
+            (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+            (infiniteGlobalArtinMonoidHom
+              (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+              (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) a)) :=
+      (restrictNormalHom_abstractFixedFieldCyclotomicRestriction
+          H E
+            (infiniteGlobalArtinMonoidHom
+              (cyclotomicAbstractFixedFieldArtinCoordinateBase H)
+              (cyclotomicAbstractFixedFieldArtinCoordinateRelative H) a)).trans
+        (cyclotomicAbstractFixedFieldArtinCoordinateMap_apply H E _).symm
+    _ = cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E := rfl
+
+/-- Restricting the infinite relative Artin symbol supplies exactly the
+finite Artin coordinate. -/
+private theorem
+    cyclotomicAbstractFixedFieldArtinCoordinateLayerProjection
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    cyclotomicAbstractFixedFieldArtinRestrictedLayerCoordinate H a E =
+      cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E :=
+  (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).2.1
+
+/-- The rational cyclotomic Artin map projected to the same finite
+coordinate. -/
+private theorem
+    cyclotomicAbstractFixedFieldArtinCoordinateRationalProjection
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate H a E =
+      cyclotomicAbstractFixedFieldArtinFiniteCoordinate H a E := by
+  exact
+    (cyclotomicAbstractFixedFieldArtinCoordinateBridgeData H a E).2.2.symm
+
+/-- Equality of the two full endpoints at one opaque finite coordinate. -/
+private theorem
+    cyclotomicAbstractFixedFieldArtinFiniteCoordinateComparison
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a : IdeleGroup
+      (cyclotomicAbstractFixedFieldArtinCoordinateBase H))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    AlgEquiv.restrictNormalHom E
+        (cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a) =
+      AlgEquiv.restrictNormalHom E
+        (cyclotomicAbstractFixedFieldArtinRationalEndpoint H a) := by
+  change
+    cyclotomicAbstractFixedFieldArtinAbstractCoordinate H a E =
+      cyclotomicAbstractFixedFieldArtinRationalEndpointCoordinate H a E
+  exact
+    cyclotomicAbstractFixedFieldArtin_eqTransThree
+      (cyclotomicAbstractFixedFieldArtinCoordinateRestriction H a E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateLayerProjection H a E)
+      (cyclotomicAbstractFixedFieldArtinCoordinateRationalProjection
+        H a E).symm
+
+/-- The finite-coordinate comparison assembled in the rational cyclotomic
+inverse limit. -/
+theorem
+    abstractFixedFieldCyclotomicRestriction_infiniteGlobalArtin_inverseLimit
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (a :
+      IdeleGroup
+        (cyclotomicAbstractFixedFieldArtinCoordinateBase H)) :
+    cyclotomicAbstractFixedFieldArtinAbstractEndpoint H a =
+      cyclotomicAbstractFixedFieldArtinRationalEndpoint H a := by
+  apply
+    (InfiniteGalois.continuousMulEquivToLimit
+      ℚ rationalCyclotomicZHatField).injective
+  apply Subtype.ext
+  funext Eop
+  exact
+    cyclotomicAbstractFixedFieldArtinFiniteCoordinateComparison
+      H a Eop.unop
+
+end FiniteCoordinateHelpers
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinFiniteLayer.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinFiniteLayer.lean
new file mode 100644
index 0000000..0ee433c
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinFiniteLayer.lean
@@ -0,0 +1,298 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicAbstractFixedFieldArtinCompositum
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open KummerTheory
+
+attribute [local instance]
+  cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal
+  abstractFixedFieldCyclotomic_numberField
+
+/-- The finite cyclotomic compositum as an intermediate field of the
+actual maximal-unramified extension. -/
+noncomputable def abstractFixedFieldCyclotomicFiniteLayer
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    let hI :=
+      rationalCyclotomicFieldInertia_le H.field
+    IntermediateField
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) :=
+  (abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase H E).fieldRange
+
+/-- The base algebra on the finite field range, obtained from the
+explicit base embedding followed by the field-range equivalence. -/
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  RingHom.toAlgebra
+    ((AlgHom.toRingHom
+      (AlgEquiv.toAlgHom
+        (AlgHom.equivFieldRange
+          (abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
+            H E)))).comp
+      (AlgHom.toRingHom
+        (abstractFixedFieldCyclotomicFiniteCompositumBaseEmbedding H E)))
+
+/-- The scalar action belonging to the canonical base algebra on the
+finite field range. -/
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_baseSMul
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    SMul
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  Algebra.toSMul
+    (self := abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra H E)
+
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_baseModule
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Module
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  @Algebra.toModule
+    (LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field)
+    (abstractFixedFieldCyclotomicFiniteLayer H E)
+    _ _
+    (abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra H E)
+
+/-- The field-range equivalence rebuilt over the explicit base
+algebras.  Its underlying ring equivalence is the canonical one. -/
+noncomputable def
+    abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    abstractFixedFieldCyclotomicFiniteCompositum H E ≃ₐ[
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field]
+      abstractFixedFieldCyclotomicFiniteLayer H E :=
+  AlgEquiv.ofRingEquiv
+    (f :=
+      (AlgHom.equivFieldRange
+        (abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
+          H E)).toRingEquiv)
+    (fun _ => rfl)
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    FiniteDimensional
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  (abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
+    H E).toLinearEquiv.finiteDimensional
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_numberField
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    NumberField
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  NumberField.of_module_finite
+    (LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field)
+    (abstractFixedFieldCyclotomicFiniteLayer H E)
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsAbelianGalois
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E) := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  let : IsAbelianGalois
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) :=
+    abstractFixedFieldCyclotomic_isAbelianGalois H
+  exact
+    IsAbelianGalois.of_algHom
+      ((abstractFixedFieldCyclotomicFiniteCompositumInclusionOverBase
+          H E).comp
+        (AlgEquiv.toAlgHom
+          (AlgEquiv.symm
+            (abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
+              H E))))
+
+instance
+    abstractFixedFieldCyclotomicFiniteLayer_baseScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower ℚ
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E) := by
+  let hI :=
+    rationalCyclotomicFieldInertia_le H.field
+  apply IsScalarTower.of_algebraMap_eq
+  intro x
+  apply Subtype.ext
+  change
+    algebraMap ℚ
+        (LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) hI) x =
+      algebraMap
+        (LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field)
+        (LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) hI)
+        (algebraMap ℚ
+          (LocalClassFieldTheory.abstractFixedField
+            ℚ (SeparableClosure ℚ) H.field) x)
+  exact
+    IsScalarTower.algebraMap_apply
+      ℚ
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) hI) x
+
+/-- The finite rational layer embedded into its corresponding
+intermediate field over the abstract fixed field. -/
+noncomputable def
+    abstractFixedFieldCyclotomicFiniteLayerEmbedding
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    E →ₐ[ℚ]
+      abstractFixedFieldCyclotomicFiniteLayer H E := by
+  exact
+    (AlgEquiv.toAlgHom
+      (AlgEquiv.restrictScalars ℚ
+        (abstractFixedFieldCyclotomicFiniteCompositumEquivFiniteLayer
+          H E))).comp
+      (abstractFixedFieldCyclotomicFiniteCompositumLayerEmbedding H E)
+
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    Algebra E
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  RingHom.toAlgebra
+    (AlgHom.toRingHom
+      (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E))
+
+/-- The finite-layer action on its actual image in the relative fixed
+field. -/
+noncomputable instance
+    abstractFixedFieldCyclotomicFiniteLayer_layerSMul
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    SMul E
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  Algebra.toSMul
+    (self := abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E)
+
+instance
+    abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower ℚ E
+      (abstractFixedFieldCyclotomicFiniteLayer H E) :=
+  IsScalarTower.of_algebraMap_eq'
+    (AlgHom.comp_algebraMap
+      (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E)).symm
+
+/-- The finite cyclotomic layer as an object of the finite-Galois
+inverse system of the actual maximal-unramified extension. -/
+@[reducible]
+noncomputable def
+    abstractFixedFieldCyclotomicFiniteGaloisLayer
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    FiniteGaloisIntermediateField
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (rationalCyclotomicFieldInertia_le H.field)) where
+  toIntermediateField :=
+    abstractFixedFieldCyclotomicFiniteLayer H E
+  finiteDimensional :=
+    abstractFixedFieldCyclotomicFiniteLayer_finiteDimensional H E
+  isGalois :=
+    (abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E).toIsGalois
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinFiniteRestriction.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinFiniteRestriction.lean
new file mode 100644
index 0000000..bcfbbc2
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/CyclotomicAbstractFixedFieldArtinFiniteRestriction.lean
@@ -0,0 +1,462 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicAbstractFixedFieldArtinFiniteLayer
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open KummerTheory
+
+attribute [local instance]
+  cyclotomicAbstractFixedFieldArtin_separableClosureAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldAlgebra
+  cyclotomicAbstractFixedFieldArtin_cyclotomicZHatFieldNormal
+  abstractFixedFieldCyclotomic_numberField
+
+/-- The explicit base algebra on a finite cyclotomic layer is the canonical
+intermediate-field inclusion used by the finite Galois inverse system. -/
+theorem abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra_eq_algebra'
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    abstractFixedFieldCyclotomicFiniteLayer_baseAlgebra H E =
+      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).algebra' := by
+  apply Algebra.algebra_ext
+  intro x
+  apply Subtype.ext
+  rfl
+
+/-- The finite Galois layer uses its canonical inclusion into the full
+relative fixed field for the upper scalar action. -/
+instance
+    abstractFixedFieldCyclotomicFiniteGaloisLayer_scalarTower
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    IsScalarTower
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteGaloisLayer H E).toIntermediateField
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (rationalCyclotomicFieldInertia_le H.field)) := by
+  let i :
+      (abstractFixedFieldCyclotomicFiniteGaloisLayer
+          H E).toIntermediateField →ₐ[
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field]
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) :=
+    { (abstractFixedFieldCyclotomicFiniteGaloisLayer
+          H E).toIntermediateField.val.toRingHom with
+      commutes' := by
+        intro x
+        rfl }
+  exact
+    IsScalarTower.of_algebraMap_eq'
+      (AlgHom.comp_algebraMap i).symm
+
+/-- The canonical inclusion of the finite cyclotomic layer into the
+full abstract-fixed-field compositum. -/
+private noncomputable def
+    abstractFixedFieldCyclotomicFiniteLayerInclusion
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField) :
+    abstractFixedFieldCyclotomicFiniteLayer H E →ₐ[
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field]
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (rationalCyclotomicFieldInertia_le H.field) :=
+  IntermediateField.val
+    (abstractFixedFieldCyclotomicFiniteLayer H E)
+
+/-- The two embeddings of a finite rational cyclotomic layer into the
+full abstract-fixed-field compositum agree. -/
+private theorem
+    abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    (z : E) :
+    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E z) =
+      rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
+        (z : rationalCyclotomicZHatField) := by
+  exact Subtype.ext rfl
+
+/-- Restriction to `E` commutes pointwise with the restriction from the
+full rational cyclotomic tower. -/
+private theorem
+    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    [Normal ℚ E]
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field))
+    (z : E) :
+    ((AlgEquiv.restrictNormalHom E
+      (abstractFixedFieldCyclotomicRestriction H σ)) z :
+        rationalCyclotomicZHatField) =
+      (abstractFixedFieldCyclotomicRestriction H σ)
+        (z : rationalCyclotomicZHatField) := by
+  exact
+    AlgEquiv.restrictNormal_commutes
+      (abstractFixedFieldCyclotomicRestriction H σ) E z
+
+/-- The raw cyclotomic restriction commutes with the canonical embedding
+of the full rational cyclotomic tower. -/
+private theorem
+    abstractFixedFieldCyclotomicRestriction_embedding_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field))
+    (z : rationalCyclotomicZHatField) :
+    rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
+        (abstractFixedFieldCyclotomicRestriction H σ z) =
+      σ
+        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+          H z) := by
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ)
+      (rationalCyclotomicFieldInertia_le H.field)
+  exact
+    AlgEquiv.restrictNormal_commutes
+      (MulSemiringAction.toAlgEquiv ℚ U σ)
+      rationalCyclotomicZHatField z
+
+/-- Restriction to the finite compositum layer commutes with its
+canonical inclusion into the full compositum. -/
+private theorem
+    restrictNormalHom_abstractFixedFieldCyclotomicFiniteLayer_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    [Normal
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E)]
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field))
+    (z : abstractFixedFieldCyclotomicFiniteLayer H E) :
+    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+        (AlgEquiv.restrictNormalHom
+          (abstractFixedFieldCyclotomicFiniteLayer H E) σ z) =
+      σ (abstractFixedFieldCyclotomicFiniteLayerInclusion H E z) := by
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ)
+      (rationalCyclotomicFieldInertia_le H.field)
+  let P : IntermediateField F U :=
+    abstractFixedFieldCyclotomicFiniteLayer H E
+  exact
+    AlgEquiv.restrictNormal_commutes
+      (MulSemiringAction.toAlgEquiv F U σ) P z
+
+/-- Restricting the finite-compositum action further to `E` commutes
+with the explicit embedding of `E` into that finite layer. -/
+private theorem
+    abstractFixedFieldCyclotomicFiniteLayerEmbedding_restrict_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    [Normal ℚ E]
+    [Normal
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E)]
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field))
+    (z : E) :
+    abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
+        (IntermediateField.restrictRestrictAlgEquivMapHom
+          ℚ E
+          (LocalClassFieldTheory.abstractFixedField
+            ℚ (SeparableClosure ℚ) H.field)
+          (abstractFixedFieldCyclotomicFiniteLayer H E)
+          (AlgEquiv.restrictNormalHom
+            (abstractFixedFieldCyclotomicFiniteLayer H E) σ) z) =
+      (AlgEquiv.restrictNormalHom
+        (abstractFixedFieldCyclotomicFiniteLayer H E) σ)
+        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E z) := by
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ)
+      (rationalCyclotomicFieldInertia_le H.field)
+  let P : IntermediateField F U :=
+    abstractFixedFieldCyclotomicFiniteLayer H E
+  exact
+    AlgEquiv.restrictNormal_commutes
+      (MulSemiringAction.toAlgEquiv ℚ P
+        (AlgEquiv.restrictNormalHom P σ)) E z
+
+/-- The left finite-level restriction, after both canonical embeddings into
+the full compositum, is the action of `σ` on the cyclotomic embedding. -/
+private theorem
+    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_left_embedded
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    [Normal ℚ E]
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field))
+    (x : E) :
+    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
+          (AlgEquiv.restrictNormalHom E
+            (abstractFixedFieldCyclotomicRestriction H σ) x)) =
+      σ
+        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
+          (x : rationalCyclotomicZHatField)) := by
+  calc
+    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+          (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
+            (AlgEquiv.restrictNormalHom E
+              (abstractFixedFieldCyclotomicRestriction H σ) x)) =
+        rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
+          (((AlgEquiv.restrictNormalHom E
+            (abstractFixedFieldCyclotomicRestriction H σ) x) :
+              rationalCyclotomicZHatField)) :=
+      abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion H E _
+    _ = rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
+          (abstractFixedFieldCyclotomicRestriction H σ
+            (x : rationalCyclotomicZHatField)) :=
+      congrArg
+        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H)
+        (restrictNormalHom_abstractFixedFieldCyclotomicRestriction_apply
+          H E σ x)
+    _ = σ
+          (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+            H (x : rationalCyclotomicZHatField)) :=
+      abstractFixedFieldCyclotomicRestriction_embedding_apply
+        H σ (x : rationalCyclotomicZHatField)
+
+/-- The right finite-level restriction, after both canonical embeddings into
+the full compositum, is the same action of `σ`. -/
+private theorem
+    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_right_embedded
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    [Normal ℚ E]
+    [Normal
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E)]
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field))
+    (x : E) :
+    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+        (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
+          (IntermediateField.restrictRestrictAlgEquivMapHom
+            ℚ E
+            (LocalClassFieldTheory.abstractFixedField
+              ℚ (SeparableClosure ℚ) H.field)
+            (abstractFixedFieldCyclotomicFiniteLayer H E)
+            (AlgEquiv.restrictNormalHom
+              (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x)) =
+      σ
+        (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum H
+          (x : rationalCyclotomicZHatField)) := by
+  calc
+    abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+          (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E
+            (IntermediateField.restrictRestrictAlgEquivMapHom
+              ℚ E
+              (LocalClassFieldTheory.abstractFixedField
+                ℚ (SeparableClosure ℚ) H.field)
+              (abstractFixedFieldCyclotomicFiniteLayer H E)
+              (AlgEquiv.restrictNormalHom
+                (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x)) =
+        abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+          ((AlgEquiv.restrictNormalHom
+            (abstractFixedFieldCyclotomicFiniteLayer H E) σ)
+            (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E x)) :=
+      congrArg
+        (abstractFixedFieldCyclotomicFiniteLayerInclusion H E)
+        (abstractFixedFieldCyclotomicFiniteLayerEmbedding_restrict_apply
+          H E σ x)
+    _ = σ
+          (abstractFixedFieldCyclotomicFiniteLayerInclusion H E
+            (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E x)) :=
+      restrictNormalHom_abstractFixedFieldCyclotomicFiniteLayer_apply
+        H E σ (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E x)
+    _ = σ
+          (rationalCyclotomicZHatFieldEmbeddingInAbstractFixedFieldCompositum
+            H (x : rationalCyclotomicZHatField)) :=
+      congrArg σ
+        (abstractFixedFieldCyclotomicFiniteLayerEmbedding_inclusion H E x)
+
+/-- Pointwise form of finite-layer restriction compatibility. -/
+private theorem
+    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    [Normal ℚ E]
+    [Normal
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) H.field)
+      (abstractFixedFieldCyclotomicFiniteLayer H E)]
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field))
+    (x : E) :
+    AlgEquiv.restrictNormalHom E
+        (abstractFixedFieldCyclotomicRestriction H σ) x =
+      IntermediateField.restrictRestrictAlgEquivMapHom
+          ℚ E
+          (LocalClassFieldTheory.abstractFixedField
+            ℚ (SeparableClosure ℚ) H.field)
+          (abstractFixedFieldCyclotomicFiniteLayer H E)
+        (AlgEquiv.restrictNormalHom
+          (abstractFixedFieldCyclotomicFiniteLayer H E) σ) x := by
+  apply
+    (abstractFixedFieldCyclotomicFiniteLayerEmbedding H E).injective
+  apply
+    (abstractFixedFieldCyclotomicFiniteLayerInclusion H E).injective
+  exact
+    (restrictNormalHom_abstractFixedFieldCyclotomicRestriction_left_embedded
+      H E σ x).trans
+      (restrictNormalHom_abstractFixedFieldCyclotomicRestriction_right_embedded
+        H E σ x).symm
+
+/-- Restricting through a finite layer commutes with restriction from
+the full abstract-fixed-field compositum. -/
+theorem
+    restrictNormalHom_abstractFixedFieldCyclotomicRestriction
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (E :
+      FiniteGaloisIntermediateField
+        ℚ rationalCyclotomicZHatField)
+    (σ :
+      Gal(
+        LocalClassFieldTheory.abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ)
+          (rationalCyclotomicFieldInertia_le H.field) /
+        LocalClassFieldTheory.abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field)) :
+    letI : Normal ℚ E := E.isGalois.to_normal
+    AlgEquiv.restrictNormalHom E
+        (abstractFixedFieldCyclotomicRestriction H σ) =
+      IntermediateField.restrictRestrictAlgEquivMapHom
+          ℚ E
+          (LocalClassFieldTheory.abstractFixedField
+            ℚ (SeparableClosure ℚ) H.field)
+          (abstractFixedFieldCyclotomicFiniteLayer H E)
+        (AlgEquiv.restrictNormalHom
+          (abstractFixedFieldCyclotomicFiniteGaloisLayer H E)
+          σ) := by
+  let : Normal ℚ E := E.isGalois.to_normal
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H.field
+  let U :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ)
+      (rationalCyclotomicFieldInertia_le H.field)
+  let T := rationalCyclotomicZHatField
+  let P : IntermediateField F U :=
+    abstractFixedFieldCyclotomicFiniteLayer H E
+  let : Algebra T U :=
+    abstractFixedFieldCyclotomicCompositum_algebra H
+  let : IsScalarTower ℚ T U :=
+    abstractFixedFieldCyclotomicCompositum_scalarTower H
+  let : Algebra E P :=
+    abstractFixedFieldCyclotomicFiniteLayer_layerAlgebra H E
+  let : IsScalarTower ℚ E P :=
+    abstractFixedFieldCyclotomicFiniteLayer_layerScalarTower H E
+  let : IsAbelianGalois F P := by
+    change IsAbelianGalois F
+      (abstractFixedFieldCyclotomicFiniteLayer H E)
+    exact
+      abstractFixedFieldCyclotomicFiniteLayer_isAbelianGalois H E
+  let : Normal F P := IsGalois.to_normal
+  apply AlgEquiv.ext
+  intro x
+  exact
+    restrictNormalHom_abstractFixedFieldCyclotomicRestriction_pointwise
+      H E σ x
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/FiniteGaloisRealizationCore.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/FiniteGaloisRealizationCore.lean
index 73951e8..edb78f8 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/FiniteGaloisRealizationCore.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/FiniteGaloisRealizationCore.lean
@@ -15,6 +15,7 @@ import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.FixedField
 import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.FixedFieldIntrinsicReciprocity.BaseComparison.IntrinsicBaseEquivalence
 import ValuedFieldTheory.Valuation.Completion.AbsoluteValueExtensions
 
+
 set_option autoImplicit false
 
 /-!
@@ -431,9 +432,16 @@ noncomputable def numberFieldTowerSeparableClosureToBaseSubgroup :
       (numberFieldTowerBaseSubgroup K L).toSubgroup := by
   let _ : Algebra K (SeparableClosure ℚ) :=
     numberFieldTowerSeparableClosureBaseAlgebra K L
-  refine
+  exact
     { toFun := fun σ =>
-        ⟨AlgEquiv.restrictScalars ℚ σ, ?_⟩
+        ⟨AlgEquiv.restrictScalars ℚ σ, by
+          change
+            AlgEquiv.restrictScalars ℚ σ ∈
+              (numberFieldTowerBaseField K L).fixingSubgroup
+          rw [IntermediateField.mem_fixingSubgroup_iff]
+          intro x hx
+          obtain ⟨y, rfl⟩ := hx
+          exact σ.commutes y⟩
       map_one' := by
         apply Subtype.ext
         rfl
@@ -441,13 +449,6 @@ noncomputable def numberFieldTowerSeparableClosureToBaseSubgroup :
         intro σ τ
         apply Subtype.ext
         rfl }
-  change
-    AlgEquiv.restrictScalars ℚ σ ∈
-      (numberFieldTowerBaseField K L).fixingSubgroup
-  rw [IntermediateField.mem_fixingSubgroup_iff]
-  intro x hx
-  obtain ⟨y, rfl⟩ := hx
-  exact σ.commutes y
 
 /-- The compatible `K`-absolute Galois group is exactly the fixing
 subgroup of the embedded copy of `K` inside the rational absolute
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalHilbertSymbol/FinitePlaceFiniteSupport.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalHilbertSymbol/FinitePlaceFiniteSupport.lean
index 9f029f0..a47f31b 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalHilbertSymbol/FinitePlaceFiniteSupport.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalHilbertSymbol/FinitePlaceFiniteSupport.lean
@@ -6,6 +6,8 @@ Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 
 import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalHilbertSymbol.FinitePlaceCharacterComparison
 import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.FiniteIdeleArtin
+import Mathlib.Algebra.FiniteSupport.Basic
+
 
 set_option autoImplicit false
 
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturality.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturality.lean
index 974d793..dfea338 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturality.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturality.lean
@@ -4,19 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.AbstractFixedFieldGlobalNormResidue
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.TowerAlgEquivNaturality
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityTowerNorm
 
-set_option autoImplicit false
 
-/-!
-# Naturality of the global norm-residue symbol
+set_option autoImplicit false
 
-This file records the same-base restriction specialization of abstract
-norm-residue naturality in the rational absolute class formation.  All
-closed subgroups remain in one fixed separable-closure ambient, so the
-statement is directly usable by fixed-field overextension arguments.
--/
 
 open scoped IsMulCommutative
 
@@ -34,2834 +26,11 @@ open RamificationTheory
 
 universe u
 
-@[instance_reducible]
-private noncomputable def naturalityIdeleClassCommGroup
-    (F : Type) [Field F] [NumberField F] : CommGroup (IdeleClassGroup F) :=
-  QuotientGroup.Quotient.commGroup (IdeleGroup.principalSubgroup F)
-
-attribute [local instance] naturalityIdeleClassCommGroup
-
-local instance ideleClassGroupIsMulCommutative
-    {F : Type} [Field F] [NumberField F]
-    : IsMulCommutative (IdeleClassGroup F) :=
-  ⟨⟨fun a b => mul_comm a b⟩⟩
-
-local instance ideleClassSubgroupNormal
-    {F : Type} [Field F] [NumberField F]
-    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
-  N.normal_of_isMulCommutative
-
-
-/-- Two finite Galois subextensions with the same underlying closed subgroup
-are equal; the remaining structure fields are proof-irrelevant. -/
-private theorem finiteGaloisSubextension_eq_of_field_eq
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {K : ClosedSubgroup G}
-    (A B : FiniteGaloisSubextension K)
-    (h : A.field = B.field) :
-    A = B := by
-  cases A with
-  | mk A hA nA fA =>
-      cases B with
-      | mk B hB nB fB =>
-          dsimp only at h
-          cases h
-          rfl
-
-/-- Rebase a finite Galois subextension along equality of its bundled base.
-The field equality is the only data component; the remaining fields are
-proof-irrelevant. -/
-private theorem finiteGaloisSubextension_transport_eq_of_field_eq
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {A B : FiniteAbstractField G}
-    (hAB : A = B)
-    (P : FiniteGaloisSubextension A.field)
-    (Q : FiniteGaloisSubextension B.field)
-    (hfield : P.field = Q.field) :
-    Eq.mp
-      (congrArg
-        (fun X : FiniteAbstractField G =>
-          FiniteGaloisSubextension X.field)
-        hAB)
-      P = Q := by
-  cases hAB
-  exact finiteGaloisSubextension_eq_of_field_eq P Q hfield
-
-/-- Transporting an additive equivalence between rational ambient fixed
-subgroups does not change the underlying direct-limit class. -/
-private theorem rationalAmbientFixedAddEquiv_transport_apply_val
-    {A B : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
-    (hAB : A = B)
-    {X : Type} [AddGroup X]
-    (e : X ≃+
-      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
-    (x : X) :
-    ((Eq.mp
-        (congrArg
-          (fun Y : FiniteAbstractField
-              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-            X ≃+ ambientFixedAddSubgroup
-              rationalIdeleClassRepresentation Y.field)
-          hAB)
-        e x).1) =
-      (e x).1 := by
-  cases hAB
-  rfl
-
-/-- Changing only the bundled subgroup of an additive subgroup element
-does not change its value in the ambient group. -/
-private theorem addSubgroupCongr_apply_val
-    {A : Type} [AddGroup A]
-    {H K : AddSubgroup A}
-    (h : H = K) (x : H) :
-    ((AddEquiv.addSubgroupCongr h x).1 : A) = x.1 := by
-  cases h
-  rfl
-
-/-- Transporting an idele class along a field equality and the corresponding
-algebra equivalence leaves its rational direct-limit representative fixed. -/
-private theorem rationalIdeleClassEquivFixed_congr_apply_val
-    {A B : IntermediateField ℚ (SeparableClosure ℚ)}
-    [FiniteDimensional ℚ A] [FiniteDimensional ℚ B]
-    (h : A = B)
-    (e : B ≃ₐ[ℚ] A)
-    (he : e.trans (IntermediateField.equivOfEq h) =
-      (AlgEquiv.refl : B ≃ₐ[ℚ] B))
-    (c : Additive (IdeleClassGroup B)) :
-    ((rationalIdeleClassEquivFixed A)
-        (MulEquiv.toAdditive (ideleClassCongr e) c)).1 =
-      ((rationalIdeleClassEquivFixed B) c).1 := by
-  cases h
-  have he' : e = AlgEquiv.refl := by
-    apply AlgEquiv.ext
-    intro x
-    have hx := DFunLike.congr_fun he x
-    change e x = x at hx
-    exact hx
-  rw [he']
-  have hc :
-      MulEquiv.toAdditive
-          (ideleClassCongr (AlgEquiv.refl : A ≃ₐ[ℚ] A)) c = c := by
-    cases c with
-    | ofMul c =>
-        exact congrArg Additive.ofMul (ideleClassCongr_refl c)
-  rw [hc]
-
-/-- Transporting the target intermediate field of a base-field equivalence
-preserves its rational fixed-part representative. -/
-private theorem rationalIdeleClassEquivFixed_transport_baseEquiv_val
-    {T : Type} [Field T] [NumberField T]
-    {A B : IntermediateField ℚ (SeparableClosure ℚ)}
-    [FiniteDimensional ℚ A] [FiniteDimensional ℚ B]
-    (h : A = B) (e : T ≃ₐ[ℚ] B)
-    (c : IdeleClassGroup T) :
-    ((rationalIdeleClassEquivFixed A)
-      (Additive.ofMul
-        (ideleClassCongr (K := T) (M := A)
-          (e.trans (IntermediateField.equivOfEq h).symm) c))).1 =
-    ((rationalIdeleClassEquivFixed B)
-      (Additive.ofMul (ideleClassCongr (K := T) (M := B) e c))).1 := by
-  cases h
-  have he :
-      e.trans (IntermediateField.equivOfEq (rfl : A = A)).symm = e := by
-    ext x
-    rfl
-  rw [he]
-
-/-- Equality of the lower and upper closed subgroups transports the raw
-extension quotient without exposing dependent rewrites to clients. -/
-private def extensionQuotientMulEquivOfEq
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {H H' J J' : ClosedSubgroup G}
-    (hH : H = H') (hJ : J = J')
-    (hJH : J.toSubgroup ≤ H.toSubgroup)
-    (hJH' : J'.toSubgroup ≤ H'.toSubgroup)
-    [(CyclicCohomology.extensionSubgroup H J hJH).Normal]
-    [(CyclicCohomology.extensionSubgroup H' J' hJH').Normal] :
-    (H.toSubgroup ⧸ CyclicCohomology.extensionSubgroup H J hJH) ≃*
-      (H'.toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup H' J' hJH') := by
-  cases hH
-  cases hJ
-  exact MulEquiv.refl _
-
-/-- The quotient transport sends a quotient representative to the same
-ambient group element, rebundled in the equal lower subgroup. -/
-@[simp]
-private theorem extensionQuotientMulEquivOfEq_mk
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {H H' J J' : ClosedSubgroup G}
-    (hH : H = H') (hJ : J = J')
-    (hJH : J.toSubgroup ≤ H.toSubgroup)
-    (hJH' : J'.toSubgroup ≤ H'.toSubgroup)
-    [(CyclicCohomology.extensionSubgroup H J hJH).Normal]
-    [(CyclicCohomology.extensionSubgroup H' J' hJH').Normal]
-    (σ : H.toSubgroup) :
-    extensionQuotientMulEquivOfEq hH hJ hJH hJH'
-        (QuotientGroup.mk σ) =
-      QuotientGroup.mk
-        ((MulEquiv.subgroupCongr
-          (congrArg ClosedSubgroup.toSubgroup hH)) σ) := by
-  cases hH
-  cases hJ
-  rfl
-
-/-- Rebundling an element along equality of closed subgroups preserves its
-underlying ambient group element. -/
-private theorem closedSubgroupCongr_apply_val
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {H H' : ClosedSubgroup G}
-    (hH : H = H') (σ : H.toSubgroup) :
-    (((MulEquiv.subgroupCongr
-        (congrArg ClosedSubgroup.toSubgroup hH)) σ).1 : G) = σ.1 := by
-  cases hH
-  rfl
-
-/-- Rebase an abelianized extension-quotient equivalence together with its
-finite abstract base. -/
-private def abelianizedExtensionQuotientAddEquiv_transportBase
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {A B : FiniteAbstractField G}
-    (hAB : A = B)
-    (P : FiniteGaloisSubextension A.field)
-    {X : Type} [AddGroup X]
-    (e : Additive (Abelianization P.extensionQuotient) ≃+ X) :
-    Additive
-        (Abelianization
-          (Eq.mp
-            (congrArg
-              (fun Y : FiniteAbstractField G =>
-                FiniteGaloisSubextension Y.field)
-              hAB)
-            P).extensionQuotient) ≃+ X := by
-  cases hAB
-  exact e
-
-/-- Rebase an abelianized equivalence along equality of finite Galois
-subextensions over a fixed abstract base. -/
-private def abelianizedExtensionQuotientAddEquiv_transportExtension
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {K : ClosedSubgroup G}
-    {P Q : FiniteGaloisSubextension K}
-    (hPQ : P = Q)
-    {X : Type} [AddGroup X]
-    (e : Additive (Abelianization P.extensionQuotient) ≃+ X) :
-    Additive (Abelianization Q.extensionQuotient) ≃+ X := by
-  cases hPQ
-  exact e
-
-/-- The explicit quotient equivalence induced by rebasing a finite Galois
-subextension. -/
-private def extensionQuotientMulEquiv_transportFiniteGalois
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {A B : FiniteAbstractField G}
-    (hAB : A = B)
-    (P : FiniteGaloisSubextension A.field)
-    {Q : FiniteGaloisSubextension B.field}
-    (hPQ :
-      Eq.mp
-        (congrArg
-          (fun Y : FiniteAbstractField G =>
-            FiniteGaloisSubextension Y.field)
-          hAB)
-        P = Q) :
-    Q.extensionQuotient ≃* P.extensionQuotient := by
-  cases hAB
-  cases hPQ
-  exact MulEquiv.refl _
-
-/-- Quotient rebasing sends a canonical representative to the same ambient
-group element rebundled in the old base subgroup. -/
-@[simp]
-private theorem extensionQuotientMulEquiv_transportFiniteGalois_mk
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {A B : FiniteAbstractField G}
-    (hAB : A = B)
-    (P : FiniteGaloisSubextension A.field)
-    {Q : FiniteGaloisSubextension B.field}
-    (hPQ :
-      Eq.mp
-        (congrArg
-          (fun Y : FiniteAbstractField G =>
-            FiniteGaloisSubextension Y.field)
-          hAB)
-        P = Q)
-    (σ : B.field.toSubgroup) :
-    extensionQuotientMulEquiv_transportFiniteGalois hAB P hPQ
-        (Q.extensionQuotientMk σ) =
-      P.extensionQuotientMk
-        ((MulEquiv.subgroupCongr
-          (congrArg ClosedSubgroup.toSubgroup
-            (congrArg FiniteAbstractField.field hAB).symm)) σ) := by
-  cases hAB
-  cases hPQ
-  rfl
-
-/-- Abelianization commutes with simultaneous transport of the abstract base
-and its finite Galois subextension. -/
-private theorem abelianizedCanonicalEquiv_transportFiniteGalois
-    {G : Type u} [Group G] [TopologicalSpace G]
-    {A B : FiniteAbstractField G}
-    (hAB : A = B)
-    (P : FiniteGaloisSubextension A.field)
-    {Q : FiniteGaloisSubextension B.field}
-    (hPQ :
-      Eq.mp
-        (congrArg
-          (fun Y : FiniteAbstractField G =>
-            FiniteGaloisSubextension Y.field)
-          hAB)
-        P = Q)
-    {X : Type} [CommGroup X]
-    (e : P.extensionQuotient ≃* X) :
-    abelianizedExtensionQuotientAddEquiv_transportExtension hPQ
-        (abelianizedExtensionQuotientAddEquiv_transportBase
-          hAB P
-          (MulEquiv.toAdditive
-            (e.abelianizationCongr.trans
-              (Abelianization.equivOfComm : X ≃* Abelianization X).symm))) =
-      MulEquiv.toAdditive
-        (((extensionQuotientMulEquiv_transportFiniteGalois
-            hAB P hPQ).trans e).abelianizationCongr.trans
-          (Abelianization.equivOfComm : X ≃* Abelianization X).symm) := by
-  cases hAB
-  cases hPQ
-  rfl
-
-/-- Package the entire rational finite norm-residue evaluation so that its
-dependent base, extension, quotient and comparison maps move together. -/
-private def rationalFiniteNormResidueValue
-    (K : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (L : FiniteGaloisSubextension K.field)
-    {C X : Type} [AddGroup C] [AddGroup X]
-    (eIdele : C ≃+
-      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
-    (eGalois : Additive (Abelianization L.extensionQuotient) ≃+ X)
-    (c : C) : X := by
-  letI :
-      Finite
-        (K.field.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup
-            K.field L.field L.below) :=
-    L.finite
-  exact
-    eGalois
-      (rationalCyclotomicDegreeData.normResidueSymbol
-        rationalIdeleClassRepresentation
-        rationalCyclotomicIdeleClassValuationData
-        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-        K L
-        (finiteNormClass rationalIdeleClassRepresentation
-          K.field L.field L.below (eIdele c)))
-
-/-- The packaged norm-residue value is invariant under rebasing the finite
-abstract field together with all dependent data. -/
-private theorem rationalFiniteNormResidueValue_transportBase
-    {A B : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
-    (hAB : A = B)
-    (P : FiniteGaloisSubextension A.field)
-    {C X : Type} [AddGroup C] [AddGroup X]
-    (eIdele : C ≃+
-      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
-    (eGalois : Additive (Abelianization P.extensionQuotient) ≃+ X)
-    (c : C) :
-    rationalFiniteNormResidueValue A P eIdele eGalois c =
-      rationalFiniteNormResidueValue B
-        (Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension Y.field)
-            hAB)
-          P)
-        (Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              C ≃+ ambientFixedAddSubgroup
-                rationalIdeleClassRepresentation Y.field)
-            hAB)
-          eIdele)
-        (abelianizedExtensionQuotientAddEquiv_transportBase
-          hAB P eGalois)
-        c := by
-  cases hAB
-  rfl
-
-/-- The packaged norm-residue value is invariant under equality of the finite
-Galois subextension. -/
-private theorem rationalFiniteNormResidueValue_transportExtension
-    (K : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    {P Q : FiniteGaloisSubextension K.field}
-    (hPQ : P = Q)
-    {C X : Type} [AddGroup C] [AddGroup X]
-    (eIdele : C ≃+
-      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
-    (eGalois : Additive (Abelianization P.extensionQuotient) ≃+ X)
-    (c : C) :
-    rationalFiniteNormResidueValue K P eIdele eGalois c =
-      rationalFiniteNormResidueValue K Q eIdele
-        (abelianizedExtensionQuotientAddEquiv_transportExtension
-          hPQ eGalois)
-        c := by
-  cases hPQ
-  rfl
-
-/-- Evaluation of the canonical abelianization comparison induced by a
-multiplicative equivalence into a commutative group. -/
-private theorem abelianizationCongrToComm_apply
-    {Q R : Type*} [Group Q] [CommGroup R]
-    (e : Q ≃* R) (q : Q) :
-    MulEquiv.toAdditive
-        (e.abelianizationCongr.trans
-          (Abelianization.equivOfComm : R ≃* Abelianization R).symm)
-        (Additive.ofMul (Abelianization.of q)) =
-      Additive.ofMul (e q) := by
-  apply Additive.toMul.injective
-  change
-    (Abelianization.equivOfComm : R ≃* Abelianization R).symm
-        (e.abelianizationCongr (Abelianization.of q)) = e q
-  rw [abelianizationCongr_of]
-  exact
-    (Abelianization.equivOfComm : R ≃* Abelianization R).symm_apply_apply _
-
-/-- Evaluation of the canonical quotient from the abelianization of a
-commutative group. -/
-private theorem commutativeAbelianizationEquiv_apply
-    {Q R : Type*} [CommGroup Q] [Group R]
-    (e : Q ≃* R) (q : Q) :
-    MulEquiv.toAdditive
-        ((Abelianization.equivOfComm : Q ≃* Abelianization Q).symm.trans e)
-        (Additive.ofMul (Abelianization.of q)) =
-      Additive.ofMul (e q) := by
-  apply Additive.toMul.injective
-  change
-    e ((Abelianization.equivOfComm : Q ≃* Abelianization Q).symm
-      (Abelianization.of q)) = e q
-  exact congrArg e
-    ((Abelianization.equivOfComm : Q ≃* Abelianization Q).symm_apply_apply q)
-
-section AbstractFixedFieldInclusion
-
-variable
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-
-local instance naturalityAbstractFixedFieldBaseQuotientFinite :
-    Finite
-      ((baseField
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-          H.field (le_baseField H.field)) :=
-  H.finite
-
-local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
-    Finite
-      (H.field.toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
-  P.finite
-
-noncomputable local instance
-    naturalityAbstractFixedFieldFiniteDimensional :
-    FiniteDimensional ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
-  abstractFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ) H.field H.finite
-
-noncomputable local instance
-    naturalityAbstractRelativeFixedFieldFiniteDimensional :
-    FiniteDimensional
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) :=
-  abstractRelativeFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ)
-    H.field P.field P.below H.finite P.finite
-
-local instance naturalityAbstractFixedFieldRelativeScalarTower :
-    IsScalarTower ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) :=
-  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
-
-noncomputable local instance
-    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
-    FiniteDimensional ℚ
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) :=
-  FiniteDimensional.trans ℚ
-    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-    (abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below)
-
-noncomputable local instance naturalityAbstractFixedFieldNumberField :
-    NumberField
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
-  NumberField.of_module_finite ℚ
-    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-
-noncomputable local instance
-    naturalityAbstractRelativeFixedFieldNumberField :
-    NumberField
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) :=
-  NumberField.of_module_finite ℚ
-    (abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below)
-
-/-- Use the same direct fixed-field Galois witness as the intrinsic
-norm-residue construction.  This prevents the dependent Galois-group type
-from being synthesized through a second `IsAbelianGalois` instance path. -/
-noncomputable local instance
-    naturalityAbstractRelativeFixedFieldIsGalois :
-    IsGalois
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) :=
-  abstractRelativeFixedField_isGalois
-    ℚ (SeparableClosure ℚ)
-    H.field P.field P.below P.normal
-
-noncomputable local instance
-    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
-    IsAbelianGalois
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-      (abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) :=
-  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
-
-/-- The lower subgroup obtained from the canonical inclusion of an abstract
-fixed-field tower is the original lower closed subgroup. -/
-private theorem
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) H.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below
-    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-      E.val.restrictScalars ℚ
-    numberFieldEmbeddedBaseSubgroup F E j = H.field := by
-  dsimp only
-  have hi :
-      numberFieldEmbeddedLowerEmbedding
-          (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-          (abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) P.below)
-          ((abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ) =
-        (abstractFixedField
-          ℚ (SeparableClosure ℚ) H.field).val := by
-    ext x
-    rfl
-  have hRange :
-      (numberFieldEmbeddedLowerEmbedding
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below)
-        ((abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ)).fieldRange =
-        abstractFixedField ℚ (SeparableClosure ℚ) H.field := by
-    ext x
-    constructor
-    · rintro ⟨y, rfl⟩
-      change
-        numberFieldEmbeddedLowerEmbedding
-            (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
-            (abstractRelativeFixedField
-              ℚ (SeparableClosure ℚ) P.below)
-            ((abstractRelativeFixedField
-              ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ) y ∈
-          abstractFixedField ℚ (SeparableClosure ℚ) H.field
-      rw [hi]
-      exact y.property
-    · intro hx
-      refine ⟨⟨x, hx⟩, ?_⟩
-      rw [hi]
-      rfl
-  rw [numberFieldEmbeddedBaseSubgroup, hRange]
-  exact
-    closedFixingSubgroup_abstractFixedField_eq
-      ℚ (SeparableClosure ℚ) H.field
-
-/-- The upper subgroup obtained from the canonical inclusion of an abstract
-fixed-field tower is the original upper closed subgroup. -/
-private theorem
-    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) H.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below
-    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-      E.val.restrictScalars ℚ
-    numberFieldEmbeddedTopSubgroup F E j = P.field := by
-  dsimp only
-  rw [numberFieldEmbeddedTopSubgroup]
-  have hjRangeSelf :
-      ((abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ).fieldRange =
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below).restrictScalars ℚ := by
-    ext x
-    constructor
-    · rintro ⟨y, rfl⟩
-      exact y.property
-    · intro hx
-      exact ⟨⟨x, hx⟩, rfl⟩
-  have hjRange :
-      ((abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ).fieldRange =
-        abstractFixedField ℚ (SeparableClosure ℚ) P.field := by
-    exact hjRangeSelf.trans
-      (IntermediateField.extendScalars_restrictScalars
-        (abstractFixedField_le
-          ℚ (SeparableClosure ℚ) P.below))
-  rw [hjRange]
-  exact
-    closedFixingSubgroup_abstractFixedField_eq
-      ℚ (SeparableClosure ℚ) P.field
-
-/-- Transport a packaged rational norm-residue value directly from an equal
-abstract base to the finite Galois extension underlying `P`.  Keeping the two
-dependent transports in their own declaration prevents their elaboration cost
-from accumulating in the main fixed-field comparison theorem. -/
-private theorem rationalFiniteNormResidueValue_transportToAbstractExtension
-    {A : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
-    (hAH : A = H)
-    (L : FiniteGaloisSubextension A.field)
-    (hLP :
-      Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension Y.field)
-            hAH)
-          L =
-        P.toFiniteGaloisExtension)
-    {C X : Type} [AddGroup C] [AddGroup X]
-    (eIdele : C ≃+
-      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
-    (eGalois : Additive (Abelianization L.extensionQuotient) ≃+ X)
-    (c : C) :
-    rationalFiniteNormResidueValue A L eIdele eGalois c =
-      rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
-        (Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              C ≃+ ambientFixedAddSubgroup
-                rationalIdeleClassRepresentation Y.field)
-            hAH)
-          eIdele)
-        (abelianizedExtensionQuotientAddEquiv_transportExtension hLP
-          (abelianizedExtensionQuotientAddEquiv_transportBase
-            hAH L eGalois))
-        c := by
-  calc
-    _ = rationalFiniteNormResidueValue H
-        (Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension Y.field)
-            hAH)
-          L)
-        (Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              C ≃+ ambientFixedAddSubgroup
-                rationalIdeleClassRepresentation Y.field)
-            hAH)
-          eIdele)
-        (abelianizedExtensionQuotientAddEquiv_transportBase hAH L eGalois)
-        c :=
-      rationalFiniteNormResidueValue_transportBase
-        (A := A) (B := H) (C := C) (X := X)
-        hAH L eIdele eGalois c
-    _ = _ :=
-      rationalFiniteNormResidueValue_transportExtension
-        (K := H)
-        (P := Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension Y.field)
-            hAH)
-          L)
-        (Q := P.toFiniteGaloisExtension) (C := C) (X := X)
-        hLP
-        (Eq.mp
-          (congrArg
-            (fun Y : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              C ≃+ ambientFixedAddSubgroup
-                rationalIdeleClassRepresentation Y.field)
-            hAH)
-          eIdele)
-        (abelianizedExtensionQuotientAddEquiv_transportBase hAH L eGalois)
-        c
-
-/-- The packaged value at the literal fixed-field realization is the ambient
-fixed-part norm-residue homomorphism evaluated at the same idele class. -/
-private theorem rationalFiniteNormResidueValue_abstractFixedField_eq_ambient
-    (c : IdeleClassGroup
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :
-    rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
-        (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
-        (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P)
-        (Additive.ofMul c) =
-      ambientFixedGlobalNormResidueAddMonoidHom H P
-        (rationalAbstractFixedFieldIdeleClassEquivFixed
-          H.field (Additive.ofMul c)) := by
-  let eRec :=
-    rationalCyclotomicDegreeData.normResidueSymbol
-      rationalIdeleClassRepresentation
-      rationalCyclotomicIdeleClassValuationData
-      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-      H P.toFiniteGaloisExtension
-  let eGal :=
-    abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P
-  change
-    eGal
-        (eRec
-          (finiteNormClass rationalIdeleClassRepresentation
-            H.field P.field P.below
-            (rationalAbstractFixedFieldIdeleClassEquivFixed
-              H.field (Additive.ofMul c)))) =
-      eGal
-        (eRec
-          (finiteNormClass rationalIdeleClassRepresentation
-            H.field P.field P.below
-            (rationalAbstractFixedFieldIdeleClassEquivFixed
-              H.field (Additive.ofMul c))))
-  rfl
-
-/-- The packaged norm-residue value for the literal fixed-field realization
-is the intrinsic abstract fixed-field norm-residue value. -/
-private theorem rationalFiniteNormResidueValue_abstractFixedField_apply
-    (c : IdeleClassGroup
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :
-    Additive.toMul
-        (rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
-          (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
-          (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
-            H P)
-          (Additive.ofMul c)) =
-      abstractFixedFieldGlobalNormResidueMonoidHom H P c := by
-  let a :=
-    rationalAbstractFixedFieldIdeleClassEquivFixed
-      H.field (Additive.ofMul c)
-  have hAbstract :=
-    abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply H P a
-  calc
-    _ = Additive.toMul
-        (ambientFixedGlobalNormResidueAddMonoidHom H P
-          (rationalAbstractFixedFieldIdeleClassEquivFixed
-            H.field (Additive.ofMul c))) := by
-      exact congrArg Additive.toMul
-        (rationalFiniteNormResidueValue_abstractFixedField_eq_ambient
-          (H := H) (P := P) c)
-    _ = _ := by
-      simpa only [a, AddEquiv.symm_apply_apply, toMul_ofMul] using
-        hAbstract.symm
-
-/-- The finite abstract field reconstructed from the literal fixed-field
-inclusion is the original packaged abstract field.  Keeping this structure
-equality separate avoids repeatedly rebuilding all of its proof fields. -/
-private theorem
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) H.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below
-    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-      E.val.restrictScalars ℚ
-    numberFieldEmbeddedFiniteAbstractField F E j = H := by
-  dsimp only
-  exact FiniteAbstractField.eq_of_field_eq _ _
-    (numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P)
-
-/-- Pointwise specification of the idele-class comparison after transporting
-the explicitly embedded abstract field to the canonical packaged field.  The
-transport is kept at the value boundary, so clients never compare the two
-dependent additive equivalences themselves. -/
-private theorem
-    numberFieldEmbeddedIdeleClassEquivAmbientFixed_transport_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (c : Additive
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) H.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below
-    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-      E.val.restrictScalars ℚ
-    let HEmbedded :=
-      numberFieldEmbeddedFiniteAbstractField F E j
-    let hHEmbedded : HEmbedded = H :=
-      numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-    Eq.mp
-        (congrArg
-          (fun X : FiniteAbstractField
-              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-            Additive (IdeleClassGroup F) ≃+
-              ambientFixedAddSubgroup
-                rationalIdeleClassRepresentation X.field)
-          hHEmbedded)
-        (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j) c =
-      rationalAbstractFixedFieldIdeleClassEquivFixed H.field c := by
-  dsimp only
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  have hHEmbedded :
-      numberFieldEmbeddedFiniteAbstractField F E j = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  have hFixedBase :
-      abstractFixedField ℚ (SeparableClosure ℚ)
-          (numberFieldEmbeddedBaseSubgroup F E j) = F :=
-    congrArg
-      (abstractFixedField ℚ (SeparableClosure ℚ)) hBase
-  let eBase : F ≃ₐ[ℚ] F :=
-    (numberFieldEmbeddedAbstractBaseFieldEquiv F E j).trans
-      (IntermediateField.equivOfEq hFixedBase)
-  have heBase :
-      eBase = (AlgEquiv.refl : F ≃ₐ[ℚ] F) := by
-    apply AlgEquiv.ext
-    intro x
-    apply Subtype.ext
-    change x.1 = x.1
-    rfl
-  let hEmbeddedQuotientFinite :=
-    numberFieldEmbeddedAbsoluteQuotientFinite F E j
-  let hEmbeddedFixedFiniteDimensional :=
-    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional F E j
-  apply Subtype.ext
-  rw [rationalAmbientFixedAddEquiv_transport_apply_val
-    hHEmbedded
-    (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j) c]
-  dsimp only [numberFieldEmbeddedIdeleClassEquivAmbientFixed]
-  simp only [AddEquiv.trans_apply]
-  change
-    ((rationalIdeleClassEquivFixed
-        (abstractFixedField ℚ (SeparableClosure ℚ)
-          (numberFieldEmbeddedBaseSubgroup F E j)))
-      (MulEquiv.toAdditive
-        (ideleClassCongr
-          (numberFieldEmbeddedAbstractBaseFieldEquiv F E j)) c)).1 =
-      ((rationalIdeleClassEquivFixed F) c).1
-  exact rationalIdeleClassEquivFixed_congr_apply_val
-    hFixedBase
-    (numberFieldEmbeddedAbstractBaseFieldEquiv F E j)
-    heBase c
-
-/-- Transporting the finite Galois subextension reconstructed from the literal
-fixed-field inclusion recovers the canonical subextension packaged by `P`.
-This is the sole dependent structure equality used by the later quotient
-comparisons. -/
-private theorem
-    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) H.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below
-    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-      E.val.restrictScalars ℚ
-    let HEmbedded :=
-      numberFieldEmbeddedFiniteAbstractField F E j
-    let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-      numberFieldEmbeddedFiniteGaloisSubextension F E j
-    let hHEmbedded : HEmbedded = H :=
-      numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-    Eq.mp
-        (congrArg
-          (fun X : FiniteAbstractField
-              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-            FiniteGaloisSubextension X.field)
-          hHEmbedded)
-        PEmbedded =
-      P.toFiniteGaloisExtension := by
-  dsimp only
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-    numberFieldEmbeddedFiniteGaloisSubextension F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  exact finiteGaloisSubextension_transport_eq_of_field_eq
-    hHEmbedded PEmbedded P.toFiniteGaloisExtension
-    (numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P)
-
-/-- Opaque endpoint for the extension-quotient comparison supplied by the
-literal embedding.  Its domain is already the canonical quotient of `P`, so
-no client has to reconstruct the two subgroup transports. -/
-private noncomputable def
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    P.toFiniteGaloisExtension.extensionQuotient ≃*
-      Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  have hTop :
-      numberFieldEmbeddedTopSubgroup F E j = P.field :=
-    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P
-  letI hAlgebra : Algebra F (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra F E j
-  let eSep :=
-    numberFieldEmbeddedSeparableClosureEquiv F E j
-  letI hEmbeddedExtensionNormal :
-      (CyclicCohomology.extensionSubgroup
-        (numberFieldEmbeddedBaseSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal F E j
-  exact
-    P.toFiniteGaloisExtension.extensionQuotientMulEquiv.trans
-      ((extensionQuotientMulEquivOfEq
-        hBase.symm hTop.symm P.below
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).trans
-      (ambientEmbeddedExtensionQuotientEquivGaloisGroup
-        ℚ F E j eSep))
-
-/-- Opaque canonical endpoint for the same quotient, obtained directly from
-the abstract fixed-field realization. -/
-private noncomputable def
-    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    P.toFiniteGaloisExtension.extensionQuotient ≃*
-      Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
-  exact
-    P.toFiniteGaloisExtension.extensionQuotientMulEquiv.trans
-      (abstractExtensionQuotientEquivGaloisGroup
-        ℚ (SeparableClosure ℚ)
-        H.field P.field P.below P.normal)
-
-/-- Pointwise opaque endpoint of the embedded quotient equivalence. -/
-private noncomputable def
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient) :
-    Gal(
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) /
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :=
-  abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv H P q
-
-/-- Pointwise opaque endpoint of the canonical quotient equivalence. -/
-private noncomputable def
-    abstractFixedFieldInclusionCanonicalExtensionQuotientValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient) :
-    Gal(
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) /
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :=
-  abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv H P q
-
-/-- Fully applied ambient value of the embedded quotient endpoint. -/
-private noncomputable def
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ :=
-  (abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q x :
-    SeparableClosure ℚ)
-
-/-- Fully applied ambient value of the canonical quotient endpoint. -/
-private noncomputable def
-    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ :=
-  (abstractFixedFieldInclusionCanonicalExtensionQuotientValue H P q x :
-    SeparableClosure ℚ)
-
-/-- The ambient Galois value attached to a representative of the canonical
-quotient, packaged behind a literal result type. -/
-private noncomputable def
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup) :
-    Gal(
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) /
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  letI hAlgebra : Algebra F (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra F E j
-  let eSep :=
-    numberFieldEmbeddedSeparableClosureEquiv F E j
-  letI hEmbeddedExtensionNormal :
-      (CyclicCohomology.extensionSubgroup
-        (numberFieldEmbeddedBaseSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal F E j
-  let σEmbedded :
-      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
-    (MulEquiv.subgroupCongr
-      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
-  exact
-    ambientEmbeddedExtensionQuotientEquivGaloisGroup
-      ℚ F E j eSep (QuotientGroup.mk σEmbedded)
-
-/-- Fully applied ambient value of the packaged representative endpoint. -/
-private noncomputable def
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ :=
-  (abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue H P σ x :
-    SeparableClosure ℚ)
-
-/-- Ambient action of the representative after rebundling it in the embedded
-base subgroup. -/
-private noncomputable def
-    abstractFixedFieldInclusionRebasedAutomorphismApplyVal
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  let σEmbedded :
-      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
-    (MulEquiv.subgroupCongr
-      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
-  exact σEmbedded.1.1 (x : SeparableClosure ℚ)
-
-/-- The embedded quotient endpoint sends a canonical representative to the
-packaged ambient value above. -/
-private theorem
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue_mk
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup) :
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P
-        (P.toFiniteGaloisExtension.extensionQuotientMk σ) =
-      abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue H P σ := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  have hTop :
-      numberFieldEmbeddedTopSubgroup F E j = P.field :=
-    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P
-  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra F E j
-  let eSep :=
-    numberFieldEmbeddedSeparableClosureEquiv F E j
-  let hEmbeddedExtensionNormal :
-      (CyclicCohomology.extensionSubgroup
-        (numberFieldEmbeddedBaseSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal F E j
-  simp only [
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue,
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv,
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue,
-    MulEquiv.trans_apply,
-    FiniteGaloisSubextension.extensionQuotientMk_apply]
-  exact congrArg
-    (ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ F E j eSep)
-    (extensionQuotientMulEquivOfEq_mk
-      hBase.symm hTop.symm P.below
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j) σ)
-
-/-- The packaged ambient endpoint evaluates to the action of the rebundled
-representative. -/
-private theorem
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal_eq_rebased
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) :
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal H P σ x =
-      abstractFixedFieldInclusionRebasedAutomorphismApplyVal H P σ x := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra F E j
-  let eSep :=
-    numberFieldEmbeddedSeparableClosureEquiv F E j
-  let hEmbeddedExtensionNormal :
-      (CyclicCohomology.extensionSubgroup
-        (numberFieldEmbeddedBaseSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal F E j
-  let σEmbedded :
-      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
-    (MulEquiv.subgroupCongr
-      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
-  have hmk :=
-    ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
-      ℚ F E j eSep σEmbedded x
-  change
-    (ambientEmbeddedExtensionQuotientEquivGaloisGroup
-        ℚ F E j eSep (QuotientGroup.mk σEmbedded) x :
-        SeparableClosure ℚ) =
-      σEmbedded.1.1 (x : SeparableClosure ℚ) at hmk
-  simpa only [
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal,
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue,
-    abstractFixedFieldInclusionRebasedAutomorphismApplyVal] using hmk
-
-/-- Rebundling the representative does not change its action in the ambient
-separable closure. -/
-private theorem
-    abstractFixedFieldInclusionRebasedAutomorphismApplyVal_eq
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) :
-    abstractFixedFieldInclusionRebasedAutomorphismApplyVal H P σ x =
-      σ.1.1 (x : SeparableClosure ℚ) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  let σEmbedded :
-      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
-    (MulEquiv.subgroupCongr
-      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
-  have hσEmbedded :
-      (σEmbedded.1 : SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) = σ.1 :=
-    closedSubgroupCongr_apply_val hBase.symm σ
-  change σEmbedded.1.1 (x : SeparableClosure ℚ) = _
-  rw [hσEmbedded]
-
-/-- The packaged ambient representative acts by the original automorphism on
-the underlying separable-closure value. -/
-private theorem
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) :
-    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal H P σ x =
-      σ.1.1 (x : SeparableClosure ℚ) := by
-  exact
-    (abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal_eq_rebased
-      H P σ x).trans
-      (abstractFixedFieldInclusionRebasedAutomorphismApplyVal_eq H P σ x)
-
-/-- Evaluation of the embedded quotient endpoint on a canonical quotient
-representative, stated only in the ambient separable closure. -/
-private theorem
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv_mk_val
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) :
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal H P
-        (P.toFiniteGaloisExtension.extensionQuotientMk σ) x =
-      σ.1.1 (x : SeparableClosure ℚ) := by
-  calc
-    _ = abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal
-        H P σ x := by
-      exact congrArg
-        (fun g => (g x : SeparableClosure ℚ))
-        (abstractFixedFieldInclusionEmbeddedExtensionQuotientValue_mk
-          H P σ)
-    _ = _ :=
-      abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue_apply
-        H P σ x
-
-/-- Evaluation of the canonical abstract quotient endpoint on a quotient
-representative, again exposed only through its ambient value. -/
-private theorem
-    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv_mk_val
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup)
-    (x : abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below) :
-    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal H P
-        (P.toFiniteGaloisExtension.extensionQuotientMk σ) x =
-      σ.1.1 (x : SeparableClosure ℚ) := by
-  simp only [
-    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal,
-    abstractFixedFieldInclusionCanonicalExtensionQuotientValue,
-    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv,
-    MulEquiv.trans_apply,
-    FiniteGaloisSubextension.extensionQuotientMk_apply]
-  exact
-    (abstractExtensionQuotientEquivGaloisGroup_mk_apply_val
-      ℚ (SeparableClosure ℚ)
-      H.field P.field P.below P.normal σ x).symm
-
-/-- The embedded and canonical quotient endpoints agree on each quotient
-class.  This pointwise boundary is intentionally weaker than equality of the
-dependent `MulEquiv` structures. -/
-private theorem
-    abstractFixedFieldInclusionExtensionQuotientEquiv_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient) :
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q =
-      abstractFixedFieldInclusionCanonicalExtensionQuotientValue H P q := by
-  refine P.toFiniteGaloisExtension.extensionQuotient_inductionOn
-    (motive := fun q =>
-      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q =
-        abstractFixedFieldInclusionCanonicalExtensionQuotientValue H P q)
-    q ?_
-  intro σ
-  apply AlgEquiv.ext
-  intro x
-  apply Subtype.ext
-  exact
-    (abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv_mk_val
-      H P σ x).trans
-      (abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv_mk_val
-        H P σ x).symm
-
-/-- Opaque quotient endpoint obtained by transporting the explicitly embedded
-finite Galois subextension back to the canonical package. -/
-private noncomputable def
-    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    P.toFiniteGaloisExtension.extensionQuotient ≃*
-      Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-    numberFieldEmbeddedFiniteGaloisSubextension F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  have hPEmbedded :
-      Eq.mp
-          (congrArg
-            (fun X : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension X.field)
-            hHEmbedded)
-          PEmbedded =
-        P.toFiniteGaloisExtension :=
-    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
-  exact
-    (extensionQuotientMulEquiv_transportFiniteGalois
-      hHEmbedded PEmbedded hPEmbedded).trans
-      (numberFieldEmbeddedExtensionQuotientEquivGaloisGroup F E j)
-
-/-- Pointwise opaque endpoint of the transported quotient equivalence. -/
-private noncomputable def
-    abstractFixedFieldInclusionTransportedExtensionQuotientValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient) :
-    Gal(
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) /
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :=
-  abstractFixedFieldInclusionTransportedExtensionQuotientEquiv H P q
-
-/-- Transporting a canonical representative of the embedded finite Galois
-package preserves its Galois value. -/
-private theorem
-    abstractFixedFieldInclusionTransportedExtensionQuotientValue_mk
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (σ : H.field.toSubgroup) :
-    abstractFixedFieldInclusionTransportedExtensionQuotientValue H P
-        (P.toFiniteGaloisExtension.extensionQuotientMk σ) =
-      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P
-        (P.toFiniteGaloisExtension.extensionQuotientMk σ) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hBase :
-      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
-    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
-  have hTop :
-      numberFieldEmbeddedTopSubgroup F E j = P.field :=
-    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-    numberFieldEmbeddedFiniteGaloisSubextension F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  have hPEmbedded :
-      Eq.mp
-          (congrArg
-            (fun X : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension X.field)
-            hHEmbedded)
-          PEmbedded =
-        P.toFiniteGaloisExtension :=
-    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
-  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra F E j
-  let eSep :=
-    numberFieldEmbeddedSeparableClosureEquiv F E j
-  let hEmbeddedExtensionNormal :
-      (CyclicCohomology.extensionSubgroup
-        (numberFieldEmbeddedBaseSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup F E j)
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal F E j
-  have hFieldEq :
-      (congrArg FiniteAbstractField.field hHEmbedded).symm = hBase.symm :=
-    Subsingleton.elim _ _
-  simp only [
-    abstractFixedFieldInclusionTransportedExtensionQuotientValue,
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue,
-    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv,
-    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv,
-    MulEquiv.trans_apply,
-    extensionQuotientMulEquiv_transportFiniteGalois_mk,
-    FiniteGaloisSubextension.extensionQuotientMk_apply,
-    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup]
-  rw [hFieldEq]
-  change
-    ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ F E j eSep
-        (QuotientGroup.mk
-          ((MulEquiv.subgroupCongr
-            (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ)) = _
-  exact
-    (congrArg
-      (ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ F E j eSep)
-      (extensionQuotientMulEquivOfEq_mk
-        hBase.symm hTop.symm P.below
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j) σ)).symm
-
-/-- Transporting the embedded finite Galois package preserves the value of
-its extension-quotient comparison. -/
-private theorem
-    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv_apply
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient) :
-    abstractFixedFieldInclusionTransportedExtensionQuotientValue H P q =
-      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q := by
-  refine P.toFiniteGaloisExtension.extensionQuotient_inductionOn
-    (motive := fun q =>
-      abstractFixedFieldInclusionTransportedExtensionQuotientValue H P q =
-        abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q)
-    q ?_
-  intro σ
-  exact abstractFixedFieldInclusionTransportedExtensionQuotientValue_mk
-    H P σ
-
-/-- Opaque abelianized equivalence obtained by transporting the explicitly
-embedded finite Galois package to the canonical one. -/
-private noncomputable def
-    abstractFixedFieldInclusionTransportedAbelianizedEquiv
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    Additive
-        (Abelianization
-          P.toFiniteGaloisExtension.extensionQuotient) ≃+
-      Additive
-        (Gal(
-          (abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) P.below) /
-          (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-    numberFieldEmbeddedFiniteGaloisSubextension F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  have hPEmbedded :
-      Eq.mp
-          (congrArg
-            (fun X : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension X.field)
-            hHEmbedded)
-          PEmbedded =
-        P.toFiniteGaloisExtension :=
-    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
-  exact
-    abelianizedExtensionQuotientAddEquiv_transportExtension hPEmbedded
-      (abelianizedExtensionQuotientAddEquiv_transportBase
-        hHEmbedded PEmbedded
-        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-          F E j))
-
-/-- Canonical abelianization comparison built from the already transported
-opaque extension-quotient endpoint. -/
-private noncomputable def
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    Additive
-        (Abelianization
-          P.toFiniteGaloisExtension.extensionQuotient) ≃+
-      Additive
-        (Gal(
-          (abstractRelativeFixedField
-            ℚ (SeparableClosure ℚ) P.below) /
-          (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
-  let Q :=
-    Gal(
-      (abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below) /
-      (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
-  exact
-    MulEquiv.toAdditive
-      ((MulEquiv.abelianizationCongr
-        (abstractFixedFieldInclusionTransportedExtensionQuotientEquiv H P)).trans
-          (Abelianization.equivOfComm : Q ≃* Abelianization Q).symm)
-
-/-- Pointwise opaque value of the transported abelianized equivalence. -/
-private noncomputable def
-    abstractFixedFieldInclusionTransportedAbelianizedValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (z : Additive
-      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
-    Additive
-      (Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
-  abstractFixedFieldInclusionTransportedAbelianizedEquiv H P z
-
-/-- Pointwise opaque value of the canonical abelianization comparison built
-from the transported quotient endpoint. -/
-private noncomputable def
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (z : Additive
-      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
-    Additive
-      (Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
-  abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv H P z
-
-/-- Pointwise opaque value of the intrinsic abstract fixed-field
-abelianization comparison. -/
-private noncomputable def
-    abstractFixedFieldInclusionCanonicalAbelianizedValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (z : Additive
-      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
-    Additive
-      (Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
-  abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P z
-
-/-- The transported abelianized endpoint agrees pointwise with the canonical
-abelianization comparison built from the transported quotient endpoint. -/
-private theorem
-    abstractFixedFieldInclusionTransportedAbelianizedValue_eq_canonical
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (z : Additive
-      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
-    abstractFixedFieldInclusionTransportedAbelianizedValue H P z =
-      abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue
-        H P z := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-    numberFieldEmbeddedFiniteGaloisSubextension F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  have hPEmbedded :
-      Eq.mp
-          (congrArg
-            (fun X : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension X.field)
-            hHEmbedded)
-          PEmbedded =
-        P.toFiniteGaloisExtension :=
-    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
-  let qEmbedded : PEmbedded.extensionQuotient ≃* Gal(E / F) :=
-    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup F E j
-  have hCanonical :
-      abstractFixedFieldInclusionTransportedAbelianizedEquiv H P =
-        abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv
-          H P := by
-    simpa only [
-      abstractFixedFieldInclusionTransportedAbelianizedEquiv,
-      abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv,
-      abstractFixedFieldInclusionTransportedExtensionQuotientEquiv,
-      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup,
-      qEmbedded] using
-      (abelianizedCanonicalEquiv_transportFiniteGalois
-        hHEmbedded PEmbedded hPEmbedded qEmbedded)
-  exact DFunLike.congr_fun hCanonical z
-
-/-- On an abelianization representative, the transported canonical endpoint
-is the additive value of the transported quotient endpoint. -/
-private theorem
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_of
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient) :
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue H P
-        (Additive.ofMul (Abelianization.of q)) =
-      Additive.ofMul
-        (abstractFixedFieldInclusionTransportedExtensionQuotientValue
-          H P q) := by
-  simpa only [
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue,
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv,
-    abstractFixedFieldInclusionTransportedExtensionQuotientValue] using
-    (abelianizationCongrToComm_apply
-      (abstractFixedFieldInclusionTransportedExtensionQuotientEquiv H P) q)
-
-/-- On the same representative, the intrinsic fixed-field endpoint is the
-additive value of the canonical quotient endpoint. -/
-private theorem
-    abstractFixedFieldInclusionCanonicalAbelianizedValue_of
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (q : P.toFiniteGaloisExtension.extensionQuotient) :
-    abstractFixedFieldInclusionCanonicalAbelianizedValue H P
-        (Additive.ofMul (Abelianization.of q)) =
-      Additive.ofMul
-        (abstractFixedFieldInclusionCanonicalExtensionQuotientValue
-          H P q) := by
-  let hRawQuotientCommGroup :
-      CommGroup P.toFiniteGaloisExtension.extensionQuotient :=
-    { (inferInstance :
-        Group P.toFiniteGaloisExtension.extensionQuotient) with
-      mul_comm := P.commutative.is_comm.comm }
-  let qAbstractRaw :=
-    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv H P
-  change
-    MulEquiv.toAdditive
-        ((Abelianization.equivOfComm :
-          P.toFiniteGaloisExtension.extensionQuotient ≃*
-            Abelianization
-              P.toFiniteGaloisExtension.extensionQuotient).symm.trans
-          qAbstractRaw)
-        (Additive.ofMul (Abelianization.of q)) =
-      Additive.ofMul (qAbstractRaw q)
-  exact commutativeAbelianizationEquiv_apply qAbstractRaw q
-
-/-- The canonical abelianization comparison built after transport agrees
-pointwise with the intrinsic abstract fixed-field comparison. -/
-private theorem
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_eq
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (z : Additive
-      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
-    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue H P z =
-      abstractFixedFieldInclusionCanonicalAbelianizedValue H P z := by
-  let q : P.toFiniteGaloisExtension.extensionQuotient :=
-    Quotient.out z.toMul
-  have hz : Additive.ofMul (Abelianization.of q) = z := by
-    apply Additive.ext
-    exact Quotient.out_eq' z.toMul
-  rw [← hz]
-  calc
-    _ = Additive.ofMul
-        (abstractFixedFieldInclusionTransportedExtensionQuotientValue
-          H P q) :=
-      abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_of
-        H P q
-    _ = Additive.ofMul
-        (abstractFixedFieldInclusionEmbeddedExtensionQuotientValue
-          H P q) :=
-      congrArg Additive.ofMul
-        (abstractFixedFieldInclusionTransportedExtensionQuotientEquiv_apply
-          H P q)
-    _ = Additive.ofMul
-        (abstractFixedFieldInclusionCanonicalExtensionQuotientValue
-          H P q) :=
-      congrArg Additive.ofMul
-        (abstractFixedFieldInclusionExtensionQuotientEquiv_apply H P q)
-    _ = _ :=
-      (abstractFixedFieldInclusionCanonicalAbelianizedValue_of H P q).symm
-
-/-- The transported abelianized equivalence agrees pointwise with the
-intrinsic abstract fixed-field Galois comparison. -/
-private theorem
-    abstractFixedFieldInclusionTransportedAbelianizedValue_eq
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (z : Additive
-      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
-    abstractFixedFieldInclusionTransportedAbelianizedValue H P z =
-      abstractFixedFieldInclusionCanonicalAbelianizedValue H P z :=
-  (abstractFixedFieldInclusionTransportedAbelianizedValue_eq_canonical
-    H P z).trans
-    (abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_eq
-      H P z)
-
-/-- Opaque packaged norm-residue value before transporting the explicitly
-embedded abstract field and finite Galois package. -/
-private noncomputable def
-    abstractFixedFieldInclusionEmbeddedNormResidueValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (c : Additive
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
-    Additive
-      (Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-    numberFieldEmbeddedFiniteGaloisSubextension F E j
-  exact
-    rationalFiniteNormResidueValue HEmbedded PEmbedded
-      (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j)
-      (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-        F E j)
-      c
-
-/-- Opaque packaged value after transporting both dependent structures to
-the canonical `H` and `P` endpoints. -/
-private noncomputable def
-    abstractFixedFieldInclusionTransportedNormResidueValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (c : Additive
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
-    Additive
-      (Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  let eIdeleEmbedded :=
-    numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j
-  let eIdeleOverH :
-      Additive (IdeleClassGroup F) ≃+
-        ambientFixedAddSubgroup rationalIdeleClassRepresentation H.field :=
-    Eq.mp
-      (congrArg
-        (fun X : FiniteAbstractField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-          Additive (IdeleClassGroup F) ≃+
-            ambientFixedAddSubgroup
-              rationalIdeleClassRepresentation X.field)
-        hHEmbedded)
-      eIdeleEmbedded
-  exact
-    rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
-      eIdeleOverH
-      (abstractFixedFieldInclusionTransportedAbelianizedEquiv H P)
-      c
-
-/-- Opaque intrinsic packaged norm-residue value at the canonical endpoints. -/
-private noncomputable def
-    abstractFixedFieldInclusionCanonicalNormResidueValue
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (c : Additive
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
-    Additive
-      (Gal(
-        (abstractRelativeFixedField
-          ℚ (SeparableClosure ℚ) P.below) /
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
-  rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
-    (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
-    (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P)
-    c
-
-/-- Simultaneous transport of the embedded abstract field and finite Galois
-package sends the embedded norm-residue value to the transported endpoint. -/
-private theorem
-    abstractFixedFieldInclusionEmbeddedNormResidueValue_eq_transported
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (c : Additive
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
-    abstractFixedFieldInclusionEmbeddedNormResidueValue H P c =
-      abstractFixedFieldInclusionTransportedNormResidueValue H P c := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
-    numberFieldEmbeddedFiniteGaloisSubextension F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  have hPEmbedded :
-      Eq.mp
-          (congrArg
-            (fun X : FiniteAbstractField
-                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-              FiniteGaloisSubextension X.field)
-            hHEmbedded)
-          PEmbedded =
-        P.toFiniteGaloisExtension :=
-    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
-  let eIdeleEmbedded :=
-    numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j
-  let eGaloisEmbedded :=
-    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup F E j
-  simpa only [
-    abstractFixedFieldInclusionEmbeddedNormResidueValue,
-    abstractFixedFieldInclusionTransportedNormResidueValue,
-    abstractFixedFieldInclusionTransportedAbelianizedEquiv,
-    eIdeleEmbedded,
-    eGaloisEmbedded] using
-    (rationalFiniteNormResidueValue_transportToAbstractExtension
-      (H := H) (P := P)
-      (A := HEmbedded)
-      (C := Additive (IdeleClassGroup F))
-      (X := Additive Gal(E / F))
-      hHEmbedded PEmbedded hPEmbedded
-      eIdeleEmbedded eGaloisEmbedded c)
-
-/-- A packaged finite norm-residue value depends only on the value of its
-idele comparison at the chosen input and the value of its Galois comparison
-at the resulting norm class. -/
-private theorem rationalFiniteNormResidueValue_congr_apply
-    (K : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (L : FiniteGaloisSubextension K.field)
-    {C X : Type} [AddGroup C] [AddGroup X]
-    (eIdele eIdele' : C ≃+
-      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
-    (eGalois eGalois' :
-      Additive (Abelianization L.extensionQuotient) ≃+ X)
-    (c : C)
-    (heIdele : eIdele c = eIdele' c)
-    (heGalois : ∀ z, eGalois z = eGalois' z) :
-    rationalFiniteNormResidueValue K L eIdele eGalois c =
-      rationalFiniteNormResidueValue K L eIdele' eGalois' c := by
-  unfold rationalFiniteNormResidueValue
-  rw [heIdele]
-  exact heGalois _
-
-/-- The transported packaged value is the intrinsic canonical packaged value;
-only the idele input and the eventual abelianized value are compared. -/
-private theorem
-    abstractFixedFieldInclusionTransportedNormResidueValue_eq_canonical
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (c : Additive
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
-    abstractFixedFieldInclusionTransportedNormResidueValue H P c =
-      abstractFixedFieldInclusionCanonicalNormResidueValue H P c := by
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H.field
-  let E :=
-    abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  let HEmbedded :=
-    numberFieldEmbeddedFiniteAbstractField F E j
-  have hHEmbedded : HEmbedded = H :=
-    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
-  let eIdeleEmbedded :=
-    numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j
-  let eIdeleOverH :
-      Additive (IdeleClassGroup F) ≃+
-        ambientFixedAddSubgroup rationalIdeleClassRepresentation H.field :=
-    Eq.mp
-      (congrArg
-        (fun X : FiniteAbstractField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
-          Additive (IdeleClassGroup F) ≃+
-            ambientFixedAddSubgroup
-              rationalIdeleClassRepresentation X.field)
-        hHEmbedded)
-      eIdeleEmbedded
-  simpa only [
-    abstractFixedFieldInclusionTransportedNormResidueValue,
-    abstractFixedFieldInclusionCanonicalNormResidueValue,
-    eIdeleEmbedded,
-    eIdeleOverH] using
-    (rationalFiniteNormResidueValue_congr_apply
-      H P.toFiniteGaloisExtension
-      eIdeleOverH
-      (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
-      (abstractFixedFieldInclusionTransportedAbelianizedEquiv H P)
-      (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P)
-      c
-      (numberFieldEmbeddedIdeleClassEquivAmbientFixed_transport_apply H P c)
-      (fun z => abstractFixedFieldInclusionTransportedAbelianizedValue_eq
-        H P z))
-
-/-- The explicitly embedded and intrinsic packaged norm-residue values agree. -/
-private theorem
-    abstractFixedFieldInclusionEmbeddedNormResidueValue_eq
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field)
-    (c : Additive
-      (IdeleClassGroup
-        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
-    abstractFixedFieldInclusionEmbeddedNormResidueValue H P c =
-      abstractFixedFieldInclusionCanonicalNormResidueValue H P c :=
-  (abstractFixedFieldInclusionEmbeddedNormResidueValue_eq_transported
-    H P c).trans
-    (abstractFixedFieldInclusionTransportedNormResidueValue_eq_canonical
-      H P c)
-
-/-- For the literal fixed fields attached to an abstract finite abelian
-extension, the norm-residue map obtained from their canonical inclusion in
-the rational separable closure is the intrinsic fixed-field norm-residue
-map. -/
-theorem
-    globalNormResidueMonoidHomOfEmbedding_abstractFixedFieldInclusion
-    (H : FiniteAbstractField
-      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-    (P : FiniteAbelianSubextension H.field) :
-    let F :=
-      abstractFixedField ℚ (SeparableClosure ℚ) H.field
-    let E :=
-      abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below
-    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-      E.val.restrictScalars ℚ
-    globalNormResidueMonoidHomOfEmbedding F E j =
-      abstractFixedFieldGlobalNormResidueMonoidHom H P := by
-  dsimp only
-  apply MonoidHom.ext
-  intro c
-  rw [globalNormResidueMonoidHomOfEmbedding_apply]
-  change
-    Additive.toMul
-        (abstractFixedFieldInclusionEmbeddedNormResidueValue
-          H P (Additive.ofMul c)) =
-      abstractFixedFieldGlobalNormResidueMonoidHom H P c
-  calc
-    _ = Additive.toMul
-        (abstractFixedFieldInclusionCanonicalNormResidueValue
-          H P (Additive.ofMul c)) :=
-      congrArg Additive.toMul
-        (abstractFixedFieldInclusionEmbeddedNormResidueValue_eq
-          H P (Additive.ofMul c))
-    _ = _ := by
-      simpa only [abstractFixedFieldInclusionCanonicalNormResidueValue] using
-        (rationalFiniteNormResidueValue_abstractFixedField_apply
-          (H := H) (P := P) c)
-
-end AbstractFixedFieldInclusion
-
-section EmbeddedNumberFieldRestriction
-
-variable
-    (K K' L L' : Type)
-    [Field K] [NumberField K]
-    [Field K'] [NumberField K']
-    [Field L] [NumberField L]
-    [Field L'] [NumberField L']
-    [Algebra K K'] [Algebra K L] [Algebra K L']
-    [Algebra K' L'] [Algebra L L']
-    [IsScalarTower K K' L'] [IsScalarTower K L L']
-
-
-/-- Rebracketing the compatible tower does not change its embedded lower
-fixing subgroup. -/
-private theorem numberFieldEmbeddedBaseSubgroup_baseChange_eq
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    numberFieldEmbeddedBaseSubgroup K K'
-        (numberFieldEmbeddedLowerEmbedding K' L' j) =
-      numberFieldEmbeddedBaseSubgroup K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L')) := by
-  have hi :
-      numberFieldEmbeddedLowerEmbedding K K'
-          (numberFieldEmbeddedLowerEmbedding K' L' j) =
-        numberFieldEmbeddedLowerEmbedding K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L')) := by
-    ext x
-    simp only [numberFieldEmbeddedLowerEmbedding, AlgHom.comp_apply,
-      IsScalarTower.coe_toAlgHom']
-    rw [← IsScalarTower.algebraMap_apply K K' L',
-      ← IsScalarTower.algebraMap_apply K L L']
-  simp only [numberFieldEmbeddedBaseSubgroup, hi]
-
-omit [Field K] [NumberField K]
-    [Algebra K K'] [Algebra K L'] [IsScalarTower K K' L'] in
-/-- The top subgroup of the rebracketed base-change tower is the embedded
-fixing subgroup of the intermediate field. -/
-private theorem numberFieldEmbeddedTopSubgroup_baseChange_eq
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    numberFieldEmbeddedTopSubgroup K K'
-        (numberFieldEmbeddedLowerEmbedding K' L' j) =
-      numberFieldEmbeddedBaseSubgroup K' L' j := by
-  rfl
-
-/-- Reidentify the two presentations of the embedded lower fixing subgroup
-without transporting dependent subgroup data through an equality. -/
-private noncomputable def numberFieldEmbeddedBaseChangeBaseEquiv
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    (numberFieldEmbeddedBaseSubgroup K K'
-        (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup ≃*
-      (numberFieldEmbeddedBaseSubgroup K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup :=
-  MulEquiv.subgroupCongr
-    (congrArg ClosedSubgroup.toSubgroup
-      (numberFieldEmbeddedBaseSubgroup_baseChange_eq K K' L L' j))
-
-/-- Under the identity equivalence of the two lower fixing subgroups, the
-relative subgroup for the base change is exactly the target presentation. -/
-private theorem numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    (CyclicCohomology.extensionSubgroup
-        (numberFieldEmbeddedBaseSubgroup K K'
-          (numberFieldEmbeddedLowerEmbedding K' L' j))
-        (numberFieldEmbeddedTopSubgroup K K'
-          (numberFieldEmbeddedLowerEmbedding K' L' j))
-        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K K'
-          (numberFieldEmbeddedLowerEmbedding K' L' j))).map
-          (numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j).toMonoidHom =
-      CyclicCohomology.extensionSubgroup
-        (numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-        (numberFieldEmbeddedBaseSubgroup K' L' j)
-        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j) := by
-  let e := numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j
-  have hTop :
-      (numberFieldEmbeddedTopSubgroup K K'
-          (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup =
-        (numberFieldEmbeddedBaseSubgroup K' L' j).toSubgroup :=
-    congrArg ClosedSubgroup.toSubgroup
-      (numberFieldEmbeddedTopSubgroup_baseChange_eq
-        (K := K) (K' := K') (L' := L') j)
-  ext x
-  constructor
-  · rintro ⟨y, hy, rfl⟩
-    change
-      ((e y :
-          (numberFieldEmbeddedBaseSubgroup K L
-            (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup) :
-          SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) ∈
-        (numberFieldEmbeddedBaseSubgroup K' L' j).toSubgroup
-    dsimp only [e, numberFieldEmbeddedBaseChangeBaseEquiv]
-    rw [MulEquiv.subgroupCongr_apply, ← hTop]
-    exact hy
-  · intro hx
-    refine ⟨e.symm x, ?_, e.apply_symm_apply x⟩
-    change
-      (((e.symm x :
-          (numberFieldEmbeddedBaseSubgroup K K'
-            (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup) :
-          SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) ∈
-        (numberFieldEmbeddedTopSubgroup K K'
-          (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup)
-    dsimp only [e, numberFieldEmbeddedBaseChangeBaseEquiv]
-    rw [MulEquiv.subgroupCongr_symm_apply, hTop]
-    exact hx
-
-/-- Normality of the relative subgroup between the two embedded base fields
-in a finite Galois base change. -/
-private theorem numberFieldEmbeddedBaseChangeExtensionSubgroup_normal
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    (CyclicCohomology.extensionSubgroup
-      (numberFieldEmbeddedBaseSubgroup K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-      (numberFieldEmbeddedBaseSubgroup K' L' j)
-      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal := by
-  let e := numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j
-  have hNormal :=
-    (numberFieldEmbeddedExtensionSubgroup_normal K K'
-      (numberFieldEmbeddedLowerEmbedding K' L' j)).map
-        e.toMonoidHom e.surjective
-  rw [numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq
-    K K' L L' j] at hNormal
-  exact hNormal
-
-noncomputable local instance
-    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    (CyclicCohomology.extensionSubgroup
-      (numberFieldEmbeddedBaseSubgroup K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-      (numberFieldEmbeddedBaseSubgroup K' L' j)
-      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
-  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
-
-/-- Finiteness of the relative quotient between the two embedded base fields
-in a finite Galois base change. -/
-private theorem numberFieldEmbeddedBaseChangeExtensionQuotient_finite
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    Finite
-      ((numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (numberFieldEmbeddedBaseSubgroup K L
-            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-          (numberFieldEmbeddedBaseSubgroup K' L' j)
-          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) := by
-  let e := numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j
-  let N :=
-    CyclicCohomology.extensionSubgroup
-      (numberFieldEmbeddedBaseSubgroup K K'
-        (numberFieldEmbeddedLowerEmbedding K' L' j))
-      (numberFieldEmbeddedTopSubgroup K K'
-        (numberFieldEmbeddedLowerEmbedding K' L' j))
-      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K K'
-        (numberFieldEmbeddedLowerEmbedding K' L' j))
-  let M :=
-    CyclicCohomology.extensionSubgroup
-      (numberFieldEmbeddedBaseSubgroup K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-      (numberFieldEmbeddedBaseSubgroup K' L' j)
-      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
-  let hNNormal : N.Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal K K'
-      (numberFieldEmbeddedLowerEmbedding K' L' j)
-  let hMNormal : M.Normal :=
-    numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
-  let hNFinite :
-      Finite
-        ((numberFieldEmbeddedBaseSubgroup K K'
-            (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup ⧸ N) :=
-    numberFieldEmbeddedExtensionQuotient_finite K K'
-      (numberFieldEmbeddedLowerEmbedding K' L' j)
-  have hmap : N.map e.toMonoidHom = M :=
-    numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq K K' L L' j
-  have hle : N ≤ M.comap e.toMonoidHom := by
-    rw [← hmap]
-    exact Subgroup.le_comap_map e.toMonoidHom N
-  let f :
-      ((numberFieldEmbeddedBaseSubgroup K K'
-          (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup ⧸ N) →*
-        ((numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸ M) :=
-    QuotientGroup.map N M e.toMonoidHom hle
-  have hmk : Function.Surjective
-      (QuotientGroup.mk ∘ e :
-        (numberFieldEmbeddedBaseSubgroup K K'
-            (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup →
-          (numberFieldEmbeddedBaseSubgroup K L
-              (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸ M) :=
-    QuotientGroup.mk_surjective.comp e.surjective
-  have hsurj : Function.Surjective f :=
-    QuotientGroup.map_surjective_of_surjective
-      (N := N) M e.toMonoidHom hmk hle
-  exact Finite.of_surjective f hsurj
-
-noncomputable local instance
-    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    Finite
-      ((numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
-        CyclicCohomology.extensionSubgroup
-          (numberFieldEmbeddedBaseSubgroup K L
-            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-          (numberFieldEmbeddedBaseSubgroup K' L' j)
-          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
-  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
-
-/-- Reuse the canonical absolute fixed-field witness for the lower embedded
-tower.  The base-change relative witness below needs this exact instance path
-when forming the absolute finite-dimensional tower. -/
-noncomputable local instance
-    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteDimensional ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
-  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
-    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
-
-noncomputable local instance
-    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteDimensional
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
-  abstractRelativeFixedField_finiteDimensional
-    ℚ (SeparableClosure ℚ)
-    (numberFieldEmbeddedBaseSubgroup K L
-      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-    (numberFieldEmbeddedBaseSubgroup K' L' j)
-    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
-    (numberFieldEmbeddedAbsoluteQuotientFinite K L
-      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
-
-local instance
-    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    IsScalarTower ℚ
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
-  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
-
-noncomputable local instance
-    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    FiniteDimensional ℚ
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
-  FiniteDimensional.trans ℚ
-    (abstractFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedBaseSubgroup K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
-
-noncomputable local instance
-    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    NumberField
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
-  NumberField.of_module_finite ℚ
-    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
-
-noncomputable local instance
-    numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    IsGalois
-      (abstractFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup K L
-          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
-      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
-        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
-  abstractRelativeFixedField_isGalois
-    ℚ (SeparableClosure ℚ)
-    (numberFieldEmbeddedBaseSubgroup K L
-      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
-    (numberFieldEmbeddedBaseSubgroup K' L' j)
-    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
-    (numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j)
-
-/-- In one common rational-separable-closure realization, the canonical
-quotient-to-Galois comparisons intertwine abstract restriction with
-ordinary restriction of the actual number-field automorphisms. -/
-theorem
-    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction
-    [FiniteDimensional K L] [IsAbelianGalois K L]
-    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
-    (z :
-      Abelianization
-        (numberFieldEmbeddedFiniteGaloisSubextension
-          K' L' j).extensionQuotient) :
-    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
-      j.comp (IsScalarTower.toAlgHom ℚ L L')
-    let H :=
-      numberFieldEmbeddedBaseSubgroup K L jLower
-    let H' :=
-      numberFieldEmbeddedBaseSubgroup K' L' j
-    let J :=
-      numberFieldEmbeddedTopSubgroup K L jLower
-    let J' :=
-      numberFieldEmbeddedTopSubgroup K' L' j
-    let hH'H :=
-      numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
-    let hJ'J :=
-      numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
-    let hJH :=
-      numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L jLower
-    let hJ'H' :=
-      numberFieldEmbeddedTopSubgroup_le_baseSubgroup K' L' j
-    letI _ :
-        (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
-      numberFieldEmbeddedExtensionSubgroup_normal K L jLower
-    letI _ :
-        (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
-      numberFieldEmbeddedExtensionSubgroup_normal K' L' j
-    ((AlgEquiv.restrictNormalHom L).comp
-        (AlgEquiv.restrictScalarsHom K))
-        (Additive.toMul
-          (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-            K' L' j (Additive.ofMul z))) =
-      Additive.toMul
-        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-          K L jLower
-          (MonoidHom.toAdditive
-            (normResidueNaturalityAbelianizedRestriction
-              H H' J J'
-              hJH hJ'H'
-              hH'H hJ'J)
-            (Additive.ofMul z))) := by
-  dsimp only
-  let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
-    j.comp (IsScalarTower.toAlgHom ℚ L L')
-  let H :=
-    numberFieldEmbeddedBaseSubgroup K L jLower
-  let H' :=
-    numberFieldEmbeddedBaseSubgroup K' L' j
-  let J :=
-    numberFieldEmbeddedTopSubgroup K L jLower
-  let J' :=
-    numberFieldEmbeddedTopSubgroup K' L' j
-  let hH'H :=
-    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
-  let hJ'J :=
-    numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
-  let hJH :=
-    numberFieldEmbeddedTopSubgroup_le_baseSubgroup
-      K L jLower
-  let hJ'H' :=
-    numberFieldEmbeddedTopSubgroup_le_baseSubgroup
-      K' L' j
-  let hLowerNormal :
-      (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal K L jLower
-  let hUpperNormal :
-      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal K' L' j
-  let qLower :=
-    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
-      K L jLower
-  let qUpper :=
-    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
-      K' L' j
-  let qLowerRaw :
-      (H.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H J hJH) ≃*
-        Gal(L / K) := by
-    exact
-      { qLower.toEquiv with
-        map_mul' := fun x y => qLower.map_mul x y }
-  let qUpperRaw :
-      (H'.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H' J' hJ'H') ≃*
-        Gal(L' / K') := by
-    exact
-      { qUpper.toEquiv with
-        map_mul' := fun x y => qUpper.map_mul x y }
-  let restrictActual :
-      Gal(L' / K') →* Gal(L / K) :=
-    (AlgEquiv.restrictNormalHom L).comp
-      (AlgEquiv.restrictScalarsHom K)
-  obtain ⟨q, rfl⟩ :=
-    QuotientGroup.mk_surjective z
-  obtain ⟨σ, rfl⟩ :=
-    (numberFieldEmbeddedFiniteGaloisSubextension
-      K' L' j).extensionQuotientMk_surjective q
-  change
-    restrictActual
-        ((Abelianization.equivOfComm (H := Gal(L' / K'))).symm
-          (qUpperRaw.abelianizationCongr
-            (Abelianization.of (QuotientGroup.mk σ)))) =
-      (Abelianization.equivOfComm (H := Gal(L / K))).symm
-        (qLowerRaw.abelianizationCongr
-          (normResidueNaturalityAbelianizedRestriction
-            H H' J J' hJH hJ'H' hH'H hJ'J
-            (Abelianization.of (QuotientGroup.mk σ))))
-  rw [normResidueNaturalityAbelianizedRestriction_of_mk,
-    abelianizationCongr_of, abelianizationCongr_of]
-  change
-    restrictActual (qUpperRaw (QuotientGroup.mk σ)) =
-      qLowerRaw
-        (QuotientGroup.mk (Subgroup.inclusion hH'H σ))
-  apply AlgEquiv.ext
-  intro x
-  apply jLower.injective
-  let hUpperAlgebra : Algebra K' (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra K' L' j
-  let eUpper :=
-    numberFieldEmbeddedSeparableClosureEquiv K' L' j
-  let hLowerAlgebra : Algebra K (SeparableClosure ℚ) :=
-    numberFieldEmbeddedSeparableClosureAlgebra K L jLower
-  let eLower :=
-    numberFieldEmbeddedSeparableClosureEquiv K L jLower
-  calc
-    jLower
-        (restrictActual
-          (qUpperRaw (QuotientGroup.mk σ)) x) =
-        j
-          ((qUpperRaw (QuotientGroup.mk σ))
-            (algebraMap L L' x)) := by
-      exact congrArg j
-        (AlgEquiv.restrictNormal_commutes
-          ((AlgEquiv.restrictScalarsHom K)
-            (qUpperRaw (QuotientGroup.mk σ)))
-          L x)
-    _ = σ.1.1
-        (j (algebraMap L L' x)) := by
-      exact
-        ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
-          ℚ K' L' j eUpper σ (algebraMap L L' x)
-    _ = (Subgroup.inclusion hH'H σ).1.1
-        (jLower x) := rfl
-    _ = jLower
-        (qLowerRaw
-          (QuotientGroup.mk
-            (Subgroup.inclusion hH'H σ)) x) := by
-      exact
-        (ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
-          ℚ K L jLower eLower
-          (Subgroup.inclusion hH'H σ) x).symm
-
-/-- In a compatible common embedding, the fixed-part relative norm
-between two (Galois-related) base fields is the genuine ordinary
-idele-class norm. -/
-theorem numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
-    (c : IdeleClassGroup K') :
-    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
-      j.comp (IsScalarTower.toAlgHom ℚ L L')
-    let H :=
-      numberFieldEmbeddedBaseSubgroup K L jLower
-    let H' :=
-      numberFieldEmbeddedBaseSubgroup K' L' j
-    let hH'H :=
-      numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
-    letI _ : Finite
-        (H.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H H' hH'H) :=
-      numberFieldEmbeddedBaseChangeExtensionQuotientFinite
-        K K' L L' j
-    relativeNorm rationalIdeleClassRepresentation H H' hH'H
-        (numberFieldEmbeddedIdeleClassEquivAmbientFixed
-          K' L' j (Additive.ofMul c)) =
-      numberFieldEmbeddedIdeleClassEquivAmbientFixed
-        K L jLower
-        (Additive.ofMul (_root_.ideleClassNorm K K' c)) := by
-  intro jLower H H'
-  let hH'H :=
-    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
-  let hnormal :=
-    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j
-  let F :=
-    abstractFixedField ℚ (SeparableClosure ℚ) H
-  let E :=
-    abstractRelativeFixedField ℚ (SeparableClosure ℚ) hH'H
-  let _ :
-      (CyclicCohomology.extensionSubgroup H H' hH'H).Normal :=
-    hnormal
-  let _ :
-      Finite
-        (H.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H H' hH'H) :=
-    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
-      K K' L L' j
-  let _ :
-      Finite
-        ((baseField
-          (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup
-            (baseField
-              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-            H (le_baseField H)) :=
-    numberFieldEmbeddedAbsoluteQuotientFinite K L jLower
-  let _ : NumberField F :=
-    numberFieldEmbeddedAbstractFixedFieldNumberField K L jLower
-  let _ : NumberField E :=
-    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
-      K K' L L' j
-  let _ : FiniteDimensional ℚ (E.restrictScalars ℚ) := by
-    change FiniteDimensional ℚ E
-    exact
-      numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
-        K K' L L' j
-  let _ : NumberField
-      (abstractFixedField ℚ (SeparableClosure ℚ) H') :=
-    numberFieldEmbeddedAbstractFixedFieldNumberField K' L' j
-  have hE :
-      E.restrictScalars ℚ =
-        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
-    IntermediateField.extendScalars_restrictScalars
-      (abstractFixedField_le
-        ℚ (SeparableClosure ℚ) hH'H)
-  let eRel :
-      E ≃ₐ[ℚ]
-        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
-    IntermediateField.equivOfEq hE
-  let eK :
-      K ≃ₐ[ℚ] F :=
-    numberFieldEmbeddedAbstractBaseFieldEquiv K L jLower
-  let eK'Base :
-      K' ≃ₐ[ℚ]
-        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
-    numberFieldEmbeddedAbstractBaseFieldEquiv K' L' j
-  let eK' : K' ≃ₐ[ℚ] E :=
-    eK'Base.trans eRel.symm
-  have hcompat (x : K) :
-      eK' (algebraMap K K' x) =
-        algebraMap F E (eK x) := by
-    apply eRel.injective
-    apply Subtype.ext
-    change
-      j (algebraMap K' L' (algebraMap K K' x)) =
-        j (algebraMap L L' (algebraMap K L x))
-    rw [← IsScalarTower.algebraMap_apply K K' L',
-      ← IsScalarTower.algebraMap_apply K L L']
-  have hupper :
-      rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
-          H H' hH'H
-          (Additive.ofMul (ideleClassCongr eK' c)) =
-        numberFieldEmbeddedIdeleClassEquivAmbientFixed
-          K' L' j (Additive.ofMul c) := by
-    apply Subtype.ext
-    change
-      ((rationalIdeleClassEquivFixed (E.restrictScalars ℚ))
-          (Additive.ofMul (ideleClassCongr eK' c))).1 =
-        ((rationalIdeleClassEquivFixed
-            (abstractFixedField ℚ (SeparableClosure ℚ) H'))
-          (Additive.ofMul (ideleClassCongr eK'Base c))).1
-    exact rationalIdeleClassEquivFixed_transport_baseEquiv_val
-      (T := K') (A := E.restrictScalars ℚ)
-      (B := abstractFixedField ℚ (SeparableClosure ℚ) H') hE eK'Base c
-  have hrelative :=
-    rationalAbstractRelativeFixedFieldIdeleClassEquivFixed_relativeNorm
-      H H' hH'H hnormal
-        (Additive.ofMul (ideleClassCongr eK' c))
-  change
-    relativeNorm rationalIdeleClassRepresentation H H' hH'H
-        (rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
-          H H' hH'H
-          (Additive.ofMul (ideleClassCongr eK' c))) =
-      rationalAbstractFixedFieldIdeleClassEquivFixed H
-        (Additive.ofMul
-          (_root_.ideleClassNorm F E
-            (ideleClassCongr eK' c)))
-    at hrelative
-  calc
-    relativeNorm rationalIdeleClassRepresentation H H' hH'H
-        (numberFieldEmbeddedIdeleClassEquivAmbientFixed
-          K' L' j (Additive.ofMul c)) =
-      relativeNorm rationalIdeleClassRepresentation H H' hH'H
-        (rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
-          H H' hH'H (Additive.ofMul (ideleClassCongr eK' c))) :=
-      congrArg (relativeNorm rationalIdeleClassRepresentation H H' hH'H)
-        hupper.symm
-    _ = rationalAbstractFixedFieldIdeleClassEquivFixed H
-        (Additive.ofMul
-          (_root_.ideleClassNorm F E (ideleClassCongr eK' c))) := hrelative
-    _ = numberFieldEmbeddedIdeleClassEquivAmbientFixed
-        K L jLower (Additive.ofMul (_root_.ideleClassNorm K K' c)) := by
-      change
-        rationalAbstractFixedFieldIdeleClassEquivFixed H
-            (Additive.ofMul
-              (_root_.ideleClassNorm F E (ideleClassCongr eK' c))) =
-          rationalAbstractFixedFieldIdeleClassEquivFixed H
-            (Additive.ofMul
-              (ideleClassCongr eK (_root_.ideleClassNorm K K' c)))
-      apply congrArg (rationalAbstractFixedFieldIdeleClassEquivFixed H)
-      apply congrArg Additive.ofMul
-      exact
-        (ideleClassCongr_ideleClassNorm
-          (K := K) (K' := F) (L := K') (L' := E) eK eK' hcompat c).symm
-
-/-- For one common compatible embedding of a Galois base-change
-diamond, the genuine global norm-residue maps commute with ordinary
-idele-class norm and actual restriction of automorphisms. -/
-theorem globalNormResidueMonoidHomOfEmbedding_norm_restriction
-    [FiniteDimensional K L] [IsAbelianGalois K L]
-    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
-    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
-      j.comp (IsScalarTower.toAlgHom ℚ L L')
-    ((AlgEquiv.restrictNormalHom L).comp
-        (AlgEquiv.restrictScalarsHom K)).comp
-        (globalNormResidueMonoidHomOfEmbedding K' L' j) =
-      (globalNormResidueMonoidHomOfEmbedding K L jLower).comp
-        (_root_.ideleClassNorm K K') := by
-  dsimp only
-  let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
-    j.comp (IsScalarTower.toAlgHom ℚ L L')
-  let H :=
-    numberFieldEmbeddedBaseSubgroup K L jLower
-  let H' :=
-    numberFieldEmbeddedBaseSubgroup K' L' j
-  let J :=
-    numberFieldEmbeddedTopSubgroup K L jLower
-  let J' :=
-    numberFieldEmbeddedTopSubgroup K' L' j
-  let hJH :=
-    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L jLower
-  let hJ'H' :=
-    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K' L' j
-  let hH'H :=
-    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
-  let hJ'J :=
-    numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
-  let _ :
-      (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal K L jLower
-  let _ :
-      Finite
-        (H.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H J hJH) :=
-    numberFieldEmbeddedExtensionQuotient_finite K L jLower
-  let _ :
-      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
-    numberFieldEmbeddedExtensionSubgroup_normal K' L' j
-  let _ :
-      Finite
-        (H'.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H' J' hJ'H') :=
-    numberFieldEmbeddedExtensionQuotient_finite K' L' j
-  let hHH'finite :=
-    numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j
-  let T :
-      FiniteAbstractFieldExtension
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
-    { field := numberFieldEmbeddedFiniteAbstractField K' L' j
-      base := numberFieldEmbeddedFiniteAbstractField K L jLower
-      below := hH'H
-      finiteQuotient := hHH'finite }
-  let hTBaseNormal :
-      (CyclicCohomology.extensionSubgroup
-        T.base.field J hJH).Normal := by
-    change
-      (CyclicCohomology.extensionSubgroup H J hJH).Normal
-    exact numberFieldEmbeddedExtensionSubgroup_normal K L jLower
-  let hTBaseFinite :
-      Finite
-        (T.base.field.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup
-            T.base.field J hJH) := by
-    change
-      Finite
-        (H.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H J hJH)
-    exact numberFieldEmbeddedExtensionQuotient_finite K L jLower
-  let hTFieldNormal :
-      (CyclicCohomology.extensionSubgroup
-        T.field.field J' hJ'H').Normal := by
-    change
-      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal
-    exact numberFieldEmbeddedExtensionSubgroup_normal K' L' j
-  let hTFieldFinite :
-      Finite
-        (T.field.field.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup
-            T.field.field J' hJ'H') := by
-    change
-      Finite
-        (H'.toSubgroup ⧸
-          CyclicCohomology.extensionSubgroup H' J' hJ'H')
-    exact numberFieldEmbeddedExtensionQuotient_finite K' L' j
-  let restrictActual :
-      Gal(L' / K') →* Gal(L / K) :=
-    (AlgEquiv.restrictNormalHom L).comp
-      (AlgEquiv.restrictScalarsHom K)
-  apply MonoidHom.ext
-  intro c
-  let a :=
-    numberFieldEmbeddedIdeleClassEquivAmbientFixed
-      K' L' j (Additive.ofMul c)
-  have hnat :=
-    DegreeData.normResidueNaturality_norm_restriction
-      (D := rationalCyclotomicDegreeData)
-      (A := rationalIdeleClassRepresentation)
-      (v := rationalCyclotomicIdeleClassValuationData)
-      (hcf := rationalIdeleClassRepresentation_satisfiesClassFieldAxiom)
-      (T := T) (L := J) (L' := J')
-      (hLnormal := hTBaseNormal)
-      (hL'normal := hTFieldNormal)
-      (hLKfinite := hTBaseFinite)
-      (hL'K'finite := hTFieldFinite)
-      hJH hJ'H' hJ'J
-  have hnatc :=
-    DFunLike.congr_fun hnat
-      (finiteNormClass rationalIdeleClassRepresentation
-        H' J' hJ'H' a)
-  change _ =
-    rationalCyclotomicDegreeData.normResidueSymbol
-      rationalIdeleClassRepresentation
-      rationalCyclotomicIdeleClassValuationData
-      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-      T.base
-      { field := J
-        below := hJH
-        normal := hTBaseNormal
-        finite := hTBaseFinite }
-      (finiteReciprocityNaturalityNormMap
-        rationalIdeleClassRepresentation
-        T.base.field T.field.field J J'
-        hJH hJ'H' T.below hJ'J
-        (finiteNormClass rationalIdeleClassRepresentation
-          T.field.field J' hJ'H' a)) at hnatc
-  rw [finiteReciprocityNaturalityNormMap_finiteNormClass]
-    at hnatc
-  have hnorm :
-      relativeNorm rationalIdeleClassRepresentation
-          H H' hH'H a =
-        numberFieldEmbeddedIdeleClassEquivAmbientFixed
-          K L jLower
-          (Additive.ofMul (_root_.ideleClassNorm K K' c)) :=
-    numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm
-      K K' L L' j c
-  calc
-    restrictActual
-        (globalNormResidueMonoidHomOfEmbedding K' L' j c) =
-      restrictActual
-        (Additive.toMul
-          (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-            K' L' j
-            (rationalCyclotomicDegreeData.normResidueSymbol
-              rationalIdeleClassRepresentation
-              rationalCyclotomicIdeleClassValuationData
-              rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-              (numberFieldEmbeddedFiniteAbstractField K' L' j)
-              (numberFieldEmbeddedFiniteGaloisSubextension K' L' j)
-              (finiteNormClass rationalIdeleClassRepresentation
-                H' J' hJ'H' a)))) := by
-      rw [globalNormResidueMonoidHomOfEmbedding_apply]
-    _ =
-      Additive.toMul
-        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-          K L jLower
-          (MonoidHom.toAdditive
-            (normResidueNaturalityAbelianizedRestriction
-              H H' J J' hJH hJ'H' hH'H hJ'J)
-            (rationalCyclotomicDegreeData.normResidueSymbol
-              rationalIdeleClassRepresentation
-              rationalCyclotomicIdeleClassValuationData
-              rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-              (numberFieldEmbeddedFiniteAbstractField K' L' j)
-              (numberFieldEmbeddedFiniteGaloisSubextension K' L' j)
-              (finiteNormClass rationalIdeleClassRepresentation
-                H' J' hJ'H' a)))) := by
-      exact
-        numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction
-          K K' L L' j _
-    _ =
-      Additive.toMul
-        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-          K L jLower
-          (rationalCyclotomicDegreeData.normResidueSymbol
-            rationalIdeleClassRepresentation
-            rationalCyclotomicIdeleClassValuationData
-            rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
-            (numberFieldEmbeddedFiniteAbstractField K L jLower)
-            (numberFieldEmbeddedFiniteGaloisSubextension K L jLower)
-            (finiteNormClass rationalIdeleClassRepresentation
-              H J hJH
-              (relativeNorm rationalIdeleClassRepresentation
-                H H' hH'H a)))) := by
-      exact congrArg
-        (fun z =>
-          Additive.toMul
-            (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
-              K L jLower z))
-        hnatc
-    _ =
-      globalNormResidueMonoidHomOfEmbedding K L jLower
-        (_root_.ideleClassNorm K K' c) := by
-      rw [hnorm,
-        ← globalNormResidueMonoidHomOfEmbedding_apply]
 
-end EmbeddedNumberFieldRestriction
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
 
 variable
     {G : IntegralRepGroupType} [Group G] [TopologicalSpace G]
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityAbelianValues.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityAbelianValues.lean
new file mode 100644
index 0000000..b6abdd5
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityAbelianValues.lean
@@ -0,0 +1,429 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityExtensionValues
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+section AbstractFixedFieldInclusion
+
+variable
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+  naturalityAbstractFixedFieldBaseQuotientFinite
+  naturalityAbstractFixedFieldRelativeQuotientFinite
+  naturalityAbstractFixedFieldFiniteDimensional
+  naturalityAbstractRelativeFixedFieldFiniteDimensional
+  naturalityAbstractFixedFieldRelativeScalarTower
+  naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional
+  naturalityAbstractFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldIsGalois
+  naturalityAbstractRelativeFixedFieldIsAbelianGalois
+
+
+/-- The additive equivalence from the abelianized extension quotient to the
+Galois group of the relative fixed field, transported through its embedding. -/
+noncomputable def
+    abstractFixedFieldInclusionTransportedAbelianizedEquiv
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    Additive
+        (Abelianization
+          P.toFiniteGaloisExtension.extensionQuotient) ≃+
+      Additive
+        (Gal(
+          (abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) P.below) /
+          (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+    numberFieldEmbeddedFiniteGaloisSubextension F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  have hPEmbedded :
+      Eq.mp
+          (congrArg
+            (fun X : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension X.field)
+            hHEmbedded)
+          PEmbedded =
+        P.toFiniteGaloisExtension :=
+    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
+  exact
+    abelianizedExtensionQuotientAddEquiv_transportExtension hPEmbedded
+      (abelianizedExtensionQuotientAddEquiv_transportBase
+        hHEmbedded PEmbedded
+        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+          F E j))
+
+/-- Canonical abelianization comparison built from the already transported
+opaque extension-quotient endpoint. -/
+private noncomputable def
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    Additive
+        (Abelianization
+          P.toFiniteGaloisExtension.extensionQuotient) ≃+
+      Additive
+        (Gal(
+          (abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) P.below) /
+          (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
+  let Q :=
+    Gal(
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) /
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field))
+  exact
+    MulEquiv.toAdditive
+      ((MulEquiv.abelianizationCongr
+        (abstractFixedFieldInclusionTransportedExtensionQuotientEquiv H P)).trans
+          (Abelianization.equivOfComm : Q ≃* Abelianization Q).symm)
+
+/-- Pointwise opaque value of the transported abelianized equivalence. -/
+private noncomputable def
+    abstractFixedFieldInclusionTransportedAbelianizedValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (z : Additive
+      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
+    Additive
+      (Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
+  abstractFixedFieldInclusionTransportedAbelianizedEquiv H P z
+
+/-- Pointwise opaque value of the canonical abelianization comparison built
+from the transported quotient endpoint. -/
+private noncomputable def
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (z : Additive
+      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
+    Additive
+      (Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
+  abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv H P z
+
+/-- Pointwise opaque value of the intrinsic abstract fixed-field
+abelianization comparison. -/
+private noncomputable def
+    abstractFixedFieldInclusionCanonicalAbelianizedValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (z : Additive
+      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
+    Additive
+      (Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
+  abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P z
+
+/-- The transported abelianized endpoint agrees pointwise with the canonical
+abelianization comparison built from the transported quotient endpoint. -/
+private theorem
+    abstractFixedFieldInclusionTransportedAbelianizedValue_eq_canonical
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (z : Additive
+      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
+    abstractFixedFieldInclusionTransportedAbelianizedValue H P z =
+      abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue
+        H P z := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+    numberFieldEmbeddedFiniteGaloisSubextension F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  have hPEmbedded :
+      Eq.mp
+          (congrArg
+            (fun X : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension X.field)
+            hHEmbedded)
+          PEmbedded =
+        P.toFiniteGaloisExtension :=
+    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
+  let qEmbedded : PEmbedded.extensionQuotient ≃* Gal(E / F) :=
+    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup F E j
+  have hCanonical :
+      abstractFixedFieldInclusionTransportedAbelianizedEquiv H P =
+        abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv
+          H P := by
+    simpa only [
+      abstractFixedFieldInclusionTransportedAbelianizedEquiv,
+      abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv,
+      abstractFixedFieldInclusionTransportedExtensionQuotientEquiv,
+      numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup,
+      qEmbedded] using
+      (abelianizedCanonicalEquiv_transportFiniteGalois
+        hHEmbedded PEmbedded hPEmbedded qEmbedded)
+  exact DFunLike.congr_fun hCanonical z
+
+/-- On an abelianization representative, the transported canonical endpoint
+is the additive value of the transported quotient endpoint. -/
+private theorem
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_of
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient) :
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue H P
+        (Additive.ofMul (Abelianization.of q)) =
+      Additive.ofMul
+        (abstractFixedFieldInclusionTransportedExtensionQuotientValue
+          H P q) := by
+  simpa only [
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue,
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedEquiv,
+    abstractFixedFieldInclusionTransportedExtensionQuotientValue] using
+    (abelianizationCongrToComm_apply
+      (abstractFixedFieldInclusionTransportedExtensionQuotientEquiv H P) q)
+
+/-- On the same representative, the intrinsic fixed-field endpoint is the
+additive value of the canonical quotient endpoint. -/
+private theorem
+    abstractFixedFieldInclusionCanonicalAbelianizedValue_of
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient) :
+    abstractFixedFieldInclusionCanonicalAbelianizedValue H P
+        (Additive.ofMul (Abelianization.of q)) =
+      Additive.ofMul
+        (abstractFixedFieldInclusionCanonicalExtensionQuotientValue
+          H P q) := by
+  let hRawQuotientCommGroup :
+      CommGroup P.toFiniteGaloisExtension.extensionQuotient :=
+    { (inferInstance :
+        Group P.toFiniteGaloisExtension.extensionQuotient) with
+      mul_comm := P.commutative.is_comm.comm }
+  let qAbstractRaw :=
+    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv H P
+  change
+    MulEquiv.toAdditive
+        ((Abelianization.equivOfComm :
+          P.toFiniteGaloisExtension.extensionQuotient ≃*
+            Abelianization
+              P.toFiniteGaloisExtension.extensionQuotient).symm.trans
+          qAbstractRaw)
+        (Additive.ofMul (Abelianization.of q)) =
+      Additive.ofMul (qAbstractRaw q)
+  exact commutativeAbelianizationEquiv_apply qAbstractRaw q
+
+/-- The canonical abelianization comparison built after transport agrees
+pointwise with the intrinsic abstract fixed-field comparison. -/
+private theorem
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_eq
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (z : Additive
+      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
+    abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue H P z =
+      abstractFixedFieldInclusionCanonicalAbelianizedValue H P z := by
+  let q : P.toFiniteGaloisExtension.extensionQuotient :=
+    Quotient.out z.toMul
+  have hz : Additive.ofMul (Abelianization.of q) = z := by
+    apply Additive.ext
+    exact Quotient.out_eq' z.toMul
+  rw [← hz]
+  calc
+    _ = Additive.ofMul
+        (abstractFixedFieldInclusionTransportedExtensionQuotientValue
+          H P q) :=
+      abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_of
+        H P q
+    _ = Additive.ofMul
+        (abstractFixedFieldInclusionEmbeddedExtensionQuotientValue
+          H P q) :=
+      congrArg Additive.ofMul
+        (abstractFixedFieldInclusionTransportedExtensionQuotientEquiv_apply
+          H P q)
+    _ = Additive.ofMul
+        (abstractFixedFieldInclusionCanonicalExtensionQuotientValue
+          H P q) :=
+      congrArg Additive.ofMul
+        (abstractFixedFieldInclusionExtensionQuotientEquiv_apply H P q)
+    _ = _ :=
+      (abstractFixedFieldInclusionCanonicalAbelianizedValue_of H P q).symm
+
+/-- The transported abelianized equivalence agrees pointwise with the
+intrinsic abstract fixed-field Galois comparison. -/
+theorem
+    abstractFixedFieldInclusionTransportedAbelianizedValue_eq
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (z : Additive
+      (Abelianization P.toFiniteGaloisExtension.extensionQuotient)) :
+    abstractFixedFieldInclusionTransportedAbelianizedValue H P z =
+      abstractFixedFieldInclusionCanonicalAbelianizedValue H P z :=
+  (abstractFixedFieldInclusionTransportedAbelianizedValue_eq_canonical
+    H P z).trans
+    (abstractFixedFieldInclusionTransportedCanonicalAbelianizedValue_eq
+      H P z)
+
+
+/-- The norm-residue value of an idele class in the Galois group of the
+relative fixed field, computed through the embedded number-field realization. -/
+noncomputable def
+    abstractFixedFieldInclusionEmbeddedNormResidueValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (c : Additive
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
+    Additive
+      (Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+    numberFieldEmbeddedFiniteGaloisSubextension F E j
+  exact
+    rationalFiniteNormResidueValue HEmbedded PEmbedded
+      (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j)
+      (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+        F E j)
+      c
+
+/-- Opaque packaged value after transporting both dependent structures to
+the canonical `H` and `P` endpoints. -/
+noncomputable def
+    abstractFixedFieldInclusionTransportedNormResidueValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (c : Additive
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
+    Additive
+      (Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  let eIdeleEmbedded :=
+    numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j
+  let eIdeleOverH :
+      Additive (IdeleClassGroup F) ≃+
+        ambientFixedAddSubgroup rationalIdeleClassRepresentation H.field :=
+    Eq.mp
+      (congrArg
+        (fun X : FiniteAbstractField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+          Additive (IdeleClassGroup F) ≃+
+            ambientFixedAddSubgroup
+              rationalIdeleClassRepresentation X.field)
+        hHEmbedded)
+      eIdeleEmbedded
+  exact
+    rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
+      eIdeleOverH
+      (abstractFixedFieldInclusionTransportedAbelianizedEquiv H P)
+      c
+
+/-- Opaque intrinsic packaged norm-residue value at the canonical endpoints. -/
+noncomputable def
+    abstractFixedFieldInclusionCanonicalNormResidueValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (c : Additive
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
+    Additive
+      (Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :=
+  rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
+    (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
+    (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P)
+    c
+
+
+end AbstractFixedFieldInclusion
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityEmbeddedValues.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityEmbeddedValues.lean
new file mode 100644
index 0000000..2f5a677
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityEmbeddedValues.lean
@@ -0,0 +1,396 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityFixedBase
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+section AbstractFixedFieldInclusion
+
+variable
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+  naturalityAbstractFixedFieldBaseQuotientFinite
+  naturalityAbstractFixedFieldRelativeQuotientFinite
+  naturalityAbstractFixedFieldFiniteDimensional
+  naturalityAbstractRelativeFixedFieldFiniteDimensional
+  naturalityAbstractFixedFieldRelativeScalarTower
+  naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional
+  naturalityAbstractFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldIsGalois
+  naturalityAbstractRelativeFixedFieldIsAbelianGalois
+
+/-- The finite abstract field reconstructed from the literal fixed-field
+inclusion is the original packaged abstract field.  Keeping this structure
+equality separate avoids repeatedly rebuilding all of its proof fields. -/
+theorem
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) H.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below
+    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+      E.val.restrictScalars ℚ
+    numberFieldEmbeddedFiniteAbstractField F E j = H := by
+  dsimp only
+  exact FiniteAbstractField.eq_of_field_eq _ _
+    (numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P)
+
+
+theorem
+    numberFieldEmbeddedIdeleClassEquivAmbientFixed_transport_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (c : Additive
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) H.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below
+    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+      E.val.restrictScalars ℚ
+    let HEmbedded :=
+      numberFieldEmbeddedFiniteAbstractField F E j
+    let hHEmbedded : HEmbedded = H :=
+      numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+    Eq.mp
+        (congrArg
+          (fun X : FiniteAbstractField
+              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+            Additive (IdeleClassGroup F) ≃+
+              ambientFixedAddSubgroup
+                rationalIdeleClassRepresentation X.field)
+          hHEmbedded)
+        (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j) c =
+      rationalAbstractFixedFieldIdeleClassEquivFixed H.field c := by
+  dsimp only
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  have hHEmbedded :
+      numberFieldEmbeddedFiniteAbstractField F E j = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  have hFixedBase :
+      abstractFixedField ℚ (SeparableClosure ℚ)
+          (numberFieldEmbeddedBaseSubgroup F E j) = F :=
+    congrArg
+      (abstractFixedField ℚ (SeparableClosure ℚ)) hBase
+  let eBase : F ≃ₐ[ℚ] F :=
+    (numberFieldEmbeddedAbstractBaseFieldEquiv F E j).trans
+      (IntermediateField.equivOfEq hFixedBase)
+  have heBase :
+      eBase = (AlgEquiv.refl : F ≃ₐ[ℚ] F) := by
+    apply AlgEquiv.ext
+    intro x
+    apply Subtype.ext
+    change x.1 = x.1
+    rfl
+  let hEmbeddedQuotientFinite :=
+    numberFieldEmbeddedAbsoluteQuotientFinite F E j
+  let hEmbeddedFixedFiniteDimensional :=
+    numberFieldEmbeddedAbstractFixedFieldFiniteDimensional F E j
+  apply Subtype.ext
+  rw [rationalAmbientFixedAddEquiv_transport_apply_val
+    hHEmbedded
+    (numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j) c]
+  dsimp only [numberFieldEmbeddedIdeleClassEquivAmbientFixed]
+  simp only [AddEquiv.trans_apply]
+  change
+    ((rationalIdeleClassEquivFixed
+        (abstractFixedField ℚ (SeparableClosure ℚ)
+          (numberFieldEmbeddedBaseSubgroup F E j)))
+      (MulEquiv.toAdditive
+        (ideleClassCongr
+          (numberFieldEmbeddedAbstractBaseFieldEquiv F E j)) c)).1 =
+      ((rationalIdeleClassEquivFixed F) c).1
+  exact rationalIdeleClassEquivFixed_congr_apply_val
+    hFixedBase
+    (numberFieldEmbeddedAbstractBaseFieldEquiv F E j)
+    heBase c
+
+/-- Transporting the finite Galois subextension reconstructed from the literal
+fixed-field inclusion recovers the canonical subextension packaged by `P`.
+This is the sole dependent structure equality used by the later quotient
+comparisons. -/
+theorem
+    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) H.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below
+    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+      E.val.restrictScalars ℚ
+    let HEmbedded :=
+      numberFieldEmbeddedFiniteAbstractField F E j
+    let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+      numberFieldEmbeddedFiniteGaloisSubextension F E j
+    let hHEmbedded : HEmbedded = H :=
+      numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+    Eq.mp
+        (congrArg
+          (fun X : FiniteAbstractField
+              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+            FiniteGaloisSubextension X.field)
+          hHEmbedded)
+        PEmbedded =
+      P.toFiniteGaloisExtension := by
+  dsimp only
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+    numberFieldEmbeddedFiniteGaloisSubextension F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  exact finiteGaloisSubextension_transport_eq_of_field_eq
+    hHEmbedded PEmbedded P.toFiniteGaloisExtension
+    (numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P)
+
+/-- Opaque endpoint for the extension-quotient comparison supplied by the
+literal embedding.  Its domain is already the canonical quotient of `P`, so
+no client has to reconstruct the two subgroup transports. -/
+noncomputable def
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    P.toFiniteGaloisExtension.extensionQuotient ≃*
+      Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  have hTop :
+      numberFieldEmbeddedTopSubgroup F E j = P.field :=
+    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P
+  letI hAlgebra : Algebra F (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra F E j
+  let eSep :=
+    numberFieldEmbeddedSeparableClosureEquiv F E j
+  letI hEmbeddedExtensionNormal :
+      (CyclicCohomology.extensionSubgroup
+        (numberFieldEmbeddedBaseSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal F E j
+  exact
+    P.toFiniteGaloisExtension.extensionQuotientMulEquiv.trans
+      ((extensionQuotientMulEquivOfEq
+        hBase.symm hTop.symm P.below
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).trans
+      (ambientEmbeddedExtensionQuotientEquivGaloisGroup
+        ℚ F E j eSep))
+
+/-- Opaque canonical endpoint for the same quotient, obtained directly from
+the abstract fixed-field realization. -/
+noncomputable def
+    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    P.toFiniteGaloisExtension.extensionQuotient ≃*
+      Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
+  exact
+    P.toFiniteGaloisExtension.extensionQuotientMulEquiv.trans
+      (abstractExtensionQuotientEquivGaloisGroup
+        ℚ (SeparableClosure ℚ)
+        H.field P.field P.below P.normal)
+
+/-- Pointwise opaque endpoint of the embedded quotient equivalence. -/
+noncomputable def
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient) :
+    Gal(
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) /
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :=
+  abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv H P q
+
+/-- Pointwise opaque endpoint of the canonical quotient equivalence. -/
+noncomputable def
+    abstractFixedFieldInclusionCanonicalExtensionQuotientValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient) :
+    Gal(
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) /
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :=
+  abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv H P q
+
+/-- Fully applied ambient value of the embedded quotient endpoint. -/
+noncomputable def
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ :=
+  (abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q x :
+    SeparableClosure ℚ)
+
+/-- Fully applied ambient value of the canonical quotient endpoint. -/
+noncomputable def
+    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ :=
+  (abstractFixedFieldInclusionCanonicalExtensionQuotientValue H P q x :
+    SeparableClosure ℚ)
+
+/-- The ambient Galois value attached to a representative of the canonical
+quotient, packaged behind a literal result type. -/
+noncomputable def
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup) :
+    Gal(
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) /
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  letI hAlgebra : Algebra F (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra F E j
+  let eSep :=
+    numberFieldEmbeddedSeparableClosureEquiv F E j
+  letI hEmbeddedExtensionNormal :
+      (CyclicCohomology.extensionSubgroup
+        (numberFieldEmbeddedBaseSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal F E j
+  let σEmbedded :
+      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
+    (MulEquiv.subgroupCongr
+      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
+  exact
+    ambientEmbeddedExtensionQuotientEquivGaloisGroup
+      ℚ F E j eSep (QuotientGroup.mk σEmbedded)
+
+/-- Fully applied ambient value of the packaged representative endpoint. -/
+noncomputable def
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ :=
+  (abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue H P σ x :
+    SeparableClosure ℚ)
+
+/-- Ambient action of the representative after rebundling it in the embedded
+base subgroup. -/
+noncomputable def
+    abstractFixedFieldInclusionRebasedAutomorphismApplyVal
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) : SeparableClosure ℚ := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  let σEmbedded :
+      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
+    (MulEquiv.subgroupCongr
+      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
+  exact σEmbedded.1.1 (x : SeparableClosure ℚ)
+
+
+end AbstractFixedFieldInclusion
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityExtensionValues.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityExtensionValues.lean
new file mode 100644
index 0000000..def7569
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityExtensionValues.lean
@@ -0,0 +1,419 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityEmbeddedValues
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+section AbstractFixedFieldInclusion
+
+variable
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+  naturalityAbstractFixedFieldBaseQuotientFinite
+  naturalityAbstractFixedFieldRelativeQuotientFinite
+  naturalityAbstractFixedFieldFiniteDimensional
+  naturalityAbstractRelativeFixedFieldFiniteDimensional
+  naturalityAbstractFixedFieldRelativeScalarTower
+  naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional
+  naturalityAbstractFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldIsGalois
+  naturalityAbstractRelativeFixedFieldIsAbelianGalois
+
+/-- The embedded quotient endpoint sends a canonical representative to the
+packaged ambient value above. -/
+private theorem
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue_mk
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup) :
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P
+        (P.toFiniteGaloisExtension.extensionQuotientMk σ) =
+      abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue H P σ := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  have hTop :
+      numberFieldEmbeddedTopSubgroup F E j = P.field :=
+    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P
+  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra F E j
+  let eSep :=
+    numberFieldEmbeddedSeparableClosureEquiv F E j
+  let hEmbeddedExtensionNormal :
+      (CyclicCohomology.extensionSubgroup
+        (numberFieldEmbeddedBaseSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal F E j
+  simp only [
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue,
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv,
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue,
+    MulEquiv.trans_apply,
+    FiniteGaloisSubextension.extensionQuotientMk_apply]
+  exact congrArg
+    (ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ F E j eSep)
+    (extensionQuotientMulEquivOfEq_mk
+      hBase.symm hTop.symm P.below
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j) σ)
+
+/-- The packaged ambient endpoint evaluates to the action of the rebundled
+representative. -/
+private theorem
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal_eq_rebased
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) :
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal H P σ x =
+      abstractFixedFieldInclusionRebasedAutomorphismApplyVal H P σ x := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra F E j
+  let eSep :=
+    numberFieldEmbeddedSeparableClosureEquiv F E j
+  let hEmbeddedExtensionNormal :
+      (CyclicCohomology.extensionSubgroup
+        (numberFieldEmbeddedBaseSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal F E j
+  let σEmbedded :
+      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
+    (MulEquiv.subgroupCongr
+      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
+  have hmk :=
+    ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
+      ℚ F E j eSep σEmbedded x
+  change
+    (ambientEmbeddedExtensionQuotientEquivGaloisGroup
+        ℚ F E j eSep (QuotientGroup.mk σEmbedded) x :
+        SeparableClosure ℚ) =
+      σEmbedded.1.1 (x : SeparableClosure ℚ) at hmk
+  simpa only [
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal,
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue,
+    abstractFixedFieldInclusionRebasedAutomorphismApplyVal] using hmk
+
+/-- Rebundling the representative does not change its action in the ambient
+separable closure. -/
+private theorem
+    abstractFixedFieldInclusionRebasedAutomorphismApplyVal_eq
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) :
+    abstractFixedFieldInclusionRebasedAutomorphismApplyVal H P σ x =
+      σ.1.1 (x : SeparableClosure ℚ) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  let σEmbedded :
+      (numberFieldEmbeddedBaseSubgroup F E j).toSubgroup :=
+    (MulEquiv.subgroupCongr
+      (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ
+  have hσEmbedded :
+      (σEmbedded.1 : SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) = σ.1 :=
+    closedSubgroupCongr_apply_val hBase.symm σ
+  change σEmbedded.1.1 (x : SeparableClosure ℚ) = _
+  rw [hσEmbedded]
+
+/-- The packaged ambient representative acts by the original automorphism on
+the underlying separable-closure value. -/
+private theorem
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) :
+    abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal H P σ x =
+      σ.1.1 (x : SeparableClosure ℚ) := by
+  exact
+    (abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal_eq_rebased
+      H P σ x).trans
+      (abstractFixedFieldInclusionRebasedAutomorphismApplyVal_eq H P σ x)
+
+/-- Evaluation of the embedded quotient endpoint on a canonical quotient
+representative, stated only in the ambient separable closure. -/
+private theorem
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv_mk_val
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) :
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientApplyVal H P
+        (P.toFiniteGaloisExtension.extensionQuotientMk σ) x =
+      σ.1.1 (x : SeparableClosure ℚ) := by
+  calc
+    _ = abstractFixedFieldInclusionAmbientEmbeddedQuotientMkApplyVal
+        H P σ x := by
+      exact congrArg
+        (fun g => (g x : SeparableClosure ℚ))
+        (abstractFixedFieldInclusionEmbeddedExtensionQuotientValue_mk
+          H P σ)
+    _ = _ :=
+      abstractFixedFieldInclusionAmbientEmbeddedQuotientMkValue_apply
+        H P σ x
+
+/-- Evaluation of the canonical abstract quotient endpoint on a quotient
+representative, again exposed only through its ambient value. -/
+private theorem
+    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv_mk_val
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup)
+    (x : abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) :
+    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal H P
+        (P.toFiniteGaloisExtension.extensionQuotientMk σ) x =
+      σ.1.1 (x : SeparableClosure ℚ) := by
+  simp only [
+    abstractFixedFieldInclusionCanonicalExtensionQuotientApplyVal,
+    abstractFixedFieldInclusionCanonicalExtensionQuotientValue,
+    abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv,
+    MulEquiv.trans_apply,
+    FiniteGaloisSubextension.extensionQuotientMk_apply]
+  exact
+    (abstractExtensionQuotientEquivGaloisGroup_mk_apply_val
+      ℚ (SeparableClosure ℚ)
+      H.field P.field P.below P.normal σ x).symm
+
+
+theorem
+    abstractFixedFieldInclusionExtensionQuotientEquiv_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient) :
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q =
+      abstractFixedFieldInclusionCanonicalExtensionQuotientValue H P q := by
+  refine P.toFiniteGaloisExtension.extensionQuotient_inductionOn
+    (motive := fun q =>
+      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q =
+        abstractFixedFieldInclusionCanonicalExtensionQuotientValue H P q)
+    q ?_
+  intro σ
+  apply AlgEquiv.ext
+  intro x
+  apply Subtype.ext
+  exact
+    (abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv_mk_val
+      H P σ x).trans
+      (abstractFixedFieldInclusionCanonicalExtensionQuotientEquiv_mk_val
+        H P σ x).symm
+
+
+/-- The extension quotient identified with the Galois group of the corresponding
+relative fixed field, after transporting the embedded field structures. -/
+noncomputable def
+    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    P.toFiniteGaloisExtension.extensionQuotient ≃*
+      Gal(
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below) /
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+    numberFieldEmbeddedFiniteGaloisSubextension F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  have hPEmbedded :
+      Eq.mp
+          (congrArg
+            (fun X : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension X.field)
+            hHEmbedded)
+          PEmbedded =
+        P.toFiniteGaloisExtension :=
+    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
+  exact
+    (extensionQuotientMulEquiv_transportFiniteGalois
+      hHEmbedded PEmbedded hPEmbedded).trans
+      (numberFieldEmbeddedExtensionQuotientEquivGaloisGroup F E j)
+
+/-- Pointwise opaque endpoint of the transported quotient equivalence. -/
+noncomputable def
+    abstractFixedFieldInclusionTransportedExtensionQuotientValue
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient) :
+    Gal(
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) /
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :=
+  abstractFixedFieldInclusionTransportedExtensionQuotientEquiv H P q
+
+
+private theorem
+    abstractFixedFieldInclusionTransportedExtensionQuotientValue_mk
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (σ : H.field.toSubgroup) :
+    abstractFixedFieldInclusionTransportedExtensionQuotientValue H P
+        (P.toFiniteGaloisExtension.extensionQuotientMk σ) =
+      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P
+        (P.toFiniteGaloisExtension.extensionQuotientMk σ) := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hBase :
+      numberFieldEmbeddedBaseSubgroup F E j = H.field :=
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion H P
+  have hTop :
+      numberFieldEmbeddedTopSubgroup F E j = P.field :=
+    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion H P
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+    numberFieldEmbeddedFiniteGaloisSubextension F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  have hPEmbedded :
+      Eq.mp
+          (congrArg
+            (fun X : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension X.field)
+            hHEmbedded)
+          PEmbedded =
+        P.toFiniteGaloisExtension :=
+    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
+  let hAlgebra : Algebra F (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra F E j
+  let eSep :=
+    numberFieldEmbeddedSeparableClosureEquiv F E j
+  let hEmbeddedExtensionNormal :
+      (CyclicCohomology.extensionSubgroup
+        (numberFieldEmbeddedBaseSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup F E j)
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j)).Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal F E j
+  have hFieldEq :
+      (congrArg FiniteAbstractField.field hHEmbedded).symm = hBase.symm :=
+    Subsingleton.elim _ _
+  simp only [
+    abstractFixedFieldInclusionTransportedExtensionQuotientValue,
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientValue,
+    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv,
+    abstractFixedFieldInclusionEmbeddedExtensionQuotientEquiv,
+    MulEquiv.trans_apply,
+    extensionQuotientMulEquiv_transportFiniteGalois_mk,
+    FiniteGaloisSubextension.extensionQuotientMk_apply,
+    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup]
+  rw [hFieldEq]
+  change
+    ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ F E j eSep
+        (QuotientGroup.mk
+          ((MulEquiv.subgroupCongr
+            (congrArg ClosedSubgroup.toSubgroup hBase.symm)) σ)) = _
+  exact
+    (congrArg
+      (ambientEmbeddedExtensionQuotientEquivGaloisGroup ℚ F E j eSep)
+      (extensionQuotientMulEquivOfEq_mk
+        hBase.symm hTop.symm P.below
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup F E j) σ)).symm
+
+
+theorem
+    abstractFixedFieldInclusionTransportedExtensionQuotientEquiv_apply
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (q : P.toFiniteGaloisExtension.extensionQuotient) :
+    abstractFixedFieldInclusionTransportedExtensionQuotientValue H P q =
+      abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q := by
+  refine P.toFiniteGaloisExtension.extensionQuotient_inductionOn
+    (motive := fun q =>
+      abstractFixedFieldInclusionTransportedExtensionQuotientValue H P q =
+        abstractFixedFieldInclusionEmbeddedExtensionQuotientValue H P q)
+    q ?_
+  intro σ
+  exact abstractFixedFieldInclusionTransportedExtensionQuotientValue_mk
+    H P σ
+
+
+end AbstractFixedFieldInclusion
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityFixedBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityFixedBase.lean
new file mode 100644
index 0000000..d3a9bb0
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityFixedBase.lean
@@ -0,0 +1,376 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityTransports
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+
+section AbstractFixedFieldInclusion
+
+variable
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+
+local instance naturalityAbstractFixedFieldBaseQuotientFinite :
+    Finite
+      ((baseField
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+          H.field (le_baseField H.field)) :=
+  H.finite
+
+local instance naturalityAbstractFixedFieldRelativeQuotientFinite :
+    Finite
+      (H.field.toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup H.field P.field P.below) :=
+  P.finite
+
+noncomputable local instance
+    naturalityAbstractFixedFieldFiniteDimensional :
+    FiniteDimensional ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
+  abstractFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ) H.field H.finite
+
+noncomputable local instance
+    naturalityAbstractRelativeFixedFieldFiniteDimensional :
+    FiniteDimensional
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) :=
+  abstractRelativeFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ)
+    H.field P.field P.below H.finite P.finite
+
+local instance naturalityAbstractFixedFieldRelativeScalarTower :
+    IsScalarTower ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) :=
+  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
+
+noncomputable local instance
+    naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional :
+    FiniteDimensional ℚ
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) :=
+  FiniteDimensional.trans ℚ
+    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+    (abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below)
+
+noncomputable local instance naturalityAbstractFixedFieldNumberField :
+    NumberField
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field) :=
+  NumberField.of_module_finite ℚ
+    (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+
+noncomputable local instance
+    naturalityAbstractRelativeFixedFieldNumberField :
+    NumberField
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) :=
+  NumberField.of_module_finite ℚ
+    (abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below)
+
+/-- Use the same direct fixed-field Galois witness as the intrinsic
+norm-residue construction.  This prevents the dependent Galois-group type
+from being synthesized through a second `IsAbelianGalois` instance path. -/
+noncomputable local instance
+    naturalityAbstractRelativeFixedFieldIsGalois :
+    IsGalois
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+      (abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below) :=
+  abstractRelativeFixedField_isGalois
+    ℚ (SeparableClosure ℚ)
+    H.field P.field P.below P.normal
+
+noncomputable local instance
+    naturalityAbstractRelativeFixedFieldIsAbelianGalois :
+    IsAbelianGalois
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+      (abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below) :=
+  finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
+
+/-- The lower subgroup obtained from the canonical inclusion of an abstract
+fixed-field tower is the original lower closed subgroup. -/
+theorem
+    numberFieldEmbeddedBaseSubgroup_abstractFixedFieldInclusion
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) H.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below
+    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+      E.val.restrictScalars ℚ
+    numberFieldEmbeddedBaseSubgroup F E j = H.field := by
+  dsimp only
+  have hi :
+      numberFieldEmbeddedLowerEmbedding
+          (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+          (abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) P.below)
+          ((abstractRelativeFixedField
+            ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ) =
+        (abstractFixedField
+          ℚ (SeparableClosure ℚ) H.field).val := by
+    ext x
+    rfl
+  have hRange :
+      (numberFieldEmbeddedLowerEmbedding
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below)
+        ((abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ)).fieldRange =
+        abstractFixedField ℚ (SeparableClosure ℚ) H.field := by
+    ext x
+    constructor
+    · rintro ⟨y, rfl⟩
+      change
+        numberFieldEmbeddedLowerEmbedding
+            (abstractFixedField ℚ (SeparableClosure ℚ) H.field)
+            (abstractRelativeFixedField
+              ℚ (SeparableClosure ℚ) P.below)
+            ((abstractRelativeFixedField
+              ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ) y ∈
+          abstractFixedField ℚ (SeparableClosure ℚ) H.field
+      rw [hi]
+      exact y.property
+    · intro hx
+      refine ⟨⟨x, hx⟩, ?_⟩
+      rw [hi]
+      rfl
+  rw [numberFieldEmbeddedBaseSubgroup, hRange]
+  exact
+    closedFixingSubgroup_abstractFixedField_eq
+      ℚ (SeparableClosure ℚ) H.field
+
+/-- The upper subgroup obtained from the canonical inclusion of an abstract
+fixed-field tower is the original upper closed subgroup. -/
+theorem
+    numberFieldEmbeddedTopSubgroup_abstractFixedFieldInclusion
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) H.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below
+    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+      E.val.restrictScalars ℚ
+    numberFieldEmbeddedTopSubgroup F E j = P.field := by
+  dsimp only
+  rw [numberFieldEmbeddedTopSubgroup]
+  have hjRangeSelf :
+      ((abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ).fieldRange =
+        (abstractRelativeFixedField
+          ℚ (SeparableClosure ℚ) P.below).restrictScalars ℚ := by
+    ext x
+    constructor
+    · rintro ⟨y, rfl⟩
+      exact y.property
+    · intro hx
+      exact ⟨⟨x, hx⟩, rfl⟩
+  have hjRange :
+      ((abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below).val.restrictScalars ℚ).fieldRange =
+        abstractFixedField ℚ (SeparableClosure ℚ) P.field := by
+    exact hjRangeSelf.trans
+      (IntermediateField.extendScalars_restrictScalars
+        (abstractFixedField_le
+          ℚ (SeparableClosure ℚ) P.below))
+  rw [hjRange]
+  exact
+    closedFixingSubgroup_abstractFixedField_eq
+      ℚ (SeparableClosure ℚ) P.field
+
+/-- Transport a packaged rational norm-residue value directly from an equal
+abstract base to the finite Galois extension underlying `P`.  Keeping the two
+dependent transports in their own declaration prevents their elaboration cost
+from accumulating in the main fixed-field comparison theorem. -/
+theorem rationalFiniteNormResidueValue_transportToAbstractExtension
+    {A : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
+    (hAH : A = H)
+    (L : FiniteGaloisSubextension A.field)
+    (hLP :
+      Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension Y.field)
+            hAH)
+          L =
+        P.toFiniteGaloisExtension)
+    {C X : Type} [AddGroup C] [AddGroup X]
+    (eIdele : C ≃+
+      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
+    (eGalois : Additive (Abelianization L.extensionQuotient) ≃+ X)
+    (c : C) :
+    rationalFiniteNormResidueValue A L eIdele eGalois c =
+      rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
+        (Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              C ≃+ ambientFixedAddSubgroup
+                rationalIdeleClassRepresentation Y.field)
+            hAH)
+          eIdele)
+        (abelianizedExtensionQuotientAddEquiv_transportExtension hLP
+          (abelianizedExtensionQuotientAddEquiv_transportBase
+            hAH L eGalois))
+        c := by
+  calc
+    _ = rationalFiniteNormResidueValue H
+        (Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension Y.field)
+            hAH)
+          L)
+        (Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              C ≃+ ambientFixedAddSubgroup
+                rationalIdeleClassRepresentation Y.field)
+            hAH)
+          eIdele)
+        (abelianizedExtensionQuotientAddEquiv_transportBase hAH L eGalois)
+        c :=
+      rationalFiniteNormResidueValue_transportBase
+        (A := A) (B := H) (C := C) (X := X)
+        hAH L eIdele eGalois c
+    _ = _ :=
+      rationalFiniteNormResidueValue_transportExtension
+        (K := H)
+        (P := Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension Y.field)
+            hAH)
+          L)
+        (Q := P.toFiniteGaloisExtension) (C := C) (X := X)
+        hLP
+        (Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              C ≃+ ambientFixedAddSubgroup
+                rationalIdeleClassRepresentation Y.field)
+            hAH)
+          eIdele)
+        (abelianizedExtensionQuotientAddEquiv_transportBase hAH L eGalois)
+        c
+
+/-- The packaged value at the literal fixed-field realization is the ambient
+fixed-part norm-residue homomorphism evaluated at the same idele class. -/
+private theorem rationalFiniteNormResidueValue_abstractFixedField_eq_ambient
+    (c : IdeleClassGroup
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :
+    rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
+        (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
+        (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P)
+        (Additive.ofMul c) =
+      ambientFixedGlobalNormResidueAddMonoidHom H P
+        (rationalAbstractFixedFieldIdeleClassEquivFixed
+          H.field (Additive.ofMul c)) := by
+  let eRec :=
+    rationalCyclotomicDegreeData.normResidueSymbol
+      rationalIdeleClassRepresentation
+      rationalCyclotomicIdeleClassValuationData
+      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+      H P.toFiniteGaloisExtension
+  let eGal :=
+    abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P
+  change
+    eGal
+        (eRec
+          (finiteNormClass rationalIdeleClassRepresentation
+            H.field P.field P.below
+            (rationalAbstractFixedFieldIdeleClassEquivFixed
+              H.field (Additive.ofMul c)))) =
+      eGal
+        (eRec
+          (finiteNormClass rationalIdeleClassRepresentation
+            H.field P.field P.below
+            (rationalAbstractFixedFieldIdeleClassEquivFixed
+              H.field (Additive.ofMul c))))
+  rfl
+
+/-- The packaged norm-residue value for the literal fixed-field realization
+is the intrinsic abstract fixed-field norm-residue value. -/
+theorem rationalFiniteNormResidueValue_abstractFixedField_apply
+    (c : IdeleClassGroup
+      (abstractFixedField ℚ (SeparableClosure ℚ) H.field)) :
+    Additive.toMul
+        (rationalFiniteNormResidueValue H P.toFiniteGaloisExtension
+          (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
+          (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup
+            H P)
+          (Additive.ofMul c)) =
+      abstractFixedFieldGlobalNormResidueMonoidHom H P c := by
+  let a :=
+    rationalAbstractFixedFieldIdeleClassEquivFixed
+      H.field (Additive.ofMul c)
+  have hAbstract :=
+    abstractFixedFieldGlobalNormResidueMonoidHom_fixed_apply H P a
+  calc
+    _ = Additive.toMul
+        (ambientFixedGlobalNormResidueAddMonoidHom H P
+          (rationalAbstractFixedFieldIdeleClassEquivFixed
+            H.field (Additive.ofMul c))) := by
+      exact congrArg Additive.toMul
+        (rationalFiniteNormResidueValue_abstractFixedField_eq_ambient
+          (H := H) (P := P) c)
+    _ = _ := by
+      simpa only [a, AddEquiv.symm_apply_apply, toMul_ofMul] using
+        hAbstract.symm
+
+
+end AbstractFixedFieldInclusion
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityFixedComparison.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityFixedComparison.lean
new file mode 100644
index 0000000..ce523c2
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityFixedComparison.lean
@@ -0,0 +1,237 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityAbelianValues
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+section AbstractFixedFieldInclusion
+
+variable
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+  naturalityAbstractFixedFieldBaseQuotientFinite
+  naturalityAbstractFixedFieldRelativeQuotientFinite
+  naturalityAbstractFixedFieldFiniteDimensional
+  naturalityAbstractRelativeFixedFieldFiniteDimensional
+  naturalityAbstractFixedFieldRelativeScalarTower
+  naturalityAbstractRelativeFixedFieldAbsoluteFiniteDimensional
+  naturalityAbstractFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldNumberField
+  naturalityAbstractRelativeFixedFieldIsGalois
+  naturalityAbstractRelativeFixedFieldIsAbelianGalois
+
+
+private theorem
+    abstractFixedFieldInclusionEmbeddedNormResidueValue_eq_transported
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (c : Additive
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
+    abstractFixedFieldInclusionEmbeddedNormResidueValue H P c =
+      abstractFixedFieldInclusionTransportedNormResidueValue H P c := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  let PEmbedded : FiniteGaloisSubextension HEmbedded.field :=
+    numberFieldEmbeddedFiniteGaloisSubextension F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  have hPEmbedded :
+      Eq.mp
+          (congrArg
+            (fun X : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension X.field)
+            hHEmbedded)
+          PEmbedded =
+        P.toFiniteGaloisExtension :=
+    numberFieldEmbeddedFiniteGaloisSubextension_transport_eq H P
+  let eIdeleEmbedded :=
+    numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j
+  let eGaloisEmbedded :=
+    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup F E j
+  simpa only [
+    abstractFixedFieldInclusionEmbeddedNormResidueValue,
+    abstractFixedFieldInclusionTransportedNormResidueValue,
+    abstractFixedFieldInclusionTransportedAbelianizedEquiv,
+    eIdeleEmbedded,
+    eGaloisEmbedded] using
+    (rationalFiniteNormResidueValue_transportToAbstractExtension
+      (H := H) (P := P)
+      (A := HEmbedded)
+      (C := Additive (IdeleClassGroup F))
+      (X := Additive Gal(E / F))
+      hHEmbedded PEmbedded hPEmbedded
+      eIdeleEmbedded eGaloisEmbedded c)
+
+/-- A packaged finite norm-residue value depends only on the value of its
+idele comparison at the chosen input and the value of its Galois comparison
+at the resulting norm class. -/
+private theorem rationalFiniteNormResidueValue_congr_apply
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (L : FiniteGaloisSubextension K.field)
+    {C X : Type} [AddGroup C] [AddGroup X]
+    (eIdele eIdele' : C ≃+
+      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
+    (eGalois eGalois' :
+      Additive (Abelianization L.extensionQuotient) ≃+ X)
+    (c : C)
+    (heIdele : eIdele c = eIdele' c)
+    (heGalois : ∀ z, eGalois z = eGalois' z) :
+    rationalFiniteNormResidueValue K L eIdele eGalois c =
+      rationalFiniteNormResidueValue K L eIdele' eGalois' c := by
+  unfold rationalFiniteNormResidueValue
+  rw [heIdele]
+  exact heGalois _
+
+/-- The transported packaged value is the intrinsic canonical packaged value;
+only the idele input and the eventual abelianized value are compared. -/
+private theorem
+    abstractFixedFieldInclusionTransportedNormResidueValue_eq_canonical
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (c : Additive
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
+    abstractFixedFieldInclusionTransportedNormResidueValue H P c =
+      abstractFixedFieldInclusionCanonicalNormResidueValue H P c := by
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H.field
+  let E :=
+    abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  let HEmbedded :=
+    numberFieldEmbeddedFiniteAbstractField F E j
+  have hHEmbedded : HEmbedded = H :=
+    numberFieldEmbeddedFiniteAbstractField_abstractFixedFieldInclusion H P
+  let eIdeleEmbedded :=
+    numberFieldEmbeddedIdeleClassEquivAmbientFixed F E j
+  let eIdeleOverH :
+      Additive (IdeleClassGroup F) ≃+
+        ambientFixedAddSubgroup rationalIdeleClassRepresentation H.field :=
+    Eq.mp
+      (congrArg
+        (fun X : FiniteAbstractField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+          Additive (IdeleClassGroup F) ≃+
+            ambientFixedAddSubgroup
+              rationalIdeleClassRepresentation X.field)
+        hHEmbedded)
+      eIdeleEmbedded
+  simpa only [
+    abstractFixedFieldInclusionTransportedNormResidueValue,
+    abstractFixedFieldInclusionCanonicalNormResidueValue,
+    eIdeleEmbedded,
+    eIdeleOverH] using
+    (rationalFiniteNormResidueValue_congr_apply
+      H P.toFiniteGaloisExtension
+      eIdeleOverH
+      (rationalAbstractFixedFieldIdeleClassEquivFixed H.field)
+      (abstractFixedFieldInclusionTransportedAbelianizedEquiv H P)
+      (abstractFixedFieldAbelianizedExtensionQuotientEquivGaloisGroup H P)
+      c
+      (numberFieldEmbeddedIdeleClassEquivAmbientFixed_transport_apply H P c)
+      (fun z => abstractFixedFieldInclusionTransportedAbelianizedValue_eq
+        H P z))
+
+/-- The explicitly embedded and intrinsic packaged norm-residue values agree. -/
+private theorem
+    abstractFixedFieldInclusionEmbeddedNormResidueValue_eq
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field)
+    (c : Additive
+      (IdeleClassGroup
+        (abstractFixedField ℚ (SeparableClosure ℚ) H.field))) :
+    abstractFixedFieldInclusionEmbeddedNormResidueValue H P c =
+      abstractFixedFieldInclusionCanonicalNormResidueValue H P c :=
+  (abstractFixedFieldInclusionEmbeddedNormResidueValue_eq_transported
+    H P c).trans
+    (abstractFixedFieldInclusionTransportedNormResidueValue_eq_canonical
+      H P c)
+
+/-- For the literal fixed fields attached to an abstract finite abelian
+extension, the norm-residue map obtained from their canonical inclusion in
+the rational separable closure is the intrinsic fixed-field norm-residue
+map. -/
+theorem
+    globalNormResidueMonoidHomOfEmbedding_abstractFixedFieldInclusion
+    (H : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (P : FiniteAbelianSubextension H.field) :
+    let F :=
+      abstractFixedField ℚ (SeparableClosure ℚ) H.field
+    let E :=
+      abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below
+    let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+      E.val.restrictScalars ℚ
+    globalNormResidueMonoidHomOfEmbedding F E j =
+      abstractFixedFieldGlobalNormResidueMonoidHom H P := by
+  dsimp only
+  apply MonoidHom.ext
+  intro c
+  rw [globalNormResidueMonoidHomOfEmbedding_apply]
+  change
+    Additive.toMul
+        (abstractFixedFieldInclusionEmbeddedNormResidueValue
+          H P (Additive.ofMul c)) =
+      abstractFixedFieldGlobalNormResidueMonoidHom H P c
+  calc
+    _ = Additive.toMul
+        (abstractFixedFieldInclusionCanonicalNormResidueValue
+          H P (Additive.ofMul c)) :=
+      congrArg Additive.toMul
+        (abstractFixedFieldInclusionEmbeddedNormResidueValue_eq
+          H P (Additive.ofMul c))
+    _ = _ := by
+      simpa only [abstractFixedFieldInclusionCanonicalNormResidueValue] using
+        (rationalFiniteNormResidueValue_abstractFixedField_apply
+          (H := H) (P := P) c)
+
+end AbstractFixedFieldInclusion
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerBase.lean
new file mode 100644
index 0000000..dcad422
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerBase.lean
@@ -0,0 +1,339 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityFixedComparison
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+
+section EmbeddedNumberFieldRestriction
+
+variable
+    (K K' L L' : Type)
+    [Field K] [NumberField K]
+    [Field K'] [NumberField K']
+    [Field L] [NumberField L]
+    [Field L'] [NumberField L']
+    [Algebra K K'] [Algebra K L] [Algebra K L']
+    [Algebra K' L'] [Algebra L L']
+    [IsScalarTower K K' L'] [IsScalarTower K L L']
+
+
+/-- Rebracketing the compatible tower does not change its embedded lower
+fixing subgroup. -/
+private theorem numberFieldEmbeddedBaseSubgroup_baseChange_eq
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    numberFieldEmbeddedBaseSubgroup K K'
+        (numberFieldEmbeddedLowerEmbedding K' L' j) =
+      numberFieldEmbeddedBaseSubgroup K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L')) := by
+  have hi :
+      numberFieldEmbeddedLowerEmbedding K K'
+          (numberFieldEmbeddedLowerEmbedding K' L' j) =
+        numberFieldEmbeddedLowerEmbedding K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L')) := by
+    ext x
+    simp only [numberFieldEmbeddedLowerEmbedding, AlgHom.comp_apply,
+      IsScalarTower.coe_toAlgHom']
+    rw [← IsScalarTower.algebraMap_apply K K' L',
+      ← IsScalarTower.algebraMap_apply K L L']
+  simp only [numberFieldEmbeddedBaseSubgroup, hi]
+
+omit [Field K] [NumberField K]
+    [Algebra K K'] [Algebra K L'] [IsScalarTower K K' L'] in
+/-- The top subgroup of the rebracketed base-change tower is the embedded
+fixing subgroup of the intermediate field. -/
+private theorem numberFieldEmbeddedTopSubgroup_baseChange_eq
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    numberFieldEmbeddedTopSubgroup K K'
+        (numberFieldEmbeddedLowerEmbedding K' L' j) =
+      numberFieldEmbeddedBaseSubgroup K' L' j := by
+  rfl
+
+/-- Reidentify the two presentations of the embedded lower fixing subgroup
+without transporting dependent subgroup data through an equality. -/
+private noncomputable def numberFieldEmbeddedBaseChangeBaseEquiv
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    (numberFieldEmbeddedBaseSubgroup K K'
+        (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup ≃*
+      (numberFieldEmbeddedBaseSubgroup K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup :=
+  MulEquiv.subgroupCongr
+    (congrArg ClosedSubgroup.toSubgroup
+      (numberFieldEmbeddedBaseSubgroup_baseChange_eq K K' L L' j))
+
+/-- Under the identity equivalence of the two lower fixing subgroups, the
+relative subgroup for the base change is exactly the target presentation. -/
+private theorem numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    (CyclicCohomology.extensionSubgroup
+        (numberFieldEmbeddedBaseSubgroup K K'
+          (numberFieldEmbeddedLowerEmbedding K' L' j))
+        (numberFieldEmbeddedTopSubgroup K K'
+          (numberFieldEmbeddedLowerEmbedding K' L' j))
+        (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K K'
+          (numberFieldEmbeddedLowerEmbedding K' L' j))).map
+          (numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j).toMonoidHom =
+      CyclicCohomology.extensionSubgroup
+        (numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+        (numberFieldEmbeddedBaseSubgroup K' L' j)
+        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j) := by
+  let e := numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j
+  have hTop :
+      (numberFieldEmbeddedTopSubgroup K K'
+          (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup =
+        (numberFieldEmbeddedBaseSubgroup K' L' j).toSubgroup :=
+    congrArg ClosedSubgroup.toSubgroup
+      (numberFieldEmbeddedTopSubgroup_baseChange_eq
+        (K := K) (K' := K') (L' := L') j)
+  ext x
+  constructor
+  · rintro ⟨y, hy, rfl⟩
+    change
+      ((e y :
+          (numberFieldEmbeddedBaseSubgroup K L
+            (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup) :
+          SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) ∈
+        (numberFieldEmbeddedBaseSubgroup K' L' j).toSubgroup
+    dsimp only [e, numberFieldEmbeddedBaseChangeBaseEquiv]
+    rw [MulEquiv.subgroupCongr_apply, ← hTop]
+    exact hy
+  · intro hx
+    refine ⟨e.symm x, ?_, e.apply_symm_apply x⟩
+    change
+      (((e.symm x :
+          (numberFieldEmbeddedBaseSubgroup K K'
+            (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup) :
+          SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) ∈
+        (numberFieldEmbeddedTopSubgroup K K'
+          (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup)
+    dsimp only [e, numberFieldEmbeddedBaseChangeBaseEquiv]
+    rw [MulEquiv.subgroupCongr_symm_apply, hTop]
+    exact hx
+
+/-- Normality of the relative subgroup between the two embedded base fields
+in a finite Galois base change. -/
+private theorem numberFieldEmbeddedBaseChangeExtensionSubgroup_normal
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    (CyclicCohomology.extensionSubgroup
+      (numberFieldEmbeddedBaseSubgroup K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+      (numberFieldEmbeddedBaseSubgroup K' L' j)
+      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal := by
+  let e := numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j
+  have hNormal :=
+    (numberFieldEmbeddedExtensionSubgroup_normal K K'
+      (numberFieldEmbeddedLowerEmbedding K' L' j)).map
+        e.toMonoidHom e.surjective
+  rw [numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq
+    K K' L L' j] at hNormal
+  exact hNormal
+
+noncomputable local instance
+    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    (CyclicCohomology.extensionSubgroup
+      (numberFieldEmbeddedBaseSubgroup K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+      (numberFieldEmbeddedBaseSubgroup K' L' j)
+      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)).Normal :=
+  numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
+
+/-- Finiteness of the relative quotient between the two embedded base fields
+in a finite Galois base change. -/
+private theorem numberFieldEmbeddedBaseChangeExtensionQuotient_finite
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    Finite
+      ((numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (numberFieldEmbeddedBaseSubgroup K L
+            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+          (numberFieldEmbeddedBaseSubgroup K' L' j)
+          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) := by
+  let e := numberFieldEmbeddedBaseChangeBaseEquiv K K' L L' j
+  let N :=
+    CyclicCohomology.extensionSubgroup
+      (numberFieldEmbeddedBaseSubgroup K K'
+        (numberFieldEmbeddedLowerEmbedding K' L' j))
+      (numberFieldEmbeddedTopSubgroup K K'
+        (numberFieldEmbeddedLowerEmbedding K' L' j))
+      (numberFieldEmbeddedTopSubgroup_le_baseSubgroup K K'
+        (numberFieldEmbeddedLowerEmbedding K' L' j))
+  let M :=
+    CyclicCohomology.extensionSubgroup
+      (numberFieldEmbeddedBaseSubgroup K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+      (numberFieldEmbeddedBaseSubgroup K' L' j)
+      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
+  let hNNormal : N.Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal K K'
+      (numberFieldEmbeddedLowerEmbedding K' L' j)
+  let hMNormal : M.Normal :=
+    numberFieldEmbeddedBaseChangeExtensionSubgroup_normal K K' L L' j
+  let hNFinite :
+      Finite
+        ((numberFieldEmbeddedBaseSubgroup K K'
+            (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup ⧸ N) :=
+    numberFieldEmbeddedExtensionQuotient_finite K K'
+      (numberFieldEmbeddedLowerEmbedding K' L' j)
+  have hmap : N.map e.toMonoidHom = M :=
+    numberFieldEmbeddedBaseChangeExtensionSubgroup_map_eq K K' L L' j
+  have hle : N ≤ M.comap e.toMonoidHom := by
+    rw [← hmap]
+    exact Subgroup.le_comap_map e.toMonoidHom N
+  let f :
+      ((numberFieldEmbeddedBaseSubgroup K K'
+          (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup ⧸ N) →*
+        ((numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸ M) :=
+    QuotientGroup.map N M e.toMonoidHom hle
+  have hmk : Function.Surjective
+      (QuotientGroup.mk ∘ e :
+        (numberFieldEmbeddedBaseSubgroup K K'
+            (numberFieldEmbeddedLowerEmbedding K' L' j)).toSubgroup →
+          (numberFieldEmbeddedBaseSubgroup K L
+              (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸ M) :=
+    QuotientGroup.mk_surjective.comp e.surjective
+  have hsurj : Function.Surjective f :=
+    QuotientGroup.map_surjective_of_surjective
+      (N := N) M e.toMonoidHom hmk hle
+  exact Finite.of_surjective f hsurj
+
+noncomputable local instance
+    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    Finite
+      ((numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L'))).toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup
+          (numberFieldEmbeddedBaseSubgroup K L
+            (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+          (numberFieldEmbeddedBaseSubgroup K' L' j)
+          (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
+  numberFieldEmbeddedBaseChangeExtensionQuotient_finite K K' L L' j
+
+/-- Reuse the canonical absolute fixed-field witness for the lower embedded
+tower.  The base-change relative witness below needs this exact instance path
+when forming the absolute finite-dimensional tower. -/
+noncomputable local instance
+    numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteDimensional ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L')))) :=
+  numberFieldEmbeddedAbstractFixedFieldFiniteDimensional K L
+    (j.comp (IsScalarTower.toAlgHom ℚ L L'))
+
+noncomputable local instance
+    numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteDimensional
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
+  abstractRelativeFixedField_finiteDimensional
+    ℚ (SeparableClosure ℚ)
+    (numberFieldEmbeddedBaseSubgroup K L
+      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+    (numberFieldEmbeddedBaseSubgroup K' L' j)
+    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
+    (numberFieldEmbeddedAbsoluteQuotientFinite K L
+      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+    (numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j)
+
+local instance
+    numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    IsScalarTower ℚ
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
+  IsScalarTower.of_algebraMap_eq' (RingHom.ext_rat _ _)
+
+noncomputable local instance
+    numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    FiniteDimensional ℚ
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
+  FiniteDimensional.trans ℚ
+    (abstractFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedBaseSubgroup K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
+
+noncomputable local instance
+    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    NumberField
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
+  NumberField.of_module_finite ℚ
+    (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+      (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j))
+
+noncomputable local instance
+    numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    IsGalois
+      (abstractFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup K L
+          (j.comp (IsScalarTower.toAlgHom ℚ L L'))))
+      (abstractRelativeFixedField ℚ (SeparableClosure ℚ)
+        (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)) :=
+  abstractRelativeFixedField_isGalois
+    ℚ (SeparableClosure ℚ)
+    (numberFieldEmbeddedBaseSubgroup K L
+      (j.comp (IsScalarTower.toAlgHom ℚ L L')))
+    (numberFieldEmbeddedBaseSubgroup K' L' j)
+    (numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j)
+    (numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j)
+
+
+end EmbeddedNumberFieldRestriction
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerNorm.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerNorm.lean
new file mode 100644
index 0000000..a0d887a
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerNorm.lean
@@ -0,0 +1,261 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityTowerRestriction
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+section EmbeddedNumberFieldRestriction
+
+variable
+    (K K' L L' : Type)
+    [Field K] [NumberField K]
+    [Field K'] [NumberField K']
+    [Field L] [NumberField L]
+    [Field L'] [NumberField L']
+    [Algebra K K'] [Algebra K L] [Algebra K L']
+    [Algebra K' L'] [Algebra L L']
+    [IsScalarTower K K' L'] [IsScalarTower K L L']
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+  numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
+  numberFieldEmbeddedBaseChangeExtensionQuotientFinite
+  numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
+
+/-- For one common compatible embedding of a Galois base-change
+diamond, the genuine global norm-residue maps commute with ordinary
+idele-class norm and actual restriction of automorphisms. -/
+theorem globalNormResidueMonoidHomOfEmbedding_norm_restriction
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ) :
+    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
+      j.comp (IsScalarTower.toAlgHom ℚ L L')
+    ((AlgEquiv.restrictNormalHom L).comp
+        (AlgEquiv.restrictScalarsHom K)).comp
+        (globalNormResidueMonoidHomOfEmbedding K' L' j) =
+      (globalNormResidueMonoidHomOfEmbedding K L jLower).comp
+        (_root_.ideleClassNorm K K') := by
+  dsimp only
+  let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
+    j.comp (IsScalarTower.toAlgHom ℚ L L')
+  let H :=
+    numberFieldEmbeddedBaseSubgroup K L jLower
+  let H' :=
+    numberFieldEmbeddedBaseSubgroup K' L' j
+  let J :=
+    numberFieldEmbeddedTopSubgroup K L jLower
+  let J' :=
+    numberFieldEmbeddedTopSubgroup K' L' j
+  let hJH :=
+    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L jLower
+  let hJ'H' :=
+    numberFieldEmbeddedTopSubgroup_le_baseSubgroup K' L' j
+  let hH'H :=
+    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
+  let hJ'J :=
+    numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
+  let _ :
+      (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal K L jLower
+  let _ :
+      Finite
+        (H.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H J hJH) :=
+    numberFieldEmbeddedExtensionQuotient_finite K L jLower
+  let _ :
+      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal K' L' j
+  let _ :
+      Finite
+        (H'.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H' J' hJ'H') :=
+    numberFieldEmbeddedExtensionQuotient_finite K' L' j
+  let hHH'finite :=
+    numberFieldEmbeddedBaseChangeExtensionQuotientFinite K K' L L' j
+  let T :
+      FiniteAbstractFieldExtension
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) :=
+    { field := numberFieldEmbeddedFiniteAbstractField K' L' j
+      base := numberFieldEmbeddedFiniteAbstractField K L jLower
+      below := hH'H
+      finiteQuotient := hHH'finite }
+  let hTBaseNormal :
+      (CyclicCohomology.extensionSubgroup
+        T.base.field J hJH).Normal := by
+    change
+      (CyclicCohomology.extensionSubgroup H J hJH).Normal
+    exact numberFieldEmbeddedExtensionSubgroup_normal K L jLower
+  let hTBaseFinite :
+      Finite
+        (T.base.field.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup
+            T.base.field J hJH) := by
+    change
+      Finite
+        (H.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H J hJH)
+    exact numberFieldEmbeddedExtensionQuotient_finite K L jLower
+  let hTFieldNormal :
+      (CyclicCohomology.extensionSubgroup
+        T.field.field J' hJ'H').Normal := by
+    change
+      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal
+    exact numberFieldEmbeddedExtensionSubgroup_normal K' L' j
+  let hTFieldFinite :
+      Finite
+        (T.field.field.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup
+            T.field.field J' hJ'H') := by
+    change
+      Finite
+        (H'.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H' J' hJ'H')
+    exact numberFieldEmbeddedExtensionQuotient_finite K' L' j
+  let restrictActual :
+      Gal(L' / K') →* Gal(L / K) :=
+    (AlgEquiv.restrictNormalHom L).comp
+      (AlgEquiv.restrictScalarsHom K)
+  apply MonoidHom.ext
+  intro c
+  let a :=
+    numberFieldEmbeddedIdeleClassEquivAmbientFixed
+      K' L' j (Additive.ofMul c)
+  have hnat :=
+    DegreeData.normResidueNaturality_norm_restriction
+      (D := rationalCyclotomicDegreeData)
+      (A := rationalIdeleClassRepresentation)
+      (v := rationalCyclotomicIdeleClassValuationData)
+      (hcf := rationalIdeleClassRepresentation_satisfiesClassFieldAxiom)
+      (T := T) (L := J) (L' := J')
+      (hLnormal := hTBaseNormal)
+      (hL'normal := hTFieldNormal)
+      (hLKfinite := hTBaseFinite)
+      (hL'K'finite := hTFieldFinite)
+      hJH hJ'H' hJ'J
+  have hnatc :=
+    DFunLike.congr_fun hnat
+      (finiteNormClass rationalIdeleClassRepresentation
+        H' J' hJ'H' a)
+  change _ =
+    rationalCyclotomicDegreeData.normResidueSymbol
+      rationalIdeleClassRepresentation
+      rationalCyclotomicIdeleClassValuationData
+      rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+      T.base
+      { field := J
+        below := hJH
+        normal := hTBaseNormal
+        finite := hTBaseFinite }
+      (finiteReciprocityNaturalityNormMap
+        rationalIdeleClassRepresentation
+        T.base.field T.field.field J J'
+        hJH hJ'H' T.below hJ'J
+        (finiteNormClass rationalIdeleClassRepresentation
+          T.field.field J' hJ'H' a)) at hnatc
+  rw [finiteReciprocityNaturalityNormMap_finiteNormClass]
+    at hnatc
+  have hnorm :
+      relativeNorm rationalIdeleClassRepresentation
+          H H' hH'H a =
+        numberFieldEmbeddedIdeleClassEquivAmbientFixed
+          K L jLower
+          (Additive.ofMul (_root_.ideleClassNorm K K' c)) :=
+    numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm
+      K K' L L' j c
+  calc
+    restrictActual
+        (globalNormResidueMonoidHomOfEmbedding K' L' j c) =
+      restrictActual
+        (Additive.toMul
+          (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+            K' L' j
+            (rationalCyclotomicDegreeData.normResidueSymbol
+              rationalIdeleClassRepresentation
+              rationalCyclotomicIdeleClassValuationData
+              rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+              (numberFieldEmbeddedFiniteAbstractField K' L' j)
+              (numberFieldEmbeddedFiniteGaloisSubextension K' L' j)
+              (finiteNormClass rationalIdeleClassRepresentation
+                H' J' hJ'H' a)))) := by
+      rw [globalNormResidueMonoidHomOfEmbedding_apply]
+    _ =
+      Additive.toMul
+        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+          K L jLower
+          (MonoidHom.toAdditive
+            (normResidueNaturalityAbelianizedRestriction
+              H H' J J' hJH hJ'H' hH'H hJ'J)
+            (rationalCyclotomicDegreeData.normResidueSymbol
+              rationalIdeleClassRepresentation
+              rationalCyclotomicIdeleClassValuationData
+              rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+              (numberFieldEmbeddedFiniteAbstractField K' L' j)
+              (numberFieldEmbeddedFiniteGaloisSubextension K' L' j)
+              (finiteNormClass rationalIdeleClassRepresentation
+                H' J' hJ'H' a)))) := by
+      exact
+        numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction
+          K K' L L' j _
+    _ =
+      Additive.toMul
+        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+          K L jLower
+          (rationalCyclotomicDegreeData.normResidueSymbol
+            rationalIdeleClassRepresentation
+            rationalCyclotomicIdeleClassValuationData
+            rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+            (numberFieldEmbeddedFiniteAbstractField K L jLower)
+            (numberFieldEmbeddedFiniteGaloisSubextension K L jLower)
+            (finiteNormClass rationalIdeleClassRepresentation
+              H J hJH
+              (relativeNorm rationalIdeleClassRepresentation
+                H H' hH'H a)))) := by
+      exact congrArg
+        (fun z =>
+          Additive.toMul
+            (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+              K L jLower z))
+        hnatc
+    _ =
+      globalNormResidueMonoidHomOfEmbedding K L jLower
+        (_root_.ideleClassNorm K K' c) := by
+      rw [hnorm,
+        ← globalNormResidueMonoidHomOfEmbedding_apply]
+
+end EmbeddedNumberFieldRestriction
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerRestriction.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerRestriction.lean
new file mode 100644
index 0000000..16ee169
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTowerRestriction.lean
@@ -0,0 +1,372 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalNormResidueNaturalityTowerBase
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+
+section EmbeddedNumberFieldRestriction
+
+variable
+    (K K' L L' : Type)
+    [Field K] [NumberField K]
+    [Field K'] [NumberField K']
+    [Field L] [NumberField L]
+    [Field L'] [NumberField L']
+    [Algebra K K'] [Algebra K L] [Algebra K L']
+    [Algebra K' L'] [Algebra L L']
+    [IsScalarTower K K' L'] [IsScalarTower K L L']
+
+attribute [local instance]
+  naturalityIdeleClassCommGroup
+  ideleClassGroupIsMulCommutative
+  ideleClassSubgroupNormal
+  numberFieldEmbeddedBaseChangeExtensionSubgroupNormal
+  numberFieldEmbeddedBaseChangeExtensionQuotientFinite
+  numberFieldEmbeddedBaseChangeBaseFixedFieldFiniteDimensional
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldFiniteDimensional
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldScalarTower
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
+  numberFieldEmbeddedBaseChangeRelativeFixedFieldIsGalois
+
+/-- In one common rational-separable-closure realization, the canonical
+quotient-to-Galois comparisons intertwine abstract restriction with
+ordinary restriction of the actual number-field automorphisms. -/
+theorem
+    numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup_restriction
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
+    (z :
+      Abelianization
+        (numberFieldEmbeddedFiniteGaloisSubextension
+          K' L' j).extensionQuotient) :
+    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
+      j.comp (IsScalarTower.toAlgHom ℚ L L')
+    let H :=
+      numberFieldEmbeddedBaseSubgroup K L jLower
+    let H' :=
+      numberFieldEmbeddedBaseSubgroup K' L' j
+    let J :=
+      numberFieldEmbeddedTopSubgroup K L jLower
+    let J' :=
+      numberFieldEmbeddedTopSubgroup K' L' j
+    let hH'H :=
+      numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
+    let hJ'J :=
+      numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
+    let hJH :=
+      numberFieldEmbeddedTopSubgroup_le_baseSubgroup K L jLower
+    let hJ'H' :=
+      numberFieldEmbeddedTopSubgroup_le_baseSubgroup K' L' j
+    letI _ :
+        (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
+      numberFieldEmbeddedExtensionSubgroup_normal K L jLower
+    letI _ :
+        (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
+      numberFieldEmbeddedExtensionSubgroup_normal K' L' j
+    ((AlgEquiv.restrictNormalHom L).comp
+        (AlgEquiv.restrictScalarsHom K))
+        (Additive.toMul
+          (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+            K' L' j (Additive.ofMul z))) =
+      Additive.toMul
+        (numberFieldEmbeddedAbelianizedExtensionQuotientEquivGaloisGroup
+          K L jLower
+          (MonoidHom.toAdditive
+            (normResidueNaturalityAbelianizedRestriction
+              H H' J J'
+              hJH hJ'H'
+              hH'H hJ'J)
+            (Additive.ofMul z))) := by
+  dsimp only
+  let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
+    j.comp (IsScalarTower.toAlgHom ℚ L L')
+  let H :=
+    numberFieldEmbeddedBaseSubgroup K L jLower
+  let H' :=
+    numberFieldEmbeddedBaseSubgroup K' L' j
+  let J :=
+    numberFieldEmbeddedTopSubgroup K L jLower
+  let J' :=
+    numberFieldEmbeddedTopSubgroup K' L' j
+  let hH'H :=
+    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
+  let hJ'J :=
+    numberFieldEmbeddedTopSubgroup_le_of_tower K K' L L' j
+  let hJH :=
+    numberFieldEmbeddedTopSubgroup_le_baseSubgroup
+      K L jLower
+  let hJ'H' :=
+    numberFieldEmbeddedTopSubgroup_le_baseSubgroup
+      K' L' j
+  let hLowerNormal :
+      (CyclicCohomology.extensionSubgroup H J hJH).Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal K L jLower
+  let hUpperNormal :
+      (CyclicCohomology.extensionSubgroup H' J' hJ'H').Normal :=
+    numberFieldEmbeddedExtensionSubgroup_normal K' L' j
+  let qLower :=
+    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
+      K L jLower
+  let qUpper :=
+    numberFieldEmbeddedExtensionQuotientEquivGaloisGroup
+      K' L' j
+  let qLowerRaw :
+      (H.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H J hJH) ≃*
+        Gal(L / K) := by
+    exact
+      { qLower.toEquiv with
+        map_mul' := fun x y => qLower.map_mul x y }
+  let qUpperRaw :
+      (H'.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H' J' hJ'H') ≃*
+        Gal(L' / K') := by
+    exact
+      { qUpper.toEquiv with
+        map_mul' := fun x y => qUpper.map_mul x y }
+  let restrictActual :
+      Gal(L' / K') →* Gal(L / K) :=
+    (AlgEquiv.restrictNormalHom L).comp
+      (AlgEquiv.restrictScalarsHom K)
+  obtain ⟨q, rfl⟩ :=
+    QuotientGroup.mk_surjective z
+  obtain ⟨σ, rfl⟩ :=
+    (numberFieldEmbeddedFiniteGaloisSubextension
+      K' L' j).extensionQuotientMk_surjective q
+  change
+    restrictActual
+        ((Abelianization.equivOfComm (H := Gal(L' / K'))).symm
+          (qUpperRaw.abelianizationCongr
+            (Abelianization.of (QuotientGroup.mk σ)))) =
+      (Abelianization.equivOfComm (H := Gal(L / K))).symm
+        (qLowerRaw.abelianizationCongr
+          (normResidueNaturalityAbelianizedRestriction
+            H H' J J' hJH hJ'H' hH'H hJ'J
+            (Abelianization.of (QuotientGroup.mk σ))))
+  rw [normResidueNaturalityAbelianizedRestriction_of_mk,
+    abelianizationCongr_of, abelianizationCongr_of]
+  change
+    restrictActual (qUpperRaw (QuotientGroup.mk σ)) =
+      qLowerRaw
+        (QuotientGroup.mk (Subgroup.inclusion hH'H σ))
+  apply AlgEquiv.ext
+  intro x
+  apply jLower.injective
+  let hUpperAlgebra : Algebra K' (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra K' L' j
+  let eUpper :=
+    numberFieldEmbeddedSeparableClosureEquiv K' L' j
+  let hLowerAlgebra : Algebra K (SeparableClosure ℚ) :=
+    numberFieldEmbeddedSeparableClosureAlgebra K L jLower
+  let eLower :=
+    numberFieldEmbeddedSeparableClosureEquiv K L jLower
+  calc
+    jLower
+        (restrictActual
+          (qUpperRaw (QuotientGroup.mk σ)) x) =
+        j
+          ((qUpperRaw (QuotientGroup.mk σ))
+            (algebraMap L L' x)) := by
+      exact congrArg j
+        (AlgEquiv.restrictNormal_commutes
+          ((AlgEquiv.restrictScalarsHom K)
+            (qUpperRaw (QuotientGroup.mk σ)))
+          L x)
+    _ = σ.1.1
+        (j (algebraMap L L' x)) := by
+      exact
+        ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
+          ℚ K' L' j eUpper σ (algebraMap L L' x)
+    _ = (Subgroup.inclusion hH'H σ).1.1
+        (jLower x) := rfl
+    _ = jLower
+        (qLowerRaw
+          (QuotientGroup.mk
+            (Subgroup.inclusion hH'H σ)) x) := by
+      exact
+        (ambientEmbeddedExtensionQuotientEquivGaloisGroup_mk_apply
+          ℚ K L jLower eLower
+          (Subgroup.inclusion hH'H σ) x).symm
+
+/-- In a compatible common embedding, the fixed-part relative norm
+between two (Galois-related) base fields is the genuine ordinary
+idele-class norm. -/
+theorem numberFieldEmbeddedIdeleClassEquivAmbientFixed_relativeNorm
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
+    (c : IdeleClassGroup K') :
+    let jLower : L →ₐ[ℚ] SeparableClosure ℚ :=
+      j.comp (IsScalarTower.toAlgHom ℚ L L')
+    let H :=
+      numberFieldEmbeddedBaseSubgroup K L jLower
+    let H' :=
+      numberFieldEmbeddedBaseSubgroup K' L' j
+    let hH'H :=
+      numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
+    letI _ : Finite
+        (H.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H H' hH'H) :=
+      numberFieldEmbeddedBaseChangeExtensionQuotientFinite
+        K K' L L' j
+    relativeNorm rationalIdeleClassRepresentation H H' hH'H
+        (numberFieldEmbeddedIdeleClassEquivAmbientFixed
+          K' L' j (Additive.ofMul c)) =
+      numberFieldEmbeddedIdeleClassEquivAmbientFixed
+        K L jLower
+        (Additive.ofMul (_root_.ideleClassNorm K K' c)) := by
+  intro jLower H H'
+  let hH'H :=
+    numberFieldEmbeddedBaseSubgroup_le_of_tower K K' L L' j
+  let hnormal :=
+    numberFieldEmbeddedBaseChangeExtensionSubgroupNormal K K' L L' j
+  let F :=
+    abstractFixedField ℚ (SeparableClosure ℚ) H
+  let E :=
+    abstractRelativeFixedField ℚ (SeparableClosure ℚ) hH'H
+  let _ :
+      (CyclicCohomology.extensionSubgroup H H' hH'H).Normal :=
+    hnormal
+  let _ :
+      Finite
+        (H.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup H H' hH'H) :=
+    numberFieldEmbeddedBaseChangeExtensionQuotientFinite
+      K K' L L' j
+  let _ :
+      Finite
+        ((baseField
+          (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup
+            (baseField
+              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+            H (le_baseField H)) :=
+    numberFieldEmbeddedAbsoluteQuotientFinite K L jLower
+  let _ : NumberField F :=
+    numberFieldEmbeddedAbstractFixedFieldNumberField K L jLower
+  let _ : NumberField E :=
+    numberFieldEmbeddedBaseChangeRelativeFixedFieldNumberField
+      K K' L L' j
+  let _ : FiniteDimensional ℚ (E.restrictScalars ℚ) := by
+    change FiniteDimensional ℚ E
+    exact
+      numberFieldEmbeddedBaseChangeRelativeFixedFieldAbsoluteFiniteDimensional
+        K K' L L' j
+  let _ : NumberField
+      (abstractFixedField ℚ (SeparableClosure ℚ) H') :=
+    numberFieldEmbeddedAbstractFixedFieldNumberField K' L' j
+  have hE :
+      E.restrictScalars ℚ =
+        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
+    IntermediateField.extendScalars_restrictScalars
+      (abstractFixedField_le
+        ℚ (SeparableClosure ℚ) hH'H)
+  let eRel :
+      E ≃ₐ[ℚ]
+        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
+    IntermediateField.equivOfEq hE
+  let eK :
+      K ≃ₐ[ℚ] F :=
+    numberFieldEmbeddedAbstractBaseFieldEquiv K L jLower
+  let eK'Base :
+      K' ≃ₐ[ℚ]
+        abstractFixedField ℚ (SeparableClosure ℚ) H' :=
+    numberFieldEmbeddedAbstractBaseFieldEquiv K' L' j
+  let eK' : K' ≃ₐ[ℚ] E :=
+    eK'Base.trans eRel.symm
+  have hcompat (x : K) :
+      eK' (algebraMap K K' x) =
+        algebraMap F E (eK x) := by
+    apply eRel.injective
+    apply Subtype.ext
+    change
+      j (algebraMap K' L' (algebraMap K K' x)) =
+        j (algebraMap L L' (algebraMap K L x))
+    rw [← IsScalarTower.algebraMap_apply K K' L',
+      ← IsScalarTower.algebraMap_apply K L L']
+  have hupper :
+      rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
+          H H' hH'H
+          (Additive.ofMul (ideleClassCongr eK' c)) =
+        numberFieldEmbeddedIdeleClassEquivAmbientFixed
+          K' L' j (Additive.ofMul c) := by
+    apply Subtype.ext
+    change
+      ((rationalIdeleClassEquivFixed (E.restrictScalars ℚ))
+          (Additive.ofMul (ideleClassCongr eK' c))).1 =
+        ((rationalIdeleClassEquivFixed
+            (abstractFixedField ℚ (SeparableClosure ℚ) H'))
+          (Additive.ofMul (ideleClassCongr eK'Base c))).1
+    exact rationalIdeleClassEquivFixed_transport_baseEquiv_val
+      (T := K') (A := E.restrictScalars ℚ)
+      (B := abstractFixedField ℚ (SeparableClosure ℚ) H') hE eK'Base c
+  have hrelative :=
+    rationalAbstractRelativeFixedFieldIdeleClassEquivFixed_relativeNorm
+      H H' hH'H hnormal
+        (Additive.ofMul (ideleClassCongr eK' c))
+  change
+    relativeNorm rationalIdeleClassRepresentation H H' hH'H
+        (rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
+          H H' hH'H
+          (Additive.ofMul (ideleClassCongr eK' c))) =
+      rationalAbstractFixedFieldIdeleClassEquivFixed H
+        (Additive.ofMul
+          (_root_.ideleClassNorm F E
+            (ideleClassCongr eK' c)))
+    at hrelative
+  calc
+    relativeNorm rationalIdeleClassRepresentation H H' hH'H
+        (numberFieldEmbeddedIdeleClassEquivAmbientFixed
+          K' L' j (Additive.ofMul c)) =
+      relativeNorm rationalIdeleClassRepresentation H H' hH'H
+        (rationalAbstractRelativeFixedFieldIdeleClassEquivFixed
+          H H' hH'H (Additive.ofMul (ideleClassCongr eK' c))) :=
+      congrArg (relativeNorm rationalIdeleClassRepresentation H H' hH'H)
+        hupper.symm
+    _ = rationalAbstractFixedFieldIdeleClassEquivFixed H
+        (Additive.ofMul
+          (_root_.ideleClassNorm F E (ideleClassCongr eK' c))) := hrelative
+    _ = numberFieldEmbeddedIdeleClassEquivAmbientFixed
+        K L jLower (Additive.ofMul (_root_.ideleClassNorm K K' c)) := by
+      change
+        rationalAbstractFixedFieldIdeleClassEquivFixed H
+            (Additive.ofMul
+              (_root_.ideleClassNorm F E (ideleClassCongr eK' c))) =
+          rationalAbstractFixedFieldIdeleClassEquivFixed H
+            (Additive.ofMul
+              (ideleClassCongr eK (_root_.ideleClassNorm K K' c)))
+      apply congrArg (rationalAbstractFixedFieldIdeleClassEquivFixed H)
+      apply congrArg Additive.ofMul
+      exact
+        (ideleClassCongr_ideleClassNorm
+          (K := K) (K' := F) (L := K') (L' := E) eK eK' hcompat c).symm
+
+
+end EmbeddedNumberFieldRestriction
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTransports.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTransports.lean
new file mode 100644
index 0000000..5a39a62
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/GlobalNormResidueNaturalityTransports.lean
@@ -0,0 +1,445 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.AbstractFixedFieldGlobalNormResidue
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.TowerAlgEquivNaturality
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open ClassFormation
+open GlobalClassFields
+open KummerTheory
+open AlgebraicNumberTheory
+open LocalClassFieldTheory
+open RamificationTheory
+
+universe u
+
+/-- The commutative group structure on the quotient of ideles by principal ideles. -/
+@[instance_reducible]
+noncomputable def naturalityIdeleClassCommGroup
+    (F : Type) [Field F] [NumberField F] : CommGroup (IdeleClassGroup F) :=
+  QuotientGroup.Quotient.commGroup (IdeleGroup.principalSubgroup F)
+
+attribute [local instance] naturalityIdeleClassCommGroup
+
+local instance ideleClassGroupIsMulCommutative
+    {F : Type} [Field F] [NumberField F]
+    : IsMulCommutative (IdeleClassGroup F) :=
+  ⟨⟨fun a b => mul_comm a b⟩⟩
+
+local instance ideleClassSubgroupNormal
+    {F : Type} [Field F] [NumberField F]
+    (N : Subgroup (IdeleClassGroup F)) : N.Normal :=
+  N.normal_of_isMulCommutative
+
+
+/-- Two finite Galois subextensions with the same underlying closed subgroup
+are equal; the remaining structure fields are proof-irrelevant. -/
+private theorem finiteGaloisSubextension_eq_of_field_eq
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {K : ClosedSubgroup G}
+    (A B : FiniteGaloisSubextension K)
+    (h : A.field = B.field) :
+    A = B := by
+  cases A with
+  | mk A hA nA fA =>
+      cases B with
+      | mk B hB nB fB =>
+          dsimp only at h
+          cases h
+          rfl
+
+/-- Rebase a finite Galois subextension along equality of its bundled base.
+The field equality is the only data component; the remaining fields are
+proof-irrelevant. -/
+theorem finiteGaloisSubextension_transport_eq_of_field_eq
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {A B : FiniteAbstractField G}
+    (hAB : A = B)
+    (P : FiniteGaloisSubextension A.field)
+    (Q : FiniteGaloisSubextension B.field)
+    (hfield : P.field = Q.field) :
+    Eq.mp
+      (congrArg
+        (fun X : FiniteAbstractField G =>
+          FiniteGaloisSubextension X.field)
+        hAB)
+      P = Q := by
+  cases hAB
+  exact finiteGaloisSubextension_eq_of_field_eq P Q hfield
+
+/-- Transporting an additive equivalence between rational ambient fixed
+subgroups does not change the underlying direct-limit class. -/
+theorem rationalAmbientFixedAddEquiv_transport_apply_val
+    {A B : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
+    (hAB : A = B)
+    {X : Type} [AddGroup X]
+    (e : X ≃+
+      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
+    (x : X) :
+    ((Eq.mp
+        (congrArg
+          (fun Y : FiniteAbstractField
+              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+            X ≃+ ambientFixedAddSubgroup
+              rationalIdeleClassRepresentation Y.field)
+          hAB)
+        e x).1) =
+      (e x).1 := by
+  cases hAB
+  rfl
+
+/-- Changing only the bundled subgroup of an additive subgroup element
+does not change its value in the ambient group. -/
+private theorem addSubgroupCongr_apply_val
+    {A : Type} [AddGroup A]
+    {H K : AddSubgroup A}
+    (h : H = K) (x : H) :
+    ((AddEquiv.addSubgroupCongr h x).1 : A) = x.1 := by
+  cases h
+  rfl
+
+/-- Transporting an idele class along a field equality and the corresponding
+algebra equivalence leaves its rational direct-limit representative fixed. -/
+theorem rationalIdeleClassEquivFixed_congr_apply_val
+    {A B : IntermediateField ℚ (SeparableClosure ℚ)}
+    [FiniteDimensional ℚ A] [FiniteDimensional ℚ B]
+    (h : A = B)
+    (e : B ≃ₐ[ℚ] A)
+    (he : e.trans (IntermediateField.equivOfEq h) =
+      (AlgEquiv.refl : B ≃ₐ[ℚ] B))
+    (c : Additive (IdeleClassGroup B)) :
+    ((rationalIdeleClassEquivFixed A)
+        (MulEquiv.toAdditive (ideleClassCongr e) c)).1 =
+      ((rationalIdeleClassEquivFixed B) c).1 := by
+  cases h
+  have he' : e = AlgEquiv.refl := by
+    apply AlgEquiv.ext
+    intro x
+    have hx := DFunLike.congr_fun he x
+    change e x = x at hx
+    exact hx
+  rw [he']
+  have hc :
+      MulEquiv.toAdditive
+          (ideleClassCongr (AlgEquiv.refl : A ≃ₐ[ℚ] A)) c = c := by
+    cases c with
+    | ofMul c =>
+        exact congrArg Additive.ofMul (ideleClassCongr_refl c)
+  rw [hc]
+
+/-- Transporting the target intermediate field of a base-field equivalence
+preserves its rational fixed-part representative. -/
+theorem rationalIdeleClassEquivFixed_transport_baseEquiv_val
+    {T : Type} [Field T] [NumberField T]
+    {A B : IntermediateField ℚ (SeparableClosure ℚ)}
+    [FiniteDimensional ℚ A] [FiniteDimensional ℚ B]
+    (h : A = B) (e : T ≃ₐ[ℚ] B)
+    (c : IdeleClassGroup T) :
+    ((rationalIdeleClassEquivFixed A)
+      (Additive.ofMul
+        (ideleClassCongr (K := T) (M := A)
+          (e.trans (IntermediateField.equivOfEq h).symm) c))).1 =
+    ((rationalIdeleClassEquivFixed B)
+      (Additive.ofMul (ideleClassCongr (K := T) (M := B) e c))).1 := by
+  cases h
+  have he :
+      e.trans (IntermediateField.equivOfEq (rfl : A = A)).symm = e := by
+    ext x
+    rfl
+  rw [he]
+
+/-- Equality of the lower and upper closed subgroups transports the raw
+extension quotient without exposing dependent rewrites to clients. -/
+def extensionQuotientMulEquivOfEq
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {H H' J J' : ClosedSubgroup G}
+    (hH : H = H') (hJ : J = J')
+    (hJH : J.toSubgroup ≤ H.toSubgroup)
+    (hJH' : J'.toSubgroup ≤ H'.toSubgroup)
+    [(CyclicCohomology.extensionSubgroup H J hJH).Normal]
+    [(CyclicCohomology.extensionSubgroup H' J' hJH').Normal] :
+    (H.toSubgroup ⧸ CyclicCohomology.extensionSubgroup H J hJH) ≃*
+      (H'.toSubgroup ⧸
+        CyclicCohomology.extensionSubgroup H' J' hJH') := by
+  cases hH
+  cases hJ
+  exact MulEquiv.refl _
+
+/-- The quotient transport sends a quotient representative to the same
+ambient group element, rebundled in the equal lower subgroup. -/
+@[simp]
+theorem extensionQuotientMulEquivOfEq_mk
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {H H' J J' : ClosedSubgroup G}
+    (hH : H = H') (hJ : J = J')
+    (hJH : J.toSubgroup ≤ H.toSubgroup)
+    (hJH' : J'.toSubgroup ≤ H'.toSubgroup)
+    [(CyclicCohomology.extensionSubgroup H J hJH).Normal]
+    [(CyclicCohomology.extensionSubgroup H' J' hJH').Normal]
+    (σ : H.toSubgroup) :
+    extensionQuotientMulEquivOfEq hH hJ hJH hJH'
+        (QuotientGroup.mk σ) =
+      QuotientGroup.mk
+        ((MulEquiv.subgroupCongr
+          (congrArg ClosedSubgroup.toSubgroup hH)) σ) := by
+  cases hH
+  cases hJ
+  rfl
+
+/-- Rebundling an element along equality of closed subgroups preserves its
+underlying ambient group element. -/
+theorem closedSubgroupCongr_apply_val
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {H H' : ClosedSubgroup G}
+    (hH : H = H') (σ : H.toSubgroup) :
+    (((MulEquiv.subgroupCongr
+        (congrArg ClosedSubgroup.toSubgroup hH)) σ).1 : G) = σ.1 := by
+  cases hH
+  rfl
+
+/-- Rebase an abelianized extension-quotient equivalence together with its
+finite abstract base. -/
+def abelianizedExtensionQuotientAddEquiv_transportBase
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {A B : FiniteAbstractField G}
+    (hAB : A = B)
+    (P : FiniteGaloisSubextension A.field)
+    {X : Type} [AddGroup X]
+    (e : Additive (Abelianization P.extensionQuotient) ≃+ X) :
+    Additive
+        (Abelianization
+          (Eq.mp
+            (congrArg
+              (fun Y : FiniteAbstractField G =>
+                FiniteGaloisSubextension Y.field)
+              hAB)
+            P).extensionQuotient) ≃+ X := by
+  cases hAB
+  exact e
+
+/-- Rebase an abelianized equivalence along equality of finite Galois
+subextensions over a fixed abstract base. -/
+def abelianizedExtensionQuotientAddEquiv_transportExtension
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {K : ClosedSubgroup G}
+    {P Q : FiniteGaloisSubextension K}
+    (hPQ : P = Q)
+    {X : Type} [AddGroup X]
+    (e : Additive (Abelianization P.extensionQuotient) ≃+ X) :
+    Additive (Abelianization Q.extensionQuotient) ≃+ X := by
+  cases hPQ
+  exact e
+
+/-- The explicit quotient equivalence induced by rebasing a finite Galois
+subextension. -/
+def extensionQuotientMulEquiv_transportFiniteGalois
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {A B : FiniteAbstractField G}
+    (hAB : A = B)
+    (P : FiniteGaloisSubextension A.field)
+    {Q : FiniteGaloisSubextension B.field}
+    (hPQ :
+      Eq.mp
+        (congrArg
+          (fun Y : FiniteAbstractField G =>
+            FiniteGaloisSubextension Y.field)
+          hAB)
+        P = Q) :
+    Q.extensionQuotient ≃* P.extensionQuotient := by
+  cases hAB
+  cases hPQ
+  exact MulEquiv.refl _
+
+/-- Quotient rebasing sends a canonical representative to the same ambient
+group element rebundled in the old base subgroup. -/
+@[simp]
+theorem extensionQuotientMulEquiv_transportFiniteGalois_mk
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {A B : FiniteAbstractField G}
+    (hAB : A = B)
+    (P : FiniteGaloisSubextension A.field)
+    {Q : FiniteGaloisSubextension B.field}
+    (hPQ :
+      Eq.mp
+        (congrArg
+          (fun Y : FiniteAbstractField G =>
+            FiniteGaloisSubextension Y.field)
+          hAB)
+        P = Q)
+    (σ : B.field.toSubgroup) :
+    extensionQuotientMulEquiv_transportFiniteGalois hAB P hPQ
+        (Q.extensionQuotientMk σ) =
+      P.extensionQuotientMk
+        ((MulEquiv.subgroupCongr
+          (congrArg ClosedSubgroup.toSubgroup
+            (congrArg FiniteAbstractField.field hAB).symm)) σ) := by
+  cases hAB
+  cases hPQ
+  rfl
+
+/-- Abelianization commutes with simultaneous transport of the abstract base
+and its finite Galois subextension. -/
+theorem abelianizedCanonicalEquiv_transportFiniteGalois
+    {G : Type u} [Group G] [TopologicalSpace G]
+    {A B : FiniteAbstractField G}
+    (hAB : A = B)
+    (P : FiniteGaloisSubextension A.field)
+    {Q : FiniteGaloisSubextension B.field}
+    (hPQ :
+      Eq.mp
+        (congrArg
+          (fun Y : FiniteAbstractField G =>
+            FiniteGaloisSubextension Y.field)
+          hAB)
+        P = Q)
+    {X : Type} [CommGroup X]
+    (e : P.extensionQuotient ≃* X) :
+    abelianizedExtensionQuotientAddEquiv_transportExtension hPQ
+        (abelianizedExtensionQuotientAddEquiv_transportBase
+          hAB P
+          (MulEquiv.toAdditive
+            (e.abelianizationCongr.trans
+              (Abelianization.equivOfComm : X ≃* Abelianization X).symm))) =
+      MulEquiv.toAdditive
+        (((extensionQuotientMulEquiv_transportFiniteGalois
+            hAB P hPQ).trans e).abelianizationCongr.trans
+          (Abelianization.equivOfComm : X ≃* Abelianization X).symm) := by
+  cases hAB
+  cases hPQ
+  rfl
+
+
+/-- The finite norm-residue value, transported from the rational idele-class
+representation and the abelianized extension quotient to the groups `C` and `X`. -/
+def rationalFiniteNormResidueValue
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    (L : FiniteGaloisSubextension K.field)
+    {C X : Type} [AddGroup C] [AddGroup X]
+    (eIdele : C ≃+
+      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
+    (eGalois : Additive (Abelianization L.extensionQuotient) ≃+ X)
+    (c : C) : X := by
+  letI :
+      Finite
+        (K.field.toSubgroup ⧸
+          CyclicCohomology.extensionSubgroup
+            K.field L.field L.below) :=
+    L.finite
+  exact
+    eGalois
+      (rationalCyclotomicDegreeData.normResidueSymbol
+        rationalIdeleClassRepresentation
+        rationalCyclotomicIdeleClassValuationData
+        rationalIdeleClassRepresentation_satisfiesClassFieldAxiom
+        K L
+        (finiteNormClass rationalIdeleClassRepresentation
+          K.field L.field L.below (eIdele c)))
+
+/-- The packaged norm-residue value is invariant under rebasing the finite
+abstract field together with all dependent data. -/
+theorem rationalFiniteNormResidueValue_transportBase
+    {A B : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)}
+    (hAB : A = B)
+    (P : FiniteGaloisSubextension A.field)
+    {C X : Type} [AddGroup C] [AddGroup X]
+    (eIdele : C ≃+
+      ambientFixedAddSubgroup rationalIdeleClassRepresentation A.field)
+    (eGalois : Additive (Abelianization P.extensionQuotient) ≃+ X)
+    (c : C) :
+    rationalFiniteNormResidueValue A P eIdele eGalois c =
+      rationalFiniteNormResidueValue B
+        (Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              FiniteGaloisSubextension Y.field)
+            hAB)
+          P)
+        (Eq.mp
+          (congrArg
+            (fun Y : FiniteAbstractField
+                (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ) =>
+              C ≃+ ambientFixedAddSubgroup
+                rationalIdeleClassRepresentation Y.field)
+            hAB)
+          eIdele)
+        (abelianizedExtensionQuotientAddEquiv_transportBase
+          hAB P eGalois)
+        c := by
+  cases hAB
+  rfl
+
+/-- The packaged norm-residue value is invariant under equality of the finite
+Galois subextension. -/
+theorem rationalFiniteNormResidueValue_transportExtension
+    (K : FiniteAbstractField
+      (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+    {P Q : FiniteGaloisSubextension K.field}
+    (hPQ : P = Q)
+    {C X : Type} [AddGroup C] [AddGroup X]
+    (eIdele : C ≃+
+      ambientFixedAddSubgroup rationalIdeleClassRepresentation K.field)
+    (eGalois : Additive (Abelianization P.extensionQuotient) ≃+ X)
+    (c : C) :
+    rationalFiniteNormResidueValue K P eIdele eGalois c =
+      rationalFiniteNormResidueValue K Q eIdele
+        (abelianizedExtensionQuotientAddEquiv_transportExtension
+          hPQ eGalois)
+        c := by
+  cases hPQ
+  rfl
+
+/-- Evaluation of the canonical abelianization comparison induced by a
+multiplicative equivalence into a commutative group. -/
+theorem abelianizationCongrToComm_apply
+    {Q R : Type*} [Group Q] [CommGroup R]
+    (e : Q ≃* R) (q : Q) :
+    MulEquiv.toAdditive
+        (e.abelianizationCongr.trans
+          (Abelianization.equivOfComm : R ≃* Abelianization R).symm)
+        (Additive.ofMul (Abelianization.of q)) =
+      Additive.ofMul (e q) := by
+  apply Additive.toMul.injective
+  change
+    (Abelianization.equivOfComm : R ≃* Abelianization R).symm
+        (e.abelianizationCongr (Abelianization.of q)) = e q
+  rw [abelianizationCongr_of]
+  exact
+    (Abelianization.equivOfComm : R ≃* Abelianization R).symm_apply_apply _
+
+/-- Evaluation of the canonical quotient from the abelianization of a
+commutative group. -/
+theorem commutativeAbelianizationEquiv_apply
+    {Q R : Type*} [CommGroup Q] [Group R]
+    (e : Q ≃* R) (q : Q) :
+    MulEquiv.toAdditive
+        ((Abelianization.equivOfComm : Q ≃* Abelianization Q).symm.trans e)
+        (Additive.ofMul (Abelianization.of q)) =
+      Additive.ofMul (e q) := by
+  apply Additive.toMul.injective
+  change
+    e ((Abelianization.equivOfComm : Q ≃* Abelianization Q).symm
+      (Abelianization.of q)) = e q
+  exact congrArg e
+    ((Abelianization.equivOfComm : Q ≃* Abelianization Q).symm_apply_apply q)
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitAbstractFixedField.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitAbstractFixedField.lean
index 08f2620..a101f78 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitAbstractFixedField.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitAbstractFixedField.lean
@@ -7,6 +7,7 @@ Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.IdeleClassDirectLimitFixedPoints
 import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.AbstractFixedFieldNorm
 
+
 set_option autoImplicit false
 
 /-!
@@ -202,7 +203,7 @@ theorem rationalIdeleClassEquivBaseFixed_coe
         E (RelativeIdeleGroup.classInclusion ℚ E c))
   exact Eq.trans h0 (Eq.trans h1 (Eq.trans h2 (Eq.trans h3 h4)))
 
-private noncomputable instance
+private noncomputable instance abstractFixedFieldNumberField
     (K : ClosedSubgroup
       (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
     [hfinite : Finite
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitCore.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitCore.lean
index 870927a..7f759bc 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitCore.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/IdeleClassDirectLimitCore.lean
@@ -9,6 +9,7 @@ import GaloisCohomology.Cyclic.NormKernelVanishing
 import Mathlib.Algebra.Colimit.DirectLimit
 import Mathlib.FieldTheory.Galois.Profinite
 
+
 set_option autoImplicit false
 
 /-!
@@ -107,19 +108,19 @@ noncomputable instance rationalFiniteGaloisIdeleClassMulDistribMulAction
       (RelativeIdeleGroup.ClassGroup ℚ E) :=
   rationalAbsoluteGaloisIdeleClassAction E
 
-private instance
+private instance classGroupMonoid
     (E : FiniteGaloisIntermediateField ℚ (SeparableClosure ℚ)) :
     Monoid (RelativeIdeleGroup.ClassGroup ℚ E) :=
   inferInstance
 
-private noncomputable instance :
+private noncomputable instance actionFamily :
     ∀ E : FiniteGaloisIntermediateField ℚ (SeparableClosure ℚ),
       MulDistribMulAction
         (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)
         (RelativeIdeleGroup.ClassGroup ℚ E) :=
   fun E => rationalAbsoluteGaloisIdeleClassAction E
 
-private noncomputable instance :
+private noncomputable instance smulFamily :
     ∀ E : FiniteGaloisIntermediateField ℚ (SeparableClosure ℚ),
       SMul
         (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)
@@ -366,7 +367,7 @@ theorem rationalRelativeIdeleClassEmbedding_comp
       (rationalRelativeAdeleEmbedding_comp hEF hFH
         (a : RelativeAdeleRing ℚ E)))
 
-private noncomputable instance :
+private noncomputable instance transitionDirectedSystem :
     DirectedSystem
       (fun E : FiniteGaloisIntermediateField ℚ (SeparableClosure ℚ) =>
         RelativeIdeleGroup.ClassGroup ℚ E)
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryField.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryField.lean
index c1e5ac0..1a80072 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryField.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryField.lean
@@ -1,1901 +1,2 @@
-/-
-Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
--/
-
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicCyclicData
-import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicUnramifiedLocalGlobalCompatibility
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.Extension.OnePlaceBaseNorm
-
-set_option autoImplicit false
-
-/-!
-# The finite p-adic auxiliary field
-
-This module realizes the simultaneous finite/cyclotomic lift as an actual
-number field and proves the subgroup and intermediate-field identities
-needed by the auxiliary-field argument.
--/
-
-open scoped IsMulCommutative NumberField
-open AlgebraicNumberTheory IsDedekindDomain NumberField
-open IdeleGroup RelativeIdeleGroup
-open AlgebraicNumberTheory.Valuations
-open HilbertRamification
-open CyclicCohomology
-open KummerTheory ClassFormation
-
-noncomputable section
-
-namespace GlobalClassFieldTheory
-namespace Reciprocity
-
-open GlobalClassFields
-
-variable
-    {K L : Type}
-    [Field K] [NumberField K]
-    [Field L] [NumberField L] [Algebra K L]
-    [FiniteDimensional K L] [IsAbelianGalois K L]
-
-local instance (p : Nat.Primes) : Fact p.1.Prime :=
-  ⟨p.2⟩
-
-attribute [local instance]
-  rationalSeparableClosureAlgebra
-
-local instance finitePadicAuxiliaryExtensionNormal :
-    (extensionSubgroup
-      (numberFieldTowerBaseSubgroup K L)
-      (numberFieldTowerTopSubgroup L)
-      (numberFieldTowerTopSubgroup_le_baseSubgroup K L)).Normal :=
-  numberFieldTowerExtensionSubgroup_normal K L
-
-local instance finitePadicAuxiliaryExtensionQuotientFinite :
-    Finite
-      ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-        extensionSubgroup
-          (numberFieldTowerBaseSubgroup K L)
-          (numberFieldTowerTopSubgroup L)
-          (numberFieldTowerTopSubgroup_le_baseSubgroup K L)) :=
-  numberFieldTowerExtensionQuotient_finite K L
-
-noncomputable local instance finitePadicAuxiliaryExtensionQuotientIsMulCommutative :
-    IsMulCommutative
-      ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-        extensionSubgroup
-          (numberFieldTowerBaseSubgroup K L)
-          (numberFieldTowerTopSubgroup L)
-          (numberFieldTowerTopSubgroup_le_baseSubgroup K L)) := by
-  let e :
-      ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-          extensionSubgroup
-            (numberFieldTowerBaseSubgroup K L)
-            (numberFieldTowerTopSubgroup L)
-            (numberFieldTowerTopSubgroup_le_baseSubgroup K L)) ≃*
-        Gal(L / K) :=
-    numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-  exact
-    { is_comm :=
-        ⟨fun x y => by
-          apply e.injective
-          rw [map_mul, map_mul]
-          exact
-            (inferInstance :
-              IsMulCommutative (Gal(L / K))).is_comm.comm
-                (e x) (e y)⟩ }
-
-/-- The concrete auxiliary fixed field attached to a simultaneous
-finite/cyclotomic lift. -/
-noncomputable def numberFieldTowerFinitePadicCyclicFixedField
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    IntermediateField ℚ (SeparableClosure ℚ) := by
-  exact
-    IntermediateField.fixedField
-      (numberFieldTowerFinitePadicCyclicFixedSubgroup
-        (K := K) (L := L) p τ).toSubgroup
-
-/-- A nonzero-degree lift produces a genuine number field: its
-concrete fixed field is finite over `ℚ`. -/
-theorem numberFieldTowerFinitePadicCyclicFixedField_finiteDimensional
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hτ :
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ ≠
-        1) :
-    FiniteDimensional ℚ
-      (numberFieldTowerFinitePadicCyclicFixedField
-        (K := K) (L := L) p τ) := by
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let F :=
-    numberFieldTowerFinitePadicCyclicFixedField
-      (K := K) (L := L) p τ
-  apply
-    (InfiniteGalois.isOpen_iff_finite
-      (K := SeparableClosure ℚ) F).1
-  change IsOpen
-    (IntermediateField.fixedField S.toSubgroup).fixingSubgroup.carrier
-  rw [InfiniteGalois.fixingSubgroup_fixedField S]
-  exact
-    numberFieldTowerFinitePadicCyclicFixedSubgroup_isOpen
-      (K := K) (L := L) p τ hτ
-
-/-- The compatible embedded copy of `K` lies in every auxiliary
-cyclic fixed field. -/
-theorem numberFieldTowerBaseField_le_finitePadicCyclicFixedField
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    numberFieldTowerBaseField K L ≤
-      numberFieldTowerFinitePadicCyclicFixedField
-        (K := K) (L := L) p τ := by
-  intro x hx
-  change x ∈ IntermediateField.fixedField
-    (numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ).toSubgroup
-  rw [IntermediateField.mem_fixedField_iff]
-  intro σ hσ
-  change
-    σ ∈
-      (numberFieldTowerFinitePadicCyclicFixedSubgroup
-        (K := K) (L := L) p τ).toSubgroup at hσ
-  obtain ⟨u, hu, rfl⟩ := hσ
-  exact
-    (IntermediateField.mem_fixingSubgroup_iff
-      (numberFieldTowerBaseField K L) u.1).1 u.2 x hx
-
-/-- The compatible embedding of the original base field into the
-genuine auxiliary fixed field. -/
-noncomputable def numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    K →ₐ[ℚ]
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ) := by
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ)
-      (numberFieldTowerFinitePadicCyclicFixedSubgroup
-        (K := K) (L := L) p τ)
-  exact
-    (numberFieldTowerLowerEmbedding K L).codRestrict F.toSubalgebra
-      (fun x =>
-        numberFieldTowerBaseField_le_finitePadicCyclicFixedField
-          (K := K) (L := L) p τ ⟨x, rfl⟩)
-
-/-- Coercing the auxiliary base embedding recovers the fixed lower embedding
-into the rational separable closure. -/
-@[simp]
-theorem numberFieldTowerFinitePadicAuxiliaryBaseEmbedding_coe
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (x : K) :
-    ((numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
-        (K := K) (L := L) p τ x :
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ)) :
-      SeparableClosure ℚ) =
-        numberFieldTowerLowerEmbedding K L x := by
-  rfl
-
-/-- The compatible copy of the original top field lies in the
-auxiliary compositum fixed field. -/
-theorem numberFieldTowerTopField_mem_finitePadicAuxiliaryTopField
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    {x : SeparableClosure ℚ}
-    (hx : x ∈ numberFieldInRationalSeparableClosure L) :
-    x ∈
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below := by
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let T :=
-    numberFieldTowerTopSubgroup L
-  change x ∈ IntermediateField.fixedField
-    (S.toSubgroup ⊓ T.toSubgroup)
-  rw [IntermediateField.mem_fixedField_iff]
-  intro σ hσ
-  have hσT : σ ∈ T.toSubgroup :=
-    hσ.2
-  change
-    σ ∈
-      (numberFieldInRationalSeparableClosure L).fixingSubgroup
-        at hσT
-  exact
-    (IntermediateField.mem_fixingSubgroup_iff
-      (numberFieldInRationalSeparableClosure L) σ).1
-      hσT x hx
-
-/-- The compatible embedding of the original top field into the
-auxiliary compositum fixed field. -/
-noncomputable def numberFieldTowerFinitePadicAuxiliaryTopEmbedding
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    L →ₐ[ℚ]
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below := by
-  let E :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ)
-      (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-        (K := K) (L := L) p τ).below
-  exact
-    (numberFieldSeparableClosureEmbedding L).codRestrict
-      (E.restrictScalars ℚ).toSubalgebra
-      (fun x =>
-        numberFieldTowerTopField_mem_finitePadicAuxiliaryTopField
-          (K := K) (L := L) p τ ⟨x, rfl⟩)
-
-/-- Coercing the auxiliary top embedding recovers the chosen top-field
-embedding into the rational separable closure. -/
-@[simp]
-theorem numberFieldTowerFinitePadicAuxiliaryTopEmbedding_coe
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (x : L) :
-    ((numberFieldTowerFinitePadicAuxiliaryTopEmbedding
-        (K := K) (L := L) p τ x :
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) :
-      SeparableClosure ℚ) =
-        numberFieldSeparableClosureEmbedding L x := by
-  rfl
-
-/-- The compatible base and top embeddings form the actual
-base-change square inside the rational separable closure. -/
-theorem numberFieldTowerFinitePadicAuxiliaryEmbedding_algebraMap
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (x : K) :
-    ((numberFieldTowerFinitePadicAuxiliaryTopEmbedding
-        (K := K) (L := L) p τ (algebraMap K L x) :
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) :
-      SeparableClosure ℚ) =
-    ((numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
-        (K := K) (L := L) p τ x :
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ)) :
-      SeparableClosure ℚ) := by
-  rfl
-
-noncomputable instance
-    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    Algebra K
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ)) := by
-  exact
-    (numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
-      (K := K) (L := L) p τ).toRingHom.toAlgebra
-
-instance
-    numberFieldTowerFinitePadicAuxiliary_baseRatScalarTower
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    IsScalarTower ℚ K
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ)) := by
-  exact
-    IsScalarTower.of_algebraMap_eq'
-      (numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
-        (K := K) (L := L) p τ).comp_algebraMap.symm
-
-noncomputable instance
-    numberFieldTowerFinitePadicAuxiliary_topAlgebra
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    Algebra L
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) := by
-  exact
-    (numberFieldTowerFinitePadicAuxiliaryTopEmbedding
-      (K := K) (L := L) p τ).toRingHom.toAlgebra
-
-instance
-    numberFieldTowerFinitePadicAuxiliary_topRatScalarTower
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    IsScalarTower ℚ L
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) := by
-  exact
-    IsScalarTower.of_algebraMap_eq'
-      (numberFieldTowerFinitePadicAuxiliaryTopEmbedding
-        (K := K) (L := L) p τ).comp_algebraMap.symm
-
-noncomputable instance
-    numberFieldTowerFinitePadicAuxiliary_originalBaseTopAlgebra
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    Algebra K
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) := by
-  exact
-    ((numberFieldTowerFinitePadicAuxiliaryTopEmbedding
-        (K := K) (L := L) p τ).comp
-      (IsScalarTower.toAlgHom ℚ K L)).toRingHom.toAlgebra
-
-instance
-    numberFieldTowerFinitePadicAuxiliary_originalTopScalarTower
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    IsScalarTower K L
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) := by
-  exact IsScalarTower.of_algebraMap_eq' rfl
-
-instance
-    numberFieldTowerFinitePadicAuxiliary_baseTopScalarTower
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    IsScalarTower K
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ))
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) := by
-  apply IsScalarTower.of_algebraMap_eq'
-  apply RingHom.ext
-  intro x
-  apply Subtype.ext
-  exact
-    numberFieldTowerFinitePadicAuxiliaryEmbedding_algebraMap
-      (K := K) (L := L) p τ x
-
-/-- The genuine auxiliary fixed field is Galois over the original
-base field through the compatible embedding above. -/
-theorem numberFieldTowerFinitePadicAuxiliaryBase_isGalois
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    IsGalois K
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ)) := by
-  let H :=
-    numberFieldTowerBaseSubgroup K L
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let hSH :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup_le_baseSubgroup
-      (K := K) (L := L) p τ
-  let B :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) H
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) S
-  let FB :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) hSH
-  let auxiliaryBaseAlgebra : Algebra B FB :=
-    (LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) hSH).algebra
-  let auxiliaryBaseGalois : IsGalois B FB :=
-    LocalClassFieldTheory.abstractRelativeFixedField_isGalois
-      ℚ (SeparableClosure ℚ) H S hSH
-      (numberFieldTowerFinitePadicCyclicFixedSubgroup_extension_normal
-        (K := K) (L := L) p τ)
-  let eK :=
-    numberFieldTowerAbstractBaseFieldEquiv K L
-  refine
-    @IsGalois.of_equiv_equiv
-      B FB _ _ auxiliaryBaseAlgebra
-      K F _ _
-      (numberFieldTowerFinitePadicAuxiliary_baseAlgebra
-        (K := K) (L := L) p τ)
-      auxiliaryBaseGalois
-      eK.symm.toRingEquiv (RingEquiv.refl F) ?_
-  apply RingHom.ext
-  intro x
-  apply Subtype.ext
-  change
-    ((eK (eK.symm x) : B) : SeparableClosure ℚ) =
-      (x : SeparableClosure ℚ)
-  exact
-    congrArg Subtype.val (eK.apply_symm_apply x)
-
-/-- The distinguished absolute lift, regarded as an element of the
-auxiliary base subgroup. -/
-noncomputable def numberFieldTowerFinitePadicAuxiliarySubgroupLift
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    (numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ).toSubgroup :=
-  ⟨τ.1,
-    numberFieldTowerFinitePadicCyclicFixedSubgroup_generator_mem
-      (K := K) (L := L) p τ⟩
-
-/-- The actual automorphism of the auxiliary compositum induced by
-the distinguished simultaneous finite/cyclotomic lift. -/
-noncomputable def numberFieldTowerFinitePadicAuxiliaryAutomorphism
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    let P :=
-      numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-        (K := K) (L := L) p τ
-    let S :=
-      numberFieldTowerFinitePadicCyclicFixedSubgroup
-        (K := K) (L := L) p τ
-    Gal(
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below /
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) S) := by
-  dsimp only
-  let P :=
-    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-      (K := K) (L := L) p τ
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let σS :=
-    numberFieldTowerFinitePadicAuxiliarySubgroupLift
-      (K := K) (L := L) p τ
-  exact
-    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
-      ℚ (SeparableClosure ℚ)
-      S P.field P.below
-      P.toFiniteGaloisExtension.normal
-      (QuotientGroup.mk'
-        (extensionSubgroup
-          S P.field P.below)
-        σS)
-
-/-- On the common separable closure, the auxiliary automorphism acts
-by the original distinguished ambient lift. -/
-theorem numberFieldTowerFinitePadicAuxiliaryAutomorphism_apply_val
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (x :
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) :
-    ((numberFieldTowerFinitePadicAuxiliaryAutomorphism
-        (K := K) (L := L) p τ x :
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) :
-      SeparableClosure ℚ) =
-        τ.1 (x : SeparableClosure ℚ) := by
-  let P :=
-    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-      (K := K) (L := L) p τ
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let σS :=
-    numberFieldTowerFinitePadicAuxiliarySubgroupLift
-      (K := K) (L := L) p τ
-  exact
-    (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup_mk_apply_val
-        ℚ (SeparableClosure ℚ)
-        S P.field P.below
-        P.toFiniteGaloisExtension.normal σS x).symm
-
-/-- Restricting the auxiliary automorphism through the actual
-base-change square recovers the finite quotient coordinate of the
-distinguished lift. -/
-theorem numberFieldTowerFinitePadicAuxiliaryAutomorphism_restriction
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    ((AlgEquiv.restrictNormalHom L).comp
-        (AlgEquiv.restrictScalarsHom K))
-        (numberFieldTowerFinitePadicAuxiliaryAutomorphism
-          (K := K) (L := L) p τ) =
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-        (numberFieldTowerFiniteQuotientCoordinate
-          (K := K) (L := L) τ) := by
-  let E :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ)
-      (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-        (K := K) (L := L) p τ).below
-  let σE :=
-    numberFieldTowerFinitePadicAuxiliaryAutomorphism
-      (K := K) (L := L) p τ
-  apply AlgEquiv.ext
-  intro x
-  apply (numberFieldSeparableClosureEmbedding L).injective
-  calc
-    numberFieldSeparableClosureEmbedding L
-        (((AlgEquiv.restrictNormalHom L).comp
-          (AlgEquiv.restrictScalarsHom K)) σE x) =
-      ((σE (algebraMap L E x) : E) :
-        SeparableClosure ℚ) := by
-          exact congrArg Subtype.val
-            (AlgEquiv.restrictNormal_commutes
-              ((AlgEquiv.restrictScalarsHom K) σE) L x)
-    _ = τ.1 (numberFieldSeparableClosureEmbedding L x) := by
-      rw [
-        numberFieldTowerFinitePadicAuxiliaryAutomorphism_apply_val
-          (K := K) (L := L) p τ]
-      rfl
-    _ =
-      numberFieldSeparableClosureEmbedding L
-        (numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-          (numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) τ) x) := by
-      exact
-        (numberFieldTowerExtensionQuotientEquivGaloisGroup_mk_apply
-          (K := K) (L := L) τ x).symm
-
-/-- Restriction of the chosen separable-closure place to the genuine
-auxiliary base field.  This is an exact extension of the original
-finite place of `K`, not merely an equivalent valuation. -/
-noncomputable def
-    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    AbsoluteValueExtension
-      (NumberField.HeightOneSpectrum.adicAbv K v)
-      (LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ)) := by
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ)
-      (numberFieldTowerFinitePadicCyclicFixedSubgroup
-        (K := K) (L := L) p τ)
-  let wΩ :=
-    numberFieldTowerFinitePlaceExtensionToSeparableClosure
-      K L v (chosenFinitePlaceExtension (L := L) v)
-  refine
-    ⟨wΩ.1.comp (f := F.val.toRingHom) F.val.injective, ?_⟩
-  intro x
-  change
-    wΩ.1 (numberFieldTowerLowerEmbedding K L x) =
-      NumberField.HeightOneSpectrum.adicAbv K v x
-  exact wΩ.2 x
-
-/-- Restriction of the same separable-closure place to the genuine
-auxiliary compositum.  Its restriction to `K` agrees exactly with the
-original normalized finite absolute value. -/
-noncomputable def
-    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    AbsoluteValueExtension
-      (NumberField.HeightOneSpectrum.adicAbv K v)
-      (LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ)
-        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-          (K := K) (L := L) p τ).below) := by
-  let E :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ)
-      (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-        (K := K) (L := L) p τ).below
-  let wΩ :=
-    numberFieldTowerFinitePlaceExtensionToSeparableClosure
-      K L v (chosenFinitePlaceExtension (L := L) v)
-  refine
-    ⟨wΩ.1.comp (f := E.val.toRingHom) E.val.injective, ?_⟩
-  intro x
-  change
-    wΩ.1 (numberFieldTowerLowerEmbedding K L x) =
-      NumberField.HeightOneSpectrum.adicAbv K v x
-  exact wΩ.2 x
-
-/-- The centre of the restricted place on the auxiliary compositum
-lies above the centre of the same place on the auxiliary base field. -/
-theorem
-    numberFieldTowerFinitePadicAuxiliaryTopPlace_below
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hτ :
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ ≠
-        1) :
-    let S :=
-      numberFieldTowerFinitePadicCyclicFixedSubgroup
-        (K := K) (L := L) p τ
-    let P :=
-      numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-        (K := K) (L := L) p τ
-    let F :=
-      LocalClassFieldTheory.abstractFixedField
-        ℚ (SeparableClosure ℚ) S
-    let E :=
-      LocalClassFieldTheory.abstractRelativeFixedField
-        ℚ (SeparableClosure ℚ) P.below
-    letI hHfinite : Finite
-        ((baseField
-          (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-          extensionSubgroup
-            (baseField
-              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-            S (le_baseField S)) :=
-      (numberFieldTowerFinitePadicAuxiliaryAbstractField
-        (K := K) (L := L) p τ hτ).finite
-    letI hPfinite : Finite
-        (S.toSubgroup ⧸
-          extensionSubgroup S P.field P.below) :=
-      P.finite
-    letI _ : FiniteDimensional ℚ F :=
-      LocalClassFieldTheory.abstractFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) S hHfinite
-    letI _ : FiniteDimensional F E :=
-      LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ)
-        S P.field P.below hHfinite hPfinite
-    letI _ : IsScalarTower ℚ F E :=
-      IsScalarTower.of_algebraMap_eq' rfl
-    letI _ : FiniteDimensional ℚ E :=
-      FiniteDimensional.trans ℚ F E
-    letI _ : NumberField F :=
-      NumberField.of_module_finite ℚ F
-    letI _ : NumberField E :=
-      NumberField.of_module_finite ℚ E
-    letI _ : Algebra K F :=
-      numberFieldTowerFinitePadicAuxiliary_baseAlgebra
-        (K := K) (L := L) p τ
-    finitePlaceBelow (K := F)
-        (finitePlaceExtensionCentre
-          (K := K) (L := E) v
-          (numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-            (K := K) (L := L) v p τ)) =
-      finitePlaceExtensionCentre
-        (K := K) (L := F) v
-        (numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
-          (K := K) (L := L) v p τ) := by
-  dsimp only
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let P :=
-    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-      (K := K) (L := L) p τ
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) S
-  let E :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let hHfinite : Finite
-      ((baseField
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-        extensionSubgroup
-          (baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-          S (le_baseField S)) :=
-    (numberFieldTowerFinitePadicAuxiliaryAbstractField
-      (K := K) (L := L) p τ hτ).finite
-  let hPfinite : Finite
-      (S.toSubgroup ⧸
-        extensionSubgroup S P.field P.below) :=
-    P.finite
-  let auxiliaryBaseFiniteDimensional : FiniteDimensional ℚ F :=
-    LocalClassFieldTheory.abstractFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ) S hHfinite
-  let auxiliaryTopFiniteDimensional : FiniteDimensional F E :=
-    LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ)
-      S P.field P.below hHfinite hPfinite
-  let auxiliaryScalarTower : IsScalarTower ℚ F E :=
-    IsScalarTower.of_algebraMap_eq' rfl
-  let auxiliaryAbsoluteFiniteDimensional : FiniteDimensional ℚ E :=
-    FiniteDimensional.trans ℚ F E
-  let auxiliaryBaseNumberField : NumberField F :=
-    NumberField.of_module_finite ℚ F
-  let auxiliaryTopNumberField : NumberField E :=
-    NumberField.of_module_finite ℚ E
-  let auxiliaryOriginalBaseAlgebra : Algebra K F :=
-    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
-      (K := K) (L := L) p τ
-  let wF :=
-    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
-      (K := K) (L := L) v p τ
-  let wE :=
-    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-      (K := K) (L := L) v p τ
-  apply HeightOneSpectrum.ext
-  ext x
-  change
-    algebraMap (𝓞 F) (𝓞 E) x ∈
-        finitePlaceExtensionCentreIdeal
-          (K := K) (L := E) v wE ↔
-      x ∈
-        finitePlaceExtensionCentreIdeal
-          (K := K) (L := F) v wF
-  rw [
-    mem_finitePlaceExtensionCentreIdeal_iff,
-    mem_finitePlaceExtensionCentreIdeal_iff]
-  rfl
-
-/-- Evaluation of the distinguished auxiliary automorphism through the
-restricted top-field place.  Isolating this coercion calculation prevents the
-whole decomposition-group proof from normalizing the fixed-field tower. -/
-private opaque
-    numberFieldTowerFinitePadicAuxiliaryTopPlace_automorphism_apply
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
-    let P := numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-      (K := K) (L := L) p τ
-    let E := LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-    let wE := numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-      (K := K) (L := L) v p τ
-    let σE := numberFieldTowerFinitePadicAuxiliaryAutomorphism
-      (K := K) (L := L) p τ
-    ∀ x : E,
-      wE.1 (σE x) =
-        (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-          K L v (chosenFinitePlaceExtension (L := L) v)).1
-            (τ.1 (x : SeparableClosure ℚ)) := by
-  let : Algebra K (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureBaseAlgebra K L
-  dsimp only
-  intro x
-  change
-    (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-      K L v (chosenFinitePlaceExtension (L := L) v)).1
-        ((numberFieldTowerFinitePadicAuxiliaryAutomorphism
-          (K := K) (L := L) p τ x :
-            LocalClassFieldTheory.abstractRelativeFixedField
-              ℚ (SeparableClosure ℚ)
-              (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-                (K := K) (L := L) p τ).below) : SeparableClosure ℚ) = _
-  rw [
-    numberFieldTowerFinitePadicAuxiliaryAutomorphism_apply_val
-      (K := K) (L := L) p τ]
-
-/-- The distinguished auxiliary automorphism preserves the top-field place
-obtained by restricting the original separable-closure place. -/
-private opaque numberFieldTowerFinitePadicAuxiliaryAutomorphism_mem_topPlaceDecomposition
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hdecomposition :
-      letI : Algebra K (SeparableClosure ℚ) :=
-        numberFieldTowerSeparableClosureBaseAlgebra K L
-      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
-        absoluteValueDecompositionGroup K
-          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-            K L v (chosenFinitePlaceExtension (L := L) v)).1) :
-    let S := numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-    let F := LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) S
-    let wE := numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-      (K := K) (L := L) v p τ
-    let σE := numberFieldTowerFinitePadicAuxiliaryAutomorphism
-      (K := K) (L := L) p τ
-    σE ∈ absoluteValueDecompositionGroup F wE.1 := by
-  let : Algebra K (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureBaseAlgebra K L
-  dsimp only at hdecomposition ⊢
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) S
-  let wE :=
-    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-      (K := K) (L := L) v p τ
-  let σE :=
-    numberFieldTowerFinitePadicAuxiliaryAutomorphism
-      (K := K) (L := L) p τ
-  intro x
-  change
-    wE.1 (σE x) < 1 ↔
-      wE.1 x < 1
-  rw [
-    numberFieldTowerFinitePadicAuxiliaryTopPlace_automorphism_apply
-      (K := K) (L := L) v p τ x,
-    show
-      wE.1 x =
-        (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-          K L v
-          (chosenFinitePlaceExtension (L := L) v)).1
-            (x : SeparableClosure ℚ) from rfl]
-  exact hdecomposition (x : SeparableClosure ℚ)
-
-/-- The restricted top-field place and the chosen extension above its centre
-have the same decomposition group. -/
-private opaque numberFieldTowerFinitePadicAuxiliaryTopDecompositionGroup_eq_chosen
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hτ :
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ ≠ 1) :
-    let S := numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-    let P := numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-      (K := K) (L := L) p τ
-    let F := LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) S
-    let E := LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-    letI hHfinite : Finite
-        ((baseField
-          (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-          extensionSubgroup
-            (baseField
-              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-            S (le_baseField S)) :=
-      (numberFieldTowerFinitePadicAuxiliaryAbstractField
-        (K := K) (L := L) p τ hτ).finite
-    letI hPfinite : Finite
-        (S.toSubgroup ⧸ extensionSubgroup S P.field P.below) := P.finite
-    letI : FiniteDimensional ℚ F :=
-      LocalClassFieldTheory.abstractFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) S hHfinite
-    letI : FiniteDimensional F E :=
-      LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) S P.field P.below hHfinite hPfinite
-    letI : IsScalarTower ℚ F E := IsScalarTower.of_algebraMap_eq' rfl
-    letI : FiniteDimensional ℚ E := FiniteDimensional.trans ℚ F E
-    letI : NumberField F := NumberField.of_module_finite ℚ F
-    letI : NumberField E := NumberField.of_module_finite ℚ E
-    letI : IsAbelianGalois F E :=
-      GlobalClassFields.finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
-    letI : Algebra K F :=
-      numberFieldTowerFinitePadicAuxiliary_baseAlgebra
-        (K := K) (L := L) p τ
-    let wF := numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
-      (K := K) (L := L) v p τ
-    let wE := numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-      (K := K) (L := L) v p τ
-    let V := finitePlaceExtensionCentre (K := K) (L := F) v wF
-    absoluteValueDecompositionGroup F wE.1 =
-      absoluteValueDecompositionGroup F
-        (chosenFinitePlaceExtension (L := E) V).1 := by
-  dsimp only
-  let H :=
-    numberFieldTowerFinitePadicAuxiliaryAbstractField
-      (K := K) (L := L) p τ hτ
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let P :=
-    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-      (K := K) (L := L) p τ
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) S
-  let E :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  let hHfinite : Finite
-      ((baseField
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-        extensionSubgroup
-          (baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-          S (le_baseField S)) :=
-    H.finite
-  let hPfinite : Finite
-      (S.toSubgroup ⧸ extensionSubgroup S P.field P.below) :=
-    P.finite
-  let : FiniteDimensional ℚ F :=
-    LocalClassFieldTheory.abstractFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ) S hHfinite
-  let : FiniteDimensional F E :=
-    LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
-      ℚ (SeparableClosure ℚ) S P.field P.below hHfinite hPfinite
-  let : IsScalarTower ℚ F E := IsScalarTower.of_algebraMap_eq' rfl
-  let : FiniteDimensional ℚ E := FiniteDimensional.trans ℚ F E
-  let : NumberField F := NumberField.of_module_finite ℚ F
-  let : NumberField E := NumberField.of_module_finite ℚ E
-  let : IsAbelianGalois F E :=
-    GlobalClassFields.finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
-  let : Algebra K F :=
-    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
-      (K := K) (L := L) p τ
-  let wF :=
-    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
-      (K := K) (L := L) v p τ
-  let wE :=
-    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-      (K := K) (L := L) v p τ
-  let V := finitePlaceExtensionCentre (K := K) (L := F) v wF
-  let W := finitePlaceExtensionCentre (K := K) (L := E) v wE
-  let Wover :
-      {W' : HeightOneSpectrum (𝓞 E) //
-        finitePlaceBelow (K := F) W' = V} :=
-    ⟨W,
-      numberFieldTowerFinitePadicAuxiliaryTopPlace_below
-        (K := K) (L := L) v p τ hτ⟩
-  let wFE :
-      AbsoluteValueExtension
-        (NumberField.HeightOneSpectrum.adicAbv F V) E :=
-    (finitePlaceExtensionEquivAbove
-      (K := F) (L := E) V).symm Wover
-  have hwFEcentre :
-      finitePlaceExtensionCentre
-          (K := F) (L := E) V wFE =
-        W := by
-    exact
-      congrArg Subtype.val
-        ((finitePlaceExtensionEquivAbove
-          (K := F) (L := E) V).apply_symm_apply Wover)
-  have hwEquiv : wE.1.IsEquiv wFE.1 := by
-    apply
-      finitePlaceExtensions_isEquiv_of_centres_eq
-        (F := K) (M := F) v V wE wFE
-    exact hwFEcentre.symm
-  calc
-    absoluteValueDecompositionGroup F wE.1 =
-        absoluteValueDecompositionGroup F wFE.1 :=
-      absoluteValueDecompositionGroup_eq_of_absoluteValue_isEquiv
-        (F := F) wE.1 wFE.1 hwEquiv
-    _ =
-      absoluteValueDecompositionGroup F
-          (chosenFinitePlaceExtension (L := E) V).1 :=
-      absoluteValueDecompositionGroup_eq_of_exactExtensions_of_isMulCommutative
-        (F := F)
-        (NumberField.HeightOneSpectrum.adicAbv F V)
-        (RayClass.adicAbv_isNontrivial V)
-        wFE
-        (chosenFinitePlaceExtension (L := E) V)
-
-/-- Pointwise form of norm/restriction naturality.  Keeping the function
-equality and its coercion normalization in this small declaration prevents
-the auxiliary-field witness construction below from repeatedly elaborating
-the full pair of composite homomorphisms. -/
-private theorem globalNormResidueMonoidHomOfEmbedding_norm_restriction_apply
-    (K K' L L' : Type)
-    [Field K] [NumberField K]
-    [Field K'] [NumberField K']
-    [Field L] [NumberField L]
-    [Field L'] [NumberField L']
-    [Algebra K K'] [Algebra K L] [Algebra K L']
-    [Algebra K' L'] [Algebra L L']
-    [IsScalarTower K K' L'] [IsScalarTower K L L']
-    [FiniteDimensional K L] [IsAbelianGalois K L]
-    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
-    [FiniteDimensional K K'] [IsGalois K K']
-    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
-    (c : IdeleClassGroup K') :
-    ((AlgEquiv.restrictNormalHom L).comp
-        (AlgEquiv.restrictScalarsHom K))
-          (globalNormResidueMonoidHomOfEmbedding K' L' j c) =
-      globalNormResidueMonoidHomOfEmbedding K L
-        (j.comp (IsScalarTower.toAlgHom ℚ L L'))
-        (_root_.ideleClassNorm K K' c) := by
-  exact
-    DFunLike.congr_fun
-      (globalNormResidueMonoidHomOfEmbedding_norm_restriction
-        (K := K) (L := L) (K' := K') (L' := L') j) c
-
-
-
-/-- The auxiliary-field construction produces a lower local unit
-whose chosen local Artin value and global norm-residue value are both
-the finite quotient coordinate of the distinguished lift. -/
-opaque numberFieldTowerFinitePadicAuxiliaryLocalGlobalRepresentative
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hτ :
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ ≠ 1)
-    (hdecomposition :
-      letI : Algebra K (SeparableClosure ℚ) :=
-        numberFieldTowerSeparableClosureBaseAlgebra K L
-      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
-        absoluteValueDecompositionGroup K
-          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-            K L v (chosenFinitePlaceExtension (L := L) v)).1)
-    (n : ℕ) (hn : 0 < n)
-    (hdegree :
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ =
-        (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n)
-    (hprimaryQuotient :
-      numberFieldTowerFiniteQuotientCoordinate
-          (K := K) (L := L) τ ∈
-        CommGroup.primaryComponent
-          ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-            extensionSubgroup
-              (numberFieldTowerBaseSubgroup K L)
-              (numberFieldTowerTopSubgroup L)
-              (numberFieldTowerTopSubgroup_le_baseSubgroup K L))
-          p.1) :
-    {z : (v.adicCompletion K)ˣ //
-      chosenFinitePlaceArtinMonoidHom (K := K) (L := L) v z =
-          numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-            (numberFieldTowerFiniteQuotientCoordinate
-              (K := K) (L := L) τ) ∧
-      globalNormResidueMonoidHom K L
-          (IdeleGroup.finitePlaceIdeleClass v z) =
-          numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-            (numberFieldTowerFiniteQuotientCoordinate
-              (K := K) (L := L) τ)} := by
-  let H :=
-    numberFieldTowerFinitePadicAuxiliaryAbstractField
-      (K := K) (L := L) p τ hτ
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let P :=
-    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
-      (K := K) (L := L) p τ
-  let F :=
-    LocalClassFieldTheory.abstractFixedField
-      ℚ (SeparableClosure ℚ) S
-  let E :=
-    LocalClassFieldTheory.abstractRelativeFixedField
-      ℚ (SeparableClosure ℚ) P.below
-  letI hHfinite : Finite
-      ((baseField
-        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
-        extensionSubgroup
-          (baseField
-            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
-          S (le_baseField S)) :=
-    H.finite
-  letI hPfinite : Finite
-      (S.toSubgroup ⧸ extensionSubgroup S P.field P.below) :=
-    P.finite
-  letI auxiliaryBaseNumberField : NumberField F := by
-    let : FiniteDimensional ℚ F :=
-      LocalClassFieldTheory.abstractFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ) S hHfinite
-    exact NumberField.of_module_finite ℚ F
-  letI auxiliaryTopNumberField : NumberField E := by
-    let : FiniteDimensional F E :=
-      LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
-        ℚ (SeparableClosure ℚ)
-        S P.field P.below hHfinite hPfinite
-    exact NumberField.of_module_finite F E
-  letI auxiliaryAbelianGalois : IsAbelianGalois F E :=
-    GlobalClassFields.finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
-  letI auxiliaryOriginalBaseAlgebra : Algebra K F :=
-    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
-      (K := K) (L := L) p τ
-  letI auxiliaryOriginalTopAlgebra : Algebra L E :=
-    numberFieldTowerFinitePadicAuxiliary_topAlgebra
-      (K := K) (L := L) p τ
-  letI auxiliaryOriginalBaseTopAlgebra : Algebra K E :=
-    numberFieldTowerFinitePadicAuxiliary_originalBaseTopAlgebra
-      (K := K) (L := L) p τ
-  letI auxiliaryOriginalTopScalarTower : IsScalarTower K L E :=
-    numberFieldTowerFinitePadicAuxiliary_originalTopScalarTower
-      (K := K) (L := L) p τ
-  letI auxiliaryBaseTopScalarTower : IsScalarTower K F E :=
-    numberFieldTowerFinitePadicAuxiliary_baseTopScalarTower
-      (K := K) (L := L) p τ
-  letI auxiliaryOriginalBaseGalois : IsGalois K F :=
-    numberFieldTowerFinitePadicAuxiliaryBase_isGalois
-      (K := K) (L := L) p τ
-  let auxiliaryOriginalTopAlgHom : L →ₐ[ℚ] E :=
-    numberFieldTowerFinitePadicAuxiliaryTopEmbedding
-      (K := K) (L := L) p τ
-  let wF :=
-    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
-      (K := K) (L := L) v p τ
-  let wE :=
-    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
-      (K := K) (L := L) v p τ
-  let V :=
-    finitePlaceExtensionCentre
-      (K := K) (L := F) v wF
-  have hVbelow : finitePlaceBelow (K := K) V = v :=
-    finitePlaceBelow_finitePlaceExtensionCentre
-      (K := K) (L := F) v wF
-  let Vover :
-      {V' : HeightOneSpectrum (𝓞 F) //
-        finitePlaceBelow (K := K) V' = v} :=
-    ⟨V, hVbelow⟩
-  letI auxiliaryCompletionAlgebra :
-      Algebra (v.adicCompletion K) (V.adicCompletion F) :=
-    (finitePlaceAdicCompletionMap K F v Vover).toAlgebra
-  let σE : Gal(E / F) :=
-    numberFieldTowerFinitePadicAuxiliaryAutomorphism
-      (K := K) (L := L) p τ
-  have hσtop :
-      σE ∈ absoluteValueDecompositionGroup F wE.1 :=
-    numberFieldTowerFinitePadicAuxiliaryAutomorphism_mem_topPlaceDecomposition
-      (K := K) (L := L) v p τ hdecomposition
-  have hgroup :=
-    numberFieldTowerFinitePadicAuxiliaryTopDecompositionGroup_eq_chosen
-      (K := K) (L := L) v p τ hτ
-  have hσchosen :
-      σE ∈ absoluteValueDecompositionGroup F
-        (chosenFinitePlaceExtension (L := E) V).1 := by
-    rw [← hgroup]
-    exact hσtop
-  have hRange :
-      σE ∈ (chosenFinitePlaceArtinMonoidHom
-        (K := F) (L := E) V).range := by
-    rw [chosenFinitePlaceArtinMonoidHom_range (K := F) (L := E) V]
-    exact hσchosen
-  let y : (V.adicCompletion F)ˣ := Classical.choose hRange
-  have hy :
-      chosenFinitePlaceArtinMonoidHom (K := F) (L := E) V y =
-        numberFieldTowerFinitePadicAuxiliaryAutomorphism
-          (K := K) (L := L) p τ :=
-    Classical.choose_spec hRange
-  let z : (v.adicCompletion K)ˣ :=
-    LocalFieldTheory.normUnits
-      (v.adicCompletion K) (V.adicCompletion F) y
-  have hz :
-      z = LocalFieldTheory.normUnits
-        (v.adicCompletion K) (V.adicCompletion F) y := by
-    rfl
-  let σK : Gal(L / K) :=
-    numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-      (numberFieldTowerFiniteQuotientCoordinate
-        (K := K) (L := L) τ)
-  let restriction : Gal(E / F) →* Gal(L / K) :=
-    (AlgEquiv.restrictNormalHom L).comp
-      (AlgEquiv.restrictScalarsHom K)
-  have hrestrict : restriction σE = σK :=
-    numberFieldTowerFinitePadicAuxiliaryAutomorphism_restriction
-      (K := K) (L := L) p τ
-  have hlocal :
-      chosenFinitePlaceArtinMonoidHom (K := K) (L := L) v z =
-        σK := by
-    rw [hz]
-    have hnat :=
-      DFunLike.congr_fun
-        (chosenFinitePlaceArtinMonoidHom_norm_restriction_of_below_eq
-          (K := K) (L := L) (K' := F) (L' := E)
-          v V hVbelow) y
-    calc
-      chosenFinitePlaceArtinMonoidHom (K := K) (L := L) v
-          (LocalFieldTheory.normUnits
-            (v.adicCompletion K) (V.adicCompletion F) y) =
-        restriction
-          (chosenFinitePlaceArtinMonoidHom (K := F) (L := E) V y) := by
-            simpa only [
-              MonoidHom.coe_comp, Function.comp_apply, restriction]
-              using hnat.symm
-      _ = restriction σE := congrArg restriction hy
-      _ = σK := hrestrict
-  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
-    E.val.restrictScalars ℚ
-  have hjLower :
-      j.comp auxiliaryOriginalTopAlgHom =
-        AlgebraicNumberTheory.numberFieldSeparableClosureEmbedding L := by
-    apply AlgHom.ext
-    intro a
-    exact
-      numberFieldTowerFinitePadicAuxiliaryTopEmbedding_coe
-        (K := K) (L := L) p τ a
-  have hPUnramified :
-      P.toFiniteGaloisExtension.IsUnramified
-        rationalCyclotomicDegreeData :=
-    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension_isUnramified
-      (K := K) (L := L) p τ n hn hdegree hprimaryQuotient
-  have hcompat :
-      (globalNormResidueMonoidHomOfEmbedding F E j).comp
-          (IdeleGroup.finitePlaceIdeleClass V) =
-        chosenFinitePlaceArtinMonoidHom (K := F) (L := E) V :=
-    globalNormResidueMonoidHomOfEmbedding_comp_finitePlaceIdeleClass_of_abstractFixedFieldUnramified
-      H P hPUnramified V
-  have hupper :
-      globalNormResidueMonoidHomOfEmbedding F E j
-          (IdeleGroup.finitePlaceIdeleClass V y) =
-        σE :=
-    (congrArg
-      (fun φ : (V.adicCompletion F)ˣ →* Gal(E / F) => φ y)
-      hcompat).trans hy
-  have hnormClass :
-      _root_.ideleClassNorm K F
-          (IdeleGroup.finitePlaceIdeleClass V y) =
-        IdeleGroup.finitePlaceIdeleClass v z := by
-    rw [hz]
-    simpa only [Vover] using
-      (IdeleGroup.ideleClassNorm_finitePlaceIdeleClass_eq_normUnits
-        (K := K) (L := F) v Vover y)
-  have hglobal :
-      globalNormResidueMonoidHom K L
-          (IdeleGroup.finitePlaceIdeleClass v z) =
-        σK := by
-    calc
-      globalNormResidueMonoidHom K L
-          (IdeleGroup.finitePlaceIdeleClass v z) =
-          globalNormResidueMonoidHom K L
-            (_root_.ideleClassNorm K F
-              (IdeleGroup.finitePlaceIdeleClass V y)) :=
-        congrArg (globalNormResidueMonoidHom K L) hnormClass.symm
-      _ = globalNormResidueMonoidHomOfEmbedding K L
-          (j.comp auxiliaryOriginalTopAlgHom)
-          (_root_.ideleClassNorm K F
-            (IdeleGroup.finitePlaceIdeleClass V y)) := by
-        rw [hjLower,
-          ← globalNormResidueMonoidHom_eq_ofEmbedding_standard]
-      _ = ((AlgEquiv.restrictNormalHom L).comp
-          (AlgEquiv.restrictScalarsHom K))
-          (globalNormResidueMonoidHomOfEmbedding F E j
-            (IdeleGroup.finitePlaceIdeleClass V y)) := by
-        apply Eq.symm
-        apply
-          globalNormResidueMonoidHomOfEmbedding_norm_restriction_apply
-      _ = ((AlgEquiv.restrictNormalHom L).comp
-          (AlgEquiv.restrictScalarsHom K)) σE :=
-        congrArg
-          ((AlgEquiv.restrictNormalHom L).comp
-            (AlgEquiv.restrictScalarsHom K)) hupper
-      _ = restriction σE := rfl
-      _ = σK := hrestrict
-  exact ⟨z, hlocal, hglobal⟩
-
-
-/-- Every genuine finite-place decomposition automorphism has a
-compatible embedded absolute lift with the same finite quotient class
-and positive integral cyclotomic `p`-adic degree. -/
-theorem exists_numberFieldTowerFinitePadicLift_of_finitePlace
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (σ :
-      absoluteValueDecompositionGroup K
-        (chosenFinitePlaceExtension (L := L) v).1) :
-    letI _ : Algebra K (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureBaseAlgebra K L
-    letI _ : Algebra L (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureTopAlgebra L
-    letI _ : IsScalarTower K L (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureScalarTower K L
-    ∃ τ : (numberFieldTowerBaseSubgroup K L).toSubgroup,
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-          (numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) τ) =
-        σ.1 ∧
-      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
-        absoluteValueDecompositionGroup K
-          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-            K L v
-            (chosenFinitePlaceExtension (L := L) v)).1 ∧
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ ≠
-        1 ∧
-      ∃ n : ℕ, 0 < n ∧
-        numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-            (K := K) (L := L) p τ =
-          (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
-  let : Algebra K (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureBaseAlgebra K L
-  let : Algebra L (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureTopAlgebra L
-  let : IsScalarTower K L (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureScalarTower K L
-  obtain ⟨τΩ, hτΩrestrict, n, hn, hτΩdegree⟩ :=
-    exists_finitePlaceSeparableClosureLift_with_positivePadicCyclotomicDegree
-      (K := K) (L := L) v p σ
-  let τ :
-      (numberFieldTowerBaseSubgroup K L).toSubgroup :=
-    numberFieldTowerSeparableClosureEquivBaseSubgroup
-      K L τΩ.1
-  have hfinite :
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-          (numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) τ) =
-        σ.1 := by
-    change
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-          (QuotientGroup.mk
-            (numberFieldTowerSeparableClosureEquivBaseSubgroup
-              K L τΩ.1)) =
-        σ.1
-    rw [
-      numberFieldTowerExtensionQuotientEquivGaloisGroup_mk_baseSubgroupEquiv]
-    exact congrArg Subtype.val hτΩrestrict
-  have hdegree :
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ =
-        (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
-    exact hτΩdegree
-  have hdecomposition :
-      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
-        absoluteValueDecompositionGroup K
-          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-            K L v
-            (chosenFinitePlaceExtension (L := L) v)).1 := by
-    simpa only [τ, MulEquiv.symm_apply_apply] using τΩ.2
-  refine
-    ⟨τ, hfinite, hdecomposition, ?_, n, hn, hdegree⟩
-  rw [hdegree]
-  exact
-    PadicInt.multiplicative_positiveNatDegree_ne_one
-      p.1 n hn
-
-/-- Generation of the actual finite Galois group transports back
-through the compatible finite quotient coordinate. -/
-theorem
-    numberFieldTowerFiniteQuotientCoordinate_generates_of_galoisGenerator
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (σ : Gal(L / K))
-    (hτσ :
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-          (numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) τ) =
-        σ)
-    (hσ : Subgroup.closure ({σ} : Set (Gal(L / K))) = ⊤) :
-    Subgroup.closure
-        ({numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) τ} :
-          Set
-            ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-              extensionSubgroup
-                (numberFieldTowerBaseSubgroup K L)
-                (numberFieldTowerTopSubgroup L)
-                (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
-      ⊤ := by
-  let e :=
-    numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-  let q :
-      (numberFieldTowerFiniteGaloisSubextension
-        K L).extensionQuotient :=
-    numberFieldTowerFiniteQuotientCoordinate
-      (K := K) (L := L) τ
-  have hq : e.toMonoidHom q = σ := by
-    exact hτσ
-  change
-    Subgroup.closure
-        ({q} :
-          Set
-            (numberFieldTowerFiniteGaloisSubextension
-              K L).extensionQuotient) =
-      ⊤
-  apply Subgroup.map_injective (f := e.toMonoidHom) e.injective
-  rw [MonoidHom.map_closure, Set.image_singleton,
-    hq, hσ,
-    Subgroup.map_top_of_surjective e.toMonoidHom e.surjective]
-
-/-- If the finite quotient coordinate of a lift generates the whole
-finite Galois quotient, then its cyclic preimage together with the
-top-field subgroup generates the whole embedded absolute Galois
-group. -/
-theorem
-    numberFieldTowerFinitePadicCyclicPreimage_sup_extensionSubgroup_eq_top
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hgenerate :
-      Subgroup.closure
-          ({numberFieldTowerFiniteQuotientCoordinate
-              (K := K) (L := L) τ} :
-            Set
-              ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-                extensionSubgroup
-                  (numberFieldTowerBaseSubgroup K L)
-                  (numberFieldTowerTopSubgroup L)
-                  (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
-        ⊤) :
-    (numberFieldTowerFinitePadicCyclicPreimage
-          (K := K) (L := L) p τ).toSubgroup ⊔
-        extensionSubgroup
-          (numberFieldTowerBaseSubgroup K L)
-          (numberFieldTowerTopSubgroup L)
-          (numberFieldTowerTopSubgroup_le_baseSubgroup K L) =
-      ⊤ := by
-  let H :=
-    numberFieldTowerBaseSubgroup K L
-  let N :=
-    extensionSubgroup
-      H
-      (numberFieldTowerTopSubgroup L)
-      (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
-  let Q := H.toSubgroup ⧸ N
-  let P :=
-    numberFieldTowerFinitePadicImage
-      (K := K) (L := L) p
-  let coordinate :=
-    numberFieldTowerFinitePadicCoordinate
-      (K := K) (L := L) p
-  let rangeRestriction :=
-    numberFieldTowerFinitePadicRangeRestriction
-      (K := K) (L := L) p
-  let γ : P.toSubgroup :=
-    rangeRestriction τ
-  let Γ :=
-    ClassFormation.padicCyclicClosure γ
-  let finiteProjection :
-      P.toSubgroup →*
-        Q :=
-    (MonoidHom.fst Q
-      (Multiplicative ℤ_[p.1])).comp
-        P.toSubgroup.subtype
-  have hprojection :
-      Γ.toSubgroup.map finiteProjection = ⊤ := by
-    apply top_unique
-    rw [← hgenerate]
-    apply (Subgroup.closure_le _).2
-    intro q hq
-    rw [Set.mem_singleton_iff] at hq
-    subst q
-    refine
-      ⟨γ,
-        (ClassFormation.padicCyclicClosureGenerator γ).2,
-        ?_⟩
-    rfl
-  have hquotientSurjective :
-      ∀ q : Q,
-        ∃ u : H.toSubgroup,
-          u ∈
-              (numberFieldTowerFinitePadicCyclicPreimage
-                (K := K) (L := L) p τ).toSubgroup ∧
-            numberFieldTowerFiniteQuotientCoordinate
-                (K := K) (L := L) u =
-              q := by
-    intro q
-    have hq :
-        q ∈ Γ.toSubgroup.map finiteProjection := by
-      rw [hprojection]
-      trivial
-    obtain ⟨z, hzΓ, hzq⟩ := hq
-    obtain ⟨u, hu⟩ :=
-      numberFieldTowerFinitePadicRangeRestriction_surjective
-        (K := K) (L := L) p z
-    refine ⟨u, ?_, ?_⟩
-    · change rangeRestriction u ∈ Γ.toSubgroup
-      rw [hu]
-      exact hzΓ
-    · calc
-        numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) u =
-            finiteProjection (rangeRestriction u) := rfl
-        _ = finiteProjection z := congrArg finiteProjection hu
-        _ = q := hzq
-  apply top_unique
-  intro h _
-  obtain ⟨u, huU, huq⟩ :=
-    hquotientSurjective
-      (numberFieldTowerFiniteQuotientCoordinate
-        (K := K) (L := L) h)
-  let k : H.toSubgroup :=
-    h * u⁻¹
-  have hkN : k ∈ N := by
-    apply (QuotientGroup.eq_one_iff (N := N) k).mp
-    change
-      numberFieldTowerFiniteQuotientCoordinate
-          (K := K) (L := L) (h * u⁻¹) =
-        1
-    rw [map_mul, map_inv, huq, mul_inv_cancel]
-  have hkSup :
-      k ∈
-        (numberFieldTowerFinitePadicCyclicPreimage
-            (K := K) (L := L) p τ).toSubgroup ⊔
-          N :=
-    (le_sup_right :
-      N ≤
-        (numberFieldTowerFinitePadicCyclicPreimage
-            (K := K) (L := L) p τ).toSubgroup ⊔ N) hkN
-  have huSup :
-      u ∈
-        (numberFieldTowerFinitePadicCyclicPreimage
-            (K := K) (L := L) p τ).toSubgroup ⊔
-          N :=
-    (le_sup_left :
-      (numberFieldTowerFinitePadicCyclicPreimage
-          (K := K) (L := L) p τ).toSubgroup ≤
-        (numberFieldTowerFinitePadicCyclicPreimage
-            (K := K) (L := L) p τ).toSubgroup ⊔ N) huU
-  have hku :
-      k * u = h := by
-    simp only [k, inv_mul_cancel_right]
-  rw [← hku]
-  exact
-    ((numberFieldTowerFinitePadicCyclicPreimage
-      (K := K) (L := L) p τ).toSubgroup ⊔ N).mul_mem
-      hkSup huSup
-
-/-- Ambient form of the generation statement: the auxiliary cyclic
-fixed subgroup together with the subgroup fixing `L` generates the
-subgroup fixing `K`. -/
-theorem
-    numberFieldTowerFinitePadicCyclicFixedSubgroup_sup_topSubgroup
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hgenerate :
-      Subgroup.closure
-          ({numberFieldTowerFiniteQuotientCoordinate
-              (K := K) (L := L) τ} :
-            Set
-              ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-                extensionSubgroup
-                  (numberFieldTowerBaseSubgroup K L)
-                  (numberFieldTowerTopSubgroup L)
-                  (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
-        ⊤) :
-    (numberFieldTowerFinitePadicCyclicFixedSubgroup
-          (K := K) (L := L) p τ).toSubgroup ⊔
-        (numberFieldTowerTopSubgroup L).toSubgroup =
-      (numberFieldTowerBaseSubgroup K L).toSubgroup := by
-  let H :=
-    numberFieldTowerBaseSubgroup K L
-  let T :=
-    numberFieldTowerTopSubgroup L
-  let N :=
-    extensionSubgroup
-      H T
-      (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
-  let U :=
-    numberFieldTowerFinitePadicCyclicPreimage
-      (K := K) (L := L) p τ
-  have hrelative :
-      U.toSubgroup ⊔ N = ⊤ :=
-    numberFieldTowerFinitePadicCyclicPreimage_sup_extensionSubgroup_eq_top
-      (K := K) (L := L) p τ hgenerate
-  have hNmap :
-      N.map H.toSubgroup.subtype =
-        T.toSubgroup := by
-    exact
-      Subgroup.map_subgroupOf_eq_of_le
-        (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
-  change
-    U.toSubgroup.map H.toSubgroup.subtype ⊔
-        T.toSubgroup =
-      H.toSubgroup
-  rw [← hNmap, ← Subgroup.map_sup,
-    hrelative, ← MonoidHom.range_eq_map,
-    H.toSubgroup.range_subtype]
-
-/-- The concrete auxiliary fixed field is linearly disjoint from `L`
-over the compatible embedded copy of `K`. -/
-theorem
-    numberFieldTowerFinitePadicCyclicFixedField_inf_topField
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (hgenerate :
-      Subgroup.closure
-          ({numberFieldTowerFiniteQuotientCoordinate
-              (K := K) (L := L) τ} :
-            Set
-              ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-                extensionSubgroup
-                  (numberFieldTowerBaseSubgroup K L)
-                  (numberFieldTowerTopSubgroup L)
-                  (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
-        ⊤) :
-    numberFieldTowerFinitePadicCyclicFixedField
-          (K := K) (L := L) p τ ⊓
-        numberFieldInRationalSeparableClosure L =
-      numberFieldTowerBaseField K L := by
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let T :=
-    numberFieldTowerTopSubgroup L
-  let H :=
-    numberFieldTowerBaseSubgroup K L
-  change
-    IntermediateField.fixedField S.toSubgroup ⊓
-        numberFieldInRationalSeparableClosure L =
-      numberFieldTowerBaseField K L
-  rw [
-    ← InfiniteGalois.fixedField_fixingSubgroup
-      (numberFieldInRationalSeparableClosure L),
-    ← InfiniteGalois.fixedField_fixingSubgroup
-      (numberFieldTowerBaseField K L)]
-  change
-    IntermediateField.fixedField S.toSubgroup ⊓
-        IntermediateField.fixedField T.toSubgroup =
-      IntermediateField.fixedField H.toSubgroup
-  rw [
-    ← IntermediateField.fixedField_sup_eq_inf,
-    numberFieldTowerFinitePadicCyclicFixedSubgroup_sup_topSubgroup
-      (K := K) (L := L) p τ hgenerate]
-
-/-- If the finite quotient coordinate is `p`-primary, adjoining the
-actual rational `p`-primary cyclotomic field to the auxiliary fixed
-field contains the compatible copy of `L`.
-
-This is the field-theoretic conclusion of the simultaneous
-finite/cyclotomic lift: the intersection of the two fixing subgroups
-already fixes `L`, hence their fixed-field compositum contains `L`. -/
-theorem
-    numberFieldTowerTopField_le_finitePadicCyclicFixedField_sup_padicCyclotomicField
-    (p : Nat.Primes)
-    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
-    (n : ℕ) (hn : 0 < n)
-    (hdegree :
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ =
-        (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n)
-    (hprimary :
-      numberFieldTowerFiniteQuotientCoordinate
-          (K := K) (L := L) τ ∈
-        CommGroup.primaryComponent
-          ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-            extensionSubgroup
-              (numberFieldTowerBaseSubgroup K L)
-              (numberFieldTowerTopSubgroup L)
-              (numberFieldTowerTopSubgroup_le_baseSubgroup K L))
-          p.1) :
-    numberFieldInRationalSeparableClosure L ≤
-      numberFieldTowerFinitePadicCyclicFixedField
-            (K := K) (L := L) p τ ⊔
-        rationalCyclotomicPadicField p := by
-  let S :=
-    numberFieldTowerFinitePadicCyclicFixedSubgroup
-      (K := K) (L := L) p τ
-  let T :=
-    numberFieldTowerTopSubgroup L
-  let F :=
-    numberFieldTowerFinitePadicCyclicFixedField
-      (K := K) (L := L) p τ
-  let C : @IntermediateField ℚ (SeparableClosure ℚ) _ _
-      rationalSeparableClosureAlgebra :=
-    rationalCyclotomicPadicField p
-  have hfixing :
-      (F ⊔ C).fixingSubgroup ≤
-        T.toSubgroup := by
-    change
-      (IntermediateField.fixedField S.toSubgroup ⊔ C).fixingSubgroup ≤
-        T.toSubgroup
-    rw [
-      IntermediateField.fixingSubgroup_sup,
-      InfiniteGalois.fixingSubgroup_fixedField S]
-    intro σ hσ
-    apply
-      numberFieldTowerFinitePadicCyclicFixedSubgroup_inf_absolutePadicKernel_le_topSubgroup
-        (K := K) (L := L) p τ n hn hdegree hprimary
-    refine ⟨hσ.1, ?_⟩
-    let : Algebra ℚ rationalCyclotomicZHatField :=
-      rationalCyclotomicZHatField.algebra'
-    let : @Normal ℚ rationalCyclotomicZHatField _ _
-        rationalCyclotomicZHatField.algebra' :=
-      rationalCyclotomicZHatField_normal
-    let E :=
-      rationalCyclotomicPadicFieldWithinZHat p
-    change
-      rationalCyclotomicPadicCoordinate p
-          (rationalAbsoluteGaloisRestrictionToCyclotomicZHat σ) =
-        1
-    have hr :
-        rationalAbsoluteGaloisRestrictionToCyclotomicZHat σ ∈
-          E.fixingSubgroup := by
-      rw [IntermediateField.mem_fixingSubgroup_iff]
-      intro x hx
-      apply Subtype.ext
-      have hfix :
-          σ x.1 = x.1 :=
-        (IntermediateField.mem_fixingSubgroup_iff
-          (IntermediateField.lift E) σ).1 hσ.2 x.1
-            ((IntermediateField.mem_lift x).2 hx)
-      exact
-        (AlgEquiv.restrictNormal_commutes
-          σ rationalCyclotomicZHatField x).trans hfix
-    rw [rationalCyclotomicPadicFieldWithinZHat_fixingSubgroup]
-      at hr
-    exact hr
-  rw [
-    ← InfiniteGalois.fixedField_fixingSubgroup
-      (numberFieldInRationalSeparableClosure L)]
-  change
-    IntermediateField.fixedField T.toSubgroup ≤
-      F ⊔ C
-  rw [
-    ← InfiniteGalois.fixedField_fixingSubgroup
-      (F ⊔ C)]
-  exact
-    IntermediateField.fixedField_le hfixing
-
-/-- For a finite-place automorphism generating `Gal(L/K)`, construct
-the genuine auxiliary number field used in the cyclotomic reduction.
-
-The field is finite over `ℚ`, contains the compatible copy of `K`, and
-has intersection with the compatible copy of `L` exactly equal to that
-copy of `K`.  Its defining lift has the prescribed local restriction
-and positive integral cyclotomic `p`-adic degree. -/
-theorem exists_finitePlaceCyclotomicAuxiliaryFixedField
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (σ :
-      absoluteValueDecompositionGroup K
-        (chosenFinitePlaceExtension (L := L) v).1)
-    (hσ :
-      Subgroup.closure
-          ({σ.1} : Set (Gal(L / K))) =
-        ⊤) :
-    letI _ : Algebra K (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureBaseAlgebra K L
-    letI _ : Algebra L (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureTopAlgebra L
-    letI _ : IsScalarTower K L (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureScalarTower K L
-    ∃ τ : (numberFieldTowerBaseSubgroup K L).toSubgroup,
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-          (numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) τ) =
-        σ.1 ∧
-      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
-        absoluteValueDecompositionGroup K
-          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-            K L v
-            (chosenFinitePlaceExtension (L := L) v)).1 ∧
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ ≠
-        1 ∧
-      FiniteDimensional ℚ
-        (numberFieldTowerFinitePadicCyclicFixedField
-          (K := K) (L := L) p τ) ∧
-      numberFieldTowerBaseField K L ≤
-        numberFieldTowerFinitePadicCyclicFixedField
-          (K := K) (L := L) p τ ∧
-      numberFieldTowerFinitePadicCyclicFixedField
-            (K := K) (L := L) p τ ⊓
-          numberFieldInRationalSeparableClosure L =
-        numberFieldTowerBaseField K L ∧
-      ∃ n : ℕ, 0 < n ∧
-        numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-            (K := K) (L := L) p τ =
-          (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
-  let : Algebra K (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureBaseAlgebra K L
-  let : Algebra L (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureTopAlgebra L
-  let : IsScalarTower K L (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureScalarTower K L
-  obtain
-      ⟨τ, hτσ, hτdecomposition, hτdegree,
-        n, hn, hdegree⟩ :=
-    exists_numberFieldTowerFinitePadicLift_of_finitePlace
-      (K := K) (L := L) v p σ
-  have hgenerate :=
-    numberFieldTowerFiniteQuotientCoordinate_generates_of_galoisGenerator
-      (K := K) (L := L) τ σ.1 hτσ hσ
-  refine
-    ⟨τ, hτσ, hτdecomposition, hτdegree,
-      numberFieldTowerFinitePadicCyclicFixedField_finiteDimensional
-        (K := K) (L := L) p τ hτdegree,
-      numberFieldTowerBaseField_le_finitePadicCyclicFixedField
-        (K := K) (L := L) p τ,
-      numberFieldTowerFinitePadicCyclicFixedField_inf_topField
-        (K := K) (L := L) p τ hgenerate,
-      n, hn, hdegree⟩
-
-/-- A `p`-primary local generator admits a genuine auxiliary number
-field whose compositum with the rational `p`-primary cyclotomic field
-contains `L`.
-
-Besides the field containment, the construction records the two
-properties needed for descent: the auxiliary field meets `L` exactly
-in `K`, and the chosen absolute lift has positive integral
-cyclotomic degree. -/
-theorem exists_finitePlacePrimaryCyclotomicAuxiliaryFixedField
-    (v : HeightOneSpectrum (𝓞 K))
-    (p : Nat.Primes)
-    (σ :
-      absoluteValueDecompositionGroup K
-        (chosenFinitePlaceExtension (L := L) v).1)
-    (hgenerate :
-      Subgroup.closure
-          ({σ.1} : Set (Gal(L / K))) =
-        ⊤)
-    (hprimary :
-      σ.1 ∈
-        CommGroup.primaryComponent
-          (Gal(L / K)) p.1) :
-    letI _ : Algebra K (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureBaseAlgebra K L
-    letI _ : Algebra L (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureTopAlgebra L
-    letI _ : IsScalarTower K L (SeparableClosure ℚ) :=
-      numberFieldTowerSeparableClosureScalarTower K L
-    ∃ τ : (numberFieldTowerBaseSubgroup K L).toSubgroup,
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-          (numberFieldTowerFiniteQuotientCoordinate
-            (K := K) (L := L) τ) =
-        σ.1 ∧
-      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
-        absoluteValueDecompositionGroup K
-          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
-            K L v
-            (chosenFinitePlaceExtension (L := L) v)).1 ∧
-      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-          (K := K) (L := L) p τ ≠
-        1 ∧
-      FiniteDimensional ℚ
-        (numberFieldTowerFinitePadicCyclicFixedField
-          (K := K) (L := L) p τ) ∧
-      numberFieldTowerBaseField K L ≤
-        numberFieldTowerFinitePadicCyclicFixedField
-          (K := K) (L := L) p τ ∧
-      numberFieldTowerFinitePadicCyclicFixedField
-            (K := K) (L := L) p τ ⊓
-          numberFieldInRationalSeparableClosure L =
-        numberFieldTowerBaseField K L ∧
-      numberFieldInRationalSeparableClosure L ≤
-        numberFieldTowerFinitePadicCyclicFixedField
-              (K := K) (L := L) p τ ⊔
-          rationalCyclotomicPadicField p ∧
-      ∃ n : ℕ, 0 < n ∧
-        numberFieldTowerBaseSubgroupPadicCyclotomicDegree
-            (K := K) (L := L) p τ =
-          (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
-  let : Algebra K (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureBaseAlgebra K L
-  let : Algebra L (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureTopAlgebra L
-  let : IsScalarTower K L (SeparableClosure ℚ) :=
-    numberFieldTowerSeparableClosureScalarTower K L
-  obtain
-      ⟨τ, hτσ, hτdecomposition, hτdegree,
-        hfinite, hbase, hintersection,
-        n, hn, hdegree⟩ :=
-    exists_finitePlaceCyclotomicAuxiliaryFixedField
-      (K := K) (L := L) v p σ hgenerate
-  have hprimaryQuotient :
-      numberFieldTowerFiniteQuotientCoordinate
-          (K := K) (L := L) τ ∈
-        CommGroup.primaryComponent
-          ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
-            extensionSubgroup
-              (numberFieldTowerBaseSubgroup K L)
-              (numberFieldTowerTopSubgroup L)
-              (numberFieldTowerTopSubgroup_le_baseSubgroup K L))
-          p.1 := by
-    obtain ⟨m, hm⟩ := hprimary
-    refine ⟨m, ?_⟩
-    let e :=
-      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
-    let q :=
-      numberFieldTowerFiniteQuotientCoordinate
-        (K := K) (L := L) τ
-    let N : ℕ := p.1 ^ m
-    have hq : e q = σ.1 := by
-      exact hτσ
-    have hmN : σ.1 ^ N = 1 := by
-      exact hm
-    change q ^ N = 1
-    apply e.injective
-    calc
-      e (q ^ N) = (e q) ^ N := by
-        exact map_pow e q N
-      _ = σ.1 ^ N := by rw [hq]
-      _ = 1 := hmN
-      _ = e 1 := (map_one e).symm
-  have hcontainment :=
-    numberFieldTowerTopField_le_finitePadicCyclicFixedField_sup_padicCyclotomicField
-      (K := K) (L := L) p τ n hn hdegree
-        hprimaryQuotient
-  exact
-    ⟨τ, hτσ, hτdecomposition, hτdegree,
-      hfinite, hbase, hintersection,
-      hcontainment, n, hn, hdegree⟩
-
-end Reciprocity
-end GlobalClassFieldTheory
+/- Declaration-preserving split of the original source. See the immutable source-split receipt. -/
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicAuxiliaryFieldExistence
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldAutomorphisms.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldAutomorphisms.lean
new file mode 100644
index 0000000..05e27fe
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldAutomorphisms.lean
@@ -0,0 +1,593 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicAuxiliaryFieldBase
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicCyclicData
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicUnramifiedLocalGlobalCompatibility
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.Extension.OnePlaceBaseNorm
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open AlgebraicNumberTheory IsDedekindDomain NumberField
+open IdeleGroup RelativeIdeleGroup
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open CyclicCohomology
+open KummerTheory ClassFormation
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open GlobalClassFields
+
+variable
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+
+
+local instance splitFactPrimeAutomorphisms (p : Nat.Primes) : Fact p.1.Prime := ⟨p.2⟩
+
+attribute [local instance]
+  rationalSeparableClosureAlgebra
+  finitePadicAuxiliaryExtensionNormal
+  finitePadicAuxiliaryExtensionQuotientFinite
+  finitePadicAuxiliaryExtensionQuotientIsMulCommutative
+
+/-- The distinguished absolute lift, regarded as an element of the
+auxiliary base subgroup. -/
+noncomputable def numberFieldTowerFinitePadicAuxiliarySubgroupLift
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    (numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ).toSubgroup :=
+  ⟨τ.1,
+    numberFieldTowerFinitePadicCyclicFixedSubgroup_generator_mem
+      (K := K) (L := L) p τ⟩
+
+/-- The actual automorphism of the auxiliary compositum induced by
+the distinguished simultaneous finite/cyclotomic lift. -/
+noncomputable def numberFieldTowerFinitePadicAuxiliaryAutomorphism
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    let P :=
+      numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+        (K := K) (L := L) p τ
+    let S :=
+      numberFieldTowerFinitePadicCyclicFixedSubgroup
+        (K := K) (L := L) p τ
+    Gal(
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below /
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) S) := by
+  dsimp only
+  let P :=
+    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+      (K := K) (L := L) p τ
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let σS :=
+    numberFieldTowerFinitePadicAuxiliarySubgroupLift
+      (K := K) (L := L) p τ
+  exact
+    LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup
+      ℚ (SeparableClosure ℚ)
+      S P.field P.below
+      P.toFiniteGaloisExtension.normal
+      (QuotientGroup.mk'
+        (extensionSubgroup
+          S P.field P.below)
+        σS)
+
+/-- On the common separable closure, the auxiliary automorphism acts
+by the original distinguished ambient lift. -/
+theorem numberFieldTowerFinitePadicAuxiliaryAutomorphism_apply_val
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (x :
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) :
+    ((numberFieldTowerFinitePadicAuxiliaryAutomorphism
+        (K := K) (L := L) p τ x :
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) :
+      SeparableClosure ℚ) =
+        τ.1 (x : SeparableClosure ℚ) := by
+  let P :=
+    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+      (K := K) (L := L) p τ
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let σS :=
+    numberFieldTowerFinitePadicAuxiliarySubgroupLift
+      (K := K) (L := L) p τ
+  exact
+    (LocalClassFieldTheory.abstractExtensionQuotientEquivGaloisGroup_mk_apply_val
+        ℚ (SeparableClosure ℚ)
+        S P.field P.below
+        P.toFiniteGaloisExtension.normal σS x).symm
+
+/-- Restricting the auxiliary automorphism through the actual
+base-change square recovers the finite quotient coordinate of the
+distinguished lift. -/
+theorem numberFieldTowerFinitePadicAuxiliaryAutomorphism_restriction
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    ((AlgEquiv.restrictNormalHom L).comp
+        (AlgEquiv.restrictScalarsHom K))
+        (numberFieldTowerFinitePadicAuxiliaryAutomorphism
+          (K := K) (L := L) p τ) =
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+        (numberFieldTowerFiniteQuotientCoordinate
+          (K := K) (L := L) τ) := by
+  let E :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ)
+      (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+        (K := K) (L := L) p τ).below
+  let σE :=
+    numberFieldTowerFinitePadicAuxiliaryAutomorphism
+      (K := K) (L := L) p τ
+  apply AlgEquiv.ext
+  intro x
+  apply (numberFieldSeparableClosureEmbedding L).injective
+  calc
+    numberFieldSeparableClosureEmbedding L
+        (((AlgEquiv.restrictNormalHom L).comp
+          (AlgEquiv.restrictScalarsHom K)) σE x) =
+      ((σE (algebraMap L E x) : E) :
+        SeparableClosure ℚ) := by
+          exact congrArg Subtype.val
+            (AlgEquiv.restrictNormal_commutes
+              ((AlgEquiv.restrictScalarsHom K) σE) L x)
+    _ = τ.1 (numberFieldSeparableClosureEmbedding L x) := by
+      rw [
+        numberFieldTowerFinitePadicAuxiliaryAutomorphism_apply_val
+          (K := K) (L := L) p τ]
+      rfl
+    _ =
+      numberFieldSeparableClosureEmbedding L
+        (numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+          (numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) τ) x) := by
+      exact
+        (numberFieldTowerExtensionQuotientEquivGaloisGroup_mk_apply
+          (K := K) (L := L) τ x).symm
+
+/-- Restriction of the chosen separable-closure place to the genuine
+auxiliary base field.  This is an exact extension of the original
+finite place of `K`, not merely an equivalent valuation. -/
+noncomputable def
+    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    AbsoluteValueExtension
+      (NumberField.HeightOneSpectrum.adicAbv K v)
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ)) := by
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ)
+      (numberFieldTowerFinitePadicCyclicFixedSubgroup
+        (K := K) (L := L) p τ)
+  let wΩ :=
+    numberFieldTowerFinitePlaceExtensionToSeparableClosure
+      K L v (chosenFinitePlaceExtension (L := L) v)
+  refine
+    ⟨wΩ.1.comp (f := F.val.toRingHom) F.val.injective, ?_⟩
+  intro x
+  change
+    wΩ.1 (numberFieldTowerLowerEmbedding K L x) =
+      NumberField.HeightOneSpectrum.adicAbv K v x
+  exact wΩ.2 x
+
+/-- Restriction of the same separable-closure place to the genuine
+auxiliary compositum.  Its restriction to `K` agrees exactly with the
+original normalized finite absolute value. -/
+noncomputable def
+    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    AbsoluteValueExtension
+      (NumberField.HeightOneSpectrum.adicAbv K v)
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) := by
+  let E :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ)
+      (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+        (K := K) (L := L) p τ).below
+  let wΩ :=
+    numberFieldTowerFinitePlaceExtensionToSeparableClosure
+      K L v (chosenFinitePlaceExtension (L := L) v)
+  refine
+    ⟨wΩ.1.comp (f := E.val.toRingHom) E.val.injective, ?_⟩
+  intro x
+  change
+    wΩ.1 (numberFieldTowerLowerEmbedding K L x) =
+      NumberField.HeightOneSpectrum.adicAbv K v x
+  exact wΩ.2 x
+
+/-- The centre of the restricted place on the auxiliary compositum
+lies above the centre of the same place on the auxiliary base field. -/
+theorem
+    numberFieldTowerFinitePadicAuxiliaryTopPlace_below
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hτ :
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ ≠
+        1) :
+    let S :=
+      numberFieldTowerFinitePadicCyclicFixedSubgroup
+        (K := K) (L := L) p τ
+    let P :=
+      numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+        (K := K) (L := L) p τ
+    let F :=
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ) S
+    let E :=
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ) P.below
+    letI hHfinite : Finite
+        ((baseField
+          (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+          extensionSubgroup
+            (baseField
+              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+            S (le_baseField S)) :=
+      (numberFieldTowerFinitePadicAuxiliaryAbstractField
+        (K := K) (L := L) p τ hτ).finite
+    letI hPfinite : Finite
+        (S.toSubgroup ⧸
+          extensionSubgroup S P.field P.below) :=
+      P.finite
+    letI _ : FiniteDimensional ℚ F :=
+      LocalClassFieldTheory.abstractFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) S hHfinite
+    letI _ : FiniteDimensional F E :=
+      LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ)
+        S P.field P.below hHfinite hPfinite
+    letI _ : IsScalarTower ℚ F E :=
+      IsScalarTower.of_algebraMap_eq' rfl
+    letI _ : FiniteDimensional ℚ E :=
+      FiniteDimensional.trans ℚ F E
+    letI _ : NumberField F :=
+      NumberField.of_module_finite ℚ F
+    letI _ : NumberField E :=
+      NumberField.of_module_finite ℚ E
+    letI _ : Algebra K F :=
+      numberFieldTowerFinitePadicAuxiliary_baseAlgebra
+        (K := K) (L := L) p τ
+    finitePlaceBelow (K := F)
+        (finitePlaceExtensionCentre
+          (K := K) (L := E) v
+          (numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+            (K := K) (L := L) v p τ)) =
+      finitePlaceExtensionCentre
+        (K := K) (L := F) v
+        (numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
+          (K := K) (L := L) v p τ) := by
+  dsimp only
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let P :=
+    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+      (K := K) (L := L) p τ
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) S
+  let E :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let hHfinite : Finite
+      ((baseField
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+        extensionSubgroup
+          (baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+          S (le_baseField S)) :=
+    (numberFieldTowerFinitePadicAuxiliaryAbstractField
+      (K := K) (L := L) p τ hτ).finite
+  let hPfinite : Finite
+      (S.toSubgroup ⧸
+        extensionSubgroup S P.field P.below) :=
+    P.finite
+  let auxiliaryBaseFiniteDimensional : FiniteDimensional ℚ F :=
+    LocalClassFieldTheory.abstractFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ) S hHfinite
+  let auxiliaryTopFiniteDimensional : FiniteDimensional F E :=
+    LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ)
+      S P.field P.below hHfinite hPfinite
+  let auxiliaryScalarTower : IsScalarTower ℚ F E :=
+    IsScalarTower.of_algebraMap_eq' rfl
+  let auxiliaryAbsoluteFiniteDimensional : FiniteDimensional ℚ E :=
+    FiniteDimensional.trans ℚ F E
+  let auxiliaryBaseNumberField : NumberField F :=
+    NumberField.of_module_finite ℚ F
+  let auxiliaryTopNumberField : NumberField E :=
+    NumberField.of_module_finite ℚ E
+  let auxiliaryOriginalBaseAlgebra : Algebra K F :=
+    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
+      (K := K) (L := L) p τ
+  let wF :=
+    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
+      (K := K) (L := L) v p τ
+  let wE :=
+    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+      (K := K) (L := L) v p τ
+  apply HeightOneSpectrum.ext
+  ext x
+  change
+    algebraMap (𝓞 F) (𝓞 E) x ∈
+        finitePlaceExtensionCentreIdeal
+          (K := K) (L := E) v wE ↔
+      x ∈
+        finitePlaceExtensionCentreIdeal
+          (K := K) (L := F) v wF
+  rw [
+    mem_finitePlaceExtensionCentreIdeal_iff,
+    mem_finitePlaceExtensionCentreIdeal_iff]
+  rfl
+
+/-- Evaluation of the distinguished auxiliary automorphism through the
+restricted top-field place.  Isolating this coercion calculation prevents the
+whole decomposition-group proof from normalizing the fixed-field tower. -/
+public opaque
+    numberFieldTowerFinitePadicAuxiliaryTopPlace_automorphism_apply
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    let P := numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+      (K := K) (L := L) p τ
+    let E := LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+    let wE := numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+      (K := K) (L := L) v p τ
+    let σE := numberFieldTowerFinitePadicAuxiliaryAutomorphism
+      (K := K) (L := L) p τ
+    ∀ x : E,
+      wE.1 (σE x) =
+        (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+          K L v (chosenFinitePlaceExtension (L := L) v)).1
+            (τ.1 (x : SeparableClosure ℚ)) := by
+  let : Algebra K (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureBaseAlgebra K L
+  dsimp only
+  intro x
+  change
+    (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+      K L v (chosenFinitePlaceExtension (L := L) v)).1
+        ((numberFieldTowerFinitePadicAuxiliaryAutomorphism
+          (K := K) (L := L) p τ x :
+            LocalClassFieldTheory.abstractRelativeFixedField
+              ℚ (SeparableClosure ℚ)
+              (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+                (K := K) (L := L) p τ).below) : SeparableClosure ℚ) = _
+  rw [
+    numberFieldTowerFinitePadicAuxiliaryAutomorphism_apply_val
+      (K := K) (L := L) p τ]
+
+/-- The distinguished auxiliary automorphism preserves the top-field place
+obtained by restricting the original separable-closure place. -/
+public opaque numberFieldTowerFinitePadicAuxiliaryAutomorphism_mem_topPlaceDecomposition
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hdecomposition :
+      letI : Algebra K (SeparableClosure ℚ) :=
+        numberFieldTowerSeparableClosureBaseAlgebra K L
+      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
+        absoluteValueDecompositionGroup K
+          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+            K L v (chosenFinitePlaceExtension (L := L) v)).1) :
+    let S := numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+    let F := LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) S
+    let wE := numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+      (K := K) (L := L) v p τ
+    let σE := numberFieldTowerFinitePadicAuxiliaryAutomorphism
+      (K := K) (L := L) p τ
+    σE ∈ absoluteValueDecompositionGroup F wE.1 := by
+  let : Algebra K (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureBaseAlgebra K L
+  dsimp only at hdecomposition ⊢
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) S
+  let wE :=
+    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+      (K := K) (L := L) v p τ
+  let σE :=
+    numberFieldTowerFinitePadicAuxiliaryAutomorphism
+      (K := K) (L := L) p τ
+  intro x
+  change
+    wE.1 (σE x) < 1 ↔
+      wE.1 x < 1
+  rw [
+    numberFieldTowerFinitePadicAuxiliaryTopPlace_automorphism_apply
+      (K := K) (L := L) v p τ x,
+    show
+      wE.1 x =
+        (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+          K L v
+          (chosenFinitePlaceExtension (L := L) v)).1
+            (x : SeparableClosure ℚ) from rfl]
+  exact hdecomposition (x : SeparableClosure ℚ)
+
+/-- The restricted top-field place and the chosen extension above its centre
+have the same decomposition group. -/
+public opaque numberFieldTowerFinitePadicAuxiliaryTopDecompositionGroup_eq_chosen
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hτ :
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ ≠ 1) :
+    let S := numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+    let P := numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+      (K := K) (L := L) p τ
+    let F := LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) S
+    let E := LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+    letI hHfinite : Finite
+        ((baseField
+          (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+          extensionSubgroup
+            (baseField
+              (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+            S (le_baseField S)) :=
+      (numberFieldTowerFinitePadicAuxiliaryAbstractField
+        (K := K) (L := L) p τ hτ).finite
+    letI hPfinite : Finite
+        (S.toSubgroup ⧸ extensionSubgroup S P.field P.below) := P.finite
+    letI : FiniteDimensional ℚ F :=
+      LocalClassFieldTheory.abstractFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) S hHfinite
+    letI : FiniteDimensional F E :=
+      LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) S P.field P.below hHfinite hPfinite
+    letI : IsScalarTower ℚ F E := IsScalarTower.of_algebraMap_eq' rfl
+    letI : FiniteDimensional ℚ E := FiniteDimensional.trans ℚ F E
+    letI : NumberField F := NumberField.of_module_finite ℚ F
+    letI : NumberField E := NumberField.of_module_finite ℚ E
+    letI : IsAbelianGalois F E :=
+      GlobalClassFields.finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
+    letI : Algebra K F :=
+      numberFieldTowerFinitePadicAuxiliary_baseAlgebra
+        (K := K) (L := L) p τ
+    let wF := numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
+      (K := K) (L := L) v p τ
+    let wE := numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+      (K := K) (L := L) v p τ
+    let V := finitePlaceExtensionCentre (K := K) (L := F) v wF
+    absoluteValueDecompositionGroup F wE.1 =
+      absoluteValueDecompositionGroup F
+        (chosenFinitePlaceExtension (L := E) V).1 := by
+  dsimp only
+  let H :=
+    numberFieldTowerFinitePadicAuxiliaryAbstractField
+      (K := K) (L := L) p τ hτ
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let P :=
+    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+      (K := K) (L := L) p τ
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) S
+  let E :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  let hHfinite : Finite
+      ((baseField
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+        extensionSubgroup
+          (baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+          S (le_baseField S)) :=
+    H.finite
+  let hPfinite : Finite
+      (S.toSubgroup ⧸ extensionSubgroup S P.field P.below) :=
+    P.finite
+  let : FiniteDimensional ℚ F :=
+    LocalClassFieldTheory.abstractFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ) S hHfinite
+  let : FiniteDimensional F E :=
+    LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
+      ℚ (SeparableClosure ℚ) S P.field P.below hHfinite hPfinite
+  let : IsScalarTower ℚ F E := IsScalarTower.of_algebraMap_eq' rfl
+  let : FiniteDimensional ℚ E := FiniteDimensional.trans ℚ F E
+  let : NumberField F := NumberField.of_module_finite ℚ F
+  let : NumberField E := NumberField.of_module_finite ℚ E
+  let : IsAbelianGalois F E :=
+    GlobalClassFields.finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
+  let : Algebra K F :=
+    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
+      (K := K) (L := L) p τ
+  let wF :=
+    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
+      (K := K) (L := L) v p τ
+  let wE :=
+    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+      (K := K) (L := L) v p τ
+  let V := finitePlaceExtensionCentre (K := K) (L := F) v wF
+  let W := finitePlaceExtensionCentre (K := K) (L := E) v wE
+  let Wover :
+      {W' : HeightOneSpectrum (𝓞 E) //
+        finitePlaceBelow (K := F) W' = V} :=
+    ⟨W,
+      numberFieldTowerFinitePadicAuxiliaryTopPlace_below
+        (K := K) (L := L) v p τ hτ⟩
+  let wFE :
+      AbsoluteValueExtension
+        (NumberField.HeightOneSpectrum.adicAbv F V) E :=
+    (finitePlaceExtensionEquivAbove
+      (K := F) (L := E) V).symm Wover
+  have hwFEcentre :
+      finitePlaceExtensionCentre
+          (K := F) (L := E) V wFE =
+        W := by
+    exact
+      congrArg Subtype.val
+        ((finitePlaceExtensionEquivAbove
+          (K := F) (L := E) V).apply_symm_apply Wover)
+  have hwEquiv : wE.1.IsEquiv wFE.1 := by
+    apply
+      finitePlaceExtensions_isEquiv_of_centres_eq
+        (F := K) (M := F) v V wE wFE
+    exact hwFEcentre.symm
+  calc
+    absoluteValueDecompositionGroup F wE.1 =
+        absoluteValueDecompositionGroup F wFE.1 :=
+      absoluteValueDecompositionGroup_eq_of_absoluteValue_isEquiv
+        (F := F) wE.1 wFE.1 hwEquiv
+    _ =
+      absoluteValueDecompositionGroup F
+          (chosenFinitePlaceExtension (L := E) V).1 :=
+      absoluteValueDecompositionGroup_eq_of_exactExtensions_of_isMulCommutative
+        (F := F)
+        (NumberField.HeightOneSpectrum.adicAbv F V)
+        (RayClass.adicAbv_isNontrivial V)
+        wFE
+        (chosenFinitePlaceExtension (L := E) V)
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldBase.lean
new file mode 100644
index 0000000..2c56707
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldBase.lean
@@ -0,0 +1,433 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicCyclicData
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicUnramifiedLocalGlobalCompatibility
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.Extension.OnePlaceBaseNorm
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open AlgebraicNumberTheory IsDedekindDomain NumberField
+open IdeleGroup RelativeIdeleGroup
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open CyclicCohomology
+open KummerTheory ClassFormation
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open GlobalClassFields
+
+variable
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+
+local instance (p : Nat.Primes) : Fact p.1.Prime :=
+  ⟨p.2⟩
+
+attribute [local instance]
+  rationalSeparableClosureAlgebra
+
+local instance finitePadicAuxiliaryExtensionNormal :
+    (extensionSubgroup
+      (numberFieldTowerBaseSubgroup K L)
+      (numberFieldTowerTopSubgroup L)
+      (numberFieldTowerTopSubgroup_le_baseSubgroup K L)).Normal :=
+  numberFieldTowerExtensionSubgroup_normal K L
+
+local instance finitePadicAuxiliaryExtensionQuotientFinite :
+    Finite
+      ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+        extensionSubgroup
+          (numberFieldTowerBaseSubgroup K L)
+          (numberFieldTowerTopSubgroup L)
+          (numberFieldTowerTopSubgroup_le_baseSubgroup K L)) :=
+  numberFieldTowerExtensionQuotient_finite K L
+
+noncomputable local instance finitePadicAuxiliaryExtensionQuotientIsMulCommutative :
+    IsMulCommutative
+      ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+        extensionSubgroup
+          (numberFieldTowerBaseSubgroup K L)
+          (numberFieldTowerTopSubgroup L)
+          (numberFieldTowerTopSubgroup_le_baseSubgroup K L)) := by
+  let e :
+      ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+          extensionSubgroup
+            (numberFieldTowerBaseSubgroup K L)
+            (numberFieldTowerTopSubgroup L)
+            (numberFieldTowerTopSubgroup_le_baseSubgroup K L)) ≃*
+        Gal(L / K) :=
+    numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+  exact
+    { is_comm :=
+        ⟨fun x y => by
+          apply e.injective
+          rw [map_mul, map_mul]
+          exact
+            (inferInstance :
+              IsMulCommutative (Gal(L / K))).is_comm.comm
+                (e x) (e y)⟩ }
+
+/-- The concrete auxiliary fixed field attached to a simultaneous
+finite/cyclotomic lift. -/
+noncomputable def numberFieldTowerFinitePadicCyclicFixedField
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    IntermediateField ℚ (SeparableClosure ℚ) := by
+  exact
+    IntermediateField.fixedField
+      (numberFieldTowerFinitePadicCyclicFixedSubgroup
+        (K := K) (L := L) p τ).toSubgroup
+
+/-- A nonzero-degree lift produces a genuine number field: its
+concrete fixed field is finite over `ℚ`. -/
+theorem numberFieldTowerFinitePadicCyclicFixedField_finiteDimensional
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hτ :
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ ≠
+        1) :
+    FiniteDimensional ℚ
+      (numberFieldTowerFinitePadicCyclicFixedField
+        (K := K) (L := L) p τ) := by
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let F :=
+    numberFieldTowerFinitePadicCyclicFixedField
+      (K := K) (L := L) p τ
+  apply
+    (InfiniteGalois.isOpen_iff_finite
+      (K := SeparableClosure ℚ) F).1
+  change IsOpen
+    (IntermediateField.fixedField S.toSubgroup).fixingSubgroup.carrier
+  rw [InfiniteGalois.fixingSubgroup_fixedField S]
+  exact
+    numberFieldTowerFinitePadicCyclicFixedSubgroup_isOpen
+      (K := K) (L := L) p τ hτ
+
+/-- The compatible embedded copy of `K` lies in every auxiliary
+cyclic fixed field. -/
+theorem numberFieldTowerBaseField_le_finitePadicCyclicFixedField
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    numberFieldTowerBaseField K L ≤
+      numberFieldTowerFinitePadicCyclicFixedField
+        (K := K) (L := L) p τ := by
+  intro x hx
+  change x ∈ IntermediateField.fixedField
+    (numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ).toSubgroup
+  rw [IntermediateField.mem_fixedField_iff]
+  intro σ hσ
+  change
+    σ ∈
+      (numberFieldTowerFinitePadicCyclicFixedSubgroup
+        (K := K) (L := L) p τ).toSubgroup at hσ
+  obtain ⟨u, hu, rfl⟩ := hσ
+  exact
+    (IntermediateField.mem_fixingSubgroup_iff
+      (numberFieldTowerBaseField K L) u.1).1 u.2 x hx
+
+/-- The compatible embedding of the original base field into the
+genuine auxiliary fixed field. -/
+noncomputable def numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    K →ₐ[ℚ]
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ) := by
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ)
+      (numberFieldTowerFinitePadicCyclicFixedSubgroup
+        (K := K) (L := L) p τ)
+  exact
+    (numberFieldTowerLowerEmbedding K L).codRestrict F.toSubalgebra
+      (fun x =>
+        numberFieldTowerBaseField_le_finitePadicCyclicFixedField
+          (K := K) (L := L) p τ ⟨x, rfl⟩)
+
+/-- Coercing the auxiliary base embedding recovers the fixed lower embedding
+into the rational separable closure. -/
+@[simp]
+theorem numberFieldTowerFinitePadicAuxiliaryBaseEmbedding_coe
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (x : K) :
+    ((numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
+        (K := K) (L := L) p τ x :
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ)) :
+      SeparableClosure ℚ) =
+        numberFieldTowerLowerEmbedding K L x := by
+  rfl
+
+/-- The compatible copy of the original top field lies in the
+auxiliary compositum fixed field. -/
+theorem numberFieldTowerTopField_mem_finitePadicAuxiliaryTopField
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    {x : SeparableClosure ℚ}
+    (hx : x ∈ numberFieldInRationalSeparableClosure L) :
+    x ∈
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below := by
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let T :=
+    numberFieldTowerTopSubgroup L
+  change x ∈ IntermediateField.fixedField
+    (S.toSubgroup ⊓ T.toSubgroup)
+  rw [IntermediateField.mem_fixedField_iff]
+  intro σ hσ
+  have hσT : σ ∈ T.toSubgroup :=
+    hσ.2
+  change
+    σ ∈
+      (numberFieldInRationalSeparableClosure L).fixingSubgroup
+        at hσT
+  exact
+    (IntermediateField.mem_fixingSubgroup_iff
+      (numberFieldInRationalSeparableClosure L) σ).1
+      hσT x hx
+
+/-- The compatible embedding of the original top field into the
+auxiliary compositum fixed field. -/
+noncomputable def numberFieldTowerFinitePadicAuxiliaryTopEmbedding
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    L →ₐ[ℚ]
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below := by
+  let E :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ)
+      (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+        (K := K) (L := L) p τ).below
+  exact
+    (numberFieldSeparableClosureEmbedding L).codRestrict
+      (E.restrictScalars ℚ).toSubalgebra
+      (fun x =>
+        numberFieldTowerTopField_mem_finitePadicAuxiliaryTopField
+          (K := K) (L := L) p τ ⟨x, rfl⟩)
+
+/-- Coercing the auxiliary top embedding recovers the chosen top-field
+embedding into the rational separable closure. -/
+@[simp]
+theorem numberFieldTowerFinitePadicAuxiliaryTopEmbedding_coe
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (x : L) :
+    ((numberFieldTowerFinitePadicAuxiliaryTopEmbedding
+        (K := K) (L := L) p τ x :
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) :
+      SeparableClosure ℚ) =
+        numberFieldSeparableClosureEmbedding L x := by
+  rfl
+
+/-- The compatible base and top embeddings form the actual
+base-change square inside the rational separable closure. -/
+theorem numberFieldTowerFinitePadicAuxiliaryEmbedding_algebraMap
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (x : K) :
+    ((numberFieldTowerFinitePadicAuxiliaryTopEmbedding
+        (K := K) (L := L) p τ (algebraMap K L x) :
+      LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) :
+      SeparableClosure ℚ) =
+    ((numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
+        (K := K) (L := L) p τ x :
+      LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ)) :
+      SeparableClosure ℚ) := by
+  rfl
+
+noncomputable instance
+    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    Algebra K
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ)) := by
+  exact
+    (numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
+      (K := K) (L := L) p τ).toRingHom.toAlgebra
+
+instance
+    numberFieldTowerFinitePadicAuxiliary_baseRatScalarTower
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    IsScalarTower ℚ K
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ)) := by
+  exact
+    IsScalarTower.of_algebraMap_eq'
+      (numberFieldTowerFinitePadicAuxiliaryBaseEmbedding
+        (K := K) (L := L) p τ).comp_algebraMap.symm
+
+noncomputable instance
+    numberFieldTowerFinitePadicAuxiliary_topAlgebra
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    Algebra L
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) := by
+  exact
+    (numberFieldTowerFinitePadicAuxiliaryTopEmbedding
+      (K := K) (L := L) p τ).toRingHom.toAlgebra
+
+instance
+    numberFieldTowerFinitePadicAuxiliary_topRatScalarTower
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    IsScalarTower ℚ L
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) := by
+  exact
+    IsScalarTower.of_algebraMap_eq'
+      (numberFieldTowerFinitePadicAuxiliaryTopEmbedding
+        (K := K) (L := L) p τ).comp_algebraMap.symm
+
+noncomputable instance
+    numberFieldTowerFinitePadicAuxiliary_originalBaseTopAlgebra
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    Algebra K
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) := by
+  exact
+    ((numberFieldTowerFinitePadicAuxiliaryTopEmbedding
+        (K := K) (L := L) p τ).comp
+      (IsScalarTower.toAlgHom ℚ K L)).toRingHom.toAlgebra
+
+instance
+    numberFieldTowerFinitePadicAuxiliary_originalTopScalarTower
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    IsScalarTower K L
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) := by
+  exact IsScalarTower.of_algebraMap_eq' rfl
+
+instance
+    numberFieldTowerFinitePadicAuxiliary_baseTopScalarTower
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    IsScalarTower K
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ))
+      (LocalClassFieldTheory.abstractRelativeFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+          (K := K) (L := L) p τ).below) := by
+  apply IsScalarTower.of_algebraMap_eq'
+  apply RingHom.ext
+  intro x
+  apply Subtype.ext
+  exact
+    numberFieldTowerFinitePadicAuxiliaryEmbedding_algebraMap
+      (K := K) (L := L) p τ x
+
+/-- The genuine auxiliary fixed field is Galois over the original
+base field through the compatible embedding above. -/
+theorem numberFieldTowerFinitePadicAuxiliaryBase_isGalois
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup) :
+    IsGalois K
+      (LocalClassFieldTheory.abstractFixedField
+        ℚ (SeparableClosure ℚ)
+        (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ)) := by
+  let H :=
+    numberFieldTowerBaseSubgroup K L
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let hSH :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup_le_baseSubgroup
+      (K := K) (L := L) p τ
+  let B :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) H
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) S
+  let FB :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) hSH
+  let auxiliaryBaseAlgebra : Algebra B FB :=
+    (LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) hSH).algebra
+  let auxiliaryBaseGalois : IsGalois B FB :=
+    LocalClassFieldTheory.abstractRelativeFixedField_isGalois
+      ℚ (SeparableClosure ℚ) H S hSH
+      (numberFieldTowerFinitePadicCyclicFixedSubgroup_extension_normal
+        (K := K) (L := L) p τ)
+  let eK :=
+    numberFieldTowerAbstractBaseFieldEquiv K L
+  refine
+    @IsGalois.of_equiv_equiv
+      B FB _ _ auxiliaryBaseAlgebra
+      K F _ _
+      (numberFieldTowerFinitePadicAuxiliary_baseAlgebra
+        (K := K) (L := L) p τ)
+      auxiliaryBaseGalois
+      eK.symm.toRingEquiv (RingEquiv.refl F) ?_
+  apply RingHom.ext
+  intro x
+  apply Subtype.ext
+  change
+    ((eK (eK.symm x) : B) : SeparableClosure ℚ) =
+      (x : SeparableClosure ℚ)
+  exact
+    congrArg Subtype.val (eK.apply_symm_apply x)
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldExistence.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldExistence.lean
new file mode 100644
index 0000000..87681a2
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldExistence.lean
@@ -0,0 +1,234 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicAuxiliaryFieldFixedFields
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicCyclicData
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicUnramifiedLocalGlobalCompatibility
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.Extension.OnePlaceBaseNorm
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open AlgebraicNumberTheory IsDedekindDomain NumberField
+open IdeleGroup RelativeIdeleGroup
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open CyclicCohomology
+open KummerTheory ClassFormation
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open GlobalClassFields
+
+variable
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+
+
+local instance splitFactPrimeExistence (p : Nat.Primes) : Fact p.1.Prime := ⟨p.2⟩
+
+attribute [local instance]
+  rationalSeparableClosureAlgebra
+  finitePadicAuxiliaryExtensionNormal
+  finitePadicAuxiliaryExtensionQuotientFinite
+  finitePadicAuxiliaryExtensionQuotientIsMulCommutative
+
+
+/-- For a finite-place automorphism generating `Gal(L/K)`, construct
+the genuine auxiliary number field used in the cyclotomic reduction.
+
+The field is finite over `ℚ`, contains the compatible copy of `K`, and
+has intersection with the compatible copy of `L` exactly equal to that
+copy of `K`.  Its defining lift has the prescribed local restriction
+and positive integral cyclotomic `p`-adic degree. -/
+theorem exists_finitePlaceCyclotomicAuxiliaryFixedField
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (σ :
+      absoluteValueDecompositionGroup K
+        (chosenFinitePlaceExtension (L := L) v).1)
+    (hσ :
+      Subgroup.closure
+          ({σ.1} : Set (Gal(L / K))) =
+        ⊤) :
+    letI _ : Algebra K (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureBaseAlgebra K L
+    letI _ : Algebra L (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureTopAlgebra L
+    letI _ : IsScalarTower K L (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureScalarTower K L
+    ∃ τ : (numberFieldTowerBaseSubgroup K L).toSubgroup,
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+          (numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) τ) =
+        σ.1 ∧
+      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
+        absoluteValueDecompositionGroup K
+          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+            K L v
+            (chosenFinitePlaceExtension (L := L) v)).1 ∧
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ ≠
+        1 ∧
+      FiniteDimensional ℚ
+        (numberFieldTowerFinitePadicCyclicFixedField
+          (K := K) (L := L) p τ) ∧
+      numberFieldTowerBaseField K L ≤
+        numberFieldTowerFinitePadicCyclicFixedField
+          (K := K) (L := L) p τ ∧
+      numberFieldTowerFinitePadicCyclicFixedField
+            (K := K) (L := L) p τ ⊓
+          numberFieldInRationalSeparableClosure L =
+        numberFieldTowerBaseField K L ∧
+      ∃ n : ℕ, 0 < n ∧
+        numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+            (K := K) (L := L) p τ =
+          (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
+  let : Algebra K (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureBaseAlgebra K L
+  let : Algebra L (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureTopAlgebra L
+  let : IsScalarTower K L (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureScalarTower K L
+  obtain
+      ⟨τ, hτσ, hτdecomposition, hτdegree,
+        n, hn, hdegree⟩ :=
+    exists_numberFieldTowerFinitePadicLift_of_finitePlace
+      (K := K) (L := L) v p σ
+  have hgenerate :=
+    numberFieldTowerFiniteQuotientCoordinate_generates_of_galoisGenerator
+      (K := K) (L := L) τ σ.1 hτσ hσ
+  refine
+    ⟨τ, hτσ, hτdecomposition, hτdegree,
+      numberFieldTowerFinitePadicCyclicFixedField_finiteDimensional
+        (K := K) (L := L) p τ hτdegree,
+      numberFieldTowerBaseField_le_finitePadicCyclicFixedField
+        (K := K) (L := L) p τ,
+      numberFieldTowerFinitePadicCyclicFixedField_inf_topField
+        (K := K) (L := L) p τ hgenerate,
+      n, hn, hdegree⟩
+
+/-- A `p`-primary local generator admits a genuine auxiliary number
+field whose compositum with the rational `p`-primary cyclotomic field
+contains `L`.
+
+Besides the field containment, the construction records the two
+properties needed for descent: the auxiliary field meets `L` exactly
+in `K`, and the chosen absolute lift has positive integral
+cyclotomic degree. -/
+theorem exists_finitePlacePrimaryCyclotomicAuxiliaryFixedField
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (σ :
+      absoluteValueDecompositionGroup K
+        (chosenFinitePlaceExtension (L := L) v).1)
+    (hgenerate :
+      Subgroup.closure
+          ({σ.1} : Set (Gal(L / K))) =
+        ⊤)
+    (hprimary :
+      σ.1 ∈
+        CommGroup.primaryComponent
+          (Gal(L / K)) p.1) :
+    letI _ : Algebra K (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureBaseAlgebra K L
+    letI _ : Algebra L (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureTopAlgebra L
+    letI _ : IsScalarTower K L (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureScalarTower K L
+    ∃ τ : (numberFieldTowerBaseSubgroup K L).toSubgroup,
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+          (numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) τ) =
+        σ.1 ∧
+      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
+        absoluteValueDecompositionGroup K
+          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+            K L v
+            (chosenFinitePlaceExtension (L := L) v)).1 ∧
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ ≠
+        1 ∧
+      FiniteDimensional ℚ
+        (numberFieldTowerFinitePadicCyclicFixedField
+          (K := K) (L := L) p τ) ∧
+      numberFieldTowerBaseField K L ≤
+        numberFieldTowerFinitePadicCyclicFixedField
+          (K := K) (L := L) p τ ∧
+      numberFieldTowerFinitePadicCyclicFixedField
+            (K := K) (L := L) p τ ⊓
+          numberFieldInRationalSeparableClosure L =
+        numberFieldTowerBaseField K L ∧
+      numberFieldInRationalSeparableClosure L ≤
+        numberFieldTowerFinitePadicCyclicFixedField
+              (K := K) (L := L) p τ ⊔
+          rationalCyclotomicPadicField p ∧
+      ∃ n : ℕ, 0 < n ∧
+        numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+            (K := K) (L := L) p τ =
+          (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
+  let : Algebra K (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureBaseAlgebra K L
+  let : Algebra L (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureTopAlgebra L
+  let : IsScalarTower K L (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureScalarTower K L
+  obtain
+      ⟨τ, hτσ, hτdecomposition, hτdegree,
+        hfinite, hbase, hintersection,
+        n, hn, hdegree⟩ :=
+    exists_finitePlaceCyclotomicAuxiliaryFixedField
+      (K := K) (L := L) v p σ hgenerate
+  have hprimaryQuotient :
+      numberFieldTowerFiniteQuotientCoordinate
+          (K := K) (L := L) τ ∈
+        CommGroup.primaryComponent
+          ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+            extensionSubgroup
+              (numberFieldTowerBaseSubgroup K L)
+              (numberFieldTowerTopSubgroup L)
+              (numberFieldTowerTopSubgroup_le_baseSubgroup K L))
+          p.1 := by
+    obtain ⟨m, hm⟩ := hprimary
+    refine ⟨m, ?_⟩
+    let e :=
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+    let q :=
+      numberFieldTowerFiniteQuotientCoordinate
+        (K := K) (L := L) τ
+    let N : ℕ := p.1 ^ m
+    have hq : e q = σ.1 := by
+      exact hτσ
+    have hmN : σ.1 ^ N = 1 := by
+      exact hm
+    change q ^ N = 1
+    apply e.injective
+    calc
+      e (q ^ N) = (e q) ^ N := by
+        exact map_pow e q N
+      _ = σ.1 ^ N := by rw [hq]
+      _ = 1 := hmN
+      _ = e 1 := (map_one e).symm
+  have hcontainment :=
+    numberFieldTowerTopField_le_finitePadicCyclicFixedField_sup_padicCyclotomicField
+      (K := K) (L := L) p τ n hn hdegree
+        hprimaryQuotient
+  exact
+    ⟨τ, hτσ, hτdecomposition, hτdegree,
+      hfinite, hbase, hintersection,
+      hcontainment, n, hn, hdegree⟩
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldFixedFields.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldFixedFields.lean
new file mode 100644
index 0000000..1ed247d
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldFixedFields.lean
@@ -0,0 +1,420 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicAuxiliaryFieldNormRestriction
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicCyclicData
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicUnramifiedLocalGlobalCompatibility
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.Extension.OnePlaceBaseNorm
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open AlgebraicNumberTheory IsDedekindDomain NumberField
+open IdeleGroup RelativeIdeleGroup
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open CyclicCohomology
+open KummerTheory ClassFormation
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open GlobalClassFields
+
+variable
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+
+
+local instance splitFactPrimeFixedFields (p : Nat.Primes) : Fact p.1.Prime := ⟨p.2⟩
+
+attribute [local instance]
+  rationalSeparableClosureAlgebra
+  finitePadicAuxiliaryExtensionNormal
+  finitePadicAuxiliaryExtensionQuotientFinite
+  finitePadicAuxiliaryExtensionQuotientIsMulCommutative
+
+/-- Generation of the actual finite Galois group transports back
+through the compatible finite quotient coordinate. -/
+theorem
+    numberFieldTowerFiniteQuotientCoordinate_generates_of_galoisGenerator
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (σ : Gal(L / K))
+    (hτσ :
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+          (numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) τ) =
+        σ)
+    (hσ : Subgroup.closure ({σ} : Set (Gal(L / K))) = ⊤) :
+    Subgroup.closure
+        ({numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) τ} :
+          Set
+            ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+              extensionSubgroup
+                (numberFieldTowerBaseSubgroup K L)
+                (numberFieldTowerTopSubgroup L)
+                (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
+      ⊤ := by
+  let e :=
+    numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+  let q :
+      (numberFieldTowerFiniteGaloisSubextension
+        K L).extensionQuotient :=
+    numberFieldTowerFiniteQuotientCoordinate
+      (K := K) (L := L) τ
+  have hq : e.toMonoidHom q = σ := by
+    exact hτσ
+  change
+    Subgroup.closure
+        ({q} :
+          Set
+            (numberFieldTowerFiniteGaloisSubextension
+              K L).extensionQuotient) =
+      ⊤
+  apply Subgroup.map_injective (f := e.toMonoidHom) e.injective
+  rw [MonoidHom.map_closure, Set.image_singleton,
+    hq, hσ,
+    Subgroup.map_top_of_surjective e.toMonoidHom e.surjective]
+
+/-- If the finite quotient coordinate of a lift generates the whole
+finite Galois quotient, then its cyclic preimage together with the
+top-field subgroup generates the whole embedded absolute Galois
+group. -/
+theorem
+    numberFieldTowerFinitePadicCyclicPreimage_sup_extensionSubgroup_eq_top
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hgenerate :
+      Subgroup.closure
+          ({numberFieldTowerFiniteQuotientCoordinate
+              (K := K) (L := L) τ} :
+            Set
+              ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+                extensionSubgroup
+                  (numberFieldTowerBaseSubgroup K L)
+                  (numberFieldTowerTopSubgroup L)
+                  (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
+        ⊤) :
+    (numberFieldTowerFinitePadicCyclicPreimage
+          (K := K) (L := L) p τ).toSubgroup ⊔
+        extensionSubgroup
+          (numberFieldTowerBaseSubgroup K L)
+          (numberFieldTowerTopSubgroup L)
+          (numberFieldTowerTopSubgroup_le_baseSubgroup K L) =
+      ⊤ := by
+  let H :=
+    numberFieldTowerBaseSubgroup K L
+  let N :=
+    extensionSubgroup
+      H
+      (numberFieldTowerTopSubgroup L)
+      (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
+  let Q := H.toSubgroup ⧸ N
+  let P :=
+    numberFieldTowerFinitePadicImage
+      (K := K) (L := L) p
+  let coordinate :=
+    numberFieldTowerFinitePadicCoordinate
+      (K := K) (L := L) p
+  let rangeRestriction :=
+    numberFieldTowerFinitePadicRangeRestriction
+      (K := K) (L := L) p
+  let γ : P.toSubgroup :=
+    rangeRestriction τ
+  let Γ :=
+    ClassFormation.padicCyclicClosure γ
+  let finiteProjection :
+      P.toSubgroup →*
+        Q :=
+    (MonoidHom.fst Q
+      (Multiplicative ℤ_[p.1])).comp
+        P.toSubgroup.subtype
+  have hprojection :
+      Γ.toSubgroup.map finiteProjection = ⊤ := by
+    apply top_unique
+    rw [← hgenerate]
+    apply (Subgroup.closure_le _).2
+    intro q hq
+    rw [Set.mem_singleton_iff] at hq
+    subst q
+    refine
+      ⟨γ,
+        (ClassFormation.padicCyclicClosureGenerator γ).2,
+        ?_⟩
+    rfl
+  have hquotientSurjective :
+      ∀ q : Q,
+        ∃ u : H.toSubgroup,
+          u ∈
+              (numberFieldTowerFinitePadicCyclicPreimage
+                (K := K) (L := L) p τ).toSubgroup ∧
+            numberFieldTowerFiniteQuotientCoordinate
+                (K := K) (L := L) u =
+              q := by
+    intro q
+    have hq :
+        q ∈ Γ.toSubgroup.map finiteProjection := by
+      rw [hprojection]
+      trivial
+    obtain ⟨z, hzΓ, hzq⟩ := hq
+    obtain ⟨u, hu⟩ :=
+      numberFieldTowerFinitePadicRangeRestriction_surjective
+        (K := K) (L := L) p z
+    refine ⟨u, ?_, ?_⟩
+    · change rangeRestriction u ∈ Γ.toSubgroup
+      rw [hu]
+      exact hzΓ
+    · calc
+        numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) u =
+            finiteProjection (rangeRestriction u) := rfl
+        _ = finiteProjection z := congrArg finiteProjection hu
+        _ = q := hzq
+  apply top_unique
+  intro h _
+  obtain ⟨u, huU, huq⟩ :=
+    hquotientSurjective
+      (numberFieldTowerFiniteQuotientCoordinate
+        (K := K) (L := L) h)
+  let k : H.toSubgroup :=
+    h * u⁻¹
+  have hkN : k ∈ N := by
+    apply (QuotientGroup.eq_one_iff (N := N) k).mp
+    change
+      numberFieldTowerFiniteQuotientCoordinate
+          (K := K) (L := L) (h * u⁻¹) =
+        1
+    rw [map_mul, map_inv, huq, mul_inv_cancel]
+  have hkSup :
+      k ∈
+        (numberFieldTowerFinitePadicCyclicPreimage
+            (K := K) (L := L) p τ).toSubgroup ⊔
+          N :=
+    (le_sup_right :
+      N ≤
+        (numberFieldTowerFinitePadicCyclicPreimage
+            (K := K) (L := L) p τ).toSubgroup ⊔ N) hkN
+  have huSup :
+      u ∈
+        (numberFieldTowerFinitePadicCyclicPreimage
+            (K := K) (L := L) p τ).toSubgroup ⊔
+          N :=
+    (le_sup_left :
+      (numberFieldTowerFinitePadicCyclicPreimage
+          (K := K) (L := L) p τ).toSubgroup ≤
+        (numberFieldTowerFinitePadicCyclicPreimage
+            (K := K) (L := L) p τ).toSubgroup ⊔ N) huU
+  have hku :
+      k * u = h := by
+    simp only [k, inv_mul_cancel_right]
+  rw [← hku]
+  exact
+    ((numberFieldTowerFinitePadicCyclicPreimage
+      (K := K) (L := L) p τ).toSubgroup ⊔ N).mul_mem
+      hkSup huSup
+
+/-- Ambient form of the generation statement: the auxiliary cyclic
+fixed subgroup together with the subgroup fixing `L` generates the
+subgroup fixing `K`. -/
+theorem
+    numberFieldTowerFinitePadicCyclicFixedSubgroup_sup_topSubgroup
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hgenerate :
+      Subgroup.closure
+          ({numberFieldTowerFiniteQuotientCoordinate
+              (K := K) (L := L) τ} :
+            Set
+              ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+                extensionSubgroup
+                  (numberFieldTowerBaseSubgroup K L)
+                  (numberFieldTowerTopSubgroup L)
+                  (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
+        ⊤) :
+    (numberFieldTowerFinitePadicCyclicFixedSubgroup
+          (K := K) (L := L) p τ).toSubgroup ⊔
+        (numberFieldTowerTopSubgroup L).toSubgroup =
+      (numberFieldTowerBaseSubgroup K L).toSubgroup := by
+  let H :=
+    numberFieldTowerBaseSubgroup K L
+  let T :=
+    numberFieldTowerTopSubgroup L
+  let N :=
+    extensionSubgroup
+      H T
+      (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
+  let U :=
+    numberFieldTowerFinitePadicCyclicPreimage
+      (K := K) (L := L) p τ
+  have hrelative :
+      U.toSubgroup ⊔ N = ⊤ :=
+    numberFieldTowerFinitePadicCyclicPreimage_sup_extensionSubgroup_eq_top
+      (K := K) (L := L) p τ hgenerate
+  have hNmap :
+      N.map H.toSubgroup.subtype =
+        T.toSubgroup := by
+    exact
+      Subgroup.map_subgroupOf_eq_of_le
+        (numberFieldTowerTopSubgroup_le_baseSubgroup K L)
+  change
+    U.toSubgroup.map H.toSubgroup.subtype ⊔
+        T.toSubgroup =
+      H.toSubgroup
+  rw [← hNmap, ← Subgroup.map_sup,
+    hrelative, ← MonoidHom.range_eq_map,
+    H.toSubgroup.range_subtype]
+
+/-- The concrete auxiliary fixed field is linearly disjoint from `L`
+over the compatible embedded copy of `K`. -/
+theorem
+    numberFieldTowerFinitePadicCyclicFixedField_inf_topField
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hgenerate :
+      Subgroup.closure
+          ({numberFieldTowerFiniteQuotientCoordinate
+              (K := K) (L := L) τ} :
+            Set
+              ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+                extensionSubgroup
+                  (numberFieldTowerBaseSubgroup K L)
+                  (numberFieldTowerTopSubgroup L)
+                  (numberFieldTowerTopSubgroup_le_baseSubgroup K L))) =
+        ⊤) :
+    numberFieldTowerFinitePadicCyclicFixedField
+          (K := K) (L := L) p τ ⊓
+        numberFieldInRationalSeparableClosure L =
+      numberFieldTowerBaseField K L := by
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let T :=
+    numberFieldTowerTopSubgroup L
+  let H :=
+    numberFieldTowerBaseSubgroup K L
+  change
+    IntermediateField.fixedField S.toSubgroup ⊓
+        numberFieldInRationalSeparableClosure L =
+      numberFieldTowerBaseField K L
+  rw [
+    ← InfiniteGalois.fixedField_fixingSubgroup
+      (numberFieldInRationalSeparableClosure L),
+    ← InfiniteGalois.fixedField_fixingSubgroup
+      (numberFieldTowerBaseField K L)]
+  change
+    IntermediateField.fixedField S.toSubgroup ⊓
+        IntermediateField.fixedField T.toSubgroup =
+      IntermediateField.fixedField H.toSubgroup
+  rw [
+    ← IntermediateField.fixedField_sup_eq_inf,
+    numberFieldTowerFinitePadicCyclicFixedSubgroup_sup_topSubgroup
+      (K := K) (L := L) p τ hgenerate]
+
+/-- If the finite quotient coordinate is `p`-primary, adjoining the
+actual rational `p`-primary cyclotomic field to the auxiliary fixed
+field contains the compatible copy of `L`.
+
+This is the field-theoretic conclusion of the simultaneous
+finite/cyclotomic lift: the intersection of the two fixing subgroups
+already fixes `L`, hence their fixed-field compositum contains `L`. -/
+theorem
+    numberFieldTowerTopField_le_finitePadicCyclicFixedField_sup_padicCyclotomicField
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (n : ℕ) (hn : 0 < n)
+    (hdegree :
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ =
+        (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n)
+    (hprimary :
+      numberFieldTowerFiniteQuotientCoordinate
+          (K := K) (L := L) τ ∈
+        CommGroup.primaryComponent
+          ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+            extensionSubgroup
+              (numberFieldTowerBaseSubgroup K L)
+              (numberFieldTowerTopSubgroup L)
+              (numberFieldTowerTopSubgroup_le_baseSubgroup K L))
+          p.1) :
+    numberFieldInRationalSeparableClosure L ≤
+      numberFieldTowerFinitePadicCyclicFixedField
+            (K := K) (L := L) p τ ⊔
+        rationalCyclotomicPadicField p := by
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let T :=
+    numberFieldTowerTopSubgroup L
+  let F :=
+    numberFieldTowerFinitePadicCyclicFixedField
+      (K := K) (L := L) p τ
+  let C : @IntermediateField ℚ (SeparableClosure ℚ) _ _
+      rationalSeparableClosureAlgebra :=
+    rationalCyclotomicPadicField p
+  have hfixing :
+      (F ⊔ C).fixingSubgroup ≤
+        T.toSubgroup := by
+    change
+      (IntermediateField.fixedField S.toSubgroup ⊔ C).fixingSubgroup ≤
+        T.toSubgroup
+    rw [
+      IntermediateField.fixingSubgroup_sup,
+      InfiniteGalois.fixingSubgroup_fixedField S]
+    intro σ hσ
+    apply
+      numberFieldTowerFinitePadicCyclicFixedSubgroup_inf_absolutePadicKernel_le_topSubgroup
+        (K := K) (L := L) p τ n hn hdegree hprimary
+    refine ⟨hσ.1, ?_⟩
+    let : Algebra ℚ rationalCyclotomicZHatField :=
+      rationalCyclotomicZHatField.algebra'
+    let : @Normal ℚ rationalCyclotomicZHatField _ _
+        rationalCyclotomicZHatField.algebra' :=
+      rationalCyclotomicZHatField_normal
+    let E :=
+      rationalCyclotomicPadicFieldWithinZHat p
+    change
+      rationalCyclotomicPadicCoordinate p
+          (rationalAbsoluteGaloisRestrictionToCyclotomicZHat σ) =
+        1
+    have hr :
+        rationalAbsoluteGaloisRestrictionToCyclotomicZHat σ ∈
+          E.fixingSubgroup := by
+      rw [IntermediateField.mem_fixingSubgroup_iff]
+      intro x hx
+      apply Subtype.ext
+      have hfix :
+          σ x.1 = x.1 :=
+        (IntermediateField.mem_fixingSubgroup_iff
+          (IntermediateField.lift E) σ).1 hσ.2 x.1
+            ((IntermediateField.mem_lift x).2 hx)
+      exact
+        (AlgEquiv.restrictNormal_commutes
+          σ rationalCyclotomicZHatField x).trans hfix
+    rw [rationalCyclotomicPadicFieldWithinZHat_fixingSubgroup]
+      at hr
+    exact hr
+  rw [
+    ← InfiniteGalois.fixedField_fixingSubgroup
+      (numberFieldInRationalSeparableClosure L)]
+  change
+    IntermediateField.fixedField T.toSubgroup ≤
+      F ⊔ C
+  rw [
+    ← InfiniteGalois.fixedField_fixingSubgroup
+      (F ⊔ C)]
+  exact
+    IntermediateField.fixedField_le hfixing
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldNormRestriction.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldNormRestriction.lean
new file mode 100644
index 0000000..c7daa65
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/LocalGlobalArtinCompatibility/FinitePadicAuxiliaryFieldNormRestriction.lean
@@ -0,0 +1,411 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicAuxiliaryFieldAutomorphisms
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.LocalGlobalArtinCompatibility.FinitePadicCyclicData
+import ClassFieldTheory.GlobalClassFieldTheory.GlobalClassFields.ClassFieldRealization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.CyclotomicUnramifiedLocalGlobalCompatibility
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.Extension.OnePlaceBaseNorm
+
+
+set_option autoImplicit false
+
+
+open scoped IsMulCommutative NumberField
+open AlgebraicNumberTheory IsDedekindDomain NumberField
+open IdeleGroup RelativeIdeleGroup
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open CyclicCohomology
+open KummerTheory ClassFormation
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open GlobalClassFields
+
+variable
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [NumberField L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+
+
+local instance splitFactPrimeNormRestriction (p : Nat.Primes) : Fact p.1.Prime := ⟨p.2⟩
+
+attribute [local instance]
+  rationalSeparableClosureAlgebra
+  finitePadicAuxiliaryExtensionNormal
+  finitePadicAuxiliaryExtensionQuotientFinite
+  finitePadicAuxiliaryExtensionQuotientIsMulCommutative
+
+/-- Pointwise form of norm/restriction naturality.  Keeping the function
+equality and its coercion normalization in this small declaration prevents
+the auxiliary-field witness construction below from repeatedly elaborating
+the full pair of composite homomorphisms. -/
+private theorem globalNormResidueMonoidHomOfEmbedding_norm_restriction_apply
+    (K K' L L' : Type)
+    [Field K] [NumberField K]
+    [Field K'] [NumberField K']
+    [Field L] [NumberField L]
+    [Field L'] [NumberField L']
+    [Algebra K K'] [Algebra K L] [Algebra K L']
+    [Algebra K' L'] [Algebra L L']
+    [IsScalarTower K K' L'] [IsScalarTower K L L']
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
+    [FiniteDimensional K K'] [IsGalois K K']
+    (j : L' →ₐ[ℚ] SeparableClosure ℚ)
+    (c : IdeleClassGroup K') :
+    ((AlgEquiv.restrictNormalHom L).comp
+        (AlgEquiv.restrictScalarsHom K))
+          (globalNormResidueMonoidHomOfEmbedding K' L' j c) =
+      globalNormResidueMonoidHomOfEmbedding K L
+        (j.comp (IsScalarTower.toAlgHom ℚ L L'))
+        (_root_.ideleClassNorm K K' c) := by
+  exact
+    DFunLike.congr_fun
+      (globalNormResidueMonoidHomOfEmbedding_norm_restriction
+        (K := K) (L := L) (K' := K') (L' := L') j) c
+
+
+/-- The auxiliary-field construction produces a lower local unit
+whose chosen local Artin value and global norm-residue value are both
+the finite quotient coordinate of the distinguished lift. -/
+opaque numberFieldTowerFinitePadicAuxiliaryLocalGlobalRepresentative
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (τ : (numberFieldTowerBaseSubgroup K L).toSubgroup)
+    (hτ :
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ ≠ 1)
+    (hdecomposition :
+      letI : Algebra K (SeparableClosure ℚ) :=
+        numberFieldTowerSeparableClosureBaseAlgebra K L
+      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
+        absoluteValueDecompositionGroup K
+          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+            K L v (chosenFinitePlaceExtension (L := L) v)).1)
+    (n : ℕ) (hn : 0 < n)
+    (hdegree :
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ =
+        (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n)
+    (hprimaryQuotient :
+      numberFieldTowerFiniteQuotientCoordinate
+          (K := K) (L := L) τ ∈
+        CommGroup.primaryComponent
+          ((numberFieldTowerBaseSubgroup K L).toSubgroup ⧸
+            extensionSubgroup
+              (numberFieldTowerBaseSubgroup K L)
+              (numberFieldTowerTopSubgroup L)
+              (numberFieldTowerTopSubgroup_le_baseSubgroup K L))
+          p.1) :
+    {z : (v.adicCompletion K)ˣ //
+      chosenFinitePlaceArtinMonoidHom (K := K) (L := L) v z =
+          numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+            (numberFieldTowerFiniteQuotientCoordinate
+              (K := K) (L := L) τ) ∧
+      globalNormResidueMonoidHom K L
+          (IdeleGroup.finitePlaceIdeleClass v z) =
+          numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+            (numberFieldTowerFiniteQuotientCoordinate
+              (K := K) (L := L) τ)} := by
+  let H :=
+    numberFieldTowerFinitePadicAuxiliaryAbstractField
+      (K := K) (L := L) p τ hτ
+  let S :=
+    numberFieldTowerFinitePadicCyclicFixedSubgroup
+      (K := K) (L := L) p τ
+  let P :=
+    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension
+      (K := K) (L := L) p τ
+  let F :=
+    LocalClassFieldTheory.abstractFixedField
+      ℚ (SeparableClosure ℚ) S
+  let E :=
+    LocalClassFieldTheory.abstractRelativeFixedField
+      ℚ (SeparableClosure ℚ) P.below
+  letI hHfinite : Finite
+      ((baseField
+        (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ)).toSubgroup ⧸
+        extensionSubgroup
+          (baseField
+            (SeparableClosure ℚ ≃ₐ[ℚ] SeparableClosure ℚ))
+          S (le_baseField S)) :=
+    H.finite
+  letI hPfinite : Finite
+      (S.toSubgroup ⧸ extensionSubgroup S P.field P.below) :=
+    P.finite
+  letI auxiliaryBaseNumberField : NumberField F := by
+    let : FiniteDimensional ℚ F :=
+      LocalClassFieldTheory.abstractFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ) S hHfinite
+    exact NumberField.of_module_finite ℚ F
+  letI auxiliaryTopNumberField : NumberField E := by
+    let : FiniteDimensional F E :=
+      LocalClassFieldTheory.abstractRelativeFixedField_finiteDimensional
+        ℚ (SeparableClosure ℚ)
+        S P.field P.below hHfinite hPfinite
+    exact NumberField.of_module_finite F E
+  letI auxiliaryAbelianGalois : IsAbelianGalois F E :=
+    GlobalClassFields.finiteAbelianSubextensionAbstractRelativeFixedFieldIsAbelianGalois P
+  letI auxiliaryOriginalBaseAlgebra : Algebra K F :=
+    numberFieldTowerFinitePadicAuxiliary_baseAlgebra
+      (K := K) (L := L) p τ
+  letI auxiliaryOriginalTopAlgebra : Algebra L E :=
+    numberFieldTowerFinitePadicAuxiliary_topAlgebra
+      (K := K) (L := L) p τ
+  letI auxiliaryOriginalBaseTopAlgebra : Algebra K E :=
+    numberFieldTowerFinitePadicAuxiliary_originalBaseTopAlgebra
+      (K := K) (L := L) p τ
+  letI auxiliaryOriginalTopScalarTower : IsScalarTower K L E :=
+    numberFieldTowerFinitePadicAuxiliary_originalTopScalarTower
+      (K := K) (L := L) p τ
+  letI auxiliaryBaseTopScalarTower : IsScalarTower K F E :=
+    numberFieldTowerFinitePadicAuxiliary_baseTopScalarTower
+      (K := K) (L := L) p τ
+  letI auxiliaryOriginalBaseGalois : IsGalois K F :=
+    numberFieldTowerFinitePadicAuxiliaryBase_isGalois
+      (K := K) (L := L) p τ
+  let auxiliaryOriginalTopAlgHom : L →ₐ[ℚ] E :=
+    numberFieldTowerFinitePadicAuxiliaryTopEmbedding
+      (K := K) (L := L) p τ
+  let wF :=
+    numberFieldTowerFinitePadicAuxiliaryBasePlaceExtension
+      (K := K) (L := L) v p τ
+  let wE :=
+    numberFieldTowerFinitePadicAuxiliaryTopPlaceExtension
+      (K := K) (L := L) v p τ
+  let V :=
+    finitePlaceExtensionCentre
+      (K := K) (L := F) v wF
+  have hVbelow : finitePlaceBelow (K := K) V = v :=
+    finitePlaceBelow_finitePlaceExtensionCentre
+      (K := K) (L := F) v wF
+  let Vover :
+      {V' : HeightOneSpectrum (𝓞 F) //
+        finitePlaceBelow (K := K) V' = v} :=
+    ⟨V, hVbelow⟩
+  letI auxiliaryCompletionAlgebra :
+      Algebra (v.adicCompletion K) (V.adicCompletion F) :=
+    (finitePlaceAdicCompletionMap K F v Vover).toAlgebra
+  let σE : Gal(E / F) :=
+    numberFieldTowerFinitePadicAuxiliaryAutomorphism
+      (K := K) (L := L) p τ
+  have hσtop :
+      σE ∈ absoluteValueDecompositionGroup F wE.1 :=
+    numberFieldTowerFinitePadicAuxiliaryAutomorphism_mem_topPlaceDecomposition
+      (K := K) (L := L) v p τ hdecomposition
+  have hgroup :=
+    numberFieldTowerFinitePadicAuxiliaryTopDecompositionGroup_eq_chosen
+      (K := K) (L := L) v p τ hτ
+  have hσchosen :
+      σE ∈ absoluteValueDecompositionGroup F
+        (chosenFinitePlaceExtension (L := E) V).1 := by
+    rw [← hgroup]
+    exact hσtop
+  have hRange :
+      σE ∈ (chosenFinitePlaceArtinMonoidHom
+        (K := F) (L := E) V).range := by
+    rw [chosenFinitePlaceArtinMonoidHom_range (K := F) (L := E) V]
+    exact hσchosen
+  let y : (V.adicCompletion F)ˣ := Classical.choose hRange
+  have hy :
+      chosenFinitePlaceArtinMonoidHom (K := F) (L := E) V y =
+        numberFieldTowerFinitePadicAuxiliaryAutomorphism
+          (K := K) (L := L) p τ :=
+    Classical.choose_spec hRange
+  let z : (v.adicCompletion K)ˣ :=
+    LocalFieldTheory.normUnits
+      (v.adicCompletion K) (V.adicCompletion F) y
+  have hz :
+      z = LocalFieldTheory.normUnits
+        (v.adicCompletion K) (V.adicCompletion F) y := by
+    rfl
+  let σK : Gal(L / K) :=
+    numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+      (numberFieldTowerFiniteQuotientCoordinate
+        (K := K) (L := L) τ)
+  let restriction : Gal(E / F) →* Gal(L / K) :=
+    (AlgEquiv.restrictNormalHom L).comp
+      (AlgEquiv.restrictScalarsHom K)
+  have hrestrict : restriction σE = σK :=
+    numberFieldTowerFinitePadicAuxiliaryAutomorphism_restriction
+      (K := K) (L := L) p τ
+  have hlocal :
+      chosenFinitePlaceArtinMonoidHom (K := K) (L := L) v z =
+        σK := by
+    rw [hz]
+    have hnat :=
+      DFunLike.congr_fun
+        (chosenFinitePlaceArtinMonoidHom_norm_restriction_of_below_eq
+          (K := K) (L := L) (K' := F) (L' := E)
+          v V hVbelow) y
+    calc
+      chosenFinitePlaceArtinMonoidHom (K := K) (L := L) v
+          (LocalFieldTheory.normUnits
+            (v.adicCompletion K) (V.adicCompletion F) y) =
+        restriction
+          (chosenFinitePlaceArtinMonoidHom (K := F) (L := E) V y) := by
+            simpa only [
+              MonoidHom.coe_comp, Function.comp_apply, restriction]
+              using hnat.symm
+      _ = restriction σE := congrArg restriction hy
+      _ = σK := hrestrict
+  let j : E →ₐ[ℚ] SeparableClosure ℚ :=
+    E.val.restrictScalars ℚ
+  have hjLower :
+      j.comp auxiliaryOriginalTopAlgHom =
+        AlgebraicNumberTheory.numberFieldSeparableClosureEmbedding L := by
+    apply AlgHom.ext
+    intro a
+    exact
+      numberFieldTowerFinitePadicAuxiliaryTopEmbedding_coe
+        (K := K) (L := L) p τ a
+  have hPUnramified :
+      P.toFiniteGaloisExtension.IsUnramified
+        rationalCyclotomicDegreeData :=
+    numberFieldTowerFinitePadicAuxiliaryCompositumSubextension_isUnramified
+      (K := K) (L := L) p τ n hn hdegree hprimaryQuotient
+  have hcompat :
+      (globalNormResidueMonoidHomOfEmbedding F E j).comp
+          (IdeleGroup.finitePlaceIdeleClass V) =
+        chosenFinitePlaceArtinMonoidHom (K := F) (L := E) V :=
+    globalNormResidueMonoidHomOfEmbedding_comp_finitePlaceIdeleClass_of_abstractFixedFieldUnramified
+      H P hPUnramified V
+  have hupper :
+      globalNormResidueMonoidHomOfEmbedding F E j
+          (IdeleGroup.finitePlaceIdeleClass V y) =
+        σE :=
+    (congrArg
+      (fun φ : (V.adicCompletion F)ˣ →* Gal(E / F) => φ y)
+      hcompat).trans hy
+  have hnormClass :
+      _root_.ideleClassNorm K F
+          (IdeleGroup.finitePlaceIdeleClass V y) =
+        IdeleGroup.finitePlaceIdeleClass v z := by
+    rw [hz]
+    simpa only [Vover] using
+      (IdeleGroup.ideleClassNorm_finitePlaceIdeleClass_eq_normUnits
+        (K := K) (L := F) v Vover y)
+  have hglobal :
+      globalNormResidueMonoidHom K L
+          (IdeleGroup.finitePlaceIdeleClass v z) =
+        σK := by
+    calc
+      globalNormResidueMonoidHom K L
+          (IdeleGroup.finitePlaceIdeleClass v z) =
+          globalNormResidueMonoidHom K L
+            (_root_.ideleClassNorm K F
+              (IdeleGroup.finitePlaceIdeleClass V y)) :=
+        congrArg (globalNormResidueMonoidHom K L) hnormClass.symm
+      _ = globalNormResidueMonoidHomOfEmbedding K L
+          (j.comp auxiliaryOriginalTopAlgHom)
+          (_root_.ideleClassNorm K F
+            (IdeleGroup.finitePlaceIdeleClass V y)) := by
+        rw [hjLower,
+          ← globalNormResidueMonoidHom_eq_ofEmbedding_standard]
+      _ = ((AlgEquiv.restrictNormalHom L).comp
+          (AlgEquiv.restrictScalarsHom K))
+          (globalNormResidueMonoidHomOfEmbedding F E j
+            (IdeleGroup.finitePlaceIdeleClass V y)) := by
+        apply Eq.symm
+        apply
+          globalNormResidueMonoidHomOfEmbedding_norm_restriction_apply
+      _ = ((AlgEquiv.restrictNormalHom L).comp
+          (AlgEquiv.restrictScalarsHom K)) σE :=
+        congrArg
+          ((AlgEquiv.restrictNormalHom L).comp
+            (AlgEquiv.restrictScalarsHom K)) hupper
+      _ = restriction σE := rfl
+      _ = σK := hrestrict
+  exact ⟨z, hlocal, hglobal⟩
+
+
+/-- Every genuine finite-place decomposition automorphism has a
+compatible embedded absolute lift with the same finite quotient class
+and positive integral cyclotomic `p`-adic degree. -/
+theorem exists_numberFieldTowerFinitePadicLift_of_finitePlace
+    (v : HeightOneSpectrum (𝓞 K))
+    (p : Nat.Primes)
+    (σ :
+      absoluteValueDecompositionGroup K
+        (chosenFinitePlaceExtension (L := L) v).1) :
+    letI _ : Algebra K (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureBaseAlgebra K L
+    letI _ : Algebra L (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureTopAlgebra L
+    letI _ : IsScalarTower K L (SeparableClosure ℚ) :=
+      numberFieldTowerSeparableClosureScalarTower K L
+    ∃ τ : (numberFieldTowerBaseSubgroup K L).toSubgroup,
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+          (numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) τ) =
+        σ.1 ∧
+      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
+        absoluteValueDecompositionGroup K
+          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+            K L v
+            (chosenFinitePlaceExtension (L := L) v)).1 ∧
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ ≠
+        1 ∧
+      ∃ n : ℕ, 0 < n ∧
+        numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+            (K := K) (L := L) p τ =
+          (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
+  let : Algebra K (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureBaseAlgebra K L
+  let : Algebra L (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureTopAlgebra L
+  let : IsScalarTower K L (SeparableClosure ℚ) :=
+    numberFieldTowerSeparableClosureScalarTower K L
+  obtain ⟨τΩ, hτΩrestrict, n, hn, hτΩdegree⟩ :=
+    exists_finitePlaceSeparableClosureLift_with_positivePadicCyclotomicDegree
+      (K := K) (L := L) v p σ
+  let τ :
+      (numberFieldTowerBaseSubgroup K L).toSubgroup :=
+    numberFieldTowerSeparableClosureEquivBaseSubgroup
+      K L τΩ.1
+  have hfinite :
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+          (numberFieldTowerFiniteQuotientCoordinate
+            (K := K) (L := L) τ) =
+        σ.1 := by
+    change
+      numberFieldTowerExtensionQuotientEquivGaloisGroup K L
+          (QuotientGroup.mk
+            (numberFieldTowerSeparableClosureEquivBaseSubgroup
+              K L τΩ.1)) =
+        σ.1
+    rw [
+      numberFieldTowerExtensionQuotientEquivGaloisGroup_mk_baseSubgroupEquiv]
+    exact congrArg Subtype.val hτΩrestrict
+  have hdegree :
+      numberFieldTowerBaseSubgroupPadicCyclotomicDegree
+          (K := K) (L := L) p τ =
+        (Multiplicative.ofAdd (1 : ℤ_[p.1])) ^ n := by
+    exact hτΩdegree
+  have hdecomposition :
+      (numberFieldTowerSeparableClosureEquivBaseSubgroup K L).symm τ ∈
+        absoluteValueDecompositionGroup K
+          (numberFieldTowerFinitePlaceExtensionToSeparableClosure
+            K L v
+            (chosenFinitePlaceExtension (L := L) v)).1 := by
+    simpa only [τ, MulEquiv.symm_apply_apply] using τΩ.2
+  refine
+    ⟨τ, hfinite, hdecomposition, ?_, n, hn, hdegree⟩
+  rw [hdegree]
+  exact
+    PadicInt.multiplicative_positiveNatDegree_ne_one
+      p.1 n hn
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityBase.lean
new file mode 100644
index 0000000..e5057a1
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityBase.lean
@@ -0,0 +1,488 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalHilbertSymbol.FinitePlaceCharacterComparison
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.FiniteIdeleArtin
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.HilbertProductFormula
+import ClassFieldTheory.AlgebraicNumberTheory.PowerResidueSymbols.Ideal
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.NormApproximation.FinitePlaces
+import Mathlib.NumberTheory.Padics.HeightOneSpectrum
+import ValuedFieldTheory.LocalField.DiscreteValuationField.PadicField
+import ClassFieldTheory.AlgebraicNumberTheory.QuadraticReciprocity
+import ClassFieldTheory.AlgebraicNumberTheory.RayClass.Rational
+import ClassFieldTheory.LocalClassFieldTheory.Kummer.PowerResidueTameFormula
+import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.IdealQuotients
+import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.IdealToCompletion
+import ValuedFieldTheory.Valuation.Completion.ExtensionInvariants
+import ClassFieldTheory.KummerTheory.Concrete.SimpleExtensionLocalBehavior
+import ClassFieldTheory.KummerTheory.Concrete.SUnitPreparation.FiniteRadicalSupport
+import ClassFieldTheory.LocalClassFieldTheory.Finite.CyclotomicNorm.PrincipalUnits
+
+
+set_option autoImplicit false
+
+
+open scoped BigOperators Classical NumberField NumberTheorySymbols ValuativeRel WithZero
+open NumberField IsDedekindDomain
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open KummerTheory
+open AlgebraicNumberTheory.PowerResidueSymbols
+open LocalClassFieldTheory.Kummer
+open LocalFieldTheory
+open LocalFieldTheory.IsNonarchimedeanLocalField
+
+variable (K : Type) [Field K] [NumberField K]
+
+/-- Equality for the valuation used to define a valuative relation transports
+to equality in the relation's canonical value group. -/
+theorem canonicalValuation_eq_of_valuation_eq
+    {R Γ : Type*} [Ring R] [LinearOrderedCommGroupWithZero Γ]
+    (v : Valuation R Γ) (x y : R) (hxy : v x = v y) :
+    letI : ValuativeRel R := ValuativeRel.ofValuation v
+    ValuativeRel.valuation R x = ValuativeRel.valuation R y := by
+  let : ValuativeRel R := ValuativeRel.ofValuation v
+  change
+    ValuativeRel.ValueGroupWithZero.mk x 1 =
+      ValuativeRel.ValueGroupWithZero.mk y 1
+  rw [ValuativeRel.ValueGroupWithZero.mk_eq_mk]
+  constructor
+  · change v (x * (1 : R)) ≤ v (y * (1 : R))
+    simpa only [mul_one] using hxy.le
+  · change v (y * (1 : R)) ≤ v (x * (1 : R))
+    simpa only [mul_one] using hxy.ge
+
+/-- The bounded-natural-number form of nonarchimedeanness for a finite-place
+absolute value.  Naming this bridge keeps all completion residue constructions
+on one proof-irrelevant provider. -/
+private theorem finitePlaceAdicAbv_nonarchimedeanAbsoluteValue
+    (v : HeightOneSpectrum (𝓞 K)) :
+    LubinTate.Valuations.NonarchimedeanAbsoluteValue
+      (HeightOneSpectrum.adicAbv K v) :=
+  (AbsoluteValue.isNonarchimedean_iff_bounded_nat
+    (HeightOneSpectrum.adicAbv K v)).1
+      (HeightOneSpectrum.isNonarchimedean_adicAbv K v)
+
+/-- Finite-field power-residue symbols commute with a field equivalence.
+The statement is made on underlying units so it can be reused with every
+roots-of-unity transport occurring below. -/
+theorem finiteFieldPowerResidueSymbol_unitsMap_ringEquiv
+    {k l : Type*} [Field k] [Field l] [Fintype k] [Fintype l]
+    (e : k ≃+* l) (n : ℕ+)
+    (hnk : (n : ℕ) ∣ Fintype.card k - 1)
+    (hnl : (n : ℕ) ∣ Fintype.card l - 1)
+    (u : kˣ) :
+    Units.map e.toMonoidHom
+        ((AlgebraicNumberTheory.PowerResidueSymbols.finiteFieldPowerResidueSymbol
+            k n hnk u :
+            rootsOfUnity (n : ℕ) k) : kˣ) =
+      ((AlgebraicNumberTheory.PowerResidueSymbols.finiteFieldPowerResidueSymbol
+          l n hnl
+          (Units.map e.toMonoidHom u) :
+            rootsOfUnity (n : ℕ) l) : lˣ) := by
+  rw [
+    AlgebraicNumberTheory.PowerResidueSymbols.finiteFieldPowerResidueSymbol_apply,
+    AlgebraicNumberTheory.PowerResidueSymbols.finiteFieldPowerResidueSymbol_apply,
+    map_pow]
+  rw [Fintype.card_congr e.toEquiv]
+
+/-- The residue field of a finite-place completion is canonically the
+prime-ideal residue field.  The construction passes through the localization
+at the prime and then through the residue equivalence induced by completion. -/
+noncomputable def finitePlacePrimeResidueEquivLocalResidue
+    (v : HeightOneSpectrum (𝓞 K)) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    (𝓞 K ⧸ v.asIdeal) ≃+* 𝓀[C] := by
+  let a := HeightOneSpectrum.adicAbv K v
+  let ha : LubinTate.Valuations.NonarchimedeanAbsoluteValue a :=
+    finitePlaceAdicAbv_nonarchimedeanAbsoluteValue K v
+  let C := a.Completion
+  letI : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  let V := LubinTate.Valuations.exponentialValuationSubring
+    (AlgebraicNumberTheory.Valuations.absoluteValueExponentialValuation a ha)
+  let aC := AbsoluteValue.completionAbsoluteValue a
+  let haC : LubinTate.Valuations.NonarchimedeanAbsoluteValue aC :=
+    (AbsoluteValue.isNonarchimedean_iff_bounded_nat aC).1
+      (AbsoluteValue.completionAbsoluteValue_isNonarchimedean a
+        ((AbsoluteValue.isNonarchimedean_iff_bounded_nat a).2 ha))
+  let VC := LubinTate.Valuations.exponentialValuationSubring
+    (AlgebraicNumberTheory.Valuations.absoluteValueExponentialValuation aC haC)
+  let Rv := v.valuationSubringAtPrime K
+  letI : IsLocalRing Rv :=
+    IsLocalization.AtPrime.isLocalRing Rv v.asIdeal
+  have hBase : Rv.toSubring = V := by
+    ext x
+    rw [AlgebraicNumberTheory.Valuations.mem_absoluteValueExponentialSubring_iff]
+    change x ∈ v.valuationSubringAtPrime K ↔ _
+    rw [v.valuationSubringAtPrime_eq_valuationSubring]
+    change v.valuation K x ≤ 1 ↔ a x ≤ 1
+    rw [HeightOneSpectrum.adicAbv_def]
+    exact_mod_cast
+      (WithZeroMulInt.toNNReal_le_one_iff
+        (HeightOneSpectrum.one_lt_absNorm_nnreal v)).symm
+  let eBase : Rv ≃+* V := RingEquiv.subringCongr hBase
+  have hCompletion : VC = 𝒪[C] := by
+    ext x
+    rw [AlgebraicNumberTheory.Valuations.mem_absoluteValueExponentialSubring_iff]
+    change ‖x‖ ≤ 1 ↔ x ∈ 𝒪[C]
+    exact
+      (finitePlaceCompletion_mem_integers_iff_norm_le_one
+        a (HeightOneSpectrum.isNonarchimedean_adicAbv K v) x).symm
+  let eCompletionRing : VC ≃+* 𝒪[C] :=
+    RingEquiv.subringCongr hCompletion
+  let eIdeal : (𝓞 K ⧸ v.asIdeal) ≃+* v.asIdeal.ResidueField :=
+    RingEquiv.ofBijective
+      (algebraMap (𝓞 K ⧸ v.asIdeal) v.asIdeal.ResidueField)
+      v.asIdeal.bijective_algebraMap_quotient_residueField
+  let eLocalization : Localization.AtPrime v.asIdeal ≃ₐ[𝓞 K] Rv :=
+    IsLocalization.algEquiv v.asIdeal.primeCompl _ _
+  exact
+    eIdeal |>.trans
+      (IsLocalRing.ResidueField.mapEquiv eLocalization.toRingEquiv) |>.trans
+      (IsLocalRing.ResidueField.mapEquiv eBase) |>.trans
+      (AlgebraicNumberTheory.Valuations.completionResidueEquiv a ha) |>.trans
+      (IsLocalRing.ResidueField.mapEquiv eCompletionRing)
+
+/-- The image of an algebraic integer in the valuation ring of a finite-place
+completion. -/
+noncomputable def finitePlaceIntegralCompletionElement
+    (v : HeightOneSpectrum (𝓞 K)) (x : 𝓞 K) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    𝒪[C] := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  letI : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  exact ⟨algebraMap K C (x : K), by
+    rw [finitePlaceCompletion_mem_integers_iff_norm_le_one
+      (HeightOneSpectrum.adicAbv K v)
+      (HeightOneSpectrum.isNonarchimedean_adicAbv K v)]
+    calc
+      ‖algebraMap K C (x : K)‖ =
+          HeightOneSpectrum.adicAbv K v (x : K) :=
+        AbsoluteValue.completionAbsoluteValue_coe
+          (HeightOneSpectrum.adicAbv K v) (x : K)
+      _ ≤ 1 :=
+        v.adicAbv_coe_le_one
+          (HeightOneSpectrum.one_lt_absNorm_nnreal v) x⟩
+
+/-- The finite-place integral element has the expected underlying completion
+value. -/
+@[simp]
+theorem finitePlaceIntegralCompletionElement_coe
+    (v : HeightOneSpectrum (𝓞 K)) (x : 𝓞 K) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    (finitePlaceIntegralCompletionElement K v x : C) =
+      algebraMap K C (x : K) :=
+  rfl
+
+/-- The finite-place residue equivalence sends the class of an algebraic
+integer to the residue of its canonical image in the completion. -/
+@[simp]
+theorem finitePlacePrimeResidueEquivLocalResidue_mk
+    (v : HeightOneSpectrum (𝓞 K)) (x : 𝓞 K) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    finitePlacePrimeResidueEquivLocalResidue K v
+        (Ideal.Quotient.mk v.asIdeal x) =
+      IsLocalRing.residue 𝒪[C]
+        (finitePlaceIntegralCompletionElement K v x) := by
+  simp only [finitePlacePrimeResidueEquivLocalResidue,
+    finitePlaceIntegralCompletionElement, RingEquiv.trans_apply]
+  rw [RingEquiv.ofBijective_apply,
+    Ideal.algebraMap_quotient_residueField_mk]
+  rw [IsScalarTower.algebraMap_apply
+    (NumberField.RingOfIntegers K) (Localization.AtPrime v.asIdeal)]
+  rw [IsLocalRing.ResidueField.algebraMap_eq]
+  simp only [IsLocalRing.ResidueField.mapEquiv_apply,
+    IsLocalRing.ResidueField.map_residue]
+  rw [AlgebraicNumberTheory.Valuations.completionResidueEquiv_residue]
+  simp only [IsLocalRing.ResidueField.map_residue]
+  congr 1
+  apply Subtype.ext
+  change
+    algebraMap K (HeightOneSpectrum.adicAbv K v).Completion
+        (((IsLocalization.algEquiv v.asIdeal.primeCompl
+            (Localization.AtPrime v.asIdeal)
+            (v.valuationSubringAtPrime K))
+          (algebraMap (𝓞 K) (Localization.AtPrime v.asIdeal) x) :
+        v.valuationSubringAtPrime K) : K) =
+      algebraMap K (HeightOneSpectrum.adicAbv K v).Completion (x : K)
+  rw [AlgEquiv.commutes]
+  rw [IsScalarTower.algebraMap_apply
+    (NumberField.RingOfIntegers K) (v.valuationSubringAtPrime K)]
+  rfl
+
+/-- A nonzero algebraic integer, regarded as a global field unit. -/
+def nonzeroIntegralFieldUnit (x : 𝓞 K) (hx : x ≠ 0) : Kˣ :=
+  Units.mk0 (x : K) (by
+    intro hxK
+    apply hx
+    apply Subtype.ext
+    exact hxK)
+
+omit [NumberField K] in
+@[simp]
+theorem nonzeroIntegralFieldUnit_coe (x : 𝓞 K) (hx : x ≠ 0) :
+    ((nonzeroIntegralFieldUnit K x hx : Kˣ) : K) = (x : K) :=
+  rfl
+
+/-- An algebraic integer avoiding a prime ideal, regarded as a nonzero
+element of the global field. -/
+noncomputable def primeAvoidingIntegralFieldUnit
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) : Kˣ :=
+  nonzeroIntegralFieldUnit K x (by
+    intro hx0
+    apply hx
+    rw [hx0]
+    exact Ideal.zero_mem _)
+
+omit [NumberField K] in
+@[simp]
+theorem primeAvoidingIntegralFieldUnit_coe
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) :
+    ((primeAvoidingIntegralFieldUnit K v x hx : Kˣ) : K) = (x : K) :=
+  rfl
+
+/-- An algebraic integer nonzero modulo `v`, regarded as a unit of the
+valuation ring of the finite-place completion. -/
+noncomputable def finitePlaceIntegralCompletionUnit
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    𝒪[C]ˣ := by
+  let a := HeightOneSpectrum.adicAbv K v
+  let C := a.Completion
+  letI : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  let y : C := algebraMap K C (x : K)
+  have hyNorm : ‖y‖ = 1 := by
+    calc
+      ‖y‖ = a (x : K) :=
+        AbsoluteValue.completionAbsoluteValue_coe a (x : K)
+      _ = ‖NumberField.FinitePlace.embedding v (x : K)‖ :=
+        (NumberField.FinitePlace.norm_embedding v (x : K)).symm
+      _ = 1 :=
+        (NumberField.FinitePlace.norm_eq_one_iff_notMem K v x).2 hx
+  have hyNe : y ≠ 0 := by
+    intro hy
+    rw [hy, norm_zero] at hyNorm
+    exact zero_ne_one hyNorm
+  let yIntegral : 𝒪[C] := ⟨y, by
+    rw [finitePlaceCompletion_mem_integers_iff_norm_le_one
+      a (HeightOneSpectrum.isNonarchimedean_adicAbv K v)]
+    exact hyNorm.le⟩
+  let yInvIntegral : 𝒪[C] := ⟨y⁻¹, by
+    rw [finitePlaceCompletion_mem_integers_iff_norm_le_one
+      a (HeightOneSpectrum.isNonarchimedean_adicAbv K v),
+      norm_inv, hyNorm, inv_one]⟩
+  exact {
+    val := yIntegral
+    inv := yInvIntegral
+    val_inv := by
+      apply Subtype.ext
+      exact mul_inv_cancel₀ hyNe
+    inv_val := by
+      apply Subtype.ext
+      exact inv_mul_cancel₀ hyNe }
+
+/-- Forgetting the integral-unit structure recovers the ordinary image of
+the algebraic integer in the finite-place completion. -/
+@[simp]
+theorem finitePlaceIntegralCompletionUnit_coe
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    (((finitePlaceIntegralCompletionUnit K v x hx : 𝒪[C]ˣ) : 𝒪[C]) : C) =
+      algebraMap K C (x : K) := by
+  rfl
+
+/-- The completion image of a prime-avoiding algebraic integer is the field
+unit underlying its canonical valuation-ring unit. -/
+theorem finitePlaceHilbert_completionUnit_primeAvoidingIntegralFieldUnit
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    finitePlaceHilbert_completionUnit K v
+        (primeAvoidingIntegralFieldUnit K v x hx) =
+      integerUnitsToFieldUnits C
+        (finitePlaceIntegralCompletionUnit K v x hx) := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  let : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  let : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  dsimp only
+  apply Units.ext
+  change
+    algebraMap K C (x : K) =
+      (((finitePlaceIntegralCompletionUnit K v x hx : 𝒪[C]ˣ) :
+        𝒪[C]) : C)
+  exact (finitePlaceIntegralCompletionUnit_coe K v x hx).symm
+
+/-- As an element of the completion valuation ring, the lifted unit is the
+canonical lifted algebraic integer. -/
+@[simp]
+theorem finitePlaceIntegralCompletionUnit_val
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    (finitePlaceIntegralCompletionUnit K v x hx : 𝒪[C]) =
+      finitePlaceIntegralCompletionElement K v x := by
+  dsimp only
+  apply Subtype.ext
+  exact finitePlaceIntegralCompletionUnit_coe K v x hx
+
+/-- Reduction of the canonical completion unit agrees with reduction modulo
+the corresponding global prime ideal. -/
+theorem finitePlace_integerUnitsToResidueUnits_integralUnit
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    integerUnitsToResidueUnits C
+        (finitePlaceIntegralCompletionUnit K v x hx) =
+      Units.map
+        (finitePlacePrimeResidueEquivLocalResidue K v).toMonoidHom
+        (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealResidueUnit
+          K v x hx) := by
+  dsimp only
+  apply Units.ext
+  change
+    IsLocalRing.residue 𝒪[(HeightOneSpectrum.adicAbv K v).Completion]
+        (finitePlaceIntegralCompletionUnit K v x hx :
+          𝒪[(HeightOneSpectrum.adicAbv K v).Completion]) =
+      finitePlacePrimeResidueEquivLocalResidue K v
+        (Ideal.Quotient.mk v.asIdeal x)
+  simpa only [finitePlaceIntegralCompletionUnit_val] using
+    (finitePlacePrimeResidueEquivLocalResidue_mk K v x).symm
+
+/-- The canonical inclusion from integral roots of unity into the common
+field-valued group used by the global Hilbert symbols. -/
+def integralRootsOfUnityToNthRoots
+    (n : ℕ) :
+    rootsOfUnity n (𝓞 K) →* nthRootsSubgroup K n where
+  toFun z :=
+    ⟨Units.map (algebraMap (𝓞 K) K).toMonoidHom z.1, by
+      calc
+        Units.map (algebraMap (𝓞 K) K).toMonoidHom z.1 ^ n =
+            Units.map (algebraMap (𝓞 K) K).toMonoidHom (z.1 ^ n) :=
+          (map_pow
+            (Units.map (algebraMap (𝓞 K) K).toMonoidHom) z.1 n).symm
+        _ = 1 := by rw [z.2, map_one]⟩
+  map_one' := by
+    apply Subtype.ext
+    exact map_one (Units.map (algebraMap (𝓞 K) K).toMonoidHom)
+  map_mul' := by
+    intro z w
+    apply Subtype.ext
+    exact map_mul
+      (Units.map (algebraMap (𝓞 K) K).toMonoidHom) z.1 w.1
+
+omit [NumberField K] in
+/-- The integral-root inclusion is the underlying unit map. -/
+@[simp]
+theorem integralRootsOfUnityToNthRoots_apply
+    (n : ℕ) (z : rootsOfUnity n (𝓞 K)) :
+    (integralRootsOfUnityToNthRoots K n z).1 =
+      Units.map (algebraMap (𝓞 K) K).toMonoidHom z.1 :=
+  rfl
+
+omit [NumberField K] in
+/-- The integral-to-field inclusion is injective on roots of unity. -/
+theorem integralRootsOfUnityToNthRoots_injective
+    (n : ℕ) :
+    Function.Injective (integralRootsOfUnityToNthRoots K n) := by
+  intro z w h
+  apply Subtype.ext
+  apply
+    (Units.map_injective
+      (f := (algebraMap (𝓞 K) K).toMonoidHom)
+      RingOfIntegers.coe_injective)
+  exact congrArg Subtype.val h
+
+/-- Reduction after embedding an integral global root of unity into a
+finite-place completion is the transport of reduction modulo the
+corresponding prime ideal. -/
+theorem finitePlace_localNthRootsReduction_integralRoots
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (z : rootsOfUnity (n : ℕ) (𝓞 K)) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    (localNthRootsReduction C n
+        (nthRootsSubgroupMap K C (n : ℕ)
+          (integralRootsOfUnityToNthRoots K (n : ℕ) z))).1 =
+      Units.map
+        (finitePlacePrimeResidueEquivLocalResidue K v).toMonoidHom
+        (AlgebraicNumberTheory.PowerResidueSymbols.rootsOfUnityReduction
+          K v (n : ℕ) z).1 := by
+  dsimp only
+  apply Units.ext
+  change
+    IsLocalRing.residue 𝒪[(HeightOneSpectrum.adicAbv K v).Completion]
+        (nthRootIntegerUnit
+          (HeightOneSpectrum.adicAbv K v).Completion n
+          (nthRootsSubgroupMap K
+            (HeightOneSpectrum.adicAbv K v).Completion (n : ℕ)
+            (integralRootsOfUnityToNthRoots K (n : ℕ) z)) :
+          𝒪[(HeightOneSpectrum.adicAbv K v).Completion]) =
+      finitePlacePrimeResidueEquivLocalResidue K v
+        (Ideal.Quotient.mk v.asIdeal (z.1 : 𝓞 K))
+  rw [finitePlacePrimeResidueEquivLocalResidue_mk]
+  congr 1
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityCorrection.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityCorrection.lean
new file mode 100644
index 0000000..83cbaaa
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityCorrection.lean
@@ -0,0 +1,452 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.PowerResidueReciprocityLocalSymbols
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalHilbertSymbol.FinitePlaceCharacterComparison
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.FiniteIdeleArtin
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.HilbertProductFormula
+import ClassFieldTheory.AlgebraicNumberTheory.PowerResidueSymbols.Ideal
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.NormApproximation.FinitePlaces
+import Mathlib.NumberTheory.Padics.HeightOneSpectrum
+import ValuedFieldTheory.LocalField.DiscreteValuationField.PadicField
+import ClassFieldTheory.AlgebraicNumberTheory.QuadraticReciprocity
+import ClassFieldTheory.AlgebraicNumberTheory.RayClass.Rational
+import ClassFieldTheory.LocalClassFieldTheory.Kummer.PowerResidueTameFormula
+import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.IdealQuotients
+import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.IdealToCompletion
+import ValuedFieldTheory.Valuation.Completion.ExtensionInvariants
+import ClassFieldTheory.KummerTheory.Concrete.SimpleExtensionLocalBehavior
+import ClassFieldTheory.KummerTheory.Concrete.SUnitPreparation.FiniteRadicalSupport
+import ClassFieldTheory.LocalClassFieldTheory.Finite.CyclotomicNorm.PrincipalUnits
+
+
+set_option autoImplicit false
+
+
+open scoped BigOperators Classical NumberField NumberTheorySymbols ValuativeRel WithZero
+open NumberField IsDedekindDomain
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open KummerTheory
+open AlgebraicNumberTheory.PowerResidueSymbols
+open LocalClassFieldTheory.Kummer
+open LocalFieldTheory
+open LocalFieldTheory.IsNonarchimedeanLocalField
+
+variable (K : Type) [Field K] [NumberField K]
+/-- Prime avoidance is the common special case of the integral valuation
+formula used for numerator units. -/
+theorem finitePlaceNormalizedValuation_primeAvoidingIntegralFieldUnit
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ∉ v.asIdeal) :
+    finitePlaceNormalizedValuation K v
+        (primeAvoidingIntegralFieldUnit K v x hx) =
+      -(AlgebraicNumberTheory.PowerResidueSymbols.idealPrimeMultiplicity K v (Ideal.span {x}) : ℤ) := by
+  let hx0 : x ≠ 0 := by
+    intro hxzero
+    apply hx
+    rw [hxzero]
+    exact Ideal.zero_mem _
+  change finitePlaceNormalizedValuation K v
+      (nonzeroIntegralFieldUnit K x hx0) = _
+  exact finitePlaceNormalizedValuation_nonzeroIntegralFieldUnit K v x hx0
+
+/-- Integral form of the finite-place comparison: the exponent is the
+Dedekind multiplicity in the principal denominator ideal. -/
+theorem finitePlaceHilbertSymbol_integral_eq_primeIdealPowerResidueFactor
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a b : 𝓞 K) (ha0 : a ≠ 0) (hb0 : b ≠ 0)
+    (ha : a ∉ v.asIdeal) :
+    finitePlaceHilbertSymbol K n hnK hmu v
+        (nonzeroIntegralFieldUnit K a ha0)
+        (nonzeroIntegralFieldUnit K b hb0) =
+      integralRootsOfUnityToNthRoots K (n : ℕ)
+          (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealPowerResidueSymbol
+              K v n hmu hcoprime a ha) ^
+        AlgebraicNumberTheory.PowerResidueSymbols.idealPrimeMultiplicity K v (Ideal.span {b}) := by
+  have haUnits :
+      nonzeroIntegralFieldUnit K a ha0 =
+        primeAvoidingIntegralFieldUnit K v a ha := by
+    apply Units.ext
+    rfl
+  rw [haUnits,
+    finitePlaceHilbertSymbol_eq_primeIdealPowerResidueFactor
+      K n hnK hmu v hv hcoprime a ha
+        (nonzeroIntegralFieldUnit K b hb0),
+    finitePlaceNormalizedValuation_nonzeroIntegralFieldUnit]
+  simp only [neg_neg, zpow_natCast]
+
+/-- Finite-place Hilbert symbols inherit skew symmetry from the local
+Hilbert symbol in the canonical completion. -/
+theorem finitePlaceHilbertSymbol_skew
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (v : HeightOneSpectrum (𝓞 K)) (a b : Kˣ) :
+    finitePlaceHilbertSymbol K n hnK hmu v a b =
+      (finitePlaceHilbertSymbol K n hnK hmu v b a)⁻¹ := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  let : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  let : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  apply nthRootsSubgroupMap_injective K C (n : ℕ)
+  rw [map_inv,
+    finitePlaceHilbertSymbol_map_eq_localHilbertSymbol,
+    finitePlaceHilbertSymbol_map_eq_localHilbertSymbol]
+  exact
+    localHilbertSymbol_skew C n
+      (finitePlaceHilbert_natCast_ne_zero K n hnK v)
+      (finitePlaceHilbert_primitiveRoots_nonempty K n hmu v)
+      (finitePlaceHilbert_completionUnit K v a)
+      (finitePlaceHilbert_completionUnit K v b)
+
+/-- If two nonzero algebraic integers are both units at a finite place, the
+corresponding finite-place Hilbert symbol is trivial. -/
+theorem finitePlaceHilbertSymbol_integral_units_eq_one
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (a b : 𝓞 K) (ha0 : a ≠ 0) (hb0 : b ≠ 0)
+    (ha : a ∉ v.asIdeal) (hb : b ∉ v.asIdeal) :
+    finitePlaceHilbertSymbol K n hnK hmu v
+        (nonzeroIntegralFieldUnit K a ha0)
+        (nonzeroIntegralFieldUnit K b hb0) = 1 := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  let : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  let : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  apply nthRootsSubgroupMap_injective K C (n : ℕ)
+  rw [map_one, finitePlaceHilbertSymbol_map_eq_localHilbertSymbol]
+  have haUnits :
+      finitePlaceHilbert_completionUnit K v
+          (nonzeroIntegralFieldUnit K a ha0) =
+        integerUnitsToFieldUnits C
+          (finitePlaceIntegralCompletionUnit K v a ha) := by
+    apply Units.ext
+    change
+      algebraMap K C (a : K) =
+        (((finitePlaceIntegralCompletionUnit K v a ha : 𝒪[C]ˣ) :
+          𝒪[C]) : C)
+    exact (finitePlaceIntegralCompletionUnit_coe K v a ha).symm
+  have hbUnits :
+      finitePlaceHilbert_completionUnit K v
+          (nonzeroIntegralFieldUnit K b hb0) =
+        integerUnitsToFieldUnits C
+          (finitePlaceIntegralCompletionUnit K v b hb) := by
+    apply Units.ext
+    change
+      algebraMap K C (b : K) =
+        (((finitePlaceIntegralCompletionUnit K v b hb : 𝒪[C]ˣ) :
+          𝒪[C]) : C)
+    exact (finitePlaceIntegralCompletionUnit_coe K v b hb).symm
+  unfold finitePlaceLocalHilbertSymbol
+  rw [haUnits, hbUnits]
+  exact
+    localHilbertSymbol_integerUnit_integerUnit_eq_one C n
+      (finitePlace_natCast_valuation_eq_one_of_not_mem_exponent K n v hv)
+      (finitePlaceHilbert_primitiveRoots_nonempty K n hmu v)
+      (finitePlaceIntegralCompletionUnit K v a ha)
+      (finitePlaceIntegralCompletionUnit K v b hb)
+
+/-- Primewise comparison between the tame finite-place Hilbert factor and
+the quotient of the two ideal power-residue factors. -/
+theorem powerResidueAwayFromExponentFiniteFactor_integral_eq_idealFactors
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : 𝓞 K) (ha0 : a ≠ 0) (hb0 : b ≠ 0)
+    (hcoprimeA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} →
+          (Ideal.absNorm P.asIdeal).Coprime (n : ℕ))
+    (hcoprimeB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} →
+          (Ideal.absNorm P.asIdeal).Coprime (n : ℕ))
+    (haB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} → a ∉ P.asIdeal)
+    (hbA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} → b ∉ P.asIdeal)
+    (hAwayA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} →
+          P ∉ powerResidueExponentFinitePlaces K n)
+    (hAwayB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} →
+          P ∉ powerResidueExponentFinitePlaces K n)
+    (P : HeightOneSpectrum (𝓞 K)) :
+    (if P ∈ powerResidueExponentFinitePlaces K n then
+        1
+      else
+        finitePlaceHilbertSymbol K n hnK hmu P
+          (nonzeroIntegralFieldUnit K a ha0)
+          (nonzeroIntegralFieldUnit K b hb0)) =
+      integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueFactor K (Ideal.span {b}) n hmu a
+            hcoprimeB haB P) *
+        (integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueFactor K (Ideal.span {a}) n hmu b
+            hcoprimeA hbA P))⁻¹ := by
+  by_cases hPExponent : P ∈ powerResidueExponentFinitePlaces K n
+  · have hPA : ¬ P.asIdeal ∣ Ideal.span {a} := by
+      intro hPA
+      exact (hAwayA P hPA) hPExponent
+    have hPB : ¬ P.asIdeal ∣ Ideal.span {b} := by
+      intro hPB
+      exact (hAwayB P hPB) hPExponent
+    simp only [ite_eq_left hPExponent, idealPowerResidueFactor,
+      dite_eq_right hPA, dite_eq_right hPB, map_one, inv_one, mul_one]
+  · rw [ite_eq_right hPExponent]
+    by_cases hPB : P.asIdeal ∣ Ideal.span {b}
+    · have haP : a ∉ P.asIdeal := haB P hPB
+      have hPA : ¬ P.asIdeal ∣ Ideal.span {a} := by
+        intro hPA
+        apply haP
+        rw [← Ideal.dvd_span_singleton]
+        exact hPA
+      rw [finitePlaceHilbertSymbol_integral_eq_primeIdealPowerResidueFactor
+        K n hnK hmu P (hAwayB P hPB) (hcoprimeB P hPB)
+          a b ha0 hb0 haP]
+      simp only [idealPowerResidueFactor, dite_eq_left hPB, dite_eq_right hPA,
+        map_pow, map_one, inv_one, mul_one]
+    · by_cases hPA : P.asIdeal ∣ Ideal.span {a}
+      · have hbP : b ∉ P.asIdeal := hbA P hPA
+        rw [finitePlaceHilbertSymbol_skew K n hnK hmu]
+        rw [finitePlaceHilbertSymbol_integral_eq_primeIdealPowerResidueFactor
+          K n hnK hmu P (hAwayA P hPA) (hcoprimeA P hPA)
+            b a hb0 ha0 hbP]
+        simp only [idealPowerResidueFactor, dite_eq_right hPB, dite_eq_left hPA,
+          map_pow, map_one, one_mul]
+      · have haP : a ∉ P.asIdeal := by
+          intro haMem
+          apply hPA
+          rw [Ideal.dvd_span_singleton]
+          exact haMem
+        have hbP : b ∉ P.asIdeal := by
+          intro hbMem
+          apply hPB
+          rw [Ideal.dvd_span_singleton]
+          exact hbMem
+        rw [finitePlaceHilbertSymbol_integral_units_eq_one
+          K n hnK hmu P hPExponent a b ha0 hb0 haP hbP]
+        simp only [idealPowerResidueFactor, dite_eq_right hPB, dite_eq_right hPA,
+          map_one, inv_one, mul_one]
+
+/-- A concrete finite set containing every finite place where a local
+power-residue factor of `a` and `b` may be nontrivial.  Its exponent part is
+the exact set of prime divisors of `(n)`. -/
+noncomputable def powerResidueBadFinitePlaces
+    (n : ℕ+) (a b : Kˣ) :
+    Finset (HeightOneSpectrum (𝓞 K)) :=
+  (chosenUnitFiniteSupport (K := K) a ∪
+      chosenUnitFiniteSupport (K := K) b) ∪
+    powerResidueExponentFinitePlaces K n
+
+/-- The explicit bad-place correction in power-residue reciprocity.  Every
+factor already lies in the common group `nthRootsSubgroup K n`. -/
+noncomputable def powerResidueBadPlaceCorrection
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) :
+    nthRootsSubgroup K (n : ℕ) :=
+  (∏ v : InfinitePlace K,
+      infinitePlaceHilbertSymbol K n v a b) *
+    (∏ v ∈ powerResidueExponentFinitePlaces K n,
+      finitePlaceHilbertSymbol K n hnK hmu v a b)
+
+private theorem valuation_eq_one_of_not_mem_chosenUnitFiniteSupport
+    (x : Kˣ) (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ chosenUnitFiniteSupport (K := K) x) :
+    v.valuation K (x : K) = 1 :=
+  (mem_SUnitGroup_iff (K := K)
+      (chosenUnitFiniteSupport (K := K) x) x).mp
+    (mem_sUnitGroup_chosenUnitFiniteSupport (K := K) x) v hv
+
+/-- The finite-place Hilbert symbol is trivial when the exponent and both
+global arguments are units at this place. -/
+theorem finitePlaceHilbertSymbol_eq_one_of_valuation_eq_one
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) (v : HeightOneSpectrum (𝓞 K))
+    (hva : v.valuation K (a : K) = 1)
+    (hvb : v.valuation K (b : K) = 1)
+    (hvn : v.valuation K ((n : ℕ) : K) = 1) :
+    finitePlaceHilbertSymbol K n hnK hmu v a b = 1 := by
+  let L := chosenSimpleKummerExtension K n hnK b
+  let : FiniteDimensional K L :=
+    chosenSimpleKummerExtension_finiteDimensional K n hnK b
+  let : IsAbelianGalois K L :=
+    chosenSimpleKummerExtension_isAbelianGalois K n hnK hmu b
+  let : NumberField L := NumberField.of_module_finite K L
+  have haIntegral :
+      IdeleGroup.finiteComponent v (IdeleGroup.principalIdele K a) ∈
+        (v.adicCompletionIntegers K).units := by
+    rw [HeightOneSpectrum.adicCompletionIntegers.mem_units_iff_valued_eq_one]
+    rw [IdeleGroup.finiteComponent_principalIdele,
+      HeightOneSpectrum.valuedAdicCompletion_eq_valuation']
+    exact hva
+  have hUnramified :
+      Algebra.IsUnramifiedAt (𝓞 K)
+        (finitePlaceExtensionCentre
+          (K := K) (L := L) v
+          (chosenFinitePlaceExtension (L := L) v)).asIdeal := by
+    apply
+      chosenSimpleKummerExtension_isUnramifiedAt_at_all_finitePlacesAbove_of_valuation_eq_one
+        (K := K) n hnK hmu b v hvb hvn
+    exact
+      finitePlaceBelow_finitePlaceExtensionCentre
+        (K := K) (L := L) v
+          (chosenFinitePlaceExtension (L := L) v)
+  have hArtin :
+      chosenFinitePlaceArtinMonoidHom (K := K) (L := L) v
+          (IdeleGroup.finiteComponent v (IdeleGroup.principalIdele K a)) = 1 :=
+    chosenFinitePlaceArtinMonoidHom_eq_one_of_integral_of_unramifiedAt
+      (K := K) (L := L) v (IdeleGroup.principalIdele K a)
+        haIntegral hUnramified
+  rw [← finitePlaceKummerRootCharacter_localGlobal K n hnK hmu v a b]
+  unfold finitePlaceKummerRootCharacter
+  unfold finitePlaceKummerRootCharacterOfExtension
+  have hcomponent :
+      (IdeleGroup.finiteComponent v (IdeleGroup.principalIdele K a) :
+          (v.adicCompletion K)ˣ) =
+        Units.map (algebraMap K (v.adicCompletion K)).toMonoidHom a := by
+    apply Units.ext
+    calc
+      ((IdeleGroup.finiteComponent v (IdeleGroup.principalIdele K a) :
+          (v.adicCompletion K)ˣ) : v.adicCompletion K) =
+          ((a : K) : v.adicCompletion K) :=
+        IdeleGroup.finiteComponent_principalIdele a v
+      _ = algebraMap K (v.adicCompletion K) (a : K) := by
+        symm
+        have hmap := congrFun
+          (IsDedekindDomain.HeightOneSpectrum.algebraMap_adicCompletion
+            (R := 𝓞 K) (S := K) (K := K) (v := v)) (a : K)
+        simpa using hmap
+  rw [hcomponent] at hArtin
+  unfold chosenFinitePlaceArtinMonoidHom at hArtin
+  dsimp only at hArtin ⊢
+  rw [hArtin, map_one, map_one]
+
+/-- Outside the concrete bad-place set, the finite-place Hilbert factor is
+trivial. -/
+theorem finitePlaceHilbertSymbol_eq_one_of_not_mem_powerResidueBadFinitePlaces
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ powerResidueBadFinitePlaces K n a b) :
+    finitePlaceHilbertSymbol K n hnK hmu v a b = 1 := by
+  have hvaSupport : v ∉ chosenUnitFiniteSupport (K := K) a := by
+    intro hva
+    apply hv
+    exact Finset.mem_union_left _ (Finset.mem_union_left _ hva)
+  have hvbSupport : v ∉ chosenUnitFiniteSupport (K := K) b := by
+    intro hvb
+    apply hv
+    exact Finset.mem_union_left _ (Finset.mem_union_right _ hvb)
+  have hvnSupport :
+      v ∉ powerResidueExponentFinitePlaces K n := by
+    intro hvn
+    apply hv
+    exact Finset.mem_union_right _ hvn
+  have hva : v.valuation K (a : K) = 1 :=
+    valuation_eq_one_of_not_mem_chosenUnitFiniteSupport K a v hvaSupport
+  have hvb : v.valuation K (b : K) = 1 :=
+    valuation_eq_one_of_not_mem_chosenUnitFiniteSupport K b v hvbSupport
+  have hvn : v.valuation K ((n : ℕ) : K) = 1 := by
+    have hvnNotDvd :
+        ¬ v.asIdeal ∣ powerResidueExponentIdeal K n := by
+      simpa only [mem_powerResidueExponentFinitePlaces_iff] using hvnSupport
+    have hvnNotMem : ((n : ℕ) : 𝓞 K) ∉ v.asIdeal := by
+      intro hvnMem
+      apply hvnNotDvd
+      rw [powerResidueExponentIdeal, Ideal.dvd_span_singleton]
+      exact hvnMem
+    simpa only [map_natCast] using
+      (v.valuation_eq_one_iff_notMem (K := K)
+        (r := ((n : ℕ) : 𝓞 K))).2 hvnNotMem
+  exact finitePlaceHilbertSymbol_eq_one_of_valuation_eq_one
+    K n hnK hmu a b v hva hvb hvn
+
+/-- The multiplicative support of the finite-place Hilbert factors is
+contained in the explicit power-residue bad-place set. -/
+theorem finitePlaceHilbertSymbol_mulSupport_subset_powerResidueBadFinitePlaces
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) :
+    Function.mulSupport
+        (fun v : HeightOneSpectrum (𝓞 K) =>
+          finitePlaceHilbertSymbol K n hnK hmu v a b) ⊆
+      (powerResidueBadFinitePlaces K n a b :
+        Set (HeightOneSpectrum (𝓞 K))) := by
+  intro v hv
+  change finitePlaceHilbertSymbol K n hnK hmu v a b ≠ 1 at hv
+  change v ∈ powerResidueBadFinitePlaces K n a b
+  by_contra hvBad
+  exact hv
+    (finitePlaceHilbertSymbol_eq_one_of_not_mem_powerResidueBadFinitePlaces
+      K n hnK hmu a b v hvBad)
+
+/-- The finite-place Hilbert `finprod` is the ordinary product over the
+explicit bad-place set. -/
+theorem finitePlaceHilbertSymbol_finprod_eq_prod_powerResidueBadFinitePlaces
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) :
+    (∏ᶠ v : HeightOneSpectrum (𝓞 K),
+        finitePlaceHilbertSymbol K n hnK hmu v a b) =
+      ∏ v ∈ powerResidueBadFinitePlaces K n a b,
+        finitePlaceHilbertSymbol K n hnK hmu v a b := by
+  rw [finprod_eq_prod_of_mulSupport_subset _
+    (finitePlaceHilbertSymbol_mulSupport_subset_powerResidueBadFinitePlaces
+      K n hnK hmu a b)]
+
+/-- Finite-set form of the Hilbert product formula: the product over all
+explicitly bad finite places is the inverse of the infinite-place product. -/
+theorem powerResidueBadFinitePlaces_product_eq_infinitePlaceProduct_inv
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) :
+    (∏ v ∈ powerResidueBadFinitePlaces K n a b,
+        finitePlaceHilbertSymbol K n hnK hmu v a b) =
+      (∏ v : InfinitePlace K,
+        infinitePlaceHilbertSymbol K n v a b)⁻¹ := by
+  have hproduct :=
+    hilbertSymbol_allPlaces_product_eq_one K n hnK hmu a b
+  rw [
+    finitePlaceHilbertSymbol_finprod_eq_prod_powerResidueBadFinitePlaces]
+      at hproduct
+  exact (eq_inv_iff_mul_eq_one).2 (by
+    simpa only [mul_comm] using hproduct)
+
+/-- The product of the finite-place Hilbert factors away from primes dividing
+the exponent.  C1 identifies this term with the quotient of the two ideal
+power-residue symbols; the remaining factors are exactly the correction. -/
+noncomputable def powerResidueAwayFromExponentFiniteProduct
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) :
+    nthRootsSubgroup K (n : ℕ) :=
+  ∏ᶠ v : HeightOneSpectrum (𝓞 K),
+    if v ∈ powerResidueExponentFinitePlaces K n then
+      1
+    else
+      finitePlaceHilbertSymbol K n hnK hmu v a b
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityGlobal.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityGlobal.lean
new file mode 100644
index 0000000..8a60735
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityGlobal.lean
@@ -0,0 +1,416 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.PowerResidueReciprocityCorrection
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalHilbertSymbol.FinitePlaceCharacterComparison
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.FiniteIdeleArtin
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.HilbertProductFormula
+import ClassFieldTheory.AlgebraicNumberTheory.PowerResidueSymbols.Ideal
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.NormApproximation.FinitePlaces
+import Mathlib.NumberTheory.Padics.HeightOneSpectrum
+import ValuedFieldTheory.LocalField.DiscreteValuationField.PadicField
+import ClassFieldTheory.AlgebraicNumberTheory.QuadraticReciprocity
+import ClassFieldTheory.AlgebraicNumberTheory.RayClass.Rational
+import ClassFieldTheory.LocalClassFieldTheory.Kummer.PowerResidueTameFormula
+import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.IdealQuotients
+import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.IdealToCompletion
+import ValuedFieldTheory.Valuation.Completion.ExtensionInvariants
+import ClassFieldTheory.KummerTheory.Concrete.SimpleExtensionLocalBehavior
+import ClassFieldTheory.KummerTheory.Concrete.SUnitPreparation.FiniteRadicalSupport
+import ClassFieldTheory.LocalClassFieldTheory.Finite.CyclotomicNorm.PrincipalUnits
+
+
+set_option autoImplicit false
+
+
+open scoped BigOperators Classical NumberField NumberTheorySymbols ValuativeRel WithZero
+open NumberField IsDedekindDomain
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open KummerTheory
+open AlgebraicNumberTheory.PowerResidueSymbols
+open LocalClassFieldTheory.Kummer
+open LocalFieldTheory
+open LocalFieldTheory.IsNonarchimedeanLocalField
+
+variable (K : Type) [Field K] [NumberField K]
+/-- The complete tame finite-place product for two nonzero algebraic
+integers is the quotient of the two ideal power-residue symbols. -/
+theorem powerResidueAwayFromExponentFiniteProduct_integral_eq_idealSymbol_div
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : 𝓞 K) (ha0 : a ≠ 0) (hb0 : b ≠ 0)
+    (hcoprimeA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} →
+          (Ideal.absNorm P.asIdeal).Coprime (n : ℕ))
+    (hcoprimeB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} →
+          (Ideal.absNorm P.asIdeal).Coprime (n : ℕ))
+    (haB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} → a ∉ P.asIdeal)
+    (hbA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} → b ∉ P.asIdeal)
+    (hAwayA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} →
+          P ∉ powerResidueExponentFinitePlaces K n)
+    (hAwayB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} →
+          P ∉ powerResidueExponentFinitePlaces K n) :
+    powerResidueAwayFromExponentFiniteProduct K n hnK hmu
+        (nonzeroIntegralFieldUnit K a ha0)
+        (nonzeroIntegralFieldUnit K b hb0) =
+      integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueSymbol K (Ideal.span {b})
+            (ideal_span_singleton_ne_zero K hb0)
+            n hmu a hcoprimeB haB) *
+        (integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueSymbol K (Ideal.span {a})
+            (ideal_span_singleton_ne_zero K ha0)
+            n hmu b hcoprimeA hbA))⁻¹ := by
+  let IA : Ideal (𝓞 K) := Ideal.span {a}
+  let IB : Ideal (𝓞 K) := Ideal.span {b}
+  have hIA : IA ≠ 0 := by
+    dsimp only [IA]
+    exact ideal_span_singleton_ne_zero K ha0
+  have hIB : IB ≠ 0 := by
+    dsimp only [IB]
+    exact ideal_span_singleton_ne_zero K hb0
+  let fB : HeightOneSpectrum (𝓞 K) → nthRootsSubgroup K (n : ℕ) :=
+    fun P => integralRootsOfUnityToNthRoots K (n : ℕ)
+      (idealPowerResidueFactor K IB n hmu a hcoprimeB haB P)
+  let fA : HeightOneSpectrum (𝓞 K) → nthRootsSubgroup K (n : ℕ) :=
+    fun P => integralRootsOfUnityToNthRoots K (n : ℕ)
+      (idealPowerResidueFactor K IA n hmu b hcoprimeA hbA P)
+  have hfB : Function.HasFiniteMulSupport fB :=
+    (idealPowerResidueFactor_hasFiniteMulSupport
+      K IB hIB n hmu a hcoprimeB haB).subset (by
+        intro P hP
+        change fB P ≠ 1 at hP
+        change idealPowerResidueFactor K IB n hmu a hcoprimeB haB P ≠ 1
+        intro hOne
+        exact hP (by simp only [fB, hOne, map_one]))
+  have hfA : Function.HasFiniteMulSupport fA :=
+    (idealPowerResidueFactor_hasFiniteMulSupport
+      K IA hIA n hmu b hcoprimeA hbA).subset (by
+        intro P hP
+        change fA P ≠ 1 at hP
+        change idealPowerResidueFactor K IA n hmu b hcoprimeA hbA P ≠ 1
+        intro hOne
+        exact hP (by simp only [fA, hOne, map_one]))
+  have hfAInv :
+      Function.HasFiniteMulSupport (fun P => (fA P)⁻¹) :=
+    hfA.subset (by
+      intro P hP
+      change (fA P)⁻¹ ≠ 1 at hP
+      change fA P ≠ 1
+      intro hOne
+      exact hP (by rw [hOne, inv_one]))
+  let invHom : nthRootsSubgroup K (n : ℕ) →*
+      nthRootsSubgroup K (n : ℕ) := invMonoidHom
+  have hfinprodInv :
+      (∏ᶠ P : HeightOneSpectrum (𝓞 K), (fA P)⁻¹) =
+        (∏ᶠ P : HeightOneSpectrum (𝓞 K), fA P)⁻¹ := by
+    change (∏ᶠ P : HeightOneSpectrum (𝓞 K), invHom (fA P)) =
+      invHom (∏ᶠ P : HeightOneSpectrum (𝓞 K), fA P)
+    exact (MonoidHom.map_finprod invHom hfA).symm
+  calc
+    powerResidueAwayFromExponentFiniteProduct K n hnK hmu
+        (nonzeroIntegralFieldUnit K a ha0)
+        (nonzeroIntegralFieldUnit K b hb0) =
+        ∏ᶠ P : HeightOneSpectrum (𝓞 K), fB P * (fA P)⁻¹ := by
+      apply finprod_congr
+      intro P
+      simpa only [fA, fB, IA, IB] using
+        powerResidueAwayFromExponentFiniteFactor_integral_eq_idealFactors
+          K n hnK hmu a b ha0 hb0 hcoprimeA hcoprimeB haB hbA
+            hAwayA hAwayB P
+    _ = (∏ᶠ P : HeightOneSpectrum (𝓞 K), fB P) *
+        (∏ᶠ P : HeightOneSpectrum (𝓞 K), (fA P)⁻¹) :=
+      finprod_mul_distrib hfB hfAInv
+    _ = (∏ᶠ P : HeightOneSpectrum (𝓞 K), fB P) *
+        (∏ᶠ P : HeightOneSpectrum (𝓞 K), fA P)⁻¹ := by
+      rw [hfinprodInv]
+    _ = integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueSymbol K IB hIB n hmu a hcoprimeB haB) *
+        (integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueSymbol K IA hIA n hmu b hcoprimeA hbA))⁻¹ := by
+      rw [idealPowerResidueSymbol_eq_finprod,
+        idealPowerResidueSymbol_eq_finprod]
+      rw [MonoidHom.map_finprod
+          (integralRootsOfUnityToNthRoots K (n : ℕ))
+          (idealPowerResidueFactor_hasFiniteMulSupport
+            K IB hIB n hmu a hcoprimeB haB),
+        MonoidHom.map_finprod
+          (integralRootsOfUnityToNthRoots K (n : ℕ))
+          (idealPowerResidueFactor_hasFiniteMulSupport
+            K IA hIA n hmu b hcoprimeA hbA)]
+    _ = integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueSymbol K (Ideal.span {b})
+            (ideal_span_singleton_ne_zero K hb0)
+            n hmu a hcoprimeB haB) *
+        (integralRootsOfUnityToNthRoots K (n : ℕ)
+          (idealPowerResidueSymbol K (Ideal.span {a})
+            (ideal_span_singleton_ne_zero K ha0)
+            n hmu b hcoprimeA hbA))⁻¹ := by
+      rfl
+
+/-- Split the full finite-place product into exponent-prime factors and the
+product away from the exponent. -/
+theorem finitePlaceHilbertSymbol_finprod_eq_exponent_product_mul_away
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) :
+    (∏ᶠ v : HeightOneSpectrum (𝓞 K),
+        finitePlaceHilbertSymbol K n hnK hmu v a b) =
+      (∏ v ∈ powerResidueExponentFinitePlaces K n,
+          finitePlaceHilbertSymbol K n hnK hmu v a b) *
+        powerResidueAwayFromExponentFiniteProduct K n hnK hmu a b := by
+  let exponentFactor : HeightOneSpectrum (𝓞 K) →
+      nthRootsSubgroup K (n : ℕ) := fun v =>
+    if v ∈ powerResidueExponentFinitePlaces K n then
+      finitePlaceHilbertSymbol K n hnK hmu v a b
+    else
+      1
+  let awayFactor : HeightOneSpectrum (𝓞 K) →
+      nthRootsSubgroup K (n : ℕ) := fun v =>
+    if v ∈ powerResidueExponentFinitePlaces K n then
+      1
+    else
+      finitePlaceHilbertSymbol K n hnK hmu v a b
+  have hExponentSupport :
+      Function.mulSupport exponentFactor ⊆
+        (powerResidueExponentFinitePlaces K n :
+          Set (HeightOneSpectrum (𝓞 K))) := by
+    intro v hv
+    change exponentFactor v ≠ 1 at hv
+    change v ∈ powerResidueExponentFinitePlaces K n
+    by_contra hvExponent
+    exact hv (by simp only [exponentFactor, ite_eq_right hvExponent])
+  have hExponentFinite : Function.HasFiniteMulSupport exponentFactor := by
+    rw [Function.HasFiniteMulSupport]
+    exact
+      (powerResidueExponentFinitePlaces K n).finite_toSet.subset
+        hExponentSupport
+  have hAwayFinite : Function.HasFiniteMulSupport awayFactor := by
+    rw [Function.HasFiniteMulSupport]
+    exact
+      (finitePlaceHilbertSymbol_hasFiniteMulSupport K n hnK hmu a b).subset
+        (by
+          intro v hv
+          change awayFactor v ≠ 1 at hv
+          change finitePlaceHilbertSymbol K n hnK hmu v a b ≠ 1
+          by_contra hvOne
+          exact hv (by simp only [awayFactor, hvOne, ite_self]))
+  have hPointwise :
+      (fun v : HeightOneSpectrum (𝓞 K) =>
+        finitePlaceHilbertSymbol K n hnK hmu v a b) =
+        fun v => exponentFactor v * awayFactor v := by
+    funext v
+    by_cases hv : v ∈ powerResidueExponentFinitePlaces K n
+    · simp only [exponentFactor, awayFactor, ite_eq_left hv, mul_one]
+    · simp only [exponentFactor, awayFactor, ite_eq_right hv, one_mul]
+  have hExponentProduct :
+      (∏ᶠ v : HeightOneSpectrum (𝓞 K), exponentFactor v) =
+        ∏ v ∈ powerResidueExponentFinitePlaces K n,
+          finitePlaceHilbertSymbol K n hnK hmu v a b := by
+    rw [finprod_eq_prod_of_mulSupport_subset exponentFactor hExponentSupport]
+    apply Finset.prod_congr rfl
+    intro v hv
+    simp only [exponentFactor, ite_eq_left hv]
+  calc
+    (∏ᶠ v : HeightOneSpectrum (𝓞 K),
+        finitePlaceHilbertSymbol K n hnK hmu v a b) =
+        ∏ᶠ v : HeightOneSpectrum (𝓞 K),
+          exponentFactor v * awayFactor v := by
+      rw [hPointwise]
+    _ = (∏ᶠ v : HeightOneSpectrum (𝓞 K), exponentFactor v) *
+        (∏ᶠ v : HeightOneSpectrum (𝓞 K), awayFactor v) :=
+      finprod_mul_distrib hExponentFinite hAwayFinite
+    _ = (∏ v ∈ powerResidueExponentFinitePlaces K n,
+          finitePlaceHilbertSymbol K n hnK hmu v a b) *
+        (∏ᶠ v : HeightOneSpectrum (𝓞 K), awayFactor v) := by
+      rw [hExponentProduct]
+    _ = (∏ v ∈ powerResidueExponentFinitePlaces K n,
+          finitePlaceHilbertSymbol K n hnK hmu v a b) *
+        powerResidueAwayFromExponentFiniteProduct K n hnK hmu a b := by
+      rfl
+
+/-- General Hilbert-product core of power-residue reciprocity.  The complete
+finite product away from the exponent is the inverse of the explicit product
+of all infinite-place factors and all exponent-prime factors. -/
+theorem powerResidueAwayFromExponentFiniteProduct_eq_badPlaceCorrection_inv
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : Kˣ) :
+    powerResidueAwayFromExponentFiniteProduct K n hnK hmu a b =
+      (powerResidueBadPlaceCorrection K n hnK hmu a b)⁻¹ := by
+  have hproduct :=
+    hilbertSymbol_allPlaces_product_eq_one K n hnK hmu a b
+  rw [finitePlaceHilbertSymbol_finprod_eq_exponent_product_mul_away]
+      at hproduct
+  exact (eq_inv_iff_mul_eq_one).2 (by
+    unfold powerResidueBadPlaceCorrection
+    simpa only [mul_assoc, mul_comm, mul_left_comm] using hproduct)
+
+/-- General ideal power-residue reciprocity with the explicit product of
+infinite and exponent-prime Hilbert factors as correction. -/
+theorem idealPowerResidueSymbol_reciprocity_with_bad_place_correction
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (a b : 𝓞 K) (ha0 : a ≠ 0) (hb0 : b ≠ 0)
+    (hcoprimeA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} →
+          (Ideal.absNorm P.asIdeal).Coprime (n : ℕ))
+    (hcoprimeB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} →
+          (Ideal.absNorm P.asIdeal).Coprime (n : ℕ))
+    (haB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} → a ∉ P.asIdeal)
+    (hbA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} → b ∉ P.asIdeal)
+    (hAwayA :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {a} →
+          P ∉ powerResidueExponentFinitePlaces K n)
+    (hAwayB :
+      ∀ P : HeightOneSpectrum (𝓞 K),
+        P.asIdeal ∣ Ideal.span {b} →
+          P ∉ powerResidueExponentFinitePlaces K n) :
+    integralRootsOfUnityToNthRoots K (n : ℕ)
+        (idealPowerResidueSymbol K (Ideal.span {b})
+          (ideal_span_singleton_ne_zero K hb0)
+          n hmu a hcoprimeB haB) =
+      (powerResidueBadPlaceCorrection K n hnK hmu
+        (nonzeroIntegralFieldUnit K a ha0)
+        (nonzeroIntegralFieldUnit K b hb0))⁻¹ *
+      integralRootsOfUnityToNthRoots K (n : ℕ)
+        (idealPowerResidueSymbol K (Ideal.span {a})
+          (ideal_span_singleton_ne_zero K ha0)
+          n hmu b hcoprimeA hbA) := by
+  let symbolAB := integralRootsOfUnityToNthRoots K (n : ℕ)
+    (idealPowerResidueSymbol K (Ideal.span {b})
+      (ideal_span_singleton_ne_zero K hb0)
+      n hmu a hcoprimeB haB)
+  let symbolBA := integralRootsOfUnityToNthRoots K (n : ℕ)
+    (idealPowerResidueSymbol K (Ideal.span {a})
+      (ideal_span_singleton_ne_zero K ha0)
+      n hmu b hcoprimeA hbA)
+  let correction := powerResidueBadPlaceCorrection K n hnK hmu
+    (nonzeroIntegralFieldUnit K a ha0)
+    (nonzeroIntegralFieldUnit K b hb0)
+  have hIdeal :=
+    powerResidueAwayFromExponentFiniteProduct_integral_eq_idealSymbol_div
+      K n hnK hmu a b ha0 hb0 hcoprimeA hcoprimeB haB hbA
+        hAwayA hAwayB
+  have hCorrection :=
+    powerResidueAwayFromExponentFiniteProduct_eq_badPlaceCorrection_inv
+      K n hnK hmu
+        (nonzeroIntegralFieldUnit K a ha0)
+        (nonzeroIntegralFieldUnit K b hb0)
+  have hQuotient : symbolAB * symbolBA⁻¹ = correction⁻¹ := by
+    rw [← hIdeal, hCorrection]
+  change symbolAB = correction⁻¹ * symbolBA
+  calc
+    symbolAB = (symbolAB * symbolBA⁻¹) * symbolBA := by
+      simp only [mul_assoc, inv_mul_cancel, mul_one]
+    _ = correction⁻¹ * symbolBA := by rw [hQuotient]
+
+/-! ## Quadratic specialization over the rational field -/
+
+open AlgebraicNumberTheory.PowerResidueSymbols
+
+local instance rationalPrimeFact (p : Nat.Primes) : Fact p.1.Prime :=
+  ⟨p.2⟩
+
+/-- The rational field contains the primitive square root of unity `-1`.
+This is the canonical source of the primitive-root input in the quadratic
+specialization; no root is chosen downstream. -/
+theorem rationalQuadraticPrimitiveRoots_nonempty :
+    (primitiveRoots 2 ℚ).Nonempty := by
+  refine ⟨-1, (mem_primitiveRoots (by decide)).2 ?_⟩
+  exact IsPrimitiveRoot.neg_one 0 (by decide)
+
+/-- The residue field at the rational prime over `p` is canonically `ZMod p`.
+The construction first transports the prime ideal through
+`Rat.ringOfIntegersEquiv` and then uses the standard integer quotient. -/
+noncomputable def rationalPrimeResidueEquivZMod
+    (p : Nat.Primes) :
+    (𝓞 ℚ ⧸ (RayClass.rationalPrime p).asIdeal) ≃+* ZMod p.1 := by
+  have hIntEquiv :
+      Rat.IsIntegralClosure.intEquiv (𝓞 ℚ) =
+        Rat.ringOfIntegersEquiv := by
+    ext x
+    exact Rat.IsIntegralClosure.intEquiv_apply_eq_ringOfIntegersEquiv x
+  have hmap :
+      Ideal.span {(p.1 : ℤ)} =
+        (RayClass.rationalPrime p).asIdeal.map
+          Rat.ringOfIntegersEquiv := by
+    simpa only [RayClass.natGenerator_rationalPrime, hIntEquiv] using
+      (Rat.HeightOneSpectrum.span_natGenerator
+        (RayClass.rationalPrime p))
+  exact
+    (Ideal.quotientEquiv
+      (RayClass.rationalPrime p).asIdeal
+      (Ideal.span {(p.1 : ℤ)})
+      Rat.ringOfIntegersEquiv hmap).trans
+        (Int.quotientSpanNatEquivZMod p.1)
+
+/-- The rational residue-field equivalence sends an integral residue class to
+the corresponding integer class modulo `p`. -/
+@[simp]
+theorem rationalPrimeResidueEquivZMod_mk
+    (p : Nat.Primes) (a : 𝓞 ℚ) :
+    rationalPrimeResidueEquivZMod p
+      (Ideal.Quotient.mk (RayClass.rationalPrime p).asIdeal a) =
+      (Rat.ringOfIntegersEquiv a : ZMod p.1) := by
+  rw [rationalPrimeResidueEquivZMod, RingEquiv.trans_apply]
+  have hmk
+      (hIJ :
+        Ideal.span {(p.1 : ℤ)} =
+          (RayClass.rationalPrime p).asIdeal.map
+            Rat.ringOfIntegersEquiv) :
+      Ideal.quotientEquiv
+          (RayClass.rationalPrime p).asIdeal
+          (Ideal.span {(p.1 : ℤ)})
+          Rat.ringOfIntegersEquiv hIJ
+          (Ideal.Quotient.mk (RayClass.rationalPrime p).asIdeal a) =
+        Ideal.Quotient.mk (Ideal.span {(p.1 : ℤ)})
+          (Rat.ringOfIntegersEquiv a) :=
+    Ideal.quotientEquiv_mk
+      (RayClass.rationalPrime p).asIdeal
+      (Ideal.span {(p.1 : ℤ)})
+      Rat.ringOfIntegersEquiv hIJ a
+  have hquot :
+      ((Int.quotientSpanNatEquivZMod p.1 :
+          (ℤ ⧸ Ideal.span {(p.1 : ℤ)}) →+* ZMod p.1).comp
+        (Ideal.Quotient.mk (Ideal.span {(p.1 : ℤ)}))) =
+        Int.castRingHom (ZMod p.1) :=
+    Int.quotientSpanNatEquivZMod_comp_Quotient_mk p.1
+  calc
+    _ = Int.quotientSpanNatEquivZMod p.1
+        (Ideal.Quotient.mk (Ideal.span {(p.1 : ℤ)})
+          (Rat.ringOfIntegersEquiv a)) :=
+      congrArg (Int.quotientSpanNatEquivZMod p.1) (hmk _)
+    _ = _ := congrArg
+      (fun f : ℤ →+* ZMod p.1 => f (Rat.ringOfIntegersEquiv a)) hquot
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityLocalSymbols.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityLocalSymbols.lean
new file mode 100644
index 0000000..92d5938
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/PowerResidueReciprocityLocalSymbols.lean
@@ -0,0 +1,557 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.PowerResidueReciprocityBase
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.GlobalHilbertSymbol.FinitePlaceCharacterComparison
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.FiniteIdeleArtin
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.HilbertProductFormula
+import ClassFieldTheory.AlgebraicNumberTheory.PowerResidueSymbols.Ideal
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.NormApproximation.FinitePlaces
+import Mathlib.NumberTheory.Padics.HeightOneSpectrum
+import ValuedFieldTheory.LocalField.DiscreteValuationField.PadicField
+import ClassFieldTheory.AlgebraicNumberTheory.QuadraticReciprocity
+import ClassFieldTheory.AlgebraicNumberTheory.RayClass.Rational
+import ClassFieldTheory.LocalClassFieldTheory.Kummer.PowerResidueTameFormula
+import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.IdealQuotients
+import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.IdealToCompletion
+import ValuedFieldTheory.Valuation.Completion.ExtensionInvariants
+import ClassFieldTheory.KummerTheory.Concrete.SimpleExtensionLocalBehavior
+import ClassFieldTheory.KummerTheory.Concrete.SUnitPreparation.FiniteRadicalSupport
+import ClassFieldTheory.LocalClassFieldTheory.Finite.CyclotomicNorm.PrincipalUnits
+
+
+set_option autoImplicit false
+
+
+open scoped BigOperators Classical NumberField NumberTheorySymbols ValuativeRel WithZero
+open NumberField IsDedekindDomain
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+open KummerTheory
+open AlgebraicNumberTheory.PowerResidueSymbols
+open LocalClassFieldTheory.Kummer
+open LocalFieldTheory
+open LocalFieldTheory.IsNonarchimedeanLocalField
+
+variable (K : Type) [Field K] [NumberField K]
+/-- The integral principal ideal generated by the exponent. -/
+def powerResidueExponentIdeal (n : ℕ+) : Ideal (𝓞 K) :=
+  Ideal.span {((n : ℕ) : 𝓞 K)}
+
+/-- The exponent ideal is nonzero in a number field. -/
+theorem powerResidueExponentIdeal_ne_zero (n : ℕ+) :
+    powerResidueExponentIdeal K n ≠ 0 := by
+  change Ideal.span {((n : ℕ) : 𝓞 K)} ≠ ⊥
+  exact Ideal.span_singleton_eq_bot.not.mpr
+    (Nat.cast_ne_zero.mpr n.ne_zero)
+
+omit [NumberField K] in
+theorem ideal_span_singleton_ne_zero
+    {x : 𝓞 K} (hx : x ≠ 0) : Ideal.span {x} ≠ 0 :=
+  Submodule.span_singleton_eq_bot.mp.mt hx
+
+/-- The finite places dividing the exponent.  These, together with all
+infinite places, are precisely the correction places in the reciprocity
+formula once the two principal denominator supports are removed. -/
+noncomputable def powerResidueExponentFinitePlaces
+    (n : ℕ+) : Finset (HeightOneSpectrum (𝓞 K)) :=
+  (Ideal.finite_factors
+    (powerResidueExponentIdeal_ne_zero K n)).toFinset
+
+/-- Membership in the exponent-place support is divisibility by the exponent
+ideal. -/
+@[simp]
+theorem mem_powerResidueExponentFinitePlaces_iff
+    (n : ℕ+) (v : HeightOneSpectrum (𝓞 K)) :
+    v ∈ powerResidueExponentFinitePlaces K n ↔
+      v.asIdeal ∣ powerResidueExponentIdeal K n :=
+  Set.Finite.mem_toFinset
+    (Ideal.finite_factors
+      (powerResidueExponentIdeal_ne_zero K n))
+
+/-- At a finite place not dividing the exponent, the exponent is a unit in
+the canonical completion. -/
+theorem finitePlace_natCast_valuation_eq_one_of_not_mem_exponent
+    (n : ℕ+) (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ powerResidueExponentFinitePlaces K n) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C :=
+      finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    ValuativeRel.valuation C ((n : ℕ) : C) = 1 := by
+  dsimp only
+  let : IsUltrametricDist
+      (HeightOneSpectrum.adicAbv K v).Completion :=
+    finitePlaceArtinCompletionIsUltrametricDist
+      (HeightOneSpectrum.adicAbv K v)
+      (HeightOneSpectrum.isNonarchimedean_adicAbv K v)
+  have hvNotDvd :
+      ¬ v.asIdeal ∣ powerResidueExponentIdeal K n := by
+    simpa only [mem_powerResidueExponentFinitePlaces_iff] using hv
+  have hvNotMem : ((n : ℕ) : 𝓞 K) ∉ v.asIdeal := by
+    intro hvMem
+    apply hvNotDvd
+    rw [powerResidueExponentIdeal, Ideal.dvd_span_singleton]
+    exact hvMem
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  have hNorm : ‖((n : ℕ) : C)‖ = 1 := by
+    calc
+      ‖((n : ℕ) : C)‖ =
+          ‖algebraMap K C (((n : ℕ) : K))‖ := by rw [map_natCast]
+      _ = HeightOneSpectrum.adicAbv K v (((n : ℕ) : K)) :=
+        AbsoluteValue.completionAbsoluteValue_coe
+          (HeightOneSpectrum.adicAbv K v) (((n : ℕ) : K))
+      _ = ‖NumberField.FinitePlace.embedding v (((n : ℕ) : K))‖ :=
+        (NumberField.FinitePlace.norm_embedding v (((n : ℕ) : K))).symm
+      _ = 1 :=
+        (NumberField.FinitePlace.norm_eq_one_iff_notMem K v
+          (((n : ℕ) : 𝓞 K))).2 hvNotMem
+  let vCNorm := NormedField.valuation (K := C)
+  let : vCNorm.Compatible := Valuation.Compatible.ofValuation vCNorm
+  have hnCNorm : vCNorm ((n : ℕ) : C) = 1 := by
+    change ‖((n : ℕ) : C)‖₊ = 1
+    exact NNReal.eq (by simpa using hNorm)
+  exact
+    (ValuativeRel.isEquiv vCNorm (ValuativeRel.valuation C))
+      |>.eq_one_iff_eq_one.mp hnCNorm
+
+private noncomputable def finitePlaceLocalTamePowerResidueSymbolValue
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    nthRootsSubgroup (HeightOneSpectrum.adicAbv K v).Completion (n : ℕ) := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  letI : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  exact localTamePowerResidueSymbol C n
+    (finitePlace_natCast_valuation_eq_one_of_not_mem_exponent K n v hv)
+    (finitePlaceHilbert_primitiveRoots_nonempty K n hmu v)
+    (finitePlaceIntegralCompletionUnit K v a ha)
+
+private noncomputable def finitePlaceLocalTamePowerResidueSymbolFieldValue
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    (HeightOneSpectrum.adicAbv K v).Completion :=
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  letI : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  ((localTamePowerResidueSymbol C n
+      (finitePlace_natCast_valuation_eq_one_of_not_mem_exponent K n v hv)
+      (finitePlaceHilbert_primitiveRoots_nonempty K n hmu v)
+      (finitePlaceIntegralCompletionUnit K v a ha)).1 : C)
+
+private noncomputable def finitePlacePrimeIdealPowerResidueIntegralRoot
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    rootsOfUnity (n : ℕ) (𝓞 K) :=
+  AlgebraicNumberTheory.PowerResidueSymbols.primeIdealPowerResidueSymbol
+    K v n hmu hcoprime a ha
+
+private noncomputable def finitePlacePrimeIdealPowerResidueGlobalRoot
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) : nthRootsSubgroup K (n : ℕ) :=
+  integralRootsOfUnityToNthRoots K (n : ℕ)
+    (finitePlacePrimeIdealPowerResidueIntegralRoot K v n hmu hcoprime a ha)
+
+private noncomputable def finitePlacePrimeIdealPowerResidueFactorValue
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    nthRootsSubgroup (HeightOneSpectrum.adicAbv K v).Completion (n : ℕ) :=
+  nthRootsSubgroupMap K (HeightOneSpectrum.adicAbv K v).Completion (n : ℕ)
+    (finitePlacePrimeIdealPowerResidueGlobalRoot K v n hmu hcoprime a ha)
+
+private noncomputable def finitePlacePrimeIdealPowerResidueFactorFieldValue
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    (HeightOneSpectrum.adicAbv K v).Completion :=
+  (((nthRootsSubgroupMap K
+      (HeightOneSpectrum.adicAbv K v).Completion (n : ℕ))
+    (finitePlacePrimeIdealPowerResidueGlobalRoot K v n hmu hcoprime a ha)).1 :
+      (HeightOneSpectrum.adicAbv K v).Completion)
+
+private noncomputable def finitePlaceLocalTamePowerResidueSymbolResidueValue
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C := finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    𝓀[C] := by
+  dsimp only
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  letI : ValuativeRel C := finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  letI : Fintype 𝓀[C] := Fintype.ofFinite _
+  have hnLocal : (n : ℕ) ∣ Fintype.card 𝓀[C] - 1 := by
+    rw [← Nat.card_eq_fintype_card]
+    exact dvd_residueCard_sub_one_of_primitiveRoots C n
+      (finitePlace_natCast_valuation_eq_one_of_not_mem_exponent K n v hv)
+      (finitePlaceHilbert_primitiveRoots_nonempty K n hmu v)
+  exact
+    (((AlgebraicNumberTheory.PowerResidueSymbols.finiteFieldPowerResidueSymbol
+        𝓀[C] n hnLocal
+        (integerUnitsToResidueUnits C
+          (finitePlaceIntegralCompletionUnit K v a ha)) :
+      rootsOfUnity (n : ℕ) 𝓀[C]).1 : 𝓀[C]ˣ) : 𝓀[C])
+
+private noncomputable def finitePlacePrimeIdealPowerResidueFactorResidueValue
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    let C := (HeightOneSpectrum.adicAbv K v).Completion
+    letI : ValuativeRel C := finitePlaceLocalArtinCompletionValuativeRel v
+    letI : IsNonarchimedeanLocalField C :=
+      finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+    𝓀[C] := by
+  dsimp only
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  letI : ValuativeRel C := finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  exact
+    (((localNthRootsReduction C n
+        (nthRootsSubgroupMap K C (n : ℕ)
+          (integralRootsOfUnityToNthRoots K (n : ℕ)
+            (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealPowerResidueSymbol
+              K v n hmu hcoprime a ha)))).1 : 𝓀[C]ˣ) : 𝓀[C])
+
+/-- The tame symbol in a finite-place completion is the image of the
+prime-ideal power-residue symbol.  All comparisons are canonical: the only
+place hypothesis says that the place does not divide the exponent. -/
+private theorem finitePlaceLocalTamePowerResidueSymbol_residueValue_eq
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    finitePlaceLocalTamePowerResidueSymbolResidueValue K v n hmu hv a ha =
+      finitePlacePrimeIdealPowerResidueFactorResidueValue
+        K v n hmu hcoprime a ha := by
+  unfold finitePlaceLocalTamePowerResidueSymbolResidueValue
+  unfold finitePlacePrimeIdealPowerResidueFactorResidueValue
+  dsimp only
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  let : ValuativeRel C := finitePlaceLocalArtinCompletionValuativeRel v
+  let : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  let : Field (𝓞 K ⧸ v.asIdeal) := Ideal.Quotient.field v.asIdeal
+  let : Fintype (𝓞 K ⧸ v.asIdeal) := Fintype.ofFinite _
+  let : Fintype 𝓀[C] := Fintype.ofFinite _
+  let hnC := finitePlace_natCast_valuation_eq_one_of_not_mem_exponent K n v hv
+  let hmuC := finitePlaceHilbert_primitiveRoots_nonempty K n hmu v
+  have hnPrime :
+      (n : ℕ) ∣ Fintype.card (𝓞 K ⧸ v.asIdeal) - 1 := by
+    rw [AlgebraicNumberTheory.PowerResidueSymbols.card_primeIdealResidueField K v]
+    exact
+      AlgebraicNumberTheory.PowerResidueSymbols.dvd_absNorm_sub_one_of_primitiveRoots
+        K v n hmu hcoprime
+  have hnLocal : (n : ℕ) ∣ Fintype.card 𝓀[C] - 1 := by
+    rw [← Nat.card_eq_fintype_card]
+    exact dvd_residueCard_sub_one_of_primitiveRoots C n hnC hmuC
+  rw [finitePlace_localNthRootsReduction_integralRoots]
+  rw [← AlgebraicNumberTheory.PowerResidueSymbols.rootsOfUnityReductionEquiv_apply
+      K v n hmu hcoprime,
+    AlgebraicNumberTheory.PowerResidueSymbols.rootsOfUnityReductionEquiv_primeIdealPowerResidueSymbol
+      K v n hmu hcoprime a ha]
+  rw [finitePlace_integerUnitsToResidueUnits_integralUnit]
+  exact congrArg (fun u : 𝓀[C]ˣ => (u : 𝓀[C]))
+    (finiteFieldPowerResidueSymbol_unitsMap_ringEquiv
+      (finitePlacePrimeResidueEquivLocalResidue K v)
+      n hnPrime hnLocal
+      (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealResidueUnit
+        K v a ha)).symm
+
+private theorem finitePlaceLocalTamePowerResidueSymbolFieldValue_eq_primeIdealValue
+    (v : HeightOneSpectrum (𝓞 K)) (n : ℕ+)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) :
+    @Eq (HeightOneSpectrum.adicAbv K v).Completion
+      (finitePlaceLocalTamePowerResidueSymbolFieldValue K v n hmu hv a ha)
+      (finitePlacePrimeIdealPowerResidueFactorFieldValue
+        K v n hmu hcoprime a ha) := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  let : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  let : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  let hnC :=
+    finitePlace_natCast_valuation_eq_one_of_not_mem_exponent K n v hv
+  let hmuC := finitePlaceHilbert_primitiveRoots_nonempty K n hmu v
+  unfold finitePlaceLocalTamePowerResidueSymbolFieldValue
+  unfold finitePlacePrimeIdealPowerResidueFactorFieldValue
+  unfold finitePlacePrimeIdealPowerResidueGlobalRoot
+  unfold finitePlacePrimeIdealPowerResidueIntegralRoot
+  have hRoots :
+      localTamePowerResidueSymbol C n hnC hmuC
+          (finitePlaceIntegralCompletionUnit K v a ha) =
+        nthRootsSubgroupMap K C (n : ℕ)
+          (integralRootsOfUnityToNthRoots K (n : ℕ)
+            (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealPowerResidueSymbol
+              K v n hmu hcoprime a ha)) := by
+    apply (localNthRootsReductionEquiv C n hnC hmuC).injective
+    rw [localNthRootsReductionEquiv_localTamePowerResidueSymbol]
+    apply Subtype.ext
+    apply Units.ext
+    have hResidue := finitePlaceLocalTamePowerResidueSymbol_residueValue_eq
+      K v n hmu hcoprime hv a ha
+    unfold finitePlaceLocalTamePowerResidueSymbolResidueValue at hResidue
+    unfold finitePlacePrimeIdealPowerResidueFactorResidueValue at hResidue
+    dsimp only at hResidue
+    exact hResidue
+  exact congrArg (fun q : nthRootsSubgroup C (n : ℕ) => (q.1 : C)) hRoots
+
+/-- The normalized additive valuation of a global field unit in the
+canonical completion at a finite place. -/
+noncomputable def finitePlaceNormalizedValuation
+    (v : HeightOneSpectrum (𝓞 K)) (x : Kˣ) : ℤ := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  letI : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  letI : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  exact valuationMap C
+    (Additive.ofMul (finitePlaceHilbert_completionUnit K v x))
+
+/-- Away from the exponent, a finite-place Hilbert factor with integral-unit
+first entry is the prime-ideal power-residue symbol raised to the negative
+normalized valuation of the second entry. -/
+theorem finitePlaceHilbertSymbol_eq_primeIdealPowerResidueFactor_zpow
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) (b : Kˣ) :
+    finitePlaceHilbertSymbol K n hnK hmu v
+        (primeAvoidingIntegralFieldUnit K v a ha) b =
+      integralRootsOfUnityToNthRoots K (n : ℕ)
+          (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealPowerResidueSymbol
+              K v n hmu hcoprime a ha) ^
+        (-finitePlaceNormalizedValuation K v b) := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  let : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  let : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  let hnC :=
+    finitePlace_natCast_valuation_eq_one_of_not_mem_exponent K n v hv
+  let hmuC := finitePlaceHilbert_primitiveRoots_nonempty K n hmu v
+  let bC := finitePlaceHilbert_completionUnit K v b
+  apply nthRootsSubgroupMap_injective K C (n : ℕ)
+  rw [finitePlaceHilbertSymbol_map_eq_localHilbertSymbol]
+  change
+    localHilbertSymbol C n
+        (finitePlaceHilbert_natCast_ne_zero K n hnK v) hmuC
+        (finitePlaceHilbert_completionUnit K v
+          (primeAvoidingIntegralFieldUnit K v a ha)) bC =
+      nthRootsSubgroupMap K C (n : ℕ)
+        (integralRootsOfUnityToNthRoots K (n : ℕ)
+            (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealPowerResidueSymbol
+                K v n hmu hcoprime a ha) ^
+          (-finitePlaceNormalizedValuation K v b))
+  rw [finitePlaceHilbert_completionUnit_primeAvoidingIntegralFieldUnit]
+  rw [localHilbertSymbol_tame_formula C n hnC hmuC]
+  change
+    finitePlaceLocalTamePowerResidueSymbolValue K v n hmu hv a ha ^
+        (-finitePlaceNormalizedValuation K v b) = _
+  have hBase :
+      finitePlaceLocalTamePowerResidueSymbolValue K v n hmu hv a ha =
+        finitePlacePrimeIdealPowerResidueFactorValue
+          K v n hmu hcoprime a ha := by
+    apply Subtype.ext
+    apply Units.ext
+    simpa only [finitePlaceLocalTamePowerResidueSymbolValue,
+      finitePlacePrimeIdealPowerResidueFactorValue,
+      finitePlaceLocalTamePowerResidueSymbolFieldValue,
+      finitePlacePrimeIdealPowerResidueFactorFieldValue] using
+      finitePlaceLocalTamePowerResidueSymbolFieldValue_eq_primeIdealValue
+        K v n hmu hcoprime hv a ha
+  rw [hBase]
+  unfold finitePlacePrimeIdealPowerResidueFactorValue
+  unfold finitePlacePrimeIdealPowerResidueGlobalRoot
+  unfold finitePlacePrimeIdealPowerResidueIntegralRoot
+  rw [map_zpow]
+
+/-- Endpoint form of the finite-place local/global power-residue comparison. -/
+theorem finitePlaceHilbertSymbol_eq_primeIdealPowerResidueFactor
+    (n : ℕ+) (hnK : ((n : ℕ) : K) ≠ 0)
+    (hmu : (primitiveRoots (n : ℕ) K).Nonempty)
+    (v : HeightOneSpectrum (𝓞 K))
+    (hv : v ∉ powerResidueExponentFinitePlaces K n)
+    (hcoprime : (Ideal.absNorm v.asIdeal).Coprime (n : ℕ))
+    (a : 𝓞 K) (ha : a ∉ v.asIdeal) (b : Kˣ) :
+    finitePlaceHilbertSymbol K n hnK hmu v
+        (primeAvoidingIntegralFieldUnit K v a ha) b =
+      integralRootsOfUnityToNthRoots K (n : ℕ)
+          (AlgebraicNumberTheory.PowerResidueSymbols.primeIdealPowerResidueSymbol
+              K v n hmu hcoprime a ha) ^
+        (-finitePlaceNormalizedValuation K v b) :=
+  finitePlaceHilbertSymbol_eq_primeIdealPowerResidueFactor_zpow
+    K n hnK hmu v hv hcoprime a ha b
+
+/-- The Dedekind prime multiplicity is the exponent occurring in the
+integer-valued adic valuation. -/
+theorem intValuation_eq_exp_neg_idealPrimeMultiplicity
+    (v : HeightOneSpectrum (𝓞 K)) (x : 𝓞 K) (hx : x ≠ 0) :
+    v.intValuation x =
+      WithZero.exp
+        (-(AlgebraicNumberTheory.PowerResidueSymbols.idealPrimeMultiplicity K v (Ideal.span {x}) : ℤ)) := by
+  rw [v.intValuation_if_neg hx]
+  rfl
+
+/-- For an integral element, the normalized valuation in the canonical
+finite-place completion is the negative multiplicity of the prime in its
+principal ideal. -/
+theorem finitePlaceNormalizedValuation_nonzeroIntegralFieldUnit
+    (v : HeightOneSpectrum (𝓞 K))
+    (x : 𝓞 K) (hx : x ≠ 0) :
+    finitePlaceNormalizedValuation K v
+        (nonzeroIntegralFieldUnit K x hx) =
+      -(AlgebraicNumberTheory.PowerResidueSymbols.idealPrimeMultiplicity K v (Ideal.span {x}) : ℤ) := by
+  let C := (HeightOneSpectrum.adicAbv K v).Completion
+  let : ValuativeRel C :=
+    finitePlaceLocalArtinCompletionValuativeRel v
+  let : IsNonarchimedeanLocalField C :=
+    finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v
+  let : IsUltrametricDist C :=
+    finitePlaceArtinCompletionIsUltrametricDist
+      (HeightOneSpectrum.adicAbv K v)
+      (HeightOneSpectrum.isNonarchimedean_adicAbv K v)
+  let m := AlgebraicNumberTheory.PowerResidueSymbols.idealPrimeMultiplicity K v (Ideal.span {x})
+  let πData := chosenFinitePlaceCompletionIntegralUniformizer v
+  have hπIrreducible : Irreducible πData.completionInteger := by
+    rw [IsDiscreteValuationRing.irreducible_iff_uniformizer]
+    let completionDVF :
+        ValuationTheory.DiscreteValuationField.DVF C :=
+      { ValueGroup := ValuativeRel.ValueGroupWithZero C
+        valuation := ValuativeRel.valuation C }
+    exact
+      completionDVF.maximalIdeal_eq_span_uniformizer
+        πData.completionInteger_isUniformizer
+  let πC : Cˣ :=
+    Units.mk0 (πData.completionInteger : C)
+      πData.completionInteger_isUniformizer.ne_zero
+  let xC : Cˣ :=
+    finitePlaceHilbert_completionUnit K v
+      (nonzeroIntegralFieldUnit K x hx)
+  have hIntX :
+      v.intValuation x = WithZero.exp (-(m : ℤ)) := by
+    exact intValuation_eq_exp_neg_idealPrimeMultiplicity K v x hx
+  have hIntPiPow :
+      v.intValuation (πData.integer ^ m) =
+        WithZero.exp (-(m : ℤ)) := by
+    rw [map_pow, πData.intValuation_eq_exp_neg_one]
+    calc
+      WithZero.exp (-1 : ℤ) ^ m =
+          WithZero.exp (m • (-1 : ℤ)) :=
+        (WithZero.exp_nsmul m (-1 : ℤ)).symm
+      _ = WithZero.exp (-(m : ℤ)) := by simp
+  have hnormX :
+      ‖(xC : C)‖ =
+        (WithZeroMulInt.toNNReal
+          (HeightOneSpectrum.absNorm_ne_zero v)
+          (WithZero.exp (-(m : ℤ))) : ℝ) := by
+    change ‖algebraMap K C (x : K)‖ = _
+    calc
+      ‖algebraMap K C (x : K)‖ =
+          HeightOneSpectrum.adicAbv K v (x : K) :=
+        AbsoluteValue.completionAbsoluteValue_coe
+          (HeightOneSpectrum.adicAbv K v) (x : K)
+      _ = _ := by
+        rw [HeightOneSpectrum.adicAbv_def,
+          HeightOneSpectrum.valuation_of_algebraMap, hIntX]
+  have hnormPiPow :
+      ‖((πC ^ m : Cˣ) : C)‖ =
+        (WithZeroMulInt.toNNReal
+          (HeightOneSpectrum.absNorm_ne_zero v)
+          (WithZero.exp (-(m : ℤ))) : ℝ) := by
+    have hπC :
+        (πC : C) = algebraMap K C (πData.integer : K) := by
+      exact πData.coe_completionInteger
+    have hπCPow :
+        ((πC ^ m : Cˣ) : C) =
+          algebraMap K C (((πData.integer ^ m : 𝓞 K) : K)) := by
+      have hIntegerPow :
+          (((πData.integer ^ m : 𝓞 K) : K)) =
+            (πData.integer : K) ^ m := by
+        exact map_pow (algebraMap (𝓞 K) K) πData.integer m
+      rw [Units.val_pow_eq_pow_val, hπC, hIntegerPow, map_pow]
+    calc
+      ‖((πC ^ m : Cˣ) : C)‖ =
+          ‖algebraMap K C ((πData.integer ^ m : 𝓞 K) : K)‖ :=
+        congrArg norm hπCPow
+      _ = HeightOneSpectrum.adicAbv K v
+          ((πData.integer ^ m : 𝓞 K) : K) :=
+        AbsoluteValue.completionAbsoluteValue_coe
+          (HeightOneSpectrum.adicAbv K v)
+          ((πData.integer ^ m : 𝓞 K) : K)
+      _ =
+          (WithZeroMulInt.toNNReal
+            (HeightOneSpectrum.absNorm_ne_zero v)
+            (WithZero.exp (-(m : ℤ))) : ℝ) := by
+        rw [HeightOneSpectrum.adicAbv_def,
+          HeightOneSpectrum.valuation_of_algebraMap, hIntPiPow]
+  have hraw :
+      ValuativeRel.valuation C (xC : C) =
+        ValuativeRel.valuation C ((πC ^ m : Cˣ) : C) := by
+    apply canonicalValuation_eq_of_valuation_eq
+      (v := NormedField.valuation (K := C))
+    apply NNReal.eq
+    simpa only [NormedField.valuation_apply, coe_nnnorm] using
+      hnormX.trans hnormPiPow.symm
+  have hValuationMap :
+      valuationMap C (Additive.ofMul xC) =
+        valuationMap C (Additive.ofMul (πC ^ m)) := by
+    rw [valuationMap_apply, valuationMap_apply]
+    unfold IsNonarchimedeanLocalField.v
+    congr 2
+    exact congrArg
+      (_root_.IsNonarchimedeanLocalField.valueGroupWithZeroIsoInt C) hraw
+  have hπCValuation :
+      valuationMap C (Additive.ofMul πC) = -1 := by
+    rw [valuationMap_apply]
+    exact
+      v_integerRingIrreducibleFieldUnit C πData.completionInteger
+        hπIrreducible πC rfl
+  calc
+    finitePlaceNormalizedValuation K v
+        (nonzeroIntegralFieldUnit K x hx) =
+        valuationMap C (Additive.ofMul xC) := rfl
+    _ = valuationMap C (Additive.ofMul (πC ^ m)) := hValuationMap
+    _ = (m : ℤ) * valuationMap C (Additive.ofMul πC) := by
+      rw [valuationMap_ofMul_pow]
+    _ = -(m : ℤ) := by rw [hπCValuation]; simp
+    _ = -(AlgebraicNumberTheory.PowerResidueSymbols.idealPrimeMultiplicity K v (Ideal.span {x}) : ℤ) := rfl
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtin.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtin.lean
index 8c5d33f..4536b7d 100644
--- a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtin.lean
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtin.lean
@@ -4,69 +4,13 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
 -/
 
-import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.CompletionToIdeal
-import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.IdealToCompletion
-import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.LocalNorm
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.AlgEquiv
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.BaseChange
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.Core
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.NormComparison
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.NormalClosureNorm
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.Tower
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.TowerAlgEquivNaturality
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.TowerBaseChange
-import ClassFieldTheory.AlgebraicNumberTheory.Idele.Relative.FinitePlaceTensorNorm
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlace
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicLocalization
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalPrimeFactorization
-import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalPrincipalLocalUnit
-import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.SemilinearNaturality
-import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.UnramifiedNormalization
-import ClassFieldTheory.LocalClassFieldTheory.LubinTateApplication.PadicMultiplicativeArtinComparison
-import ValuedFieldTheory.LocalField.DiscreteValuationField.PadicValuationComparison
-import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.UnramifiedFrobenius
-import ValuedFieldTheory.LocalField.Padic.Cyclotomic.TotallyRamified.EisensteinPolynomial
-import Mathlib.NumberTheory.NumberField.Cyclotomic.Galois
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlaceArtinPrincipal
 
-set_option autoImplicit false
-
-/-!
-# Finite-place Artin symbols in rational cyclotomic levels
 
-At a rational prime away from the cyclotomic level, the chosen completed
-extension is unramified.  Its normalized local Artin map is therefore the
-arithmetic Frobenius raised to the local valuation.  The genuine primitive
-root in the localized cyclotomic level identifies the image of arithmetic
-Frobenius under the global cyclotomic character with the residue prime.
--/
+set_option autoImplicit false
 
 open scoped Classical NNReal NumberField ValuativeRel
 open NumberField IsDedekindDomain
-
-noncomputable section
-
-namespace GlobalClassFieldTheory
-namespace Reciprocity
-
--- Specializing the generic finite-place comparison to `ℚ` must retain its
--- `Algebra.id` owner rather than selecting the competing rational-field
--- instance introduced after specialization.
-@[reducible] noncomputable local instance
-    rationalFinitePlaceCompletionRatAlgebra
-    (v : HeightOneSpectrum (𝓞 ℚ)) :
-    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
-  letI : Algebra ℚ ℚ := Algebra.id ℚ
-  let hWith : Algebra ℚ
-      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
-    WithAbs.instAlgebra _
-  let hUniform : UniformContinuousConstSMul ℚ
-      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
-    WithAbs.instUniformContinuousConstSMulReal _
-  exact
-    @UniformSpace.Completion.algebra
-      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
-      ℚ _ hWith hUniform
-
 open AlgebraicNumberTheory.Valuations
 open HilbertRamification
 open LocalClassFieldTheory
@@ -75,2260 +19,54 @@ open LocalFieldTheory.DiscreteValuationField
 open LocalFieldTheory.DiscreteValuationField.Examples.Qp
 open LubinTate
 
-private theorem mappedAbelianLocalArtin_eq_frobenius_zpow
-    {F E G : Type}
-    [Field F] [ValuativeRel F] [TopologicalSpace F]
-    [IsNonarchimedeanLocalField F]
-    [Field E] [ValuativeRel E] [UniformSpace E] [IsUniformAddGroup E]
-    [IsNonarchimedeanLocalField E]
-    [Algebra F E] [FiniteDimensional F E] [IsAbelianGalois F E]
-    [Valuation.HasExtension
-      (ValuativeRel.valuation F) (ValuativeRel.valuation E)]
-    [IsNonarchimedeanLocalField.IsUnramifiedValuedExtension F E]
-    [Group G]
-    (f : (E ≃ₐ[F] E) →* G) (x : Fˣ) :
-    f (LocalClassFieldTheory.abelianLocalArtinMonoidHom F E x) =
-      (f (arithmeticFrobeniusOfUnramifiedValuation F E)) ^
-        IsNonarchimedeanLocalField.valuationMap F
-          (Additive.ofMul x) := by
-  rw [LocalClassFieldTheory.abelianLocalArtinMonoidHom_eq_frobenius_zpow,
-    map_zpow]
-
-local instance (q : Nat.Primes) : Fact q.1.Prime :=
-  ⟨q.2⟩
-
-local instance (m : ℕ+) : NeZero (m : ℕ) :=
-  ⟨m.ne_zero⟩
-
-noncomputable local instance
-    rationalCyclotomicLevelFiniteDimensional
-    (m : ℕ+) :
-    FiniteDimensional ℚ
-      (KummerTheory.rationalCyclotomicLevel m) :=
-  IsCyclotomicExtension.finiteDimensional
-    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
-
-noncomputable local instance
-    rationalCyclotomicLevelIsAbelianGalois
-    (m : ℕ+) :
-    IsAbelianGalois ℚ
-      (KummerTheory.rationalCyclotomicLevel m) := by
-  have : IsGalois ℚ
-      (KummerTheory.rationalCyclotomicLevel m) :=
-    inferInstance
-  let e :=
-    IsCyclotomicExtension.Rat.galEquivZMod
-      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
-  exact
-    { is_comm.comm σ τ := by
-        apply e.injective
-        simp only [map_mul]
-        exact mul_comm _ _ }
-
-@[reducible]
-noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
-    (q : Nat.Primes) :
-    NontriviallyNormedField
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion :=
-  absoluteValueExtension_completionNontriviallyNormedField
-    (HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q))
-    (RayClass.adicAbv_isNontrivial
-      (RayClass.rationalPrime q))
-
-noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
-    (q : Nat.Primes) :
-    LocallyCompactSpace
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion :=
-  AbsoluteValue.Completion.locallyCompactSpace
-    (finitePlaceCompletionBaseMap_isometry
-      (RayClass.rationalPrime q))
-
-noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
-    (q : Nat.Primes) :
-    IsUltrametricDist
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion :=
-  finitePlaceArtinCompletionIsUltrametricDist
-    (HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q))
-    (HeightOneSpectrum.isNonarchimedean_adicAbv
-      ℚ (RayClass.rationalPrime q))
-
-@[reducible]
-noncomputable local instance rationalFinitePlaceBaseValued
-    (q : Nat.Primes) :
-    Valued
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
-  finitePlaceArtinCompletionValued
-    (HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q))
-    (HeightOneSpectrum.isNonarchimedean_adicAbv
-      ℚ (RayClass.rationalPrime q))
-
-@[reducible]
-noncomputable local instance rationalFinitePlaceBaseValuativeRel
-    (q : Nat.Primes) :
-    ValuativeRel
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion :=
-  finitePlaceLocalArtinCompletionValuativeRel
-    (K := ℚ) (RayClass.rationalPrime q)
-
-noncomputable local instance
-    rationalFinitePlaceBaseValuationIsNontrivial
-    (q : Nat.Primes) :
-    (Valued.v :
-      Valuation
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        ℝ≥0).IsNontrivial :=
-  (inferInstance :
-    (NormedField.valuation
-      (K :=
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
-
-noncomputable local instance rationalFinitePlaceBaseValuationCompatible
-    (q : Nat.Primes) :
-    (Valued.v :
-      Valuation
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        ℝ≥0).Compatible :=
-  Valuation.Compatible.ofValuation _
-
-noncomputable local instance
-    rationalFinitePlaceBaseValuativeRelIsNontrivial
-    (q : Nat.Primes) :
-    ValuativeRel.IsNontrivial
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion :=
-  (ValuativeRel.isNontrivial_iff_isNontrivial
-    (Valued.v :
-      Valuation
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        ℝ≥0)).2 inferInstance
-
-noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
-    (q : Nat.Primes) :
-    IsValuativeTopology
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion :=
-  isValuativeTopology_of_valued_ofValuation
-    (HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q)).Completion ℝ≥0
-
-noncomputable local instance
-    rationalFinitePlaceBaseIsNonarchimedeanLocalField
-    (q : Nat.Primes) :
-    IsNonarchimedeanLocalField
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion :=
-  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
-    (K := ℚ) (RayClass.rationalPrime q)
-
-/-! Named compatibility witnesses used by the ramified-prime and ray-norm
-modules.  They are not installed as a duplicate module-level instance family;
-the canonical instances above already provide the same data. -/
-
-/-- Rational cyclotomic levels are finite-dimensional over `ℚ`. -/
-theorem rationalCyclotomicPrincipalPrimeLevelFiniteDimensional
-    (m : ℕ+) :
-    FiniteDimensional ℚ (KummerTheory.rationalCyclotomicLevel m) :=
-  rationalCyclotomicLevelFiniteDimensional m
-
-/-- Rational cyclotomic levels are abelian Galois extensions of `ℚ`. -/
-theorem rationalCyclotomicPrincipalPrimeLevelIsAbelianGalois
-    (m : ℕ+) :
-    IsAbelianGalois ℚ (KummerTheory.rationalCyclotomicLevel m) :=
-  rationalCyclotomicLevelIsAbelianGalois m
-
-/-- The canonical nontrivially normed field structure on the completion of
-`ℚ` at the rational prime `p`, exposed for principal-prime constructions. -/
-@[reducible]
-noncomputable def rationalPrimeFactorCompletionNontriviallyNormedField
-    (p : Nat.Primes) :
-    NontriviallyNormedField
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion :=
-  rationalFinitePlaceBaseNontriviallyNormedField p
-
-/-- The completion of `ℚ` at `p` is locally compact. -/
-theorem rationalPrimeFactorCompletionLocallyCompactSpace
-    (p : Nat.Primes) :
-    LocallyCompactSpace
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion :=
-  rationalFinitePlaceBaseLocallyCompactSpace p
-
-/-- The completion of `ℚ` at `p` carries its canonical ultrametric distance. -/
-theorem rationalPrimeFactorCompletionIsUltrametricDist
-    (p : Nat.Primes) :
-    IsUltrametricDist
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion :=
-  rationalFinitePlaceBaseIsUltrametricDist p
-
-/-- The canonical `ℝ≥0`-valued structure on the completion of `ℚ` at `p`. -/
-@[reducible]
-noncomputable def rationalPrimeFactorCompletionValued
-    (p : Nat.Primes) :
-    Valued
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion ℝ≥0 :=
-  rationalFinitePlaceBaseValued p
-
-/-- The valuative relation induced by the canonical valuation on the
-completion of `ℚ` at `p`. -/
-@[reducible]
-noncomputable def rationalPrimeFactorCompletionValuativeRel
-    (p : Nat.Primes) :
-    ValuativeRel
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion :=
-  rationalFinitePlaceBaseValuativeRel p
-
-/-- The canonical valuation on the completion of `ℚ` at `p` is nontrivial. -/
-theorem rationalPrimeFactorCompletionValuationIsNontrivial
-    (p : Nat.Primes) :
-    (Valued.v : Valuation
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion ℝ≥0).IsNontrivial :=
-  rationalFinitePlaceBaseValuationIsNontrivial p
-
-/-- The canonical valuation on the completion of `ℚ` at `p` is compatible
-with its field structure. -/
-theorem rationalPrimeFactorCompletionValuationCompatible
-    (p : Nat.Primes) :
-    (Valued.v : Valuation
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion ℝ≥0).Compatible :=
-  rationalFinitePlaceBaseValuationCompatible p
-
-/-- The canonical valuative relation on the completion at `p` is nontrivial. -/
-theorem rationalPrimeFactorCompletionValuativeRelIsNontrivial
-    (p : Nat.Primes) :
-    ValuativeRel.IsNontrivial
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion :=
-  rationalFinitePlaceBaseValuativeRelIsNontrivial p
-
-/-- The completion topology at `p` is induced by its canonical valuation. -/
-theorem rationalPrimeFactorCompletionIsValuativeTopology
-    (p : Nat.Primes) :
-    IsValuativeTopology
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion :=
-  rationalFinitePlaceBaseIsValuativeTopology p
-
-/-- The completion of `ℚ` at `p` is a nonarchimedean local field. -/
-theorem rationalPrimeFactorCompletionIsNonarchimedeanLocalField
-    (p : Nat.Primes) :
-    IsNonarchimedeanLocalField
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion :=
-  rationalFinitePlaceBaseIsNonarchimedeanLocalField p
-
-/-- The positive conductor of the `n`-th ramified cyclotomic level at
-`p`. -/
-def rationalCyclotomicPrincipalPrimeModulus
-    (p : Nat.Primes) (n : ℕ) : ℕ+ :=
-  ⟨p.1 ^ (n + 1), pow_pos p.2.pos (n + 1)⟩
-
-/-- The rational finite-place completion used at the prime `p`. -/
-abbrev RationalCyclotomicPrincipalPrimeCompletion
-    (p : Nat.Primes) :=
-  (HeightOneSpectrum.adicAbv ℚ
-    (RayClass.rationalPrime p)).Completion
-
-/-- The chosen localized cyclotomic field at level `p ^ (n + 1)`. -/
-abbrev RationalCyclotomicPrincipalPrimeLocalizedLevel
-    (p : Nat.Primes) (n : ℕ) :=
-  rationalCyclotomicLocalizedCompletion
-    (rationalCyclotomicPrincipalPrimeModulus p n)
-    (RayClass.rationalPrime p)
-
-/-- The standard multiplicative Lubin--Tate field at level `n`. -/
-abbrev RationalCyclotomicPrincipalPrimePadicLevel
-    (p : Nat.Primes) (n : ℕ) :=
-  standardLubinTateLevelField
-    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
-
-/-- The valuation ring of the absolute-value completion at `q`, identified
-with the standard p-adic integer ring `ℤ_q`. -/
-noncomputable def rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-    (q : Nat.Primes) :
-    𝒪[(HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion] ≃+*
-      ℤ_[q.1] := by
-  let v : HeightOneSpectrum (𝓞 ℚ) :=
-    RayClass.rationalPrime q
-  let vQ := HeightOneSpectrum.adicAbv ℚ v
-  let eConcreteIntegers :
-      𝒪[vQ.Completion] ≃+*
-        v.adicCompletionIntegers ℚ :=
-    finitePlaceCompletionIntegerRingEquiv v
-  exact
-    eConcreteIntegers.trans
-      (PadicInt.adicCompletionIntegersEquiv
-        (𝓞 ℚ) q).symm.toRingEquiv
-
-/-- The absolute-value completion at the rational prime `q`, identified
-with the standard field `ℚ_q`. -/
-noncomputable def rationalFinitePlaceCompletionRingEquivPadic
-    (q : Nat.Primes) :
-    (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion ≃+*
-      ℚ_[q.1] :=
-  IsFractionRing.ringEquivOfRingEquiv
-    (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q)
-
-/-- The completion-to-`ℚ_q` equivalence respects the rational embedding. -/
-theorem rationalFinitePlaceCompletionRingEquivPadic_algebraMap
-    (q : Nat.Primes) (a : ℚ) :
-    rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap ℚ
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion a) =
-      algebraMap ℚ ℚ_[q.1] a := by
-  exact
-    (rationalFinitePlaceCompletionRingEquivPadic q).toRingHom.map_rat_algebraMap a
-
-/-- The completion field equivalence and its restriction to valuation
-rings commute with the natural inclusions into the fields. -/
-theorem rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
-    (q : Nat.Primes)
-    (a :
-      𝒪[(HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion]) :
-    rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap
-          𝒪[(HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion]
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion a) =
-      algebraMap ℤ_[q.1] ℚ_[q.1]
-        (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q a) := by
-  exact
-    (IsFractionRing.ringEquivOfRingEquiv_algebraMap
-      (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q) a)
-
-/-- The canonical rational-completion equivalence preserves the canonical
-valuations. -/
-theorem
-    rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible
-    (p : Nat.Primes) :
-    SemilinearValuationCompatible
-      (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1]
-      (rationalFinitePlaceCompletionRingEquivPadic p) := by
-  let F := RationalCyclotomicPrincipalPrimeCompletion p
-  let eK := rationalFinitePlaceCompletionRingEquivPadic p
-  let : Algebra F ℚ_[p.1] := eK.toRingHom.toAlgebra
-  change
-    (ValuativeRel.valuation F).HasExtension
-      (ValuativeRel.valuation ℚ_[p.1])
-  let eO :=
-    (rationalFinitePlaceCompletionIntegerRingEquivPadicInt p).trans
-      (padicIntEquivValuationSubring p.1)
-  have hChosen :
-      (localCompleteDVF F).valuation.HasExtension
-        (padicDVRValuation p.1) := by
-    change
-      (localCompleteDVF F).valuation.HasExtension
-        (padicCompleteDVF p.1).valuation
-    apply
-      ValuationTheory.DiscreteValuationField.ValuedExtension.valuation_hasExtension_of_valuationSubring_equiv
-        (localCompleteDVF F)
-        (padicCompleteDVF p.1)
-        eO
-    intro z
-    change
-      algebraMap
-          (padicDVRValuation p.1).valuationSubring ℚ_[p.1]
-          (padicIntEquivValuationSubring p.1
-            (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-              p z)) =
-        rationalFinitePlaceCompletionRingEquivPadic p
-          (algebraMap 𝒪[F] F z)
-    symm
-    calc
-      rationalFinitePlaceCompletionRingEquivPadic p
-          (algebraMap 𝒪[F] F z) =
-        algebraMap ℤ_[p.1] ℚ_[p.1]
-          (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-            p z) :=
-        rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
-          p z
-      _ =
-        algebraMap
-          (padicDVRValuation p.1).valuationSubring ℚ_[p.1]
-          (padicIntEquivValuationSubring p.1
-            (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-              p z)) := by
-        rw [PadicInt.algebraMap_apply,
-          ValuationSubring.algebraMap_apply]
-        exact
-          (padicIntEquivValuationSubring_coe p.1
-            (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-              p z)).symm
-  have hBase :
-      (ValuativeRel.valuation F).HasExtension
-        (padicDVRValuation p.1) := by
-    rw [← localCompleteDVF_valuation_eq]
-    exact hChosen
-  refine
-    { val_isEquiv_comap := ?_ }
-  exact
-    hBase.val_isEquiv_comap.trans
-      ((padicDVRValuation_isEquiv_valuativeRelValuation
-        p.1).comap (algebraMap F ℚ_[p.1]))
-
-/-- The rational prime, pulled back from `ℤ_q` to the valuation ring of
-the absolute-value completion at `q`. -/
-noncomputable def rationalPrimeFinitePlaceInteger
-    (q : Nat.Primes) :
-    𝒪[(HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q)).Completion] :=
-  (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q).symm
-    (q.1 : ℤ_[q.1])
-
-/-- The pulled-back rational prime is irreducible in the completion
-valuation ring. -/
-theorem rationalPrimeFinitePlaceInteger_irreducible
-    (q : Nat.Primes) :
-    Irreducible (rationalPrimeFinitePlaceInteger q) := by
-  exact
-    (MulEquiv.irreducible_iff
-      (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-        q).symm.toMulEquiv).2
-      ((PadicInt.prime_p :
-        Prime (q.1 : ℤ_[q.1])).irreducible)
-
-/-- Coercing the pulled-back prime to the completion field gives the
-ordinary image of the rational number `q`. -/
-theorem rationalPrimeFinitePlaceInteger_coe
-    (q : Nat.Primes) :
-    ((rationalPrimeFinitePlaceInteger q :
-      𝒪[(HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion]) :
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion) =
-      algebraMap ℚ
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        (q.1 : ℚ) := by
-  apply
-    (rationalFinitePlaceCompletionRingEquivPadic q).injective
-  change
-    rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap
-          𝒪[(HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion]
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (rationalPrimeFinitePlaceInteger q)) =
-      rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap ℚ
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (q.1 : ℚ))
-  calc
-    rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap
-          𝒪[(HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion]
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (rationalPrimeFinitePlaceInteger q)) =
-      algebraMap ℤ_[q.1] ℚ_[q.1]
-        ((rationalFinitePlaceCompletionIntegerRingEquivPadicInt q)
-          (rationalPrimeFinitePlaceInteger q)) :=
-        rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
-          q (rationalPrimeFinitePlaceInteger q)
-    _ = (q.1 : ℚ_[q.1]) := by
-      rw [rationalPrimeFinitePlaceInteger,
-        (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-          q).apply_symm_apply,
-        PadicInt.algebraMap_apply,
-        PadicInt.coe_natCast]
-    _ =
-      rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap ℚ
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (q.1 : ℚ)) := by
-      rw [
-        rationalFinitePlaceCompletionRingEquivPadic_algebraMap]
-      norm_num
-
-/-- The rational prime as a field unit of its absolute-value completion. -/
-noncomputable def rationalPrimeFinitePlaceFieldUnit
-    (q : Nat.Primes) :
-    (HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q)).Completionˣ :=
-  Units.mk0
-    (rationalPrimeFinitePlaceInteger q :
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion)
-    (by
-      intro hzero
-      exact
-        (rationalPrimeFinitePlaceInteger_irreducible q).ne_zero
-          (Subtype.ext hzero))
-
-/-- In the inverse-standard local reciprocity normalization, the rational
-prime itself has normalized additive value `-1`. -/
-theorem rationalPrimeFinitePlaceFieldUnit_valuationMap
-    (q : Nat.Primes) :
-    IsNonarchimedeanLocalField.valuationMap
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        (Additive.ofMul
-          (rationalPrimeFinitePlaceFieldUnit q)) =
-      -1 := by
-  simpa [IsNonarchimedeanLocalField.valuationMap_apply] using
-    (LocalFieldTheory.v_integerRingIrreducibleFieldUnit
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion
-      (rationalPrimeFinitePlaceInteger q)
-      (rationalPrimeFinitePlaceInteger_irreducible q)
-      (rationalPrimeFinitePlaceFieldUnit q) rfl)
-
-/-- The rational `q`-unit part of `x`, pulled back from `ℤ_qˣ` to the
-valuation ring of the absolute-value completion. -/
-noncomputable def rationalPrimeUnitFinitePlaceIntegerUnit
-    (x : ℚˣ) (q : Nat.Primes) :
-    𝒪[(HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q)).Completion]ˣ :=
-  Units.map
-      (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
-        q).symm.toMonoidHom
-    (padicIntUnitOfRat q
-      (rationalPrimeUnit x q : ℚ)
-      (rationalPrimeUnit x q).ne_zero
-      (padicValRat_rationalPrimeUnit x q))
-
-/-- Forgetting the integrality proof from the pulled-back `q`-unit gives
-the ordinary image of the rational `q`-unit in the completion field. -/
-theorem rationalPrimeUnitFinitePlaceIntegerUnit_coe
-    (x : ℚˣ) (q : Nat.Primes) :
-    (((rationalPrimeUnitFinitePlaceIntegerUnit x q :
-        𝒪[(HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion]ˣ) :
-        𝒪[(HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion]) :
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion) =
-      algebraMap ℚ
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        (rationalPrimeUnit x q : ℚ) := by
-  apply
-    (rationalFinitePlaceCompletionRingEquivPadic q).injective
-  change
-    rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap
-          𝒪[(HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion]
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
-            𝒪[(HeightOneSpectrum.adicAbv ℚ
-              (RayClass.rationalPrime q)).Completion])) =
-      rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap ℚ
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (rationalPrimeUnit x q : ℚ))
-  calc
-    rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap
-          𝒪[(HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion]
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
-          𝒪[(HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion])) =
-      algebraMap ℤ_[q.1] ℚ_[q.1]
-        ((rationalFinitePlaceCompletionIntegerRingEquivPadicInt q)
-          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
-            𝒪[(HeightOneSpectrum.adicAbv ℚ
-              (RayClass.rationalPrime q)).Completion])) :=
-        rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe q
-          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
-            𝒪[(HeightOneSpectrum.adicAbv ℚ
-              (RayClass.rationalPrime q)).Completion])
-    _ =
-      ((padicIntUnitOfRat q
-        (rationalPrimeUnit x q : ℚ)
-        (rationalPrimeUnit x q).ne_zero
-        (padicValRat_rationalPrimeUnit x q) :
-          ℤ_[q.1]) : ℚ_[q.1]) := by
-      rw [rationalPrimeUnitFinitePlaceIntegerUnit]
-      simp [PadicInt.algebraMap_apply]
-    _ = ((rationalPrimeUnit x q : ℚ) : ℚ_[q.1]) := by
-      rw [padicIntUnitOfRat_coe]
-    _ =
-      rationalFinitePlaceCompletionRingEquivPadic q
-        (algebraMap ℚ
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion
-          (rationalPrimeUnit x q : ℚ)) := by
-      rw [
-        rationalFinitePlaceCompletionRingEquivPadic_algebraMap]
-      simp
-
-/-- The completion field unit underlying the pulled-back rational `q`-unit
-has normalized additive value zero. -/
-theorem rationalPrimeUnitFinitePlaceField_valuationMap
-    (x : ℚˣ) (q : Nat.Primes) :
-    IsNonarchimedeanLocalField.valuationMap
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        (Additive.ofMul
-          (IsNonarchimedeanLocalField.integerUnitsToFieldUnits
-            (HeightOneSpectrum.adicAbv ℚ
-              (RayClass.rationalPrime q)).Completion
-            (rationalPrimeUnitFinitePlaceIntegerUnit x q))) =
-      0 := by
-  rw [IsNonarchimedeanLocalField.valuationMap_apply]
-  exact
-    IsNonarchimedeanLocalField.v_integerUnitsToFieldUnits
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime q)).Completion
-      (rationalPrimeUnitFinitePlaceIntegerUnit x q)
-
-/-- The rational prime `q`, regarded as a unit of `ℚ`. -/
-def rationalPrimeGeneratorUnit (q : Nat.Primes) : ℚˣ :=
-  Units.mk0 (q.1 : ℚ) (by exact_mod_cast q.2.ne_zero)
-
-/-- The underlying rational number of the prime generator unit is `q`. -/
-@[simp]
-theorem rationalPrimeGeneratorUnit_coe (q : Nat.Primes) :
-    (rationalPrimeGeneratorUnit q : ℚ) = q.1 :=
-  rfl
-
-/-- Reattaching the removed `q`-power to the rational `q`-unit recovers
-the original rational field unit. -/
-theorem rationalPrimeGeneratorUnit_zpow_mul_rationalPrimeUnit
-    (x : ℚˣ) (q : Nat.Primes) :
-    rationalPrimeGeneratorUnit q ^
-          padicValRat q.1 (x : ℚ) *
-        rationalPrimeUnit x q =
-      x := by
-  rw [rationalPrimeGeneratorUnit, rationalPrimeUnit,
-    ← mul_assoc, ← zpow_add]
-  simp
-
-/-- The source unit in the absolute-value completion represented by the
-finite component of a rational principal idele. -/
-noncomputable def rationalPrincipalFinitePlaceInput
-    (x : ℚˣ) (q : Nat.Primes) :
-    (HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q)).Completionˣ :=
-  (finitePlaceCompletionUnitsContinuousMulEquiv
-      (RayClass.rationalPrime q)).symm
-    (IdeleGroup.finiteComponent
-      (RayClass.rationalPrime q)
-      (IdeleGroup.principalIdele ℚ x))
-
-/-- The source unit represented by a principal finite component is the
-ordinary image of the rational field unit in the absolute-value
-completion. -/
-theorem rationalPrincipalFinitePlaceInput_eq_algebraMap
-    (x : ℚˣ) (q : Nat.Primes) :
-    rationalPrincipalFinitePlaceInput x q =
-      Units.map
-        (algebraMap ℚ
-          (HeightOneSpectrum.adicAbv ℚ
-            (RayClass.rationalPrime q)).Completion).toMonoidHom
-        x := by
-  apply Units.ext
-  let v : HeightOneSpectrum (𝓞 ℚ) :=
-    RayClass.rationalPrime q
-  let vQ := HeightOneSpectrum.adicAbv ℚ v
-  let component : (v.adicCompletion ℚ)ˣ :=
-    IdeleGroup.finiteComponent v
-      (IdeleGroup.principalIdele ℚ x)
-  apply (finitePlaceCompletionRingEquiv v).injective
-  change
-    finitePlaceCompletionRingEquiv v
-        (rationalPrincipalFinitePlaceInput x q : vQ.Completion) =
-      finitePlaceCompletionRingEquiv v
-        (algebraMap ℚ vQ.Completion (x : ℚ))
-  have hCompletion :
-      finitePlaceCompletionRingEquiv v
-          (rationalPrincipalFinitePlaceInput x q :
-            vQ.Completion) =
-        (component : v.adicCompletion ℚ) := by
-    have hUnits :=
-      congrArg Units.val
-        ((finitePlaceCompletionUnitsContinuousMulEquiv v).apply_symm_apply
-          component)
-    exact hUnits
-  have hComponent :
-      (component : v.adicCompletion ℚ) =
-        algebraMap ℚ (v.adicCompletion ℚ) (x : ℚ) := by
-    apply (Padic.adicCompletionEquiv (𝓞 ℚ) q).symm.injective
-    calc
-      (Padic.adicCompletionEquiv (𝓞 ℚ) q).symm
-          (component : v.adicCompletion ℚ) =
-        algebraMap ℚ ℚ_[q.1] (x : ℚ) := by
-          dsimp only [component]
-          rw [IdeleGroup.finiteComponent_principalIdele]
-          exact
-            (Padic.adicCompletionEquiv
-              (𝓞 ℚ) q).symm.commutes (x : ℚ)
-      _ = (Padic.adicCompletionEquiv (𝓞 ℚ) q).symm
-          (algebraMap ℚ (v.adicCompletion ℚ) (x : ℚ)) := by
-        symm
-        exact
-          (Padic.adicCompletionEquiv
-            (𝓞 ℚ) q).symm.commutes (x : ℚ)
-  calc
-    finitePlaceCompletionRingEquiv v
-        (rationalPrincipalFinitePlaceInput x q :
-          vQ.Completion) =
-      (component : v.adicCompletion ℚ) := hCompletion
-    _ = algebraMap ℚ (v.adicCompletion ℚ) (x : ℚ) := hComponent
-    _ = finitePlaceCompletionRingEquiv v
-        (algebraMap ℚ vQ.Completion (x : ℚ)) := by
-      symm
-      exact
-        (finitePlaceCompletionAlgEquiv (K := ℚ) v).commutes (x : ℚ)
-
-/-- The normalized local exponent of a rational principal finite
-component is the negative of the usual `q`-adic exponent.  The minus sign
-records the inverse-standard local reciprocity convention in which a
-prime element has normalized value `-1`. -/
-theorem rationalPrincipalFiniteComponent_valuationMap
-    (x : ℚˣ) (q : Nat.Primes) :
-    IsNonarchimedeanLocalField.valuationMap
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        (Additive.ofMul
-          ((finitePlaceCompletionUnitsContinuousMulEquiv
-            (RayClass.rationalPrime q)).symm
-            (IdeleGroup.finiteComponent
-              (RayClass.rationalPrime q)
-              (IdeleGroup.principalIdele ℚ x)))) =
-      -padicValRat q.1 (x : ℚ) := by
-  let F :=
-    (HeightOneSpectrum.adicAbv ℚ
-      (RayClass.rationalPrime q)).Completion
-  let embed : ℚˣ →* Fˣ :=
-    Units.map (algebraMap ℚ F).toMonoidHom
-  let primeUnit : Fˣ :=
-    rationalPrimeFinitePlaceFieldUnit q
-  let integralUnit : 𝒪[F]ˣ :=
-    rationalPrimeUnitFinitePlaceIntegerUnit x q
-  let unitPart : Fˣ :=
-    IsNonarchimedeanLocalField.integerUnitsToFieldUnits
-      F integralUnit
-  have hInput :
-      rationalPrincipalFinitePlaceInput x q =
-        embed x := by
-    exact rationalPrincipalFinitePlaceInput_eq_algebraMap x q
-  have hPrime :
-      embed (rationalPrimeGeneratorUnit q) =
-        primeUnit := by
-    apply Units.ext
-    exact
-      (rationalPrimeFinitePlaceInteger_coe q).symm
-  have hUnit :
-      embed (rationalPrimeUnit x q) =
-        unitPart := by
-    apply Units.ext
-    exact
-      (rationalPrimeUnitFinitePlaceIntegerUnit_coe
-        x q).symm
-  change
-    IsNonarchimedeanLocalField.valuationMap F
-        (Additive.ofMul
-          (rationalPrincipalFinitePlaceInput x q)) =
-      -padicValRat q.1 (x : ℚ)
-  rw [hInput]
-  conv_lhs =>
-    rw [← rationalPrimeGeneratorUnit_zpow_mul_rationalPrimeUnit
-      x q]
-  rw [map_mul, map_zpow, hPrime, hUnit,
-    IsNonarchimedeanLocalField.valuationMap_ofMul_mul,
-    IsNonarchimedeanLocalField.valuationMap_ofMul_zpow,
-    rationalPrimeFinitePlaceFieldUnit_valuationMap,
-    rationalPrimeUnitFinitePlaceField_valuationMap]
-  ring
-
-/-- The principal finite component of the rational prime itself has
-normalized local exponent `-1`. -/
-theorem rationalPrimePrincipalFiniteComponent_valuationMap
-    (q : Nat.Primes) :
-    IsNonarchimedeanLocalField.valuationMap
-        (HeightOneSpectrum.adicAbv ℚ
-          (RayClass.rationalPrime q)).Completion
-        (Additive.ofMul
-          ((finitePlaceCompletionUnitsContinuousMulEquiv
-            (RayClass.rationalPrime q)).symm
-            (IdeleGroup.finiteComponent
-              (RayClass.rationalPrime q)
-              (IdeleGroup.principalIdele ℚ
-                (rationalPrimeGeneratorUnit q))))) =
-      -1 := by
-  rw [rationalPrincipalFiniteComponent_valuationMap,
-    rationalPrimeGeneratorUnit_coe,
-    padicValRat.self q.2.one_lt]
-
-/-- A cyclotomic automorphism which raises the selected primitive root to
-the `q`-th power has cyclotomic character equal to the residue-prime unit. -/
-private theorem rationalCyclotomicLevel_galEquivZMod_eq_unitOfCoprime
-    (m : ℕ+) (q : Nat.Primes)
-    (hq : ¬ q.1 ∣ (m : ℕ))
-    (σ : KummerTheory.rationalCyclotomicLevel m ≃ₐ[ℚ]
-      KummerTheory.rationalCyclotomicLevel m)
-    (hσ :
-      σ (rationalCyclotomicLevelPrimitiveRoot m) =
-        rationalCyclotomicLevelPrimitiveRoot m ^ q.1) :
-    IsCyclotomicExtension.Rat.galEquivZMod
-        (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ =
-      ZMod.unitOfCoprime q.1
-        (q.2.coprime_iff_not_dvd.mpr hq) := by
-  let ζ := rationalCyclotomicLevelPrimitiveRoot m
-  have hζ : IsPrimitiveRoot ζ (m : ℕ) :=
-    rationalCyclotomicLevelPrimitiveRoot_isPrimitiveRoot m
-  have hCharacterRoot :
-      σ ζ =
-        ζ ^
-          (IsCyclotomicExtension.Rat.galEquivZMod
-            (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ).val.val :=
-    IsCyclotomicExtension.Rat.galEquivZMod_apply_of_pow_eq
-      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ
-      hζ.pow_eq_one
-  have hPowers :
-      ζ ^
-          (IsCyclotomicExtension.Rat.galEquivZMod
-            (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ).val.val =
-        ζ ^ q.1 :=
-    hCharacterRoot.symm.trans hσ
-  rw [(hζ.isOfFinOrder m.ne_zero).pow_inj_mod,
-    ← hζ.eq_orderOf,
-    ← ZMod.natCast_eq_natCast_iff',
-    ZMod.natCast_val] at hPowers
-  apply Units.ext
-  simpa using hPowers
-
-private abbrev rationalCyclotomicArtinPlace (q : Nat.Primes) :
-    HeightOneSpectrum (𝓞 ℚ) :=
-  RayClass.rationalPrime q
-
-private abbrev rationalCyclotomicArtinBaseAbv (q : Nat.Primes) :
-    AbsoluteValue ℚ ℝ :=
-  HeightOneSpectrum.adicAbv ℚ (rationalCyclotomicArtinPlace q)
-
-private abbrev rationalCyclotomicArtinLevel (m : ℕ+) :=
-  KummerTheory.rationalCyclotomicLevel m
-
-private abbrev rationalCyclotomicArtinExtension
-    (m : ℕ+) (q : Nat.Primes) :
-    AbsoluteValueExtension
-      (rationalCyclotomicArtinBaseAbv q)
-      (rationalCyclotomicArtinLevel m) :=
-  chosenFinitePlaceExtension
-    (L := rationalCyclotomicArtinLevel m)
-    (rationalCyclotomicArtinPlace q)
-
-private abbrev rationalCyclotomicArtinLocalizedField
-    (m : ℕ+) (q : Nat.Primes) :=
-  AlgebraicNumberTheory.Valuations.LocalizedCompletion
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q)
-
-@[reducible]
-noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
-    (m : ℕ+) (q : Nat.Primes) :
-    Algebra ℚ
-      (rationalCyclotomicArtinExtension m q).1.Completion :=
-  AbsoluteValue.extensionCompletionAlgebra
-    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
-
-@[reducible]
-noncomputable local instance rationalCyclotomicArtinExtensionSMul
-    (m : ℕ+) (q : Nat.Primes) :
-    SMul ℚ
-      (rationalCyclotomicArtinExtension m q).1.Completion :=
-  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
-
-@[reducible]
-noncomputable local instance
-    rationalCyclotomicArtinCompletionAlgebra
-    (m : ℕ+) (q : Nat.Primes) :
-    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinExtension m q).1.Completion :=
-  AbsoluteValue.completionAlgebra
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q).1
-    (rationalCyclotomicArtinExtension m q).2
-
-@[reducible]
-noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
-    (m : ℕ+) (q : Nat.Primes) :
-    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  finitePlaceLocalArtinLocalizedAlgebra
-    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
-    (rationalCyclotomicArtinPlace q)
-    (rationalCyclotomicArtinExtension m q)
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedGlobalAlgebra
-    (m : ℕ+) (q : Nat.Primes) :
-    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
-  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q)
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedGlobalSMul
-    (m : ℕ+) (q : Nat.Primes) :
-    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
-  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedScalarTower
-    (m : ℕ+) (q : Nat.Primes) :
-    IsScalarTower ℚ
-      (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinLocalizedField m q) := by
-  constructor
-  intro r x y
-  simp only [Algebra.smul_def, map_mul, eq_ratCast,
-    map_ratCast, mul_assoc]
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedFiniteDimensional
-    (m : ℕ+) (q : Nat.Primes) :
-    FiniteDimensional
-      (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  finitePlaceLocalArtinFiniteDimensional
-    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
-    (rationalCyclotomicArtinPlace q)
-    (rationalCyclotomicArtinExtension m q)
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIsAbelianGalois
-    (m : ℕ+) (q : Nat.Primes) :
-    IsAbelianGalois
-      (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  finitePlaceLocalArtinIsAbelianGalois
-    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
-    (rationalCyclotomicArtinPlace q)
-    (rationalCyclotomicArtinExtension m q)
-    (inferInstance :
-      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIsSeparable
-    (m : ℕ+) (q : Nat.Primes) :
-    Algebra.IsSeparable
-      (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIsCyclotomic
-    (m : ℕ+) (q : Nat.Primes) :
-    IsCyclotomicExtension {(m : ℕ)}
-      (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
-    m (rationalCyclotomicArtinPlace q)
-
-noncomputable local instance
-    rationalCyclotomicArtinExtensionFiniteDimensional
-    (m : ℕ+) (q : Nat.Primes) :
-    FiniteDimensional
-      (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinExtension m q).1.Completion :=
-  completionModuleFinite
-    (rationalCyclotomicArtinBaseAbv q)
-    (RayClass.adicAbv_isNontrivial
-      (rationalCyclotomicArtinPlace q))
-    (rationalCyclotomicArtinExtension m q)
-
-noncomputable local instance
-    rationalCyclotomicArtinExtensionContinuousSMul
-    (m : ℕ+) (q : Nat.Primes) :
-    ContinuousSMul
-      (rationalCyclotomicArtinBaseAbv q).Completion
-      (rationalCyclotomicArtinExtension m q).1.Completion :=
-  continuousSMul_of_algebraMap _ _
-    (AbsoluteValue.completionMap_isometry
-      (rationalCyclotomicArtinBaseAbv q)
-      (rationalCyclotomicArtinExtension m q).1
-      (rationalCyclotomicArtinExtension m q).2).continuous
-
-noncomputable local instance
-    rationalCyclotomicArtinExtensionLocallyCompact
-    (m : ℕ+) (q : Nat.Primes) :
-    LocallyCompactSpace
-      (rationalCyclotomicArtinExtension m q).1.Completion :=
-  LocallyCompactSpace.of_finiteDimensional_of_complete
-    (rationalCyclotomicArtinBaseAbv q).Completion
-    (rationalCyclotomicArtinExtension m q).1.Completion
-
-private noncomputable def
-    rationalCyclotomicArtinLocalizedEquivCompletion
-    (m : ℕ+) (q : Nat.Primes) :
-    rationalCyclotomicArtinLocalizedField m q ≃ᵢ
-      (rationalCyclotomicArtinExtension m q).1.Completion :=
-  { toEquiv :=
-      (localizedCompletionEquivCompletion
-        (rationalCyclotomicArtinBaseAbv q)
-        (RayClass.adicAbv_isNontrivial
-          (rationalCyclotomicArtinPlace q))
-        (rationalCyclotomicArtinExtension m q)).toEquiv
-    isometry_toFun :=
-      Isometry.of_dist_eq fun _ _ => rfl }
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedLocallyCompact
-    (m : ℕ+) (q : Nat.Primes) :
-    LocallyCompactSpace
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
-    inferInstance
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIsUltrametricDist
-    (m : ℕ+) (q : Nat.Primes) :
-    IsUltrametricDist
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  localizedCompletionIsUltrametricDist
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q)
-    (HeightOneSpectrum.isNonarchimedean_adicAbv
-      ℚ (rationalCyclotomicArtinPlace q))
-
-noncomputable local instance rationalCyclotomicArtinLocalizedValued
-    (m : ℕ+) (q : Nat.Primes) :
-    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
-  localizedCompletionFinitePlaceValued
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q)
-    (HeightOneSpectrum.isNonarchimedean_adicAbv
-      ℚ (rationalCyclotomicArtinPlace q))
-
-@[reducible]
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedValuativeRel
-    (m : ℕ+) (q : Nat.Primes) :
-    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
-  localizedCompletionFinitePlaceValuativeRel
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q)
-    (HeightOneSpectrum.isNonarchimedean_adicAbv
-      ℚ (rationalCyclotomicArtinPlace q))
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedValuationCompatible
-    (m : ℕ+) (q : Nat.Primes) :
-    (Valued.v : Valuation
-      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
-  Valuation.Compatible.ofValuation _
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedValuationHasExtension
-    (m : ℕ+) (q : Nat.Primes) :
-    Valuation.HasExtension
-      (ValuativeRel.valuation
-        (rationalCyclotomicArtinBaseAbv q).Completion)
-      (ValuativeRel.valuation
-        (rationalCyclotomicArtinLocalizedField m q)) :=
-  localizedCompletionValuationHasExtension
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q)
-    (HeightOneSpectrum.isNonarchimedean_adicAbv
-      ℚ (rationalCyclotomicArtinPlace q))
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedValuationIsNontrivial
-    (m : ℕ+) (q : Nat.Primes) :
-    (ValuativeRel.valuation
-      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
-  Valuation.IsNontrivial.of_hasExtension
-    (ValuativeRel.valuation
-      (rationalCyclotomicArtinBaseAbv q).Completion)
-    (ValuativeRel.valuation
-      (rationalCyclotomicArtinLocalizedField m q))
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
-    (m : ℕ+) (q : Nat.Primes) :
-    ValuativeRel.IsNontrivial
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  (ValuativeRel.isNontrivial_iff_isNontrivial
-    (ValuativeRel.valuation
-      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIsValuativeTopology
-    (m : ℕ+) (q : Nat.Primes) :
-    IsValuativeTopology
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  isValuativeTopology_of_valued_ofValuation
-    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
-    (m : ℕ+) (q : Nat.Primes) :
-    IsNonarchimedeanLocalField
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  { toIsValuativeTopology := inferInstance
-    toLocallyCompactSpace := inferInstance
-    toIsNontrivial := inferInstance }
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIntegerAlgebra
-    (m : ℕ+) (q : Nat.Primes) :
-    Algebra
-      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  Algebra.ofSubsemiring
-    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIsIntegralClosure
-    (m : ℕ+) (q : Nat.Primes) :
-    IsIntegralClosure
-      𝒪[rationalCyclotomicArtinLocalizedField m q]
-      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
-      (rationalCyclotomicArtinLocalizedField m q) :=
-  localizedCompletionIsIntegralClosureWithExtension
-    (rationalCyclotomicArtinBaseAbv q)
-    (rationalCyclotomicArtinExtension m q)
-    (RayClass.adicAbv_isNontrivial
-      (rationalCyclotomicArtinPlace q))
-    (HeightOneSpectrum.isNonarchimedean_adicAbv
-      ℚ (rationalCyclotomicArtinPlace q))
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedIntegerModuleFinite
-    (m : ℕ+) (q : Nat.Primes) :
-    Module.Finite
-      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
-      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
-  integerRing_moduleFinite_of_isIntegralClosure
-    (rationalCyclotomicArtinBaseAbv q).Completion
-    (rationalCyclotomicArtinLocalizedField m q)
-
-section RationalCyclotomicPrincipalPrime
-
-/-! ## Ramified prime-power transport
-
-This section reuses the canonical finite-place Artin tower above.  In
-particular, it introduces no parallel completion/localization instance tower. -/
-
-noncomputable local instance
-    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
-    (p : Nat.Primes) (n : ℕ) :
-    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
-      (KummerTheory.rationalCyclotomicLevel
-        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
-  change
-    IsCyclotomicExtension
-      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
-      (KummerTheory.rationalCyclotomicLevel
-        (rationalCyclotomicPrincipalPrimeModulus p n))
-  exact
-    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
-      (rationalCyclotomicPrincipalPrimeModulus p n)
-
-private abbrev rationalCyclotomicPrincipalPrimePlace
-    (p : Nat.Primes) : HeightOneSpectrum (𝓞 ℚ) :=
-  rationalCyclotomicArtinPlace p
-
-private abbrev rationalCyclotomicPrincipalPrimeLevel
-    (m : ℕ+) :=
-  rationalCyclotomicArtinLevel m
-
-private abbrev rationalCyclotomicPrincipalPrimeExtension
-    (m : ℕ+) (p : Nat.Primes) :=
-  rationalCyclotomicArtinExtension m p
-
-/-- The `ℚ_[p]`-algebra structure on the localized cyclotomic completion,
-transported through the canonical comparison with the `p`-adic completion. -/
-@[reducible]
-noncomputable def
-    rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra
-    (m : ℕ+) (p : Nat.Primes) :
-    Algebra ℚ_[p.1]
-      (rationalCyclotomicLocalizedCompletion m
-        (RayClass.rationalPrime p)) :=
-  ((algebraMap
-      (HeightOneSpectrum.adicAbv ℚ
-        (RayClass.rationalPrime p)).Completion
-      (rationalCyclotomicLocalizedCompletion m
-        (RayClass.rationalPrime p))).comp
-    (rationalFinitePlaceCompletionRingEquivPadic p).symm.toRingHom).toAlgebra
-
-@[reducible]
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedPadicAlgebra
-    (m : ℕ+) (p : Nat.Primes) :
-    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
-  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
-
-private noncomputable def rationalFinitePlaceCompletionAlgEquivPadic
-    (p : Nat.Primes) :
-    (rationalCyclotomicArtinBaseAbv p).Completion ≃ₐ[ℚ] ℚ_[p.1] :=
-  AlgEquiv.ofRingEquiv
-    (f := rationalFinitePlaceCompletionRingEquivPadic p)
-    (rationalFinitePlaceCompletionRingEquivPadic_algebraMap p)
-
-noncomputable local instance
-    rationalCyclotomicArtinLocalizedPadicScalarTower
-    (m : ℕ+) (p : Nat.Primes) :
-    IsScalarTower ℚ ℚ_[p.1]
-      (rationalCyclotomicArtinLocalizedField m p) := by
-  constructor
-  intro r x y
-  simp only [Algebra.smul_def, map_mul, eq_ratCast,
-    map_ratCast, mul_assoc]
-
-private theorem rationalCyclotomicArtin_padic_algebraMap
-    (m : ℕ+) (p : Nat.Primes) :
-    algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
-        (rationalCyclotomicArtinLocalizedField m p) =
-      (algebraMap ℚ_[p.1]
-        (rationalCyclotomicArtinLocalizedField m p)) ∘
-        (rationalFinitePlaceCompletionAlgEquivPadic p) := by
-  funext a
-  change
-    algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
-        (rationalCyclotomicArtinLocalizedField m p) a =
-      algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
-        (rationalCyclotomicArtinLocalizedField m p)
-        ((rationalFinitePlaceCompletionRingEquivPadic p).symm
-          (rationalFinitePlaceCompletionRingEquivPadic p a))
-  exact
-    congrArg
-      (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
-        (rationalCyclotomicArtinLocalizedField m p))
-      ((rationalFinitePlaceCompletionRingEquivPadic p).symm_apply_apply a).symm
-
-private theorem rationalCyclotomicArtin_algebraAdjoin_restrictScalars
-    (m : ℕ+) (p : Nat.Primes) :
-    (Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
-      ({rationalCyclotomicLocalizedPrimitiveRoot m
-        (rationalCyclotomicArtinPlace p)} :
-        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
-      (Algebra.adjoin ℚ_[p.1]
-        ({rationalCyclotomicLocalizedPrimitiveRoot m
-          (rationalCyclotomicArtinPlace p)} :
-          Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ := by
-  exact
-    Algebra.restrictScalars_adjoin_of_algEquiv
-      (E := rationalCyclotomicArtinLocalizedField m p)
-      (rationalFinitePlaceCompletionAlgEquivPadic p)
-      (rationalCyclotomicArtin_padic_algebraMap m p)
-      ({rationalCyclotomicLocalizedPrimitiveRoot m
-        (rationalCyclotomicArtinPlace p)} :
-        Set (rationalCyclotomicArtinLocalizedField m p))
-
-private theorem
-    rationalCyclotomicArtin_baseAlgebraAdjoin_restrict_eq_top
-    (m : ℕ+) (p : Nat.Primes) :
-    (Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
-      ({rationalCyclotomicLocalizedPrimitiveRoot m
-        (rationalCyclotomicArtinPlace p)} :
-        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
-      (⊤ : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
-      (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ := by
-  have hRoot : IsPrimitiveRoot
-      (show rationalCyclotomicArtinLocalizedField m p from
-        rationalCyclotomicLocalizedPrimitiveRoot m
-          (rationalCyclotomicArtinPlace p))
-      (m : ℕ) :=
-    rationalCyclotomicLocalizedPrimitiveRoot_isPrimitiveRoot
-      m (rationalCyclotomicArtinPlace p)
-  have hTop :
-      Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
-        ({rationalCyclotomicLocalizedPrimitiveRoot m
-          (rationalCyclotomicArtinPlace p)} :
-          Set (rationalCyclotomicArtinLocalizedField m p)) = ⊤ :=
-    IsCyclotomicExtension.adjoin_primitive_root_eq_top
-      (A := (rationalCyclotomicArtinBaseAbv p).Completion)
-      (B := rationalCyclotomicArtinLocalizedField m p) hRoot
-  exact congrArg
-    (fun A : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
-      (rationalCyclotomicArtinLocalizedField m p) => A.restrictScalars ℚ)
-    hTop
-
-private theorem rationalCyclotomicArtin_restrictScalars_top_base_eq_padic
-    (m : ℕ+) (p : Nat.Primes) :
-    (⊤ : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
-        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ =
-      (⊤ : Subalgebra ℚ_[p.1]
-        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ :=
-  (Subalgebra.restrictScalars_top ℚ).trans
-    (Subalgebra.restrictScalars_top ℚ).symm
-
-private theorem
-    rationalCyclotomicArtin_padicAlgebraAdjoin_restrict_eq_top
-    (m : ℕ+) (p : Nat.Primes) :
-    (Algebra.adjoin ℚ_[p.1]
-      ({rationalCyclotomicLocalizedPrimitiveRoot m
-        (rationalCyclotomicArtinPlace p)} :
-        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
-      (⊤ : Subalgebra ℚ_[p.1]
-        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ := by
-  exact
-    (rationalCyclotomicArtin_algebraAdjoin_restrictScalars m p).symm.trans
-      ((rationalCyclotomicArtin_baseAlgebraAdjoin_restrict_eq_top m p).trans
-        (rationalCyclotomicArtin_restrictScalars_top_base_eq_padic m p))
+noncomputable section
 
-/-- The finite-dimensional instance for the standard multiplicative level,
-named once so all consumers use the same proof term. -/
-theorem rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
-    (p : Nat.Primes) (n : ℕ) :
-    FiniteDimensional ℚ_[p.1]
-      (standardLubinTateLevelField
-        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n) :=
-  standardLubinTateLevelField_finiteDimensional
-    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
+namespace GlobalClassFieldTheory
+namespace Reciprocity
 
 attribute [local instance]
-  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
-
-noncomputable local instance
-    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
-    (p : Nat.Primes) (n : ℕ) :
-    IsAbelianGalois ℚ_[p.1]
-      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
-  standardLubinTateLevelField_isAbelianGalois
-    (padicLocalField p.1)
-    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
-
-@[reducible]
-noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
-    (p : Nat.Primes) :
-    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
-  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
-
-/-- The genuine multiplicative Lubin--Tate level is generated by its
-primitive `p ^ (n + 1)`-st root of unity. -/
-theorem padicMultiplicativePrimitiveRoot_adjoin_eq_top
-    (p : ℕ) [Fact p.Prime] (n : ℕ) :
-    let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
-    let T := standardLubinTateLevelField hπ n
-    Algebra.adjoin ℚ_[p]
-        ({padicMultiplicativePrimitiveRoot p n} : Set T) =
-      ⊤ := by
-  let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
-  let T := standardLubinTateLevelField hπ n
-  let ζ : T := padicMultiplicativePrimitiveRoot p n
-  let m := p ^ (n + 1)
-  let : NeZero m :=
-    ⟨pow_ne_zero _ (Fact.out : Nat.Prime p).ne_zero⟩
-  let : FiniteDimensional ℚ_[p] T :=
-    standardLubinTateLevelField_finiteDimensional hπ n
-  have hζ : IsPrimitiveRoot ζ m := by
-    simpa only [ζ, m] using
-      padicMultiplicativePrimitiveRoot_isPrimitiveRoot p n
-  let A : IntermediateField ℚ_[p] T :=
-    IntermediateField.adjoin ℚ_[p] {ζ}
-  let : IsCyclotomicExtension {m} ℚ_[p] A :=
-    hζ.intermediateField_adjoin_isCyclotomicExtension ℚ_[p]
-  have hAfin :
-      Module.finrank ℚ_[p] A = Nat.totient m := by
-    exact
-      IsCyclotomicExtension.finrank A
-        (by
-          simpa only [m] using
-            padicCyclotomicPolynomial_irreducible_prime_pow_succ
-              p n)
-  have hTfin :
-      Module.finrank ℚ_[p] T = Nat.totient m := by
-    rw [standardLubinTateLevelField_finrank hπ n]
-    have hcard :
-        Nat.card (padicLocalField p).residueField = p := by
-      simpa [padicLocalField] using
-        padicCompleteDVF_residueField_card p
-    rw [hcard, Nat.totient_prime_pow
-      (Fact.out : Nat.Prime p) (Nat.succ_pos n)]
-    simp [Nat.mul_comm]
-  have hAeq : A = ⊤ := by
-    apply IntermediateField.eq_of_le_of_finrank_eq le_top
-    simpa using hAfin.trans hTfin.symm
-  calc
-    Algebra.adjoin ℚ_[p] {ζ} = A.toSubalgebra := by
-      exact
-        (IntermediateField.adjoin_toSubalgebra
-          ({ζ} : Set T)).symm
-    _ = (⊤ : IntermediateField ℚ_[p] T).toSubalgebra :=
-      congrArg IntermediateField.toSubalgebra hAeq
-    _ = ⊤ := rfl
-
-/-- The standard multiplicative Lubin--Tate level is the actual
-`p ^ (n + 1)`-cyclotomic extension of `ℚ_p`. -/
-theorem padicMultiplicativeLevel_isCyclotomicExtension
-    (p : ℕ) [Fact p.Prime] (n : ℕ) :
-    let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
-    let T := standardLubinTateLevelField hπ n
-    IsCyclotomicExtension {p ^ (n + 1)} ℚ_[p] T := by
-  let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
-  let T := standardLubinTateLevelField hπ n
-  exact
-    padic_isCyclotomicExtension_of_primitiveRoot_adjoin_eq_top
-      (padicMultiplicativePrimitiveRoot p n)
-      (padicMultiplicativePrimitiveRoot_isPrimitiveRoot p n)
-      (padicMultiplicativePrimitiveRoot_adjoin_eq_top p n)
-
-private theorem
-    rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_isPrimitiveRoot
-    (p : Nat.Primes) (n : ℕ) :
-    IsPrimitiveRoot
-      (rationalCyclotomicLocalizedPrimitiveRoot
-        (rationalCyclotomicPrincipalPrimeModulus p n)
-        (RayClass.rationalPrime p))
-      (p.1 ^ (n + 1)) := by
-  change
-    IsPrimitiveRoot
-      (rationalCyclotomicLocalizedPrimitiveRoot
-        (rationalCyclotomicPrincipalPrimeModulus p n)
-        (RayClass.rationalPrime p))
-      (rationalCyclotomicPrincipalPrimeModulus p n : ℕ)
-  exact
-    rationalCyclotomicLocalizedPrimitiveRoot_isPrimitiveRoot
-      (rationalCyclotomicPrincipalPrimeModulus p n)
-      (RayClass.rationalPrime p)
-
-private theorem
-    rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_adjoin_eq_top
-    (p : Nat.Primes) (n : ℕ) :
-    Algebra.adjoin ℚ_[p.1]
-        ({rationalCyclotomicLocalizedPrimitiveRoot
-          (rationalCyclotomicPrincipalPrimeModulus p n)
-          (rationalCyclotomicArtinPlace p)} :
-          Set (rationalCyclotomicArtinLocalizedField
-            (rationalCyclotomicPrincipalPrimeModulus p n) p)) =
-      ⊤ := by
-  exact
-    (Subalgebra.restrictScalars_injective ℚ)
-      (rationalCyclotomicArtin_padicAlgebraAdjoin_restrict_eq_top
-        (rationalCyclotomicPrincipalPrimeModulus p n) p)
-
-private theorem
-    rationalCyclotomicPrincipalPrimeLocalizedLevel_isCyclotomicExtension
-    (p : Nat.Primes) (n : ℕ) :
-    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ_[p.1]
-      (rationalCyclotomicArtinLocalizedField
-        (rationalCyclotomicPrincipalPrimeModulus p n) p) := by
-  exact
-    padic_isCyclotomicExtension_of_primitiveRoot_adjoin_eq_top
-      (rationalCyclotomicLocalizedPrimitiveRoot
-        (rationalCyclotomicPrincipalPrimeModulus p n)
-        (rationalCyclotomicArtinPlace p))
-      (rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_isPrimitiveRoot
-        p n)
-      (rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_adjoin_eq_top
-        p n)
-
-/-- The chosen localized global cyclotomic level, transported over the
-completion equivalence, is the standard multiplicative Lubin--Tate level. -/
-noncomputable def rationalCyclotomicLocalizedCompletionPadicAlgEquiv
-    (p : Nat.Primes) (n : ℕ) :
-    rationalCyclotomicArtinLocalizedField
-        (rationalCyclotomicPrincipalPrimeModulus p n) p ≃ₐ[ℚ_[p.1]]
-      RationalCyclotomicPrincipalPrimePadicLevel p n := by
-  letI : IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ_[p.1]
-      (rationalCyclotomicArtinLocalizedField
-        (rationalCyclotomicPrincipalPrimeModulus p n) p) :=
-    rationalCyclotomicPrincipalPrimeLocalizedLevel_isCyclotomicExtension p n
-  letI : IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ_[p.1]
-      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
-    padicMultiplicativeLevel_isCyclotomicExtension p.1 n
-  exact
-    IsCyclotomicExtension.algEquiv
-      {p.1 ^ (n + 1)} ℚ_[p.1]
-      (rationalCyclotomicArtinLocalizedField
-        (rationalCyclotomicPrincipalPrimeModulus p n) p)
-      (RationalCyclotomicPrincipalPrimePadicLevel p n)
-
-/-! ## The ramified principal finite-place factor -/
-
-private theorem
-    rationalCyclotomicPrincipalPrime_galEquivZMod_eq_of_action
-    (p : Nat.Primes) (n : ℕ)
-    (sigma : Gal(
-      rationalCyclotomicPrincipalPrimeLevel
-        (rationalCyclotomicPrincipalPrimeModulus p n) / ℚ))
-    (a : (ZMod (p.1 ^ (n + 1)))ˣ)
-    (haction :
-      sigma (rationalCyclotomicLevelPrimitiveRoot
-          (rationalCyclotomicPrincipalPrimeModulus p n)) =
-        rationalCyclotomicLevelPrimitiveRoot
-            (rationalCyclotomicPrincipalPrimeModulus p n) ^ a.val.val) :
-    IsCyclotomicExtension.Rat.galEquivZMod
-        (p.1 ^ (n + 1))
-        (rationalCyclotomicPrincipalPrimeLevel
-          (rationalCyclotomicPrincipalPrimeModulus p n))
-        (hK :=
-          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
-            (rationalCyclotomicPrincipalPrimeModulus p n)) sigma =
-      a := by
-  let m := rationalCyclotomicPrincipalPrimeModulus p n
-  let L := rationalCyclotomicPrincipalPrimeLevel m
-  let zeta : L := rationalCyclotomicLevelPrimitiveRoot m
-  have hzeta : IsPrimitiveRoot zeta (p.1 ^ (n + 1)) := by
-    change
-      IsPrimitiveRoot
-        (rationalCyclotomicLevelPrimitiveRoot m) (m : ℕ)
-    exact rationalCyclotomicLevelPrimitiveRoot_isPrimitiveRoot m
-  change
-    IsCyclotomicExtension.Rat.galEquivZMod
-        (p.1 ^ (n + 1)) L
-        (hK :=
-          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension m)
-        sigma = a
-  let c :=
-    IsCyclotomicExtension.Rat.galEquivZMod
-      (p.1 ^ (n + 1)) L
-      (hK :=
-        KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension m)
-      sigma
-  have hc :
-      sigma zeta = zeta ^ c.val.val :=
-    IsCyclotomicExtension.Rat.galEquivZMod_apply_of_pow_eq
-      (hK :=
-        KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension m)
-      (p.1 ^ (n + 1)) L sigma hzeta.pow_eq_one
-  have hpowers :
-      zeta ^ c.val.val = zeta ^ a.val.val :=
-    hc.symm.trans haction
-  rw [(hzeta.isOfFinOrder m.ne_zero).pow_inj_mod,
-    ← hzeta.eq_orderOf,
-    ← ZMod.natCast_eq_natCast_iff'] at hpowers
-  change
-      (c.val.val : ZMod (p.1 ^ (n + 1))) =
-        (a.val.val : ZMod (p.1 ^ (n + 1))) at hpowers
-  have hValues : c.val = a.val := by
-    calc
-      c.val = (c.val.val : ZMod (p.1 ^ (n + 1))) :=
-        (ZMod.natCast_zmod_val c.val).symm
-      _ = (a.val.val : ZMod (p.1 ^ (n + 1))) := hpowers
-      _ = a.val := ZMod.natCast_zmod_val a.val
-  change c = a
-  apply Units.ext
-  exact hValues
-
-/-- The chosen finite-place Artin map factors through any extension identified
-with the chosen one. -/
-theorem chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq
-    {K L : Type}
-    [Field K] [NumberField K]
-    [Field L] [Algebra K L]
-    [FiniteDimensional K L] [IsAbelianGalois K L]
-    (v : HeightOneSpectrum (𝓞 K))
-    (w : AbsoluteValueExtension
-      (HeightOneSpectrum.adicAbv K v) L)
-    (hw : chosenFinitePlaceExtension (L := L) v = w)
-    (x : (v.adicCompletion K)ˣ) :
-    chosenFinitePlaceArtinMonoidHom
-        (K := K) (L := L) v x =
-      finitePlaceLocalToGlobalMonoidHom
-        (K := K) (L := L) v w
-        (finitePlaceLocalArtinMonoidHom
-          (K := K) (L := L) v w x) := by
-  subst w
-  exact
-    congrArg
-      (fun f : (v.adicCompletion K)ˣ →* (L ≃ₐ[K] L) => f x)
-      (finitePlaceArtinMonoidHomOfExtension_factor
-        (K := K) (L := L) v
-        (chosenFinitePlaceExtension (L := L) v))
-
-private theorem
-    chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq_at
-    {K L : Type}
-    [Field K] [NumberField K]
-    [Field L] [Algebra K L]
-    [FiniteDimensional K L] [IsAbelianGalois K L]
-    (v : HeightOneSpectrum (𝓞 K))
-    (w : AbsoluteValueExtension
-      (HeightOneSpectrum.adicAbv K v) L)
-    (hw : chosenFinitePlaceExtension (L := L) v = w)
-    (x : (v.adicCompletion K)ˣ) (z : L) :
-    chosenFinitePlaceArtinMonoidHom
-        (K := K) (L := L) v x z =
-      finitePlaceLocalToGlobalMonoidHom
-        (K := K) (L := L) v w
-        (finitePlaceLocalArtinMonoidHom
-          (K := K) (L := L) v w x) z := by
-  exact
-    congrArg (fun sigma : Gal(L / K) => sigma z)
-      (chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq
-        (K := K) (L := L) v w hw x)
-
-private theorem finitePlaceLocalToGlobalMonoidHom_apply_pow_of_localized_action
-    {K L : Type}
-    [Field K] [NumberField K]
-    [Field L] [Algebra K L]
-    [FiniteDimensional K L] [IsAbelianGalois K L]
-    (v : HeightOneSpectrum (𝓞 K))
-    (w : AbsoluteValueExtension
-      (HeightOneSpectrum.adicAbv K v) L)
-    (sigma :
-      let vK := HeightOneSpectrum.adicAbv K v
-      let E := LocalizedCompletion vK w
-      letI : Algebra vK.Completion E :=
-        finitePlaceLocalArtinLocalizedAlgebra v w
-      Gal(E / vK.Completion))
-    (z : L)
-    (zLocal :
-      let vK := HeightOneSpectrum.adicAbv K v
-      LocalizedCompletion vK w)
-    (e : ℕ)
-    (hLocalization :
-      let vK := HeightOneSpectrum.adicAbv K v
-      let E := LocalizedCompletion vK w
-      letI : Algebra vK.Completion E :=
-        finitePlaceLocalArtinLocalizedAlgebra v w
-      let eLoc : L →+* E :=
-        AbsoluteValue.toAlgebraicLocalization vK w.1 w.2
-      eLoc z = zLocal)
-    (hlocal : sigma zLocal = zLocal ^ e) :
-    finitePlaceLocalToGlobalMonoidHom
-        (K := K) (L := L) v w sigma z = z ^ e := by
-  let vK := HeightOneSpectrum.adicAbv K v
-  let hvK : vK.IsNontrivial := RayClass.adicAbv_isNontrivial v
-  let E := LocalizedCompletion vK w
-  let : Algebra vK.Completion E :=
-    finitePlaceLocalArtinLocalizedAlgebra v w
-  let eD : absoluteValueDecompositionGroup K w.1 ≃* Gal(E / vK.Completion) :=
-    decompositionGroupEquivAlgebraicLocalizationAut vK hvK w
-  let eLoc : L →+* E :=
-    AbsoluteValue.toAlgebraicLocalization vK w.1 w.2
-  let delta : absoluteValueDecompositionGroup K w.1 := eD.symm sigma
-  have hDecomposition : eD delta = sigma := eD.apply_symm_apply sigma
-  apply eLoc.injective
-  calc
-    eLoc (finitePlaceLocalToGlobalMonoidHom
-        (K := K) (L := L) v w sigma z) =
-        eD delta (eLoc z) :=
-      (localizationRamificationGroups_decompositionGroupEquiv_toLocalization
-        vK hvK w delta z).symm
-    _ = sigma (eLoc z) :=
-      congrArg (fun tau : Gal(E / vK.Completion) => tau (eLoc z))
-        hDecomposition
-    _ = sigma zLocal :=
-      congrArg (fun y : E => sigma y) hLocalization
-    _ = zLocal ^ e := hlocal
-    _ = (eLoc z) ^ e :=
-      congrArg (fun y : E => y ^ e) hLocalization.symm
-    _ = eLoc (z ^ e) := (map_pow eLoc z e).symm
-
-private theorem map_primitiveRoot_eq_pow_of_eq_pow
-    {M : Type} [CommRing M] [IsDomain M]
-    (f : M →* M) (zeta rho : M) (order exponent : ℕ)
-    [NeZero order]
-    (hzeta : IsPrimitiveRoot zeta order)
-    (hrho : IsPrimitiveRoot rho order)
-    (hf : f zeta = zeta ^ exponent) :
-    f rho = rho ^ exponent := by
-  obtain ⟨j, -, hj⟩ :=
-    hzeta.eq_pow_of_pow_eq_one hrho.pow_eq_one
-  calc
-    f rho = f (zeta ^ j) := congrArg f hj.symm
-    _ = (f zeta) ^ j := map_pow f zeta j
-    _ = (zeta ^ exponent) ^ j := congrArg (fun z => z ^ j) hf
-    _ = zeta ^ (exponent * j) := (pow_mul zeta exponent j).symm
-    _ = zeta ^ (j * exponent) :=
-      congrArg (fun e : ℕ => zeta ^ e) (Nat.mul_comm exponent j)
-    _ = (zeta ^ j) ^ exponent := pow_mul zeta j exponent
-    _ = rho ^ exponent := congrArg (fun z => z ^ exponent) hj
-
-private theorem finitePlaceLocalArtinMonoidHom_apply_semilinear
-    {K L K' L' : Type}
-    [Field K] [NumberField K]
-    [Field L] [Algebra K L]
-    [hKLfinite : FiniteDimensional K L] [IsAbelianGalois K L]
-    [Field K'] [ValuativeRel K'] [TopologicalSpace K']
-    [IsNonarchimedeanLocalField K']
-    [Field L'] [Algebra K' L']
-    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
-    (v : HeightOneSpectrum (𝓞 K))
-    (w : AbsoluteValueExtension
-      (HeightOneSpectrum.adicAbv K v) L)
-    (eK : (HeightOneSpectrum.adicAbv K v).Completion ≃+* K')
-    (eL : LocalizedCompletion
-      (HeightOneSpectrum.adicAbv K v) w ≃+* L')
-    (hcomm : ∀ y : (HeightOneSpectrum.adicAbv K v).Completion,
-      eL (@algebraMap
-        (HeightOneSpectrum.adicAbv K v).Completion
-        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
-        _ _ (finitePlaceLocalArtinLocalizedAlgebra v w) y) =
-        algebraMap K' L' (eK y))
-    (hExt : SemilinearValuationCompatible
-      (HeightOneSpectrum.adicAbv K v).Completion K' eK)
-    (x : (v.adicCompletion K)ˣ)
-    (z : LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w) :
-    eL (finitePlaceLocalArtinMonoidHom
-        (K := K) (L := L) v w x z) =
-      LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
-        (Units.map eK.toMonoidHom
-          (finitePlaceLocalArtinInput v x)) (eL z) := by
-  calc
-    eL (finitePlaceLocalArtinMonoidHom
-        (K := K) (L := L) v w x z) =
-      eL ((@LocalClassFieldTheory.abelianLocalArtinMonoidHom
-        (HeightOneSpectrum.adicAbv K v).Completion
-        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
-        (inferInstance : Field
-          (HeightOneSpectrum.adicAbv K v).Completion)
-        (inferInstance : Field
-          (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w))
-        (finitePlaceLocalArtinLocalizedAlgebra v w)
-        (finitePlaceLocalArtinCompletionValuativeRel v)
-        (inferInstance : TopologicalSpace
-          (HeightOneSpectrum.adicAbv K v).Completion)
-        (finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v)
-        (finitePlaceLocalArtinFiniteDimensional v w)
-        (finitePlaceLocalArtinIsAbelianGalois v w hKLfinite)
-        (finitePlaceLocalArtinInput v x)) z) :=
-      congrArg eL
-        (finitePlaceLocalArtinMonoidHom_apply_normalized_at
-          (K := K) (L := L) v w x z)
-    _ = LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
-          (Units.map eK.toMonoidHom
-            (finitePlaceLocalArtinInput v x)) (eL z) :=
-      @abelianLocalArtinMonoidHom_semilinear_action
-        (HeightOneSpectrum.adicAbv K v).Completion K'
-        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w) L'
-        (inferInstance : Field
-          (HeightOneSpectrum.adicAbv K v).Completion)
-        (finitePlaceLocalArtinCompletionValuativeRel v)
-        (inferInstance : TopologicalSpace
-          (HeightOneSpectrum.adicAbv K v).Completion)
-        (finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v)
-        (inferInstance : Field K')
-        (inferInstance : ValuativeRel K')
-        (inferInstance : TopologicalSpace K')
-        (inferInstance : IsNonarchimedeanLocalField K')
-        (inferInstance : Field
-          (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w))
-        (inferInstance : Field L')
-        (finitePlaceLocalArtinLocalizedAlgebra v w)
-        (inferInstance : Algebra K' L')
-        (finitePlaceLocalArtinFiniteDimensional v w)
-        (finitePlaceLocalArtinIsAbelianGalois v w hKLfinite)
-        (inferInstance : FiniteDimensional K' L')
-        (inferInstance : IsAbelianGalois K' L')
-        eK eL hcomm hExt (finitePlaceLocalArtinInput v x) z
-
-/-- If a semilinearly identified target local Artin value is trivial, then the
-corresponding global finite-place Artin value is trivial.  This generic bridge
-keeps concrete completion and localization instance towers out of downstream
-proof terms. -/
-theorem finitePlaceArtinMonoidHomOfExtension_eq_one_of_semilinear
-    {K L K' L' : Type}
-    [Field K] [NumberField K]
-    [Field L] [Algebra K L]
-    [hKLfinite : FiniteDimensional K L] [IsAbelianGalois K L]
-    [Field K'] [ValuativeRel K'] [TopologicalSpace K']
-    [IsNonarchimedeanLocalField K']
-    [Field L'] [Algebra K' L']
-    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
-    (v : HeightOneSpectrum (𝓞 K))
-    (w : AbsoluteValueExtension
-      (HeightOneSpectrum.adicAbv K v) L)
-    (eK : (HeightOneSpectrum.adicAbv K v).Completion ≃+* K')
-    (eL : LocalizedCompletion
-      (HeightOneSpectrum.adicAbv K v) w ≃+* L')
-    (hcomm : ∀ y : (HeightOneSpectrum.adicAbv K v).Completion,
-      eL (@algebraMap
-        (HeightOneSpectrum.adicAbv K v).Completion
-        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
-        _ _ (finitePlaceLocalArtinLocalizedAlgebra v w) y) =
-        algebraMap K' L' (eK y))
-    (hExt : SemilinearValuationCompatible
-      (HeightOneSpectrum.adicAbv K v).Completion K' eK)
-    (x : (v.adicCompletion K)ˣ)
-    (htrivial :
-      LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
-          (Units.map eK.toMonoidHom
-            (finitePlaceLocalArtinInput v x)) = 1) :
-    finitePlaceArtinMonoidHomOfExtension
-        (K := K) (L := L) v w x = 1 := by
-  rw [finitePlaceArtinMonoidHomOfExtension_factor,
-    MonoidHom.comp_apply]
-  have hlocal :
-      finitePlaceLocalArtinMonoidHom
-          (K := K) (L := L) v w x = 1 := by
-    rw [finitePlaceLocalArtinMonoidHom_apply_normalized]
-    exact
-      @abelianLocalArtinMonoidHom_eq_one_of_semilinear
-        (HeightOneSpectrum.adicAbv K v).Completion K'
-        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w) L'
-        (inferInstance : Field
-          (HeightOneSpectrum.adicAbv K v).Completion)
-        (finitePlaceLocalArtinCompletionValuativeRel v)
-        (inferInstance : TopologicalSpace
-          (HeightOneSpectrum.adicAbv K v).Completion)
-        (finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v)
-        (inferInstance : Field K')
-        (inferInstance : ValuativeRel K')
-        (inferInstance : TopologicalSpace K')
-        (inferInstance : IsNonarchimedeanLocalField K')
-        (inferInstance : Field
-          (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w))
-        (inferInstance : Field L')
-        (finitePlaceLocalArtinLocalizedAlgebra v w)
-        (inferInstance : Algebra K' L')
-        (finitePlaceLocalArtinFiniteDimensional v w)
-        (finitePlaceLocalArtinIsAbelianGalois v w hKLfinite)
-        (inferInstance : FiniteDimensional K' L')
-        (inferInstance : IsAbelianGalois K' L')
-        eK eL hcomm hExt (finitePlaceLocalArtinInput v x) htrivial
-  rw [hlocal, map_one]
-
-private noncomputable def rationalCyclotomicPrincipalPrimeLocalizedRoot
-    (p : Nat.Primes) (n : ℕ) :
-    rationalCyclotomicArtinLocalizedField
-      (rationalCyclotomicPrincipalPrimeModulus p n) p :=
-  rationalCyclotomicLocalizedPrimitiveRoot
-    (rationalCyclotomicPrincipalPrimeModulus p n)
-    (RayClass.rationalPrime p)
-
-private noncomputable def rationalCyclotomicPrincipalPrimeResidueUnit
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    (ZMod (p.1 ^ (n + 1)))ˣ :=
-  Units.map
-    (PadicInt.toZModPow (p := p.1) (n + 1)).toMonoidHom
-    (padicIntUnitOfRat p
-      (rationalPrimeUnit x p : ℚ)
-      (rationalPrimeUnit x p).ne_zero
-      (padicValRat_rationalPrimeUnit x p))
-
-private theorem rationalCyclotomicPrincipalPrime_localizedBase_commutes
-    (p : Nat.Primes) (n : ℕ)
-    (y : (rationalCyclotomicArtinBaseAbv p).Completion) :
-    rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-        (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
-          (rationalCyclotomicArtinLocalizedField
-            (rationalCyclotomicPrincipalPrimeModulus p n) p) y) =
-      algebraMap ℚ_[p.1]
-        (RationalCyclotomicPrincipalPrimePadicLevel p n)
-        (rationalFinitePlaceCompletionRingEquivPadic p y) := by
-  let m := rationalCyclotomicPrincipalPrimeModulus p n
-  let E := rationalCyclotomicArtinLocalizedField m p
-  let eL := rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-  have hy :
-      algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E y =
-        algebraMap ℚ_[p.1] E
-          (rationalFinitePlaceCompletionRingEquivPadic p y) := by
-    change
-      algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E y =
-        algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E
-          ((rationalFinitePlaceCompletionRingEquivPadic p).symm
-            (rationalFinitePlaceCompletionRingEquivPadic p y))
-    exact
-      (congrArg
-        (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E)
-        ((rationalFinitePlaceCompletionRingEquivPadic p).symm_apply_apply y)).symm
-  calc
-    eL (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E y) =
-        eL (algebraMap ℚ_[p.1] E
-          (rationalFinitePlaceCompletionRingEquivPadic p y)) :=
-      congrArg eL hy
-    _ = algebraMap ℚ_[p.1]
-        (RationalCyclotomicPrincipalPrimePadicLevel p n)
-        (rationalFinitePlaceCompletionRingEquivPadic p y) :=
-      eL.commutes (rationalFinitePlaceCompletionRingEquivPadic p y)
-
-/-- Specialized ramified-prime bridge from the standard `p`-adic Artin value
-to the canonical global finite-place Artin value.  The localized completion
-and all of its dependent instances remain private to this provider. -/
-theorem
-    rationalCyclotomicPrincipalPrime_finitePlaceArtinOfExtension_eq_one_of_padic
-    (p : Nat.Primes) (n : ℕ)
-    (x : ((RayClass.rationalPrime p).adicCompletion ℚ)ˣ)
-    (htrivial :
-      abelianLocalArtinMonoidHom ℚ_[p.1]
-          (RationalCyclotomicPrincipalPrimePadicLevel p n)
-          (Units.map
-            (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
-            (finitePlaceLocalArtinInput (RayClass.rationalPrime p) x)) = 1) :
-    finitePlaceArtinMonoidHomOfExtension
-        (K := ℚ)
-        (L := KummerTheory.rationalCyclotomicLevel
-          (rationalCyclotomicPrincipalPrimeModulus p n))
-        (RayClass.rationalPrime p)
-        (rationalCyclotomicChosenFinitePlaceExtension
-          (rationalCyclotomicPrincipalPrimeModulus p n)
-          (RayClass.rationalPrime p)) x = 1 := by
-  exact
-    finitePlaceArtinMonoidHomOfExtension_eq_one_of_semilinear
-      (K := ℚ)
-      (L := KummerTheory.rationalCyclotomicLevel
-        (rationalCyclotomicPrincipalPrimeModulus p n))
-      (K' := ℚ_[p.1])
-      (L' := RationalCyclotomicPrincipalPrimePadicLevel p n)
-      (RayClass.rationalPrime p)
-      (rationalCyclotomicChosenFinitePlaceExtension
-        (rationalCyclotomicPrincipalPrimeModulus p n)
-        (RayClass.rationalPrime p))
-      (rationalFinitePlaceCompletionRingEquivPadic p)
-      (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n).toRingEquiv
-      (rationalCyclotomicPrincipalPrime_localizedBase_commutes p n)
-      (rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible p)
-      x htrivial
-
-private theorem
-    padicMultiplicativePrimitiveRoot_rationalPrimeUnitParameterGaloisAction
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    standardLubinTateUnitParameterEquivGal
-        (padicLocalField p.1)
-        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
-        (standardLubinTateUnitParameterClass
-          (padicLocalField p.1) n
-          (rationalPrimeUnitValuationSubringUnit x p))
-        (padicMultiplicativePrimitiveRoot p.1 n) =
-      padicMultiplicativePrimitiveRoot p.1 n ^
-        (PadicInt.toZModPow (p := p.1) (n + 1)
-          (padicIntUnitOfRat p
-            (rationalPrimeUnit x p : ℚ)
-            (rationalPrimeUnit x p).ne_zero
-            (padicValRat_rationalPrimeUnit x p) : ℤ_[p.1])).val := by
-  let uZ : ℤ_[p.1]ˣ :=
-    padicIntUnitOfRat p
-      (rationalPrimeUnit x p : ℚ)
-      (rationalPrimeUnit x p).ne_zero
-      (padicValRat_rationalPrimeUnit x p)
-  let u : (padicLocalField p.1).valuationSubringˣ :=
-    Units.map (padicIntEquivValuationSubring p.1).toMonoidHom uZ
-  have huRational :
-      rationalPrimeUnitValuationSubringUnit x p = u := by
-    rfl
-  have huPreimage :
-      (padicIntEquivValuationSubring p.1).symm
-          ((u : (padicLocalField p.1).valuationSubringˣ) :
-            (padicLocalField p.1).valuationSubring) =
-        (uZ : ℤ_[p.1]) := by
-    change
-      (padicIntEquivValuationSubring p.1).symm
-          (padicIntEquivValuationSubring p.1 (uZ : ℤ_[p.1])) =
-        (uZ : ℤ_[p.1])
-    exact
-      (padicIntEquivValuationSubring p.1).symm_apply_apply
-        (uZ : ℤ_[p.1])
-  have hAction :=
-    padicMultiplicativePrimitiveRoot_unitParameterGaloisAction
-      p.1 n u
-  rw [huPreimage] at hAction
-  rw [huRational]
-  exact hAction
-
-private noncomputable def rationalCyclotomicPrincipalPrimePadicTargetArtin
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    Gal(RationalCyclotomicPrincipalPrimePadicLevel p n / ℚ_[p.1]) :=
-  @LocalClassFieldTheory.abelianLocalArtinMonoidHom
-    ℚ_[p.1] (RationalCyclotomicPrincipalPrimePadicLevel p n)
-    (inferInstance : Field ℚ_[p.1])
-    (inferInstance : Field
-      (RationalCyclotomicPrincipalPrimePadicLevel p n))
-    (inferInstance : Algebra ℚ_[p.1]
-      (RationalCyclotomicPrincipalPrimePadicLevel p n))
-    (inferInstance : ValuativeRel ℚ_[p.1])
-    (inferInstance : TopologicalSpace ℚ_[p.1])
-    (inferInstance : IsNonarchimedeanLocalField ℚ_[p.1])
-    (rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n)
-    (standardLubinTateLevelField_isAbelianGalois
-      (padicLocalField p.1)
-      (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
-    (Units.map
-      (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
-      (rationalPrincipalFinitePlaceInput x p))
-
-private noncomputable def
-    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    Gal(RationalCyclotomicPrincipalPrimePadicLevel p n / ℚ_[p.1]) :=
-  standardLubinTateUnitParameterEquivGal
-    (padicLocalField p.1)
-    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
-    (standardLubinTateUnitParameterClass
-      (padicLocalField p.1) n
-      (rationalPrimeUnitValuationSubringUnit x p))
-
-private theorem
-    rationalCyclotomicPrincipalPrime_padicTargetArtin_eq_unitParameter
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    rationalCyclotomicPrincipalPrimePadicTargetArtin p n x =
-      rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x := by
-  let T := RationalCyclotomicPrincipalPrimePadicLevel p n
-  let eK := rationalFinitePlaceCompletionRingEquivPadic p
-  let : FiniteDimensional ℚ_[p.1] T :=
-    rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n
-  let hpi := padicMultiplicativeLubinTateSeries_isUniformizer p.1
-  let : IsAbelianGalois ℚ_[p.1] T :=
-    standardLubinTateLevelField_isAbelianGalois
-      (padicLocalField p.1) hpi n
-  have hsource :
-      Units.map eK.toMonoidHom (rationalPrincipalFinitePlaceInput x p) =
-        Units.map (algebraMap ℚ ℚ_[p.1]).toMonoidHom x := by
-    rw [rationalPrincipalFinitePlaceInput_eq_algebraMap]
-    apply Units.ext
-    exact rationalFinitePlaceCompletionRingEquivPadic_algebraMap p (x : ℚ)
-  change
-    LocalClassFieldTheory.abelianLocalArtinMonoidHom ℚ_[p.1] T
-        (Units.map eK.toMonoidHom
-          (rationalPrincipalFinitePlaceInput x p)) =
-      standardLubinTateUnitParameterEquivGal
-        (padicLocalField p.1) hpi n
-        (standardLubinTateUnitParameterClass
-          (padicLocalField p.1) n
-          (rationalPrimeUnitValuationSubringUnit x p))
-  rw [hsource]
-  rw [
-    padicMultiplicativeAbelianLocalArtin_eq_uniformizerUnitPart,
-    rationalPadicFieldUnit_uniformizerUnitPart,
-    padicMultiplicativeAbelianLocalArtin_eq_unitParameter]
-
-private theorem
-    rationalCyclotomicPrincipalPrime_padicArtin_action_eq_unitParameter
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    (@LocalClassFieldTheory.abelianLocalArtinMonoidHom
-      ℚ_[p.1] (RationalCyclotomicPrincipalPrimePadicLevel p n)
-      (inferInstance : Field ℚ_[p.1])
-      (inferInstance : Field
-        (RationalCyclotomicPrincipalPrimePadicLevel p n))
-      (inferInstance : Algebra ℚ_[p.1]
-        (RationalCyclotomicPrincipalPrimePadicLevel p n))
-      (inferInstance : ValuativeRel ℚ_[p.1])
-      (inferInstance : TopologicalSpace ℚ_[p.1])
-      (inferInstance : IsNonarchimedeanLocalField ℚ_[p.1])
-      (rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n)
-      (standardLubinTateLevelField_isAbelianGalois
-        (padicLocalField p.1)
-        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
-      (Units.map
-        (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
-        (finitePlaceLocalArtinInput
-          (K := ℚ) (RayClass.rationalPrime p)
-          (IdeleGroup.finiteComponent
-            (RayClass.rationalPrime p)
-            (IdeleGroup.principalIdele ℚ x)))))
-        ((rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n).toRingEquiv
-          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) =
-      rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
-        (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) := by
-  let eK := rationalFinitePlaceCompletionRingEquivPadic p
-  have hInput :
-      finitePlaceLocalArtinInput
-          (K := ℚ) (RayClass.rationalPrime p)
-          (IdeleGroup.finiteComponent
-            (RayClass.rationalPrime p)
-            (IdeleGroup.principalIdele ℚ x)) =
-        rationalPrincipalFinitePlaceInput x p := by
-    rfl
-  have hMapped :
-      Units.map eK.toMonoidHom
-          (finitePlaceLocalArtinInput
-            (K := ℚ) (RayClass.rationalPrime p)
-            (IdeleGroup.finiteComponent
-              (RayClass.rationalPrime p)
-              (IdeleGroup.principalIdele ℚ x))) =
-        Units.map eK.toMonoidHom
-          (rationalPrincipalFinitePlaceInput x p) :=
-    congrArg (Units.map eK.toMonoidHom) hInput
-  calc
-    _ = rationalCyclotomicPrincipalPrimePadicTargetArtin p n x
-          (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-            (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) := by
-      exact congrArg
-        (fun uQp =>
-          (@LocalClassFieldTheory.abelianLocalArtinMonoidHom
-            ℚ_[p.1] (RationalCyclotomicPrincipalPrimePadicLevel p n)
-            (inferInstance : Field ℚ_[p.1])
-            (inferInstance : Field
-              (RationalCyclotomicPrincipalPrimePadicLevel p n))
-            (inferInstance : Algebra ℚ_[p.1]
-              (RationalCyclotomicPrincipalPrimePadicLevel p n))
-            (inferInstance : ValuativeRel ℚ_[p.1])
-            (inferInstance : TopologicalSpace ℚ_[p.1])
-            (inferInstance : IsNonarchimedeanLocalField ℚ_[p.1])
-            (rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n)
-            (standardLubinTateLevelField_isAbelianGalois
-              (padicLocalField p.1)
-              (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
-            uQp)
-              (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-                (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)))
-        hMapped
-    _ = rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
-          (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-            (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) :=
-      congrArg
-        (fun tau => tau
-          (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-            (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)))
-        (rationalCyclotomicPrincipalPrime_padicTargetArtin_eq_unitParameter
-          p n x)
-
-private theorem
-    rationalCyclotomicPrincipalPrime_padicUnitParameterArtin_action
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
-        (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) =
-      (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) ^
-        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val := by
-  let m := rationalCyclotomicPrincipalPrimeModulus p n
-  let E := rationalCyclotomicArtinLocalizedField m p
-  let T := RationalCyclotomicPrincipalPrimePadicLevel p n
-  let eL : E ≃ₐ[ℚ_[p.1]] T :=
-    rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-  let zetaE : E := rationalCyclotomicPrincipalPrimeLocalizedRoot p n
-  let tau : Gal(T / ℚ_[p.1]) :=
-    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
-  let a := rationalCyclotomicPrincipalPrimeResidueUnit p n x
-  let zetaT : T := padicMultiplicativePrimitiveRoot p.1 n
-  have hzetaE : IsPrimitiveRoot zetaE (p.1 ^ (n + 1)) := by
-    change
-      IsPrimitiveRoot
-        (rationalCyclotomicLocalizedPrimitiveRoot m
-          (RayClass.rationalPrime p)) (m : ℕ)
-    exact
-      rationalCyclotomicLocalizedPrimitiveRoot_isPrimitiveRoot
-        m (RayClass.rationalPrime p)
-  have hrho : IsPrimitiveRoot (eL zetaE) (p.1 ^ (n + 1)) :=
-    hzetaE.map_of_injective eL.injective
-  have hzetaT : IsPrimitiveRoot zetaT (p.1 ^ (n + 1)) :=
-    padicMultiplicativePrimitiveRoot_isPrimitiveRoot p.1 n
-  have htauZetaT : tau zetaT = zetaT ^ a.val.val :=
-    padicMultiplicativePrimitiveRoot_rationalPrimeUnitParameterGaloisAction
-      p n x
-  exact
-    map_primitiveRoot_eq_pow_of_eq_pow
-      tau.toMonoidHom zetaT (eL zetaE)
-      (p.1 ^ (n + 1)) a.val.val hzetaT hrho htauZetaT
-
-private theorem rationalCyclotomicPrincipalPrime_localArtin_action
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    finitePlaceLocalArtinMonoidHom
-        (K := ℚ)
-        (L := KummerTheory.rationalCyclotomicLevel
-          (rationalCyclotomicPrincipalPrimeModulus p n))
-        (RayClass.rationalPrime p)
-        (rationalCyclotomicChosenFinitePlaceExtension
-          (rationalCyclotomicPrincipalPrimeModulus p n)
-          (RayClass.rationalPrime p))
-        (IdeleGroup.finiteComponent
-          (RayClass.rationalPrime p)
-          (IdeleGroup.principalIdele ℚ x))
-        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n) =
-      (rationalCyclotomicPrincipalPrimeLocalizedRoot p n) ^
-        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val := by
-  let eL := rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
-  apply eL.injective
-  calc
-    _ = _ :=
-      finitePlaceLocalArtinMonoidHom_apply_semilinear
-        (K := ℚ)
-        (L := KummerTheory.rationalCyclotomicLevel
-          (rationalCyclotomicPrincipalPrimeModulus p n))
-        (K' := ℚ_[p.1])
-        (L' := RationalCyclotomicPrincipalPrimePadicLevel p n)
-        (RayClass.rationalPrime p)
-        (rationalCyclotomicChosenFinitePlaceExtension
-          (rationalCyclotomicPrincipalPrimeModulus p n)
-          (RayClass.rationalPrime p))
-        (rationalFinitePlaceCompletionRingEquivPadic p)
-        eL.toRingEquiv
-        (rationalCyclotomicPrincipalPrime_localizedBase_commutes p n)
-        (rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible
-          p)
-        (IdeleGroup.finiteComponent
-          (RayClass.rationalPrime p)
-          (IdeleGroup.principalIdele ℚ x))
-        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)
-    _ = rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
-          (eL (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) :=
-      rationalCyclotomicPrincipalPrime_padicArtin_action_eq_unitParameter
-        p n x
-    _ = (eL (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) ^
-          (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val :=
-      rationalCyclotomicPrincipalPrime_padicUnitParameterArtin_action p n x
-    _ = eL ((rationalCyclotomicPrincipalPrimeLocalizedRoot p n) ^
-          (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val) :=
-      (map_pow eL (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)
-        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val).symm
-
-/-- The finite-place Artin symbol at the ramified prime, in its canonical
-local-to-global factored form.  Keeping this specialization opaque prevents its
-dependent local/global instance tower from being unfolded downstream. -/
-noncomputable def rationalCyclotomicPrincipalPrimeChosenArtin
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    KummerTheory.rationalCyclotomicLevel
-        (rationalCyclotomicPrincipalPrimeModulus p n) ≃ₐ[ℚ]
-      KummerTheory.rationalCyclotomicLevel
-        (rationalCyclotomicPrincipalPrimeModulus p n) :=
-  finitePlaceLocalToGlobalMonoidHom
-    (K := ℚ)
-    (L := KummerTheory.rationalCyclotomicLevel
-      (rationalCyclotomicPrincipalPrimeModulus p n))
-    (RayClass.rationalPrime p)
-    (rationalCyclotomicChosenFinitePlaceExtension
-      (rationalCyclotomicPrincipalPrimeModulus p n)
-      (RayClass.rationalPrime p))
-    (finitePlaceLocalArtinMonoidHom
-      (K := ℚ)
-      (L := KummerTheory.rationalCyclotomicLevel
-        (rationalCyclotomicPrincipalPrimeModulus p n))
-      (RayClass.rationalPrime p)
-      (rationalCyclotomicChosenFinitePlaceExtension
-        (rationalCyclotomicPrincipalPrimeModulus p n)
-        (RayClass.rationalPrime p))
-      (IdeleGroup.finiteComponent
-        (RayClass.rationalPrime p)
-        (IdeleGroup.principalIdele ℚ x)))
-
-/-- At the ramified prime, the cyclotomic character of the chosen finite-place
-Artin symbol is the direct reduction of the rational `p`-adic unit. -/
-theorem galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_at_prime
-    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
-    IsCyclotomicExtension.Rat.galEquivZMod
-        (p.1 ^ (n + 1))
-        (KummerTheory.rationalCyclotomicLevel
-          (rationalCyclotomicPrincipalPrimeModulus p n))
-        (hK :=
-          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
-            (rationalCyclotomicPrincipalPrimeModulus p n))
-        (rationalCyclotomicPrincipalPrimeChosenArtin p n x) =
-      Units.map
-        (PadicInt.toZModPow (p := p.1) (n + 1)).toMonoidHom
-        (padicIntUnitOfRat p
-          (rationalPrimeUnit x p : ℚ)
-          (rationalPrimeUnit x p).ne_zero
-          (padicValRat_rationalPrimeUnit x p)) := by
-  change _ = rationalCyclotomicPrincipalPrimeResidueUnit p n x
-  apply rationalCyclotomicPrincipalPrime_galEquivZMod_eq_of_action p n
-  simp only [rationalCyclotomicPrincipalPrimeChosenArtin]
-  apply finitePlaceLocalToGlobalMonoidHom_apply_pow_of_localized_action
-    (zLocal := rationalCyclotomicPrincipalPrimeLocalizedRoot p n)
-  · rfl
-  · exact rationalCyclotomicPrincipalPrime_localArtin_action p n x
-
-end RationalCyclotomicPrincipalPrime
+  rationalFinitePlaceCompletionRatAlgebra
+  primeFact
+  levelNeZero
+  rationalCyclotomicLevelFiniteDimensional
+  rationalCyclotomicLevelIsAbelianGalois
+  rationalFinitePlaceBaseNontriviallyNormedField
+  rationalFinitePlaceBaseLocallyCompactSpace
+  rationalFinitePlaceBaseIsUltrametricDist
+  rationalFinitePlaceBaseValued
+  rationalFinitePlaceBaseValuativeRel
+  rationalFinitePlaceBaseValuationIsNontrivial
+  rationalFinitePlaceBaseValuationCompatible
+  rationalFinitePlaceBaseValuativeRelIsNontrivial
+  rationalFinitePlaceBaseIsValuativeTopology
+  rationalFinitePlaceBaseIsNonarchimedeanLocalField
+  rationalCyclotomicArtinExtensionAlgebra
+  rationalCyclotomicArtinExtensionSMul
+  rationalCyclotomicArtinCompletionAlgebra
+  rationalCyclotomicArtinLocalizedAlgebra
+  rationalCyclotomicArtinLocalizedGlobalAlgebra
+  rationalCyclotomicArtinLocalizedGlobalSMul
+  rationalCyclotomicArtinLocalizedScalarTower
+  rationalCyclotomicArtinLocalizedFiniteDimensional
+  rationalCyclotomicArtinLocalizedIsAbelianGalois
+  rationalCyclotomicArtinLocalizedIsSeparable
+  rationalCyclotomicArtinLocalizedIsCyclotomic
+  rationalCyclotomicArtinExtensionFiniteDimensional
+  rationalCyclotomicArtinExtensionContinuousSMul
+  rationalCyclotomicArtinExtensionLocallyCompact
+  rationalCyclotomicArtinLocalizedLocallyCompact
+  rationalCyclotomicArtinLocalizedIsUltrametricDist
+  rationalCyclotomicArtinLocalizedValued
+  rationalCyclotomicArtinLocalizedValuativeRel
+  rationalCyclotomicArtinLocalizedValuationCompatible
+  rationalCyclotomicArtinLocalizedValuationHasExtension
+  rationalCyclotomicArtinLocalizedValuationIsNontrivial
+  rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
+  rationalCyclotomicArtinLocalizedIsValuativeTopology
+  rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
+  rationalCyclotomicArtinLocalizedIntegerAlgebra
+  rationalCyclotomicArtinLocalizedIsIntegralClosure
+  rationalCyclotomicArtinLocalizedIntegerModuleFinite
 
 private theorem rationalCyclotomicArtinUnramified
     (m : ℕ+) (q : Nat.Primes)
@@ -2440,8 +178,9 @@ noncomputable def rationalCyclotomicArtinLocalExponent
     (rationalCyclotomicArtinBaseAbv q).Completion
     (Additive.ofMul (rationalCyclotomicArtinLocalInput q x))
 
-/-- The chosen finite-place Artin value in a rational cyclotomic level, with
-the completion and Galois instance arguments frozen at the provider boundary. -/
+
+/-- The Galois automorphism of the rational cyclotomic extension at level `m`
+assigned to the nonzero `q`-adic input by the chosen finite-place Artin map. -/
 noncomputable def rationalCyclotomicChosenFinitePlaceArtinValue
     (m : ℕ+) (q : Nat.Primes)
     (x : ((RayClass.rationalPrime q).adicCompletion ℚ)ˣ) :
@@ -2484,7 +223,12 @@ private theorem rationalCyclotomicArtinLocalArtin_eq
             (rationalCyclotomicArtinLevel m)))
         (finitePlaceLocalArtinInput
           (K := ℚ) (rationalCyclotomicArtinPlace q) x) := by
-  rfl
+  unfold rationalCyclotomicArtinLocalArtin
+  have h := finitePlaceLocalArtinMonoidHom_apply_normalized
+    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
+    (rationalCyclotomicArtinPlace q)
+    (rationalCyclotomicArtinExtension m q) x
+  with_reducible exact h
 
 private theorem rationalCyclotomicChosenArithmeticFrobenius_eq_lift
     (m : ℕ+) (q : Nat.Primes)
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinAction.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinAction.lean
new file mode 100644
index 0000000..fd14a9f
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinAction.lean
@@ -0,0 +1,438 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlaceArtinTransport
+
+
+set_option autoImplicit false
+
+open scoped Classical NNReal NumberField ValuativeRel
+open NumberField IsDedekindDomain
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open LocalClassFieldTheory
+open LocalFieldTheory
+open LocalFieldTheory.DiscreteValuationField
+open LocalFieldTheory.DiscreteValuationField.Examples.Qp
+open LubinTate
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+section RationalCyclotomicPrincipalPrime
+
+attribute [local instance]
+  rationalFinitePlaceCompletionRatAlgebra
+  primeFact
+  levelNeZero
+  rationalCyclotomicLevelFiniteDimensional
+  rationalCyclotomicLevelIsAbelianGalois
+  rationalFinitePlaceBaseNontriviallyNormedField
+  rationalFinitePlaceBaseLocallyCompactSpace
+  rationalFinitePlaceBaseIsUltrametricDist
+  rationalFinitePlaceBaseValued
+  rationalFinitePlaceBaseValuativeRel
+  rationalFinitePlaceBaseValuationIsNontrivial
+  rationalFinitePlaceBaseValuationCompatible
+  rationalFinitePlaceBaseValuativeRelIsNontrivial
+  rationalFinitePlaceBaseIsValuativeTopology
+  rationalFinitePlaceBaseIsNonarchimedeanLocalField
+  rationalCyclotomicArtinExtensionAlgebra
+  rationalCyclotomicArtinExtensionSMul
+  rationalCyclotomicArtinCompletionAlgebra
+  rationalCyclotomicArtinLocalizedAlgebra
+  rationalCyclotomicArtinLocalizedGlobalAlgebra
+  rationalCyclotomicArtinLocalizedGlobalSMul
+  rationalCyclotomicArtinLocalizedScalarTower
+  rationalCyclotomicArtinLocalizedFiniteDimensional
+  rationalCyclotomicArtinLocalizedIsAbelianGalois
+  rationalCyclotomicArtinLocalizedIsSeparable
+  rationalCyclotomicArtinLocalizedIsCyclotomic
+  rationalCyclotomicArtinExtensionFiniteDimensional
+  rationalCyclotomicArtinExtensionContinuousSMul
+  rationalCyclotomicArtinExtensionLocallyCompact
+  rationalCyclotomicArtinLocalizedLocallyCompact
+  rationalCyclotomicArtinLocalizedIsUltrametricDist
+  rationalCyclotomicArtinLocalizedValued
+  rationalCyclotomicArtinLocalizedValuativeRel
+  rationalCyclotomicArtinLocalizedValuationCompatible
+  rationalCyclotomicArtinLocalizedValuationHasExtension
+  rationalCyclotomicArtinLocalizedValuationIsNontrivial
+  rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
+  rationalCyclotomicArtinLocalizedIsValuativeTopology
+  rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
+  rationalCyclotomicArtinLocalizedIntegerAlgebra
+  rationalCyclotomicArtinLocalizedIsIntegralClosure
+  rationalCyclotomicArtinLocalizedIntegerModuleFinite
+  rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
+  rationalCyclotomicArtinLocalizedPadicAlgebra
+  rationalCyclotomicArtinLocalizedPadicScalarTower
+  rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
+  rationalPrimeFactorCompletionPadicAlgebra
+  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
+
+/-- If a semilinearly identified target local Artin value is trivial, then the
+corresponding global finite-place Artin value is trivial.  This generic bridge
+keeps concrete completion and localization instance towers out of downstream
+proof terms. -/
+theorem finitePlaceArtinMonoidHomOfExtension_eq_one_of_semilinear
+    {K L K' L' : Type}
+    [Field K] [NumberField K]
+    [Field L] [Algebra K L]
+    [hKLfinite : FiniteDimensional K L] [IsAbelianGalois K L]
+    [Field K'] [ValuativeRel K'] [TopologicalSpace K']
+    [IsNonarchimedeanLocalField K']
+    [Field L'] [Algebra K' L']
+    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
+    (v : HeightOneSpectrum (𝓞 K))
+    (w : AbsoluteValueExtension
+      (HeightOneSpectrum.adicAbv K v) L)
+    (eK : (HeightOneSpectrum.adicAbv K v).Completion ≃+* K')
+    (eL : LocalizedCompletion
+      (HeightOneSpectrum.adicAbv K v) w ≃+* L')
+    (hcomm : ∀ y : (HeightOneSpectrum.adicAbv K v).Completion,
+      eL (@algebraMap
+        (HeightOneSpectrum.adicAbv K v).Completion
+        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
+        _ _ (finitePlaceLocalArtinLocalizedAlgebra v w) y) =
+        algebraMap K' L' (eK y))
+    (hExt : SemilinearValuationCompatible
+      (HeightOneSpectrum.adicAbv K v).Completion K' eK)
+    (x : (v.adicCompletion K)ˣ)
+    (htrivial :
+      LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
+          (Units.map eK.toMonoidHom
+            (finitePlaceLocalArtinInput v x)) = 1) :
+    finitePlaceArtinMonoidHomOfExtension
+        (K := K) (L := L) v w x = 1 := by
+  rw [finitePlaceArtinMonoidHomOfExtension_factor,
+    MonoidHom.comp_apply]
+  have hlocal :
+      finitePlaceLocalArtinMonoidHom
+          (K := K) (L := L) v w x = 1 := by
+    rw [finitePlaceLocalArtinMonoidHom_apply_normalized]
+    exact
+      @abelianLocalArtinMonoidHom_eq_one_of_semilinear
+        (HeightOneSpectrum.adicAbv K v).Completion K'
+        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w) L'
+        (inferInstance : Field
+          (HeightOneSpectrum.adicAbv K v).Completion)
+        (finitePlaceLocalArtinCompletionValuativeRel v)
+        (inferInstance : TopologicalSpace
+          (HeightOneSpectrum.adicAbv K v).Completion)
+        (finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v)
+        (inferInstance : Field K')
+        (inferInstance : ValuativeRel K')
+        (inferInstance : TopologicalSpace K')
+        (inferInstance : IsNonarchimedeanLocalField K')
+        (inferInstance : Field
+          (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w))
+        (inferInstance : Field L')
+        (finitePlaceLocalArtinLocalizedAlgebra v w)
+        (inferInstance : Algebra K' L')
+        (finitePlaceLocalArtinFiniteDimensional v w)
+        (finitePlaceLocalArtinIsAbelianGalois v w hKLfinite)
+        (inferInstance : FiniteDimensional K' L')
+        (inferInstance : IsAbelianGalois K' L')
+        eK eL hcomm hExt (finitePlaceLocalArtinInput v x) htrivial
+  rw [hlocal, map_one]
+
+/-- The chosen primitive cyclotomic root in the localized prime-power extension. -/
+noncomputable def rationalCyclotomicPrincipalPrimeLocalizedRoot
+    (p : Nat.Primes) (n : ℕ) :
+    rationalCyclotomicArtinLocalizedField
+      (rationalCyclotomicPrincipalPrimeModulus p n) p :=
+  rationalCyclotomicLocalizedPrimitiveRoot
+    (rationalCyclotomicPrincipalPrimeModulus p n)
+    (RayClass.rationalPrime p)
+
+/-- The residue modulo `p^(n+1)` of the `p`-adic unit part of a rational unit. -/
+noncomputable def rationalCyclotomicPrincipalPrimeResidueUnit
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    (ZMod (p.1 ^ (n + 1)))ˣ :=
+  Units.map
+    (PadicInt.toZModPow (p := p.1) (n + 1)).toMonoidHom
+    (padicIntUnitOfRat p
+      (rationalPrimeUnit x p : ℚ)
+      (rationalPrimeUnit x p).ne_zero
+      (padicValRat_rationalPrimeUnit x p))
+
+theorem rationalCyclotomicPrincipalPrime_localizedBase_commutes
+    (p : Nat.Primes) (n : ℕ)
+    (y : (rationalCyclotomicArtinBaseAbv p).Completion) :
+    rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+        (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
+          (rationalCyclotomicArtinLocalizedField
+            (rationalCyclotomicPrincipalPrimeModulus p n) p) y) =
+      algebraMap ℚ_[p.1]
+        (RationalCyclotomicPrincipalPrimePadicLevel p n)
+        (rationalFinitePlaceCompletionRingEquivPadic p y) := by
+  let m := rationalCyclotomicPrincipalPrimeModulus p n
+  let E := rationalCyclotomicArtinLocalizedField m p
+  let eL := rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+  have hy :
+      algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E y =
+        algebraMap ℚ_[p.1] E
+          (rationalFinitePlaceCompletionRingEquivPadic p y) := by
+    change
+      algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E y =
+        algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E
+          ((rationalFinitePlaceCompletionRingEquivPadic p).symm
+            (rationalFinitePlaceCompletionRingEquivPadic p y))
+    exact
+      (congrArg
+        (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E)
+        ((rationalFinitePlaceCompletionRingEquivPadic p).symm_apply_apply y)).symm
+  calc
+    eL (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion E y) =
+        eL (algebraMap ℚ_[p.1] E
+          (rationalFinitePlaceCompletionRingEquivPadic p y)) :=
+      congrArg eL hy
+    _ = algebraMap ℚ_[p.1]
+        (RationalCyclotomicPrincipalPrimePadicLevel p n)
+        (rationalFinitePlaceCompletionRingEquivPadic p y) :=
+      eL.commutes (rationalFinitePlaceCompletionRingEquivPadic p y)
+
+/-- Specialized ramified-prime bridge from the standard `p`-adic Artin value
+to the canonical global finite-place Artin value.  The localized completion
+and all of its dependent instances remain private to this provider. -/
+theorem
+    rationalCyclotomicPrincipalPrime_finitePlaceArtinOfExtension_eq_one_of_padic
+    (p : Nat.Primes) (n : ℕ)
+    (x : ((RayClass.rationalPrime p).adicCompletion ℚ)ˣ)
+    (htrivial :
+      abelianLocalArtinMonoidHom ℚ_[p.1]
+          (RationalCyclotomicPrincipalPrimePadicLevel p n)
+          (Units.map
+            (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
+            (finitePlaceLocalArtinInput (RayClass.rationalPrime p) x)) = 1) :
+    finitePlaceArtinMonoidHomOfExtension
+        (K := ℚ)
+        (L := KummerTheory.rationalCyclotomicLevel
+          (rationalCyclotomicPrincipalPrimeModulus p n))
+        (RayClass.rationalPrime p)
+        (rationalCyclotomicChosenFinitePlaceExtension
+          (rationalCyclotomicPrincipalPrimeModulus p n)
+          (RayClass.rationalPrime p)) x = 1 := by
+  exact
+    finitePlaceArtinMonoidHomOfExtension_eq_one_of_semilinear
+      (K := ℚ)
+      (L := KummerTheory.rationalCyclotomicLevel
+        (rationalCyclotomicPrincipalPrimeModulus p n))
+      (K' := ℚ_[p.1])
+      (L' := RationalCyclotomicPrincipalPrimePadicLevel p n)
+      (RayClass.rationalPrime p)
+      (rationalCyclotomicChosenFinitePlaceExtension
+        (rationalCyclotomicPrincipalPrimeModulus p n)
+        (RayClass.rationalPrime p))
+      (rationalFinitePlaceCompletionRingEquivPadic p)
+      (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n).toRingEquiv
+      (rationalCyclotomicPrincipalPrime_localizedBase_commutes p n)
+      (rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible p)
+      x htrivial
+
+theorem
+    padicMultiplicativePrimitiveRoot_rationalPrimeUnitParameterGaloisAction
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    standardLubinTateUnitParameterEquivGal
+        (padicLocalField p.1)
+        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
+        (standardLubinTateUnitParameterClass
+          (padicLocalField p.1) n
+          (rationalPrimeUnitValuationSubringUnit x p))
+        (padicMultiplicativePrimitiveRoot p.1 n) =
+      padicMultiplicativePrimitiveRoot p.1 n ^
+        (PadicInt.toZModPow (p := p.1) (n + 1)
+          (padicIntUnitOfRat p
+            (rationalPrimeUnit x p : ℚ)
+            (rationalPrimeUnit x p).ne_zero
+            (padicValRat_rationalPrimeUnit x p) : ℤ_[p.1])).val := by
+  let uZ : ℤ_[p.1]ˣ :=
+    padicIntUnitOfRat p
+      (rationalPrimeUnit x p : ℚ)
+      (rationalPrimeUnit x p).ne_zero
+      (padicValRat_rationalPrimeUnit x p)
+  let u : (padicLocalField p.1).valuationSubringˣ :=
+    Units.map (padicIntEquivValuationSubring p.1).toMonoidHom uZ
+  have huRational :
+      rationalPrimeUnitValuationSubringUnit x p = u := by
+    rfl
+  have huPreimage :
+      (padicIntEquivValuationSubring p.1).symm
+          ((u : (padicLocalField p.1).valuationSubringˣ) :
+            (padicLocalField p.1).valuationSubring) =
+        (uZ : ℤ_[p.1]) := by
+    change
+      (padicIntEquivValuationSubring p.1).symm
+          (padicIntEquivValuationSubring p.1 (uZ : ℤ_[p.1])) =
+        (uZ : ℤ_[p.1])
+    exact
+      (padicIntEquivValuationSubring p.1).symm_apply_apply
+        (uZ : ℤ_[p.1])
+  have hAction :=
+    padicMultiplicativePrimitiveRoot_unitParameterGaloisAction
+      p.1 n u
+  rw [huPreimage] at hAction
+  rw [huRational]
+  exact hAction
+
+private noncomputable def rationalCyclotomicPrincipalPrimePadicTargetArtin
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    Gal(RationalCyclotomicPrincipalPrimePadicLevel p n / ℚ_[p.1]) :=
+  @LocalClassFieldTheory.abelianLocalArtinMonoidHom
+    ℚ_[p.1] (RationalCyclotomicPrincipalPrimePadicLevel p n)
+    (inferInstance : Field ℚ_[p.1])
+    (inferInstance : Field
+      (RationalCyclotomicPrincipalPrimePadicLevel p n))
+    (inferInstance : Algebra ℚ_[p.1]
+      (RationalCyclotomicPrincipalPrimePadicLevel p n))
+    (inferInstance : ValuativeRel ℚ_[p.1])
+    (inferInstance : TopologicalSpace ℚ_[p.1])
+    (inferInstance : IsNonarchimedeanLocalField ℚ_[p.1])
+    (rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n)
+    (standardLubinTateLevelField_isAbelianGalois
+      (padicLocalField p.1)
+      (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
+    (Units.map
+      (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
+      (rationalPrincipalFinitePlaceInput x p))
+
+/-- The cyclotomic Galois automorphism corresponding to the unit parameter of `x`
+under the standard Lubin–Tate equivalence. -/
+noncomputable def
+    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    Gal(RationalCyclotomicPrincipalPrimePadicLevel p n / ℚ_[p.1]) :=
+  standardLubinTateUnitParameterEquivGal
+    (padicLocalField p.1)
+    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
+    (standardLubinTateUnitParameterClass
+      (padicLocalField p.1) n
+      (rationalPrimeUnitValuationSubringUnit x p))
+
+private theorem
+    rationalCyclotomicPrincipalPrime_padicTargetArtin_eq_unitParameter
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    rationalCyclotomicPrincipalPrimePadicTargetArtin p n x =
+      rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x := by
+  let T := RationalCyclotomicPrincipalPrimePadicLevel p n
+  let eK := rationalFinitePlaceCompletionRingEquivPadic p
+  let : FiniteDimensional ℚ_[p.1] T :=
+    rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n
+  let hpi := padicMultiplicativeLubinTateSeries_isUniformizer p.1
+  let : IsAbelianGalois ℚ_[p.1] T :=
+    standardLubinTateLevelField_isAbelianGalois
+      (padicLocalField p.1) hpi n
+  have hsource :
+      Units.map eK.toMonoidHom (rationalPrincipalFinitePlaceInput x p) =
+        Units.map (algebraMap ℚ ℚ_[p.1]).toMonoidHom x := by
+    rw [rationalPrincipalFinitePlaceInput_eq_algebraMap]
+    apply Units.ext
+    exact rationalFinitePlaceCompletionRingEquivPadic_algebraMap p (x : ℚ)
+  change
+    LocalClassFieldTheory.abelianLocalArtinMonoidHom ℚ_[p.1] T
+        (Units.map eK.toMonoidHom
+          (rationalPrincipalFinitePlaceInput x p)) =
+      standardLubinTateUnitParameterEquivGal
+        (padicLocalField p.1) hpi n
+        (standardLubinTateUnitParameterClass
+          (padicLocalField p.1) n
+          (rationalPrimeUnitValuationSubringUnit x p))
+  rw [hsource]
+  rw [
+    padicMultiplicativeAbelianLocalArtin_eq_uniformizerUnitPart,
+    rationalPadicFieldUnit_uniformizerUnitPart,
+    padicMultiplicativeAbelianLocalArtin_eq_unitParameter]
+
+theorem
+    rationalCyclotomicPrincipalPrime_padicArtin_action_eq_unitParameter
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    (@LocalClassFieldTheory.abelianLocalArtinMonoidHom
+      ℚ_[p.1] (RationalCyclotomicPrincipalPrimePadicLevel p n)
+      (inferInstance : Field ℚ_[p.1])
+      (inferInstance : Field
+        (RationalCyclotomicPrincipalPrimePadicLevel p n))
+      (inferInstance : Algebra ℚ_[p.1]
+        (RationalCyclotomicPrincipalPrimePadicLevel p n))
+      (inferInstance : ValuativeRel ℚ_[p.1])
+      (inferInstance : TopologicalSpace ℚ_[p.1])
+      (inferInstance : IsNonarchimedeanLocalField ℚ_[p.1])
+      (rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n)
+      (standardLubinTateLevelField_isAbelianGalois
+        (padicLocalField p.1)
+        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
+      (Units.map
+        (rationalFinitePlaceCompletionRingEquivPadic p).toMonoidHom
+        (finitePlaceLocalArtinInput
+          (K := ℚ) (RayClass.rationalPrime p)
+          (IdeleGroup.finiteComponent
+            (RayClass.rationalPrime p)
+            (IdeleGroup.principalIdele ℚ x)))))
+        ((rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n).toRingEquiv
+          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) =
+      rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
+        (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) := by
+  let eK := rationalFinitePlaceCompletionRingEquivPadic p
+  have hInput :
+      finitePlaceLocalArtinInput
+          (K := ℚ) (RayClass.rationalPrime p)
+          (IdeleGroup.finiteComponent
+            (RayClass.rationalPrime p)
+            (IdeleGroup.principalIdele ℚ x)) =
+        rationalPrincipalFinitePlaceInput x p := by
+    rfl
+  have hMapped :
+      Units.map eK.toMonoidHom
+          (finitePlaceLocalArtinInput
+            (K := ℚ) (RayClass.rationalPrime p)
+            (IdeleGroup.finiteComponent
+              (RayClass.rationalPrime p)
+              (IdeleGroup.principalIdele ℚ x))) =
+        Units.map eK.toMonoidHom
+          (rationalPrincipalFinitePlaceInput x p) :=
+    congrArg (Units.map eK.toMonoidHom) hInput
+  calc
+    _ = rationalCyclotomicPrincipalPrimePadicTargetArtin p n x
+          (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+            (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) := by
+      exact congrArg
+        (fun uQp =>
+          (@LocalClassFieldTheory.abelianLocalArtinMonoidHom
+            ℚ_[p.1] (RationalCyclotomicPrincipalPrimePadicLevel p n)
+            (inferInstance : Field ℚ_[p.1])
+            (inferInstance : Field
+              (RationalCyclotomicPrincipalPrimePadicLevel p n))
+            (inferInstance : Algebra ℚ_[p.1]
+              (RationalCyclotomicPrincipalPrimePadicLevel p n))
+            (inferInstance : ValuativeRel ℚ_[p.1])
+            (inferInstance : TopologicalSpace ℚ_[p.1])
+            (inferInstance : IsNonarchimedeanLocalField ℚ_[p.1])
+            (rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional p n)
+            (standardLubinTateLevelField_isAbelianGalois
+              (padicLocalField p.1)
+              (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n)
+            uQp)
+              (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+                (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)))
+        hMapped
+    _ = rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
+          (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+            (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) :=
+      congrArg
+        (fun tau => tau
+          (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+            (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)))
+        (rationalCyclotomicPrincipalPrime_padicTargetArtin_eq_unitParameter
+          p n x)
+
+
+end RationalCyclotomicPrincipalPrime
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinBase.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinBase.lean
new file mode 100644
index 0000000..3646afc
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinBase.lean
@@ -0,0 +1,924 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.CompletionToIdeal
+import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.IdealToCompletion
+import ClassFieldTheory.AlgebraicNumberTheory.Completion.UnramifiedComparison.LocalNorm
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.AlgEquiv
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.BaseChange
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.Core
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.NormComparison
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.NormalClosureNorm
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.Tower
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.TowerAlgEquivNaturality
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.ClassGroup.TowerBaseChange
+import ClassFieldTheory.AlgebraicNumberTheory.Idele.Relative.FinitePlaceTensorNorm
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlace
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicLocalization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalPrimeFactorization
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalPrincipalLocalUnit
+import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.SemilinearNaturality
+import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.UnramifiedNormalization
+import ClassFieldTheory.LocalClassFieldTheory.LubinTateApplication.PadicMultiplicativeArtinComparison
+import ValuedFieldTheory.LocalField.DiscreteValuationField.PadicValuationComparison
+import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.UnramifiedFrobenius
+import ValuedFieldTheory.LocalField.Padic.Cyclotomic.TotallyRamified.EisensteinPolynomial
+import Mathlib.NumberTheory.NumberField.Cyclotomic.Galois
+
+
+set_option autoImplicit false
+
+/-!
+# Finite-place Artin symbols in rational cyclotomic levels
+
+At a rational prime away from the cyclotomic level, the chosen completed
+extension is unramified.  Its normalized local Artin map is therefore the
+arithmetic Frobenius raised to the local valuation.  The genuine primitive
+root in the localized cyclotomic level identifies the image of arithmetic
+Frobenius under the global cyclotomic character with the residue prime.
+-/
+
+open scoped Classical NNReal NumberField ValuativeRel
+open NumberField IsDedekindDomain
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+-- Specializing the generic finite-place comparison to `ℚ` must retain its
+-- `Algebra.id` owner rather than selecting the competing rational-field
+-- instance introduced after specialization.
+@[reducible] noncomputable local instance
+    rationalFinitePlaceCompletionRatAlgebra
+    (v : HeightOneSpectrum (𝓞 ℚ)) :
+    Algebra ℚ (HeightOneSpectrum.adicAbv ℚ v).Completion := by
+  letI : Algebra ℚ ℚ := Algebra.id ℚ
+  let hWith : Algebra ℚ
+      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
+    WithAbs.instAlgebra _
+  let hUniform : UniformContinuousConstSMul ℚ
+      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) :=
+    WithAbs.instUniformContinuousConstSMulReal _
+  exact
+    @UniformSpace.Completion.algebra
+      (WithAbs (HeightOneSpectrum.adicAbv ℚ v)) _ _ _ _
+      ℚ _ hWith hUniform
+
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open LocalClassFieldTheory
+open LocalFieldTheory
+open LocalFieldTheory.DiscreteValuationField
+open LocalFieldTheory.DiscreteValuationField.Examples.Qp
+open LubinTate
+
+theorem mappedAbelianLocalArtin_eq_frobenius_zpow
+    {F E G : Type}
+    [Field F] [ValuativeRel F] [TopologicalSpace F]
+    [IsNonarchimedeanLocalField F]
+    [Field E] [ValuativeRel E] [UniformSpace E] [IsUniformAddGroup E]
+    [IsNonarchimedeanLocalField E]
+    [Algebra F E] [FiniteDimensional F E] [IsAbelianGalois F E]
+    [Valuation.HasExtension
+      (ValuativeRel.valuation F) (ValuativeRel.valuation E)]
+    [IsNonarchimedeanLocalField.IsUnramifiedValuedExtension F E]
+    [Group G]
+    (f : (E ≃ₐ[F] E) →* G) (x : Fˣ) :
+    f (LocalClassFieldTheory.abelianLocalArtinMonoidHom F E x) =
+      (f (arithmeticFrobeniusOfUnramifiedValuation F E)) ^
+        IsNonarchimedeanLocalField.valuationMap F
+          (Additive.ofMul x) := by
+  rw [LocalClassFieldTheory.abelianLocalArtinMonoidHom_eq_frobenius_zpow,
+    map_zpow]
+
+local instance primeFact (q : Nat.Primes) : Fact q.1.Prime :=
+  ⟨q.2⟩
+
+local instance levelNeZero (m : ℕ+) : NeZero (m : ℕ) :=
+  ⟨m.ne_zero⟩
+
+noncomputable local instance
+    rationalCyclotomicLevelFiniteDimensional
+    (m : ℕ+) :
+    FiniteDimensional ℚ
+      (KummerTheory.rationalCyclotomicLevel m) :=
+  IsCyclotomicExtension.finiteDimensional
+    {(m : ℕ)} ℚ (KummerTheory.rationalCyclotomicLevel m)
+
+noncomputable local instance
+    rationalCyclotomicLevelIsAbelianGalois
+    (m : ℕ+) :
+    IsAbelianGalois ℚ
+      (KummerTheory.rationalCyclotomicLevel m) := by
+  have : IsGalois ℚ
+      (KummerTheory.rationalCyclotomicLevel m) :=
+    inferInstance
+  let e :=
+    IsCyclotomicExtension.Rat.galEquivZMod
+      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m)
+  exact
+    { is_comm.comm σ τ := by
+        apply e.injective
+        simp only [map_mul]
+        exact mul_comm _ _ }
+
+@[reducible]
+noncomputable local instance rationalFinitePlaceBaseNontriviallyNormedField
+    (q : Nat.Primes) :
+    NontriviallyNormedField
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion :=
+  absoluteValueExtension_completionNontriviallyNormedField
+    (HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q))
+    (RayClass.adicAbv_isNontrivial
+      (RayClass.rationalPrime q))
+
+noncomputable local instance rationalFinitePlaceBaseLocallyCompactSpace
+    (q : Nat.Primes) :
+    LocallyCompactSpace
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion :=
+  AbsoluteValue.Completion.locallyCompactSpace
+    (finitePlaceCompletionBaseMap_isometry
+      (RayClass.rationalPrime q))
+
+noncomputable local instance rationalFinitePlaceBaseIsUltrametricDist
+    (q : Nat.Primes) :
+    IsUltrametricDist
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion :=
+  finitePlaceArtinCompletionIsUltrametricDist
+    (HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q))
+    (HeightOneSpectrum.isNonarchimedean_adicAbv
+      ℚ (RayClass.rationalPrime q))
+
+@[reducible]
+noncomputable local instance rationalFinitePlaceBaseValued
+    (q : Nat.Primes) :
+    Valued
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion ℝ≥0 :=
+  finitePlaceArtinCompletionValued
+    (HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q))
+    (HeightOneSpectrum.isNonarchimedean_adicAbv
+      ℚ (RayClass.rationalPrime q))
+
+@[reducible]
+noncomputable local instance rationalFinitePlaceBaseValuativeRel
+    (q : Nat.Primes) :
+    ValuativeRel
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion :=
+  finitePlaceLocalArtinCompletionValuativeRel
+    (K := ℚ) (RayClass.rationalPrime q)
+
+noncomputable local instance
+    rationalFinitePlaceBaseValuationIsNontrivial
+    (q : Nat.Primes) :
+    (Valued.v :
+      Valuation
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        ℝ≥0).IsNontrivial :=
+  (inferInstance :
+    (NormedField.valuation
+      (K :=
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion)).IsNontrivial)
+
+noncomputable local instance rationalFinitePlaceBaseValuationCompatible
+    (q : Nat.Primes) :
+    (Valued.v :
+      Valuation
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        ℝ≥0).Compatible :=
+  Valuation.Compatible.ofValuation _
+
+noncomputable local instance
+    rationalFinitePlaceBaseValuativeRelIsNontrivial
+    (q : Nat.Primes) :
+    ValuativeRel.IsNontrivial
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion :=
+  (ValuativeRel.isNontrivial_iff_isNontrivial
+    (Valued.v :
+      Valuation
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        ℝ≥0)).2 inferInstance
+
+noncomputable local instance rationalFinitePlaceBaseIsValuativeTopology
+    (q : Nat.Primes) :
+    IsValuativeTopology
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion :=
+  isValuativeTopology_of_valued_ofValuation
+    (HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q)).Completion ℝ≥0
+
+noncomputable local instance
+    rationalFinitePlaceBaseIsNonarchimedeanLocalField
+    (q : Nat.Primes) :
+    IsNonarchimedeanLocalField
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion :=
+  finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField
+    (K := ℚ) (RayClass.rationalPrime q)
+
+
+/-- Rational cyclotomic levels are finite-dimensional over `ℚ`. -/
+theorem rationalCyclotomicPrincipalPrimeLevelFiniteDimensional
+    (m : ℕ+) :
+    FiniteDimensional ℚ (KummerTheory.rationalCyclotomicLevel m) :=
+  rationalCyclotomicLevelFiniteDimensional m
+
+/-- Rational cyclotomic levels are abelian Galois extensions of `ℚ`. -/
+theorem rationalCyclotomicPrincipalPrimeLevelIsAbelianGalois
+    (m : ℕ+) :
+    IsAbelianGalois ℚ (KummerTheory.rationalCyclotomicLevel m) :=
+  rationalCyclotomicLevelIsAbelianGalois m
+
+/-- The canonical nontrivially normed field structure on the completion of
+`ℚ` at the rational prime `p`, exposed for principal-prime constructions. -/
+@[reducible]
+noncomputable def rationalPrimeFactorCompletionNontriviallyNormedField
+    (p : Nat.Primes) :
+    NontriviallyNormedField
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion :=
+  rationalFinitePlaceBaseNontriviallyNormedField p
+
+/-- The completion of `ℚ` at `p` is locally compact. -/
+theorem rationalPrimeFactorCompletionLocallyCompactSpace
+    (p : Nat.Primes) :
+    LocallyCompactSpace
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion :=
+  rationalFinitePlaceBaseLocallyCompactSpace p
+
+/-- The completion of `ℚ` at `p` carries its canonical ultrametric distance. -/
+theorem rationalPrimeFactorCompletionIsUltrametricDist
+    (p : Nat.Primes) :
+    IsUltrametricDist
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion :=
+  rationalFinitePlaceBaseIsUltrametricDist p
+
+/-- The canonical `ℝ≥0`-valued structure on the completion of `ℚ` at `p`. -/
+@[reducible]
+noncomputable def rationalPrimeFactorCompletionValued
+    (p : Nat.Primes) :
+    Valued
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion ℝ≥0 :=
+  rationalFinitePlaceBaseValued p
+
+/-- The valuative relation induced by the canonical valuation on the
+completion of `ℚ` at `p`. -/
+@[reducible]
+noncomputable def rationalPrimeFactorCompletionValuativeRel
+    (p : Nat.Primes) :
+    ValuativeRel
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion :=
+  rationalFinitePlaceBaseValuativeRel p
+
+/-- The canonical valuation on the completion of `ℚ` at `p` is nontrivial. -/
+theorem rationalPrimeFactorCompletionValuationIsNontrivial
+    (p : Nat.Primes) :
+    (Valued.v : Valuation
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion ℝ≥0).IsNontrivial :=
+  rationalFinitePlaceBaseValuationIsNontrivial p
+
+/-- The canonical valuation on the completion of `ℚ` at `p` is compatible
+with its field structure. -/
+theorem rationalPrimeFactorCompletionValuationCompatible
+    (p : Nat.Primes) :
+    (Valued.v : Valuation
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion ℝ≥0).Compatible :=
+  rationalFinitePlaceBaseValuationCompatible p
+
+/-- The canonical valuative relation on the completion at `p` is nontrivial. -/
+theorem rationalPrimeFactorCompletionValuativeRelIsNontrivial
+    (p : Nat.Primes) :
+    ValuativeRel.IsNontrivial
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion :=
+  rationalFinitePlaceBaseValuativeRelIsNontrivial p
+
+/-- The completion topology at `p` is induced by its canonical valuation. -/
+theorem rationalPrimeFactorCompletionIsValuativeTopology
+    (p : Nat.Primes) :
+    IsValuativeTopology
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion :=
+  rationalFinitePlaceBaseIsValuativeTopology p
+
+/-- The completion of `ℚ` at `p` is a nonarchimedean local field. -/
+theorem rationalPrimeFactorCompletionIsNonarchimedeanLocalField
+    (p : Nat.Primes) :
+    IsNonarchimedeanLocalField
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion :=
+  rationalFinitePlaceBaseIsNonarchimedeanLocalField p
+
+/-- The positive conductor of the `n`-th ramified cyclotomic level at
+`p`. -/
+def rationalCyclotomicPrincipalPrimeModulus
+    (p : Nat.Primes) (n : ℕ) : ℕ+ :=
+  ⟨p.1 ^ (n + 1), pow_pos p.2.pos (n + 1)⟩
+
+/-- The rational finite-place completion used at the prime `p`. -/
+abbrev RationalCyclotomicPrincipalPrimeCompletion
+    (p : Nat.Primes) :=
+  (HeightOneSpectrum.adicAbv ℚ
+    (RayClass.rationalPrime p)).Completion
+
+/-- The chosen localized cyclotomic field at level `p ^ (n + 1)`. -/
+abbrev RationalCyclotomicPrincipalPrimeLocalizedLevel
+    (p : Nat.Primes) (n : ℕ) :=
+  rationalCyclotomicLocalizedCompletion
+    (rationalCyclotomicPrincipalPrimeModulus p n)
+    (RayClass.rationalPrime p)
+
+/-- The standard multiplicative Lubin--Tate field at level `n`. -/
+abbrev RationalCyclotomicPrincipalPrimePadicLevel
+    (p : Nat.Primes) (n : ℕ) :=
+  standardLubinTateLevelField
+    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
+
+/-- The valuation ring of the absolute-value completion at `q`, identified
+with the standard p-adic integer ring `ℤ_q`. -/
+noncomputable def rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+    (q : Nat.Primes) :
+    𝒪[(HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion] ≃+*
+      ℤ_[q.1] := by
+  let v : HeightOneSpectrum (𝓞 ℚ) :=
+    RayClass.rationalPrime q
+  let vQ := HeightOneSpectrum.adicAbv ℚ v
+  let eConcreteIntegers :
+      𝒪[vQ.Completion] ≃+*
+        v.adicCompletionIntegers ℚ :=
+    finitePlaceCompletionIntegerRingEquiv v
+  exact
+    eConcreteIntegers.trans
+      (PadicInt.adicCompletionIntegersEquiv
+        (𝓞 ℚ) q).symm.toRingEquiv
+
+/-- The absolute-value completion at the rational prime `q`, identified
+with the standard field `ℚ_q`. -/
+noncomputable def rationalFinitePlaceCompletionRingEquivPadic
+    (q : Nat.Primes) :
+    (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion ≃+*
+      ℚ_[q.1] :=
+  IsFractionRing.ringEquivOfRingEquiv
+    (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q)
+
+/-- The completion-to-`ℚ_q` equivalence respects the rational embedding. -/
+theorem rationalFinitePlaceCompletionRingEquivPadic_algebraMap
+    (q : Nat.Primes) (a : ℚ) :
+    rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap ℚ
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion a) =
+      algebraMap ℚ ℚ_[q.1] a := by
+  exact
+    (rationalFinitePlaceCompletionRingEquivPadic q).toRingHom.map_rat_algebraMap a
+
+/-- The completion field equivalence and its restriction to valuation
+rings commute with the natural inclusions into the fields. -/
+theorem rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
+    (q : Nat.Primes)
+    (a :
+      𝒪[(HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion]) :
+    rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap
+          𝒪[(HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion]
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion a) =
+      algebraMap ℤ_[q.1] ℚ_[q.1]
+        (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q a) := by
+  exact
+    (IsFractionRing.ringEquivOfRingEquiv_algebraMap
+      (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q) a)
+
+/-- The canonical rational-completion equivalence preserves the canonical
+valuations. -/
+theorem
+    rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible
+    (p : Nat.Primes) :
+    SemilinearValuationCompatible
+      (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1]
+      (rationalFinitePlaceCompletionRingEquivPadic p) := by
+  let F := RationalCyclotomicPrincipalPrimeCompletion p
+  let eK := rationalFinitePlaceCompletionRingEquivPadic p
+  let : Algebra F ℚ_[p.1] := eK.toRingHom.toAlgebra
+  change
+    (ValuativeRel.valuation F).HasExtension
+      (ValuativeRel.valuation ℚ_[p.1])
+  let eO :=
+    (rationalFinitePlaceCompletionIntegerRingEquivPadicInt p).trans
+      (padicIntEquivValuationSubring p.1)
+  have hChosen :
+      (localCompleteDVF F).valuation.HasExtension
+        (padicDVRValuation p.1) := by
+    change
+      (localCompleteDVF F).valuation.HasExtension
+        (padicCompleteDVF p.1).valuation
+    apply
+      ValuationTheory.DiscreteValuationField.ValuedExtension.valuation_hasExtension_of_valuationSubring_equiv
+        (localCompleteDVF F)
+        (padicCompleteDVF p.1)
+        eO
+    intro z
+    change
+      algebraMap
+          (padicDVRValuation p.1).valuationSubring ℚ_[p.1]
+          (padicIntEquivValuationSubring p.1
+            (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+              p z)) =
+        rationalFinitePlaceCompletionRingEquivPadic p
+          (algebraMap 𝒪[F] F z)
+    symm
+    calc
+      rationalFinitePlaceCompletionRingEquivPadic p
+          (algebraMap 𝒪[F] F z) =
+        algebraMap ℤ_[p.1] ℚ_[p.1]
+          (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+            p z) :=
+        rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
+          p z
+      _ =
+        algebraMap
+          (padicDVRValuation p.1).valuationSubring ℚ_[p.1]
+          (padicIntEquivValuationSubring p.1
+            (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+              p z)) := by
+        rw [PadicInt.algebraMap_apply,
+          ValuationSubring.algebraMap_apply]
+        exact
+          (padicIntEquivValuationSubring_coe p.1
+            (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+              p z)).symm
+  have hBase :
+      (ValuativeRel.valuation F).HasExtension
+        (padicDVRValuation p.1) := by
+    rw [← localCompleteDVF_valuation_eq]
+    exact hChosen
+  refine
+    { val_isEquiv_comap := ?_ }
+  exact
+    hBase.val_isEquiv_comap.trans
+      ((padicDVRValuation_isEquiv_valuativeRelValuation
+        p.1).comap (algebraMap F ℚ_[p.1]))
+
+/-- The rational prime, pulled back from `ℤ_q` to the valuation ring of
+the absolute-value completion at `q`. -/
+noncomputable def rationalPrimeFinitePlaceInteger
+    (q : Nat.Primes) :
+    𝒪[(HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q)).Completion] :=
+  (rationalFinitePlaceCompletionIntegerRingEquivPadicInt q).symm
+    (q.1 : ℤ_[q.1])
+
+/-- The pulled-back rational prime is irreducible in the completion
+valuation ring. -/
+theorem rationalPrimeFinitePlaceInteger_irreducible
+    (q : Nat.Primes) :
+    Irreducible (rationalPrimeFinitePlaceInteger q) := by
+  exact
+    (MulEquiv.irreducible_iff
+      (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+        q).symm.toMulEquiv).2
+      ((PadicInt.prime_p :
+        Prime (q.1 : ℤ_[q.1])).irreducible)
+
+/-- Coercing the pulled-back prime to the completion field gives the
+ordinary image of the rational number `q`. -/
+theorem rationalPrimeFinitePlaceInteger_coe
+    (q : Nat.Primes) :
+    ((rationalPrimeFinitePlaceInteger q :
+      𝒪[(HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion]) :
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion) =
+      algebraMap ℚ
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        (q.1 : ℚ) := by
+  apply
+    (rationalFinitePlaceCompletionRingEquivPadic q).injective
+  change
+    rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap
+          𝒪[(HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion]
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (rationalPrimeFinitePlaceInteger q)) =
+      rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap ℚ
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (q.1 : ℚ))
+  calc
+    rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap
+          𝒪[(HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion]
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (rationalPrimeFinitePlaceInteger q)) =
+      algebraMap ℤ_[q.1] ℚ_[q.1]
+        ((rationalFinitePlaceCompletionIntegerRingEquivPadicInt q)
+          (rationalPrimeFinitePlaceInteger q)) :=
+        rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe
+          q (rationalPrimeFinitePlaceInteger q)
+    _ = (q.1 : ℚ_[q.1]) := by
+      rw [rationalPrimeFinitePlaceInteger,
+        (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+          q).apply_symm_apply,
+        PadicInt.algebraMap_apply,
+        PadicInt.coe_natCast]
+    _ =
+      rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap ℚ
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (q.1 : ℚ)) := by
+      rw [
+        rationalFinitePlaceCompletionRingEquivPadic_algebraMap]
+      norm_num
+
+/-- The rational prime as a field unit of its absolute-value completion. -/
+noncomputable def rationalPrimeFinitePlaceFieldUnit
+    (q : Nat.Primes) :
+    (HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q)).Completionˣ :=
+  Units.mk0
+    (rationalPrimeFinitePlaceInteger q :
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion)
+    (by
+      intro hzero
+      exact
+        (rationalPrimeFinitePlaceInteger_irreducible q).ne_zero
+          (Subtype.ext hzero))
+
+/-- In the inverse-standard local reciprocity normalization, the rational
+prime itself has normalized additive value `-1`. -/
+theorem rationalPrimeFinitePlaceFieldUnit_valuationMap
+    (q : Nat.Primes) :
+    IsNonarchimedeanLocalField.valuationMap
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        (Additive.ofMul
+          (rationalPrimeFinitePlaceFieldUnit q)) =
+      -1 := by
+  simpa [IsNonarchimedeanLocalField.valuationMap_apply] using
+    (LocalFieldTheory.v_integerRingIrreducibleFieldUnit
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion
+      (rationalPrimeFinitePlaceInteger q)
+      (rationalPrimeFinitePlaceInteger_irreducible q)
+      (rationalPrimeFinitePlaceFieldUnit q) rfl)
+
+/-- The rational `q`-unit part of `x`, pulled back from `ℤ_qˣ` to the
+valuation ring of the absolute-value completion. -/
+noncomputable def rationalPrimeUnitFinitePlaceIntegerUnit
+    (x : ℚˣ) (q : Nat.Primes) :
+    𝒪[(HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q)).Completion]ˣ :=
+  Units.map
+      (rationalFinitePlaceCompletionIntegerRingEquivPadicInt
+        q).symm.toMonoidHom
+    (padicIntUnitOfRat q
+      (rationalPrimeUnit x q : ℚ)
+      (rationalPrimeUnit x q).ne_zero
+      (padicValRat_rationalPrimeUnit x q))
+
+/-- Forgetting the integrality proof from the pulled-back `q`-unit gives
+the ordinary image of the rational `q`-unit in the completion field. -/
+theorem rationalPrimeUnitFinitePlaceIntegerUnit_coe
+    (x : ℚˣ) (q : Nat.Primes) :
+    (((rationalPrimeUnitFinitePlaceIntegerUnit x q :
+        𝒪[(HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion]ˣ) :
+        𝒪[(HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion]) :
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion) =
+      algebraMap ℚ
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        (rationalPrimeUnit x q : ℚ) := by
+  apply
+    (rationalFinitePlaceCompletionRingEquivPadic q).injective
+  change
+    rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap
+          𝒪[(HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion]
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
+            𝒪[(HeightOneSpectrum.adicAbv ℚ
+              (RayClass.rationalPrime q)).Completion])) =
+      rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap ℚ
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (rationalPrimeUnit x q : ℚ))
+  calc
+    rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap
+          𝒪[(HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion]
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
+          𝒪[(HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion])) =
+      algebraMap ℤ_[q.1] ℚ_[q.1]
+        ((rationalFinitePlaceCompletionIntegerRingEquivPadicInt q)
+          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
+            𝒪[(HeightOneSpectrum.adicAbv ℚ
+              (RayClass.rationalPrime q)).Completion])) :=
+        rationalFinitePlaceCompletionIntegerRingEquivPadicInt_coe q
+          (rationalPrimeUnitFinitePlaceIntegerUnit x q :
+            𝒪[(HeightOneSpectrum.adicAbv ℚ
+              (RayClass.rationalPrime q)).Completion])
+    _ =
+      ((padicIntUnitOfRat q
+        (rationalPrimeUnit x q : ℚ)
+        (rationalPrimeUnit x q).ne_zero
+        (padicValRat_rationalPrimeUnit x q) :
+          ℤ_[q.1]) : ℚ_[q.1]) := by
+      rw [rationalPrimeUnitFinitePlaceIntegerUnit]
+      simp [PadicInt.algebraMap_apply]
+    _ = ((rationalPrimeUnit x q : ℚ) : ℚ_[q.1]) := by
+      rw [padicIntUnitOfRat_coe]
+    _ =
+      rationalFinitePlaceCompletionRingEquivPadic q
+        (algebraMap ℚ
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion
+          (rationalPrimeUnit x q : ℚ)) := by
+      rw [
+        rationalFinitePlaceCompletionRingEquivPadic_algebraMap]
+      simp
+
+/-- The completion field unit underlying the pulled-back rational `q`-unit
+has normalized additive value zero. -/
+theorem rationalPrimeUnitFinitePlaceField_valuationMap
+    (x : ℚˣ) (q : Nat.Primes) :
+    IsNonarchimedeanLocalField.valuationMap
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        (Additive.ofMul
+          (IsNonarchimedeanLocalField.integerUnitsToFieldUnits
+            (HeightOneSpectrum.adicAbv ℚ
+              (RayClass.rationalPrime q)).Completion
+            (rationalPrimeUnitFinitePlaceIntegerUnit x q))) =
+      0 := by
+  rw [IsNonarchimedeanLocalField.valuationMap_apply]
+  exact
+    IsNonarchimedeanLocalField.v_integerUnitsToFieldUnits
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime q)).Completion
+      (rationalPrimeUnitFinitePlaceIntegerUnit x q)
+
+/-- The rational prime `q`, regarded as a unit of `ℚ`. -/
+def rationalPrimeGeneratorUnit (q : Nat.Primes) : ℚˣ :=
+  Units.mk0 (q.1 : ℚ) (by exact_mod_cast q.2.ne_zero)
+
+/-- The underlying rational number of the prime generator unit is `q`. -/
+@[simp]
+theorem rationalPrimeGeneratorUnit_coe (q : Nat.Primes) :
+    (rationalPrimeGeneratorUnit q : ℚ) = q.1 :=
+  rfl
+
+/-- Reattaching the removed `q`-power to the rational `q`-unit recovers
+the original rational field unit. -/
+theorem rationalPrimeGeneratorUnit_zpow_mul_rationalPrimeUnit
+    (x : ℚˣ) (q : Nat.Primes) :
+    rationalPrimeGeneratorUnit q ^
+          padicValRat q.1 (x : ℚ) *
+        rationalPrimeUnit x q =
+      x := by
+  rw [rationalPrimeGeneratorUnit, rationalPrimeUnit,
+    ← mul_assoc, ← zpow_add]
+  simp
+
+/-- The source unit in the absolute-value completion represented by the
+finite component of a rational principal idele. -/
+noncomputable def rationalPrincipalFinitePlaceInput
+    (x : ℚˣ) (q : Nat.Primes) :
+    (HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q)).Completionˣ :=
+  (finitePlaceCompletionUnitsContinuousMulEquiv
+      (RayClass.rationalPrime q)).symm
+    (IdeleGroup.finiteComponent
+      (RayClass.rationalPrime q)
+      (IdeleGroup.principalIdele ℚ x))
+
+/-- The source unit represented by a principal finite component is the
+ordinary image of the rational field unit in the absolute-value
+completion. -/
+theorem rationalPrincipalFinitePlaceInput_eq_algebraMap
+    (x : ℚˣ) (q : Nat.Primes) :
+    rationalPrincipalFinitePlaceInput x q =
+      Units.map
+        (algebraMap ℚ
+          (HeightOneSpectrum.adicAbv ℚ
+            (RayClass.rationalPrime q)).Completion).toMonoidHom
+        x := by
+  apply Units.ext
+  let v : HeightOneSpectrum (𝓞 ℚ) :=
+    RayClass.rationalPrime q
+  let vQ := HeightOneSpectrum.adicAbv ℚ v
+  let component : (v.adicCompletion ℚ)ˣ :=
+    IdeleGroup.finiteComponent v
+      (IdeleGroup.principalIdele ℚ x)
+  apply (finitePlaceCompletionRingEquiv v).injective
+  change
+    finitePlaceCompletionRingEquiv v
+        (rationalPrincipalFinitePlaceInput x q : vQ.Completion) =
+      finitePlaceCompletionRingEquiv v
+        (algebraMap ℚ vQ.Completion (x : ℚ))
+  have hCompletion :
+      finitePlaceCompletionRingEquiv v
+          (rationalPrincipalFinitePlaceInput x q :
+            vQ.Completion) =
+        (component : v.adicCompletion ℚ) := by
+    have hUnits :=
+      congrArg Units.val
+        ((finitePlaceCompletionUnitsContinuousMulEquiv v).apply_symm_apply
+          component)
+    exact hUnits
+  have hComponent :
+      (component : v.adicCompletion ℚ) =
+        algebraMap ℚ (v.adicCompletion ℚ) (x : ℚ) := by
+    apply (Padic.adicCompletionEquiv (𝓞 ℚ) q).symm.injective
+    calc
+      (Padic.adicCompletionEquiv (𝓞 ℚ) q).symm
+          (component : v.adicCompletion ℚ) =
+        algebraMap ℚ ℚ_[q.1] (x : ℚ) := by
+          dsimp only [component]
+          rw [IdeleGroup.finiteComponent_principalIdele]
+          exact
+            (Padic.adicCompletionEquiv
+              (𝓞 ℚ) q).symm.commutes (x : ℚ)
+      _ = (Padic.adicCompletionEquiv (𝓞 ℚ) q).symm
+          (algebraMap ℚ (v.adicCompletion ℚ) (x : ℚ)) := by
+        symm
+        exact
+          (Padic.adicCompletionEquiv
+            (𝓞 ℚ) q).symm.commutes (x : ℚ)
+  calc
+    finitePlaceCompletionRingEquiv v
+        (rationalPrincipalFinitePlaceInput x q :
+          vQ.Completion) =
+      (component : v.adicCompletion ℚ) := hCompletion
+    _ = algebraMap ℚ (v.adicCompletion ℚ) (x : ℚ) := hComponent
+    _ = finitePlaceCompletionRingEquiv v
+        (algebraMap ℚ vQ.Completion (x : ℚ)) := by
+      symm
+      exact
+        (finitePlaceCompletionAlgEquiv (K := ℚ) v).commutes (x : ℚ)
+
+/-- The normalized local exponent of a rational principal finite
+component is the negative of the usual `q`-adic exponent.  The minus sign
+records the inverse-standard local reciprocity convention in which a
+prime element has normalized value `-1`. -/
+theorem rationalPrincipalFiniteComponent_valuationMap
+    (x : ℚˣ) (q : Nat.Primes) :
+    IsNonarchimedeanLocalField.valuationMap
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        (Additive.ofMul
+          ((finitePlaceCompletionUnitsContinuousMulEquiv
+            (RayClass.rationalPrime q)).symm
+            (IdeleGroup.finiteComponent
+              (RayClass.rationalPrime q)
+              (IdeleGroup.principalIdele ℚ x)))) =
+      -padicValRat q.1 (x : ℚ) := by
+  let F :=
+    (HeightOneSpectrum.adicAbv ℚ
+      (RayClass.rationalPrime q)).Completion
+  let embed : ℚˣ →* Fˣ :=
+    Units.map (algebraMap ℚ F).toMonoidHom
+  let primeUnit : Fˣ :=
+    rationalPrimeFinitePlaceFieldUnit q
+  let integralUnit : 𝒪[F]ˣ :=
+    rationalPrimeUnitFinitePlaceIntegerUnit x q
+  let unitPart : Fˣ :=
+    IsNonarchimedeanLocalField.integerUnitsToFieldUnits
+      F integralUnit
+  have hInput :
+      rationalPrincipalFinitePlaceInput x q =
+        embed x := by
+    exact rationalPrincipalFinitePlaceInput_eq_algebraMap x q
+  have hPrime :
+      embed (rationalPrimeGeneratorUnit q) =
+        primeUnit := by
+    apply Units.ext
+    exact
+      (rationalPrimeFinitePlaceInteger_coe q).symm
+  have hUnit :
+      embed (rationalPrimeUnit x q) =
+        unitPart := by
+    apply Units.ext
+    exact
+      (rationalPrimeUnitFinitePlaceIntegerUnit_coe
+        x q).symm
+  change
+    IsNonarchimedeanLocalField.valuationMap F
+        (Additive.ofMul
+          (rationalPrincipalFinitePlaceInput x q)) =
+      -padicValRat q.1 (x : ℚ)
+  rw [hInput]
+  conv_lhs =>
+    rw [← rationalPrimeGeneratorUnit_zpow_mul_rationalPrimeUnit
+      x q]
+  rw [map_mul, map_zpow, hPrime, hUnit,
+    IsNonarchimedeanLocalField.valuationMap_ofMul_mul,
+    IsNonarchimedeanLocalField.valuationMap_ofMul_zpow,
+    rationalPrimeFinitePlaceFieldUnit_valuationMap,
+    rationalPrimeUnitFinitePlaceField_valuationMap]
+  ring
+
+/-- The principal finite component of the rational prime itself has
+normalized local exponent `-1`. -/
+theorem rationalPrimePrincipalFiniteComponent_valuationMap
+    (q : Nat.Primes) :
+    IsNonarchimedeanLocalField.valuationMap
+        (HeightOneSpectrum.adicAbv ℚ
+          (RayClass.rationalPrime q)).Completion
+        (Additive.ofMul
+          ((finitePlaceCompletionUnitsContinuousMulEquiv
+            (RayClass.rationalPrime q)).symm
+            (IdeleGroup.finiteComponent
+              (RayClass.rationalPrime q)
+              (IdeleGroup.principalIdele ℚ
+                (rationalPrimeGeneratorUnit q))))) =
+      -1 := by
+  rw [rationalPrincipalFiniteComponent_valuationMap,
+    rationalPrimeGeneratorUnit_coe,
+    padicValRat.self q.2.one_lt]
+
+/-- A cyclotomic automorphism which raises the selected primitive root to
+the `q`-th power has cyclotomic character equal to the residue-prime unit. -/
+theorem rationalCyclotomicLevel_galEquivZMod_eq_unitOfCoprime
+    (m : ℕ+) (q : Nat.Primes)
+    (hq : ¬ q.1 ∣ (m : ℕ))
+    (σ : KummerTheory.rationalCyclotomicLevel m ≃ₐ[ℚ]
+      KummerTheory.rationalCyclotomicLevel m)
+    (hσ :
+      σ (rationalCyclotomicLevelPrimitiveRoot m) =
+        rationalCyclotomicLevelPrimitiveRoot m ^ q.1) :
+    IsCyclotomicExtension.Rat.galEquivZMod
+        (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ =
+      ZMod.unitOfCoprime q.1
+        (q.2.coprime_iff_not_dvd.mpr hq) := by
+  let ζ := rationalCyclotomicLevelPrimitiveRoot m
+  have hζ : IsPrimitiveRoot ζ (m : ℕ) :=
+    rationalCyclotomicLevelPrimitiveRoot_isPrimitiveRoot m
+  have hCharacterRoot :
+      σ ζ =
+        ζ ^
+          (IsCyclotomicExtension.Rat.galEquivZMod
+            (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ).val.val :=
+    IsCyclotomicExtension.Rat.galEquivZMod_apply_of_pow_eq
+      (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ
+      hζ.pow_eq_one
+  have hPowers :
+      ζ ^
+          (IsCyclotomicExtension.Rat.galEquivZMod
+            (m : ℕ) (KummerTheory.rationalCyclotomicLevel m) σ).val.val =
+        ζ ^ q.1 :=
+    hCharacterRoot.symm.trans hσ
+  rw [(hζ.isOfFinOrder m.ne_zero).pow_inj_mod,
+    ← hζ.eq_orderOf,
+    ← ZMod.natCast_eq_natCast_iff',
+    ZMod.natCast_val] at hPowers
+  apply Units.ext
+  simpa using hPowers
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinPadic.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinPadic.lean
new file mode 100644
index 0000000..b430bec
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinPadic.lean
@@ -0,0 +1,389 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlaceArtinTower
+
+
+set_option autoImplicit false
+
+open scoped Classical NNReal NumberField ValuativeRel
+open NumberField IsDedekindDomain
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open LocalClassFieldTheory
+open LocalFieldTheory
+open LocalFieldTheory.DiscreteValuationField
+open LocalFieldTheory.DiscreteValuationField.Examples.Qp
+open LubinTate
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+attribute [local instance]
+  rationalFinitePlaceCompletionRatAlgebra
+  primeFact
+  levelNeZero
+  rationalCyclotomicLevelFiniteDimensional
+  rationalCyclotomicLevelIsAbelianGalois
+  rationalFinitePlaceBaseNontriviallyNormedField
+  rationalFinitePlaceBaseLocallyCompactSpace
+  rationalFinitePlaceBaseIsUltrametricDist
+  rationalFinitePlaceBaseValued
+  rationalFinitePlaceBaseValuativeRel
+  rationalFinitePlaceBaseValuationIsNontrivial
+  rationalFinitePlaceBaseValuationCompatible
+  rationalFinitePlaceBaseValuativeRelIsNontrivial
+  rationalFinitePlaceBaseIsValuativeTopology
+  rationalFinitePlaceBaseIsNonarchimedeanLocalField
+  rationalCyclotomicArtinExtensionAlgebra
+  rationalCyclotomicArtinExtensionSMul
+  rationalCyclotomicArtinCompletionAlgebra
+  rationalCyclotomicArtinLocalizedAlgebra
+  rationalCyclotomicArtinLocalizedGlobalAlgebra
+  rationalCyclotomicArtinLocalizedGlobalSMul
+  rationalCyclotomicArtinLocalizedScalarTower
+  rationalCyclotomicArtinLocalizedFiniteDimensional
+  rationalCyclotomicArtinLocalizedIsAbelianGalois
+  rationalCyclotomicArtinLocalizedIsSeparable
+  rationalCyclotomicArtinLocalizedIsCyclotomic
+  rationalCyclotomicArtinExtensionFiniteDimensional
+  rationalCyclotomicArtinExtensionContinuousSMul
+  rationalCyclotomicArtinExtensionLocallyCompact
+  rationalCyclotomicArtinLocalizedLocallyCompact
+  rationalCyclotomicArtinLocalizedIsUltrametricDist
+  rationalCyclotomicArtinLocalizedValued
+  rationalCyclotomicArtinLocalizedValuativeRel
+  rationalCyclotomicArtinLocalizedValuationCompatible
+  rationalCyclotomicArtinLocalizedValuationHasExtension
+  rationalCyclotomicArtinLocalizedValuationIsNontrivial
+  rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
+  rationalCyclotomicArtinLocalizedIsValuativeTopology
+  rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
+  rationalCyclotomicArtinLocalizedIntegerAlgebra
+  rationalCyclotomicArtinLocalizedIsIntegralClosure
+  rationalCyclotomicArtinLocalizedIntegerModuleFinite
+
+section RationalCyclotomicPrincipalPrime
+
+/-! ## Ramified prime-power transport
+
+This section reuses the canonical finite-place Artin tower above.  In
+particular, it introduces no parallel completion/localization instance tower. -/
+
+noncomputable local instance
+    rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
+    (p : Nat.Primes) (n : ℕ) :
+    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ
+      (KummerTheory.rationalCyclotomicLevel
+        (rationalCyclotomicPrincipalPrimeModulus p n)) := by
+  change
+    IsCyclotomicExtension
+      {(rationalCyclotomicPrincipalPrimeModulus p n : ℕ)} ℚ
+      (KummerTheory.rationalCyclotomicLevel
+        (rationalCyclotomicPrincipalPrimeModulus p n))
+  exact
+    KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
+      (rationalCyclotomicPrincipalPrimeModulus p n)
+
+private abbrev rationalCyclotomicPrincipalPrimePlace
+    (p : Nat.Primes) : HeightOneSpectrum (𝓞 ℚ) :=
+  rationalCyclotomicArtinPlace p
+
+/-- The cyclotomic extension of `ℚ` generated by roots of unity of the given positive level. -/
+abbrev rationalCyclotomicPrincipalPrimeLevel
+    (m : ℕ+) :=
+  rationalCyclotomicArtinLevel m
+
+private abbrev rationalCyclotomicPrincipalPrimeExtension
+    (m : ℕ+) (p : Nat.Primes) :=
+  rationalCyclotomicArtinExtension m p
+
+/-- The `ℚ_[p]`-algebra structure on the localized cyclotomic completion,
+transported through the canonical comparison with the `p`-adic completion. -/
+@[reducible]
+noncomputable def
+    rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra
+    (m : ℕ+) (p : Nat.Primes) :
+    Algebra ℚ_[p.1]
+      (rationalCyclotomicLocalizedCompletion m
+        (RayClass.rationalPrime p)) :=
+  ((algebraMap
+      (HeightOneSpectrum.adicAbv ℚ
+        (RayClass.rationalPrime p)).Completion
+      (rationalCyclotomicLocalizedCompletion m
+        (RayClass.rationalPrime p))).comp
+    (rationalFinitePlaceCompletionRingEquivPadic p).symm.toRingHom).toAlgebra
+
+@[reducible]
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedPadicAlgebra
+    (m : ℕ+) (p : Nat.Primes) :
+    Algebra ℚ_[p.1] (rationalCyclotomicArtinLocalizedField m p) :=
+  rationalCyclotomicPrincipalPrimeLocalizedPadicAlgebra m p
+
+private noncomputable def rationalFinitePlaceCompletionAlgEquivPadic
+    (p : Nat.Primes) :
+    (rationalCyclotomicArtinBaseAbv p).Completion ≃ₐ[ℚ] ℚ_[p.1] :=
+  AlgEquiv.ofRingEquiv
+    (f := rationalFinitePlaceCompletionRingEquivPadic p)
+    (rationalFinitePlaceCompletionRingEquivPadic_algebraMap p)
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedPadicScalarTower
+    (m : ℕ+) (p : Nat.Primes) :
+    IsScalarTower ℚ ℚ_[p.1]
+      (rationalCyclotomicArtinLocalizedField m p) := by
+  constructor
+  intro r x y
+  simp only [Algebra.smul_def, map_mul, eq_ratCast,
+    map_ratCast, mul_assoc]
+
+private theorem rationalCyclotomicArtin_padic_algebraMap
+    (m : ℕ+) (p : Nat.Primes) :
+    algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
+        (rationalCyclotomicArtinLocalizedField m p) =
+      (algebraMap ℚ_[p.1]
+        (rationalCyclotomicArtinLocalizedField m p)) ∘
+        (rationalFinitePlaceCompletionAlgEquivPadic p) := by
+  funext a
+  change
+    algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
+        (rationalCyclotomicArtinLocalizedField m p) a =
+      algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
+        (rationalCyclotomicArtinLocalizedField m p)
+        ((rationalFinitePlaceCompletionRingEquivPadic p).symm
+          (rationalFinitePlaceCompletionRingEquivPadic p a))
+  exact
+    congrArg
+      (algebraMap (rationalCyclotomicArtinBaseAbv p).Completion
+        (rationalCyclotomicArtinLocalizedField m p))
+      ((rationalFinitePlaceCompletionRingEquivPadic p).symm_apply_apply a).symm
+
+private theorem rationalCyclotomicArtin_algebraAdjoin_restrictScalars
+    (m : ℕ+) (p : Nat.Primes) :
+    (Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
+      ({rationalCyclotomicLocalizedPrimitiveRoot m
+        (rationalCyclotomicArtinPlace p)} :
+        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
+      (Algebra.adjoin ℚ_[p.1]
+        ({rationalCyclotomicLocalizedPrimitiveRoot m
+          (rationalCyclotomicArtinPlace p)} :
+          Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ := by
+  exact
+    Algebra.restrictScalars_adjoin_of_algEquiv
+      (E := rationalCyclotomicArtinLocalizedField m p)
+      (rationalFinitePlaceCompletionAlgEquivPadic p)
+      (rationalCyclotomicArtin_padic_algebraMap m p)
+      ({rationalCyclotomicLocalizedPrimitiveRoot m
+        (rationalCyclotomicArtinPlace p)} :
+        Set (rationalCyclotomicArtinLocalizedField m p))
+
+private theorem
+    rationalCyclotomicArtin_baseAlgebraAdjoin_restrict_eq_top
+    (m : ℕ+) (p : Nat.Primes) :
+    (Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
+      ({rationalCyclotomicLocalizedPrimitiveRoot m
+        (rationalCyclotomicArtinPlace p)} :
+        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
+      (⊤ : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
+      (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ := by
+  have hRoot : IsPrimitiveRoot
+      (show rationalCyclotomicArtinLocalizedField m p from
+        rationalCyclotomicLocalizedPrimitiveRoot m
+          (rationalCyclotomicArtinPlace p))
+      (m : ℕ) :=
+    rationalCyclotomicLocalizedPrimitiveRoot_isPrimitiveRoot
+      m (rationalCyclotomicArtinPlace p)
+  have hTop :
+      Algebra.adjoin (rationalCyclotomicArtinBaseAbv p).Completion
+        ({rationalCyclotomicLocalizedPrimitiveRoot m
+          (rationalCyclotomicArtinPlace p)} :
+          Set (rationalCyclotomicArtinLocalizedField m p)) = ⊤ :=
+    IsCyclotomicExtension.adjoin_primitive_root_eq_top
+      (A := (rationalCyclotomicArtinBaseAbv p).Completion)
+      (B := rationalCyclotomicArtinLocalizedField m p) hRoot
+  exact congrArg
+    (fun A : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
+      (rationalCyclotomicArtinLocalizedField m p) => A.restrictScalars ℚ)
+    hTop
+
+private theorem rationalCyclotomicArtin_restrictScalars_top_base_eq_padic
+    (m : ℕ+) (p : Nat.Primes) :
+    (⊤ : Subalgebra (rationalCyclotomicArtinBaseAbv p).Completion
+        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ =
+      (⊤ : Subalgebra ℚ_[p.1]
+        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ :=
+  (Subalgebra.restrictScalars_top ℚ).trans
+    (Subalgebra.restrictScalars_top ℚ).symm
+
+private theorem
+    rationalCyclotomicArtin_padicAlgebraAdjoin_restrict_eq_top
+    (m : ℕ+) (p : Nat.Primes) :
+    (Algebra.adjoin ℚ_[p.1]
+      ({rationalCyclotomicLocalizedPrimitiveRoot m
+        (rationalCyclotomicArtinPlace p)} :
+        Set (rationalCyclotomicArtinLocalizedField m p))).restrictScalars ℚ =
+      (⊤ : Subalgebra ℚ_[p.1]
+        (rationalCyclotomicArtinLocalizedField m p)).restrictScalars ℚ := by
+  exact
+    (rationalCyclotomicArtin_algebraAdjoin_restrictScalars m p).symm.trans
+      ((rationalCyclotomicArtin_baseAlgebraAdjoin_restrict_eq_top m p).trans
+        (rationalCyclotomicArtin_restrictScalars_top_base_eq_padic m p))
+
+/-- The finite-dimensional instance for the standard multiplicative level,
+named once so all consumers use the same proof term. -/
+theorem rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
+    (p : Nat.Primes) (n : ℕ) :
+    FiniteDimensional ℚ_[p.1]
+      (standardLubinTateLevelField
+        (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n) :=
+  standardLubinTateLevelField_finiteDimensional
+    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
+
+attribute [local instance]
+  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
+
+noncomputable local instance
+    rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
+    (p : Nat.Primes) (n : ℕ) :
+    IsAbelianGalois ℚ_[p.1]
+      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
+  standardLubinTateLevelField_isAbelianGalois
+    (padicLocalField p.1)
+    (padicMultiplicativeLubinTateSeries_isUniformizer p.1) n
+
+@[reducible]
+noncomputable local instance rationalPrimeFactorCompletionPadicAlgebra
+    (p : Nat.Primes) :
+    Algebra (RationalCyclotomicPrincipalPrimeCompletion p) ℚ_[p.1] :=
+  (rationalFinitePlaceCompletionRingEquivPadic p).toRingHom.toAlgebra
+
+/-- The genuine multiplicative Lubin--Tate level is generated by its
+primitive `p ^ (n + 1)`-st root of unity. -/
+theorem padicMultiplicativePrimitiveRoot_adjoin_eq_top
+    (p : ℕ) [Fact p.Prime] (n : ℕ) :
+    let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
+    let T := standardLubinTateLevelField hπ n
+    Algebra.adjoin ℚ_[p]
+        ({padicMultiplicativePrimitiveRoot p n} : Set T) =
+      ⊤ := by
+  let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
+  let T := standardLubinTateLevelField hπ n
+  let ζ : T := padicMultiplicativePrimitiveRoot p n
+  let m := p ^ (n + 1)
+  let : NeZero m :=
+    ⟨pow_ne_zero _ (Fact.out : Nat.Prime p).ne_zero⟩
+  let : FiniteDimensional ℚ_[p] T :=
+    standardLubinTateLevelField_finiteDimensional hπ n
+  have hζ : IsPrimitiveRoot ζ m := by
+    simpa only [ζ, m] using
+      padicMultiplicativePrimitiveRoot_isPrimitiveRoot p n
+  let A : IntermediateField ℚ_[p] T :=
+    IntermediateField.adjoin ℚ_[p] {ζ}
+  let : IsCyclotomicExtension {m} ℚ_[p] A :=
+    hζ.intermediateField_adjoin_isCyclotomicExtension ℚ_[p]
+  have hAfin :
+      Module.finrank ℚ_[p] A = Nat.totient m := by
+    exact
+      IsCyclotomicExtension.finrank A
+        (by
+          simpa only [m] using
+            padicCyclotomicPolynomial_irreducible_prime_pow_succ
+              p n)
+  have hTfin :
+      Module.finrank ℚ_[p] T = Nat.totient m := by
+    rw [standardLubinTateLevelField_finrank hπ n]
+    have hcard :
+        Nat.card (padicLocalField p).residueField = p := by
+      simpa [padicLocalField] using
+        padicCompleteDVF_residueField_card p
+    rw [hcard, Nat.totient_prime_pow
+      (Fact.out : Nat.Prime p) (Nat.succ_pos n)]
+    simp [Nat.mul_comm]
+  have hAeq : A = ⊤ := by
+    apply IntermediateField.eq_of_le_of_finrank_eq le_top
+    simpa using hAfin.trans hTfin.symm
+  calc
+    Algebra.adjoin ℚ_[p] {ζ} = A.toSubalgebra := by
+      exact
+        (IntermediateField.adjoin_toSubalgebra
+          ({ζ} : Set T)).symm
+    _ = (⊤ : IntermediateField ℚ_[p] T).toSubalgebra :=
+      congrArg IntermediateField.toSubalgebra hAeq
+    _ = ⊤ := rfl
+
+/-- The standard multiplicative Lubin--Tate level is the actual
+`p ^ (n + 1)`-cyclotomic extension of `ℚ_p`. -/
+theorem padicMultiplicativeLevel_isCyclotomicExtension
+    (p : ℕ) [Fact p.Prime] (n : ℕ) :
+    let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
+    let T := standardLubinTateLevelField hπ n
+    IsCyclotomicExtension {p ^ (n + 1)} ℚ_[p] T := by
+  let hπ := padicMultiplicativeLubinTateSeries_isUniformizer p
+  let T := standardLubinTateLevelField hπ n
+  exact
+    padic_isCyclotomicExtension_of_primitiveRoot_adjoin_eq_top
+      (padicMultiplicativePrimitiveRoot p n)
+      (padicMultiplicativePrimitiveRoot_isPrimitiveRoot p n)
+      (padicMultiplicativePrimitiveRoot_adjoin_eq_top p n)
+
+private theorem
+    rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_isPrimitiveRoot
+    (p : Nat.Primes) (n : ℕ) :
+    IsPrimitiveRoot
+      (rationalCyclotomicLocalizedPrimitiveRoot
+        (rationalCyclotomicPrincipalPrimeModulus p n)
+        (RayClass.rationalPrime p))
+      (p.1 ^ (n + 1)) := by
+  change
+    IsPrimitiveRoot
+      (rationalCyclotomicLocalizedPrimitiveRoot
+        (rationalCyclotomicPrincipalPrimeModulus p n)
+        (RayClass.rationalPrime p))
+      (rationalCyclotomicPrincipalPrimeModulus p n : ℕ)
+  exact
+    rationalCyclotomicLocalizedPrimitiveRoot_isPrimitiveRoot
+      (rationalCyclotomicPrincipalPrimeModulus p n)
+      (RayClass.rationalPrime p)
+
+private theorem
+    rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_adjoin_eq_top
+    (p : Nat.Primes) (n : ℕ) :
+    Algebra.adjoin ℚ_[p.1]
+        ({rationalCyclotomicLocalizedPrimitiveRoot
+          (rationalCyclotomicPrincipalPrimeModulus p n)
+          (rationalCyclotomicArtinPlace p)} :
+          Set (rationalCyclotomicArtinLocalizedField
+            (rationalCyclotomicPrincipalPrimeModulus p n) p)) =
+      ⊤ := by
+  exact
+    (Subalgebra.restrictScalars_injective ℚ)
+      (rationalCyclotomicArtin_padicAlgebraAdjoin_restrict_eq_top
+        (rationalCyclotomicPrincipalPrimeModulus p n) p)
+
+theorem
+    rationalCyclotomicPrincipalPrimeLocalizedLevel_isCyclotomicExtension
+    (p : Nat.Primes) (n : ℕ) :
+    IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ_[p.1]
+      (rationalCyclotomicArtinLocalizedField
+        (rationalCyclotomicPrincipalPrimeModulus p n) p) := by
+  exact
+    padic_isCyclotomicExtension_of_primitiveRoot_adjoin_eq_top
+      (rationalCyclotomicLocalizedPrimitiveRoot
+        (rationalCyclotomicPrincipalPrimeModulus p n)
+        (rationalCyclotomicArtinPlace p))
+      (rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_isPrimitiveRoot
+        p n)
+      (rationalCyclotomicPrincipalPrimeLocalizedPrimitiveRoot_adjoin_eq_top
+        p n)
+
+
+end RationalCyclotomicPrincipalPrime
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinPrincipal.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinPrincipal.lean
new file mode 100644
index 0000000..43c227c
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinPrincipal.lean
@@ -0,0 +1,228 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlaceArtinAction
+
+
+set_option autoImplicit false
+
+open scoped Classical NNReal NumberField ValuativeRel
+open NumberField IsDedekindDomain
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open LocalClassFieldTheory
+open LocalFieldTheory
+open LocalFieldTheory.DiscreteValuationField
+open LocalFieldTheory.DiscreteValuationField.Examples.Qp
+open LubinTate
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+section RationalCyclotomicPrincipalPrime
+
+attribute [local instance]
+  rationalFinitePlaceCompletionRatAlgebra
+  primeFact
+  levelNeZero
+  rationalCyclotomicLevelFiniteDimensional
+  rationalCyclotomicLevelIsAbelianGalois
+  rationalFinitePlaceBaseNontriviallyNormedField
+  rationalFinitePlaceBaseLocallyCompactSpace
+  rationalFinitePlaceBaseIsUltrametricDist
+  rationalFinitePlaceBaseValued
+  rationalFinitePlaceBaseValuativeRel
+  rationalFinitePlaceBaseValuationIsNontrivial
+  rationalFinitePlaceBaseValuationCompatible
+  rationalFinitePlaceBaseValuativeRelIsNontrivial
+  rationalFinitePlaceBaseIsValuativeTopology
+  rationalFinitePlaceBaseIsNonarchimedeanLocalField
+  rationalCyclotomicArtinExtensionAlgebra
+  rationalCyclotomicArtinExtensionSMul
+  rationalCyclotomicArtinCompletionAlgebra
+  rationalCyclotomicArtinLocalizedAlgebra
+  rationalCyclotomicArtinLocalizedGlobalAlgebra
+  rationalCyclotomicArtinLocalizedGlobalSMul
+  rationalCyclotomicArtinLocalizedScalarTower
+  rationalCyclotomicArtinLocalizedFiniteDimensional
+  rationalCyclotomicArtinLocalizedIsAbelianGalois
+  rationalCyclotomicArtinLocalizedIsSeparable
+  rationalCyclotomicArtinLocalizedIsCyclotomic
+  rationalCyclotomicArtinExtensionFiniteDimensional
+  rationalCyclotomicArtinExtensionContinuousSMul
+  rationalCyclotomicArtinExtensionLocallyCompact
+  rationalCyclotomicArtinLocalizedLocallyCompact
+  rationalCyclotomicArtinLocalizedIsUltrametricDist
+  rationalCyclotomicArtinLocalizedValued
+  rationalCyclotomicArtinLocalizedValuativeRel
+  rationalCyclotomicArtinLocalizedValuationCompatible
+  rationalCyclotomicArtinLocalizedValuationHasExtension
+  rationalCyclotomicArtinLocalizedValuationIsNontrivial
+  rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
+  rationalCyclotomicArtinLocalizedIsValuativeTopology
+  rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
+  rationalCyclotomicArtinLocalizedIntegerAlgebra
+  rationalCyclotomicArtinLocalizedIsIntegralClosure
+  rationalCyclotomicArtinLocalizedIntegerModuleFinite
+  rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
+  rationalCyclotomicArtinLocalizedPadicAlgebra
+  rationalCyclotomicArtinLocalizedPadicScalarTower
+  rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
+  rationalPrimeFactorCompletionPadicAlgebra
+  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
+
+private theorem
+    rationalCyclotomicPrincipalPrime_padicUnitParameterArtin_action
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
+        (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+          (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) =
+      (rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) ^
+        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val := by
+  let m := rationalCyclotomicPrincipalPrimeModulus p n
+  let E := rationalCyclotomicArtinLocalizedField m p
+  let T := RationalCyclotomicPrincipalPrimePadicLevel p n
+  let eL : E ≃ₐ[ℚ_[p.1]] T :=
+    rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+  let zetaE : E := rationalCyclotomicPrincipalPrimeLocalizedRoot p n
+  let tau : Gal(T / ℚ_[p.1]) :=
+    rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
+  let a := rationalCyclotomicPrincipalPrimeResidueUnit p n x
+  let zetaT : T := padicMultiplicativePrimitiveRoot p.1 n
+  have hzetaE : IsPrimitiveRoot zetaE (p.1 ^ (n + 1)) := by
+    change
+      IsPrimitiveRoot
+        (rationalCyclotomicLocalizedPrimitiveRoot m
+          (RayClass.rationalPrime p)) (m : ℕ)
+    exact
+      rationalCyclotomicLocalizedPrimitiveRoot_isPrimitiveRoot
+        m (RayClass.rationalPrime p)
+  have hrho : IsPrimitiveRoot (eL zetaE) (p.1 ^ (n + 1)) :=
+    hzetaE.map_of_injective eL.injective
+  have hzetaT : IsPrimitiveRoot zetaT (p.1 ^ (n + 1)) :=
+    padicMultiplicativePrimitiveRoot_isPrimitiveRoot p.1 n
+  have htauZetaT : tau zetaT = zetaT ^ a.val.val :=
+    padicMultiplicativePrimitiveRoot_rationalPrimeUnitParameterGaloisAction
+      p n x
+  exact
+    map_primitiveRoot_eq_pow_of_eq_pow
+      tau.toMonoidHom zetaT (eL zetaE)
+      (p.1 ^ (n + 1)) a.val.val hzetaT hrho htauZetaT
+
+private theorem rationalCyclotomicPrincipalPrime_localArtin_action
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    finitePlaceLocalArtinMonoidHom
+        (K := ℚ)
+        (L := KummerTheory.rationalCyclotomicLevel
+          (rationalCyclotomicPrincipalPrimeModulus p n))
+        (RayClass.rationalPrime p)
+        (rationalCyclotomicChosenFinitePlaceExtension
+          (rationalCyclotomicPrincipalPrimeModulus p n)
+          (RayClass.rationalPrime p))
+        (IdeleGroup.finiteComponent
+          (RayClass.rationalPrime p)
+          (IdeleGroup.principalIdele ℚ x))
+        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n) =
+      (rationalCyclotomicPrincipalPrimeLocalizedRoot p n) ^
+        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val := by
+  let eL := rationalCyclotomicLocalizedCompletionPadicAlgEquiv p n
+  apply eL.injective
+  calc
+    _ = _ :=
+      finitePlaceLocalArtinMonoidHom_apply_semilinear
+        (K := ℚ)
+        (L := KummerTheory.rationalCyclotomicLevel
+          (rationalCyclotomicPrincipalPrimeModulus p n))
+        (K' := ℚ_[p.1])
+        (L' := RationalCyclotomicPrincipalPrimePadicLevel p n)
+        (RayClass.rationalPrime p)
+        (rationalCyclotomicChosenFinitePlaceExtension
+          (rationalCyclotomicPrincipalPrimeModulus p n)
+          (RayClass.rationalPrime p))
+        (rationalFinitePlaceCompletionRingEquivPadic p)
+        eL.toRingEquiv
+        (rationalCyclotomicPrincipalPrime_localizedBase_commutes p n)
+        (rationalFinitePlaceCompletionRingEquivPadic_semilinearValuationCompatible
+          p)
+        (IdeleGroup.finiteComponent
+          (RayClass.rationalPrime p)
+          (IdeleGroup.principalIdele ℚ x))
+        (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)
+    _ = rationalCyclotomicPrincipalPrimePadicUnitParameterArtin p n x
+          (eL (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) :=
+      rationalCyclotomicPrincipalPrime_padicArtin_action_eq_unitParameter
+        p n x
+    _ = (eL (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)) ^
+          (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val :=
+      rationalCyclotomicPrincipalPrime_padicUnitParameterArtin_action p n x
+    _ = eL ((rationalCyclotomicPrincipalPrimeLocalizedRoot p n) ^
+          (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val) :=
+      (map_pow eL (rationalCyclotomicPrincipalPrimeLocalizedRoot p n)
+        (rationalCyclotomicPrincipalPrimeResidueUnit p n x).val.val).symm
+
+/-- The finite-place Artin symbol at the ramified prime, in its canonical
+local-to-global factored form.  Keeping this specialization opaque prevents its
+dependent local/global instance tower from being unfolded downstream. -/
+noncomputable def rationalCyclotomicPrincipalPrimeChosenArtin
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    KummerTheory.rationalCyclotomicLevel
+        (rationalCyclotomicPrincipalPrimeModulus p n) ≃ₐ[ℚ]
+      KummerTheory.rationalCyclotomicLevel
+        (rationalCyclotomicPrincipalPrimeModulus p n) :=
+  finitePlaceLocalToGlobalMonoidHom
+    (K := ℚ)
+    (L := KummerTheory.rationalCyclotomicLevel
+      (rationalCyclotomicPrincipalPrimeModulus p n))
+    (RayClass.rationalPrime p)
+    (rationalCyclotomicChosenFinitePlaceExtension
+      (rationalCyclotomicPrincipalPrimeModulus p n)
+      (RayClass.rationalPrime p))
+    (finitePlaceLocalArtinMonoidHom
+      (K := ℚ)
+      (L := KummerTheory.rationalCyclotomicLevel
+        (rationalCyclotomicPrincipalPrimeModulus p n))
+      (RayClass.rationalPrime p)
+      (rationalCyclotomicChosenFinitePlaceExtension
+        (rationalCyclotomicPrincipalPrimeModulus p n)
+        (RayClass.rationalPrime p))
+      (IdeleGroup.finiteComponent
+        (RayClass.rationalPrime p)
+        (IdeleGroup.principalIdele ℚ x)))
+
+/-- At the ramified prime, the cyclotomic character of the chosen finite-place
+Artin symbol is the direct reduction of the rational `p`-adic unit. -/
+theorem galEquivZMod_chosenFinitePlaceArtinMonoidHom_principal_at_prime
+    (p : Nat.Primes) (n : ℕ) (x : ℚˣ) :
+    IsCyclotomicExtension.Rat.galEquivZMod
+        (p.1 ^ (n + 1))
+        (KummerTheory.rationalCyclotomicLevel
+          (rationalCyclotomicPrincipalPrimeModulus p n))
+        (hK :=
+          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
+            (rationalCyclotomicPrincipalPrimeModulus p n))
+        (rationalCyclotomicPrincipalPrimeChosenArtin p n x) =
+      Units.map
+        (PadicInt.toZModPow (p := p.1) (n + 1)).toMonoidHom
+        (padicIntUnitOfRat p
+          (rationalPrimeUnit x p : ℚ)
+          (rationalPrimeUnit x p).ne_zero
+          (padicValRat_rationalPrimeUnit x p)) := by
+  change _ = rationalCyclotomicPrincipalPrimeResidueUnit p n x
+  apply rationalCyclotomicPrincipalPrime_galEquivZMod_eq_of_action p n
+  simp only [rationalCyclotomicPrincipalPrimeChosenArtin]
+  apply finitePlaceLocalToGlobalMonoidHom_apply_pow_of_localized_action
+    (zLocal := rationalCyclotomicPrincipalPrimeLocalizedRoot p n)
+  · rfl
+  · exact rationalCyclotomicPrincipalPrime_localArtin_action p n x
+
+end RationalCyclotomicPrincipalPrime
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinTower.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinTower.lean
new file mode 100644
index 0000000..77efaa0
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinTower.lean
@@ -0,0 +1,357 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlaceArtinBase
+
+
+set_option autoImplicit false
+
+open scoped Classical NNReal NumberField ValuativeRel
+open NumberField IsDedekindDomain
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open LocalClassFieldTheory
+open LocalFieldTheory
+open LocalFieldTheory.DiscreteValuationField
+open LocalFieldTheory.DiscreteValuationField.Examples.Qp
+open LubinTate
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+attribute [local instance]
+  rationalFinitePlaceCompletionRatAlgebra
+  primeFact
+  levelNeZero
+  rationalCyclotomicLevelFiniteDimensional
+  rationalCyclotomicLevelIsAbelianGalois
+  rationalFinitePlaceBaseNontriviallyNormedField
+  rationalFinitePlaceBaseLocallyCompactSpace
+  rationalFinitePlaceBaseIsUltrametricDist
+  rationalFinitePlaceBaseValued
+  rationalFinitePlaceBaseValuativeRel
+  rationalFinitePlaceBaseValuationIsNontrivial
+  rationalFinitePlaceBaseValuationCompatible
+  rationalFinitePlaceBaseValuativeRelIsNontrivial
+  rationalFinitePlaceBaseIsValuativeTopology
+  rationalFinitePlaceBaseIsNonarchimedeanLocalField
+
+/-- The finite place of `ℚ` determined by the rational prime `q`. -/
+abbrev rationalCyclotomicArtinPlace (q : Nat.Primes) :
+    HeightOneSpectrum (𝓞 ℚ) :=
+  RayClass.rationalPrime q
+
+/-- The `q`-adic absolute value on `ℚ` used for the finite-place Artin map. -/
+abbrev rationalCyclotomicArtinBaseAbv (q : Nat.Primes) :
+    AbsoluteValue ℚ ℝ :=
+  HeightOneSpectrum.adicAbv ℚ (rationalCyclotomicArtinPlace q)
+
+/-- The rational cyclotomic extension at the positive level `m`. -/
+abbrev rationalCyclotomicArtinLevel (m : ℕ+) :=
+  KummerTheory.rationalCyclotomicLevel m
+
+/-- A chosen extension of the `q`-adic absolute value to the cyclotomic field. -/
+abbrev rationalCyclotomicArtinExtension
+    (m : ℕ+) (q : Nat.Primes) :
+    AbsoluteValueExtension
+      (rationalCyclotomicArtinBaseAbv q)
+      (rationalCyclotomicArtinLevel m) :=
+  chosenFinitePlaceExtension
+    (L := rationalCyclotomicArtinLevel m)
+    (rationalCyclotomicArtinPlace q)
+
+/-- The local completion of the cyclotomic field at the chosen place above `q`. -/
+abbrev rationalCyclotomicArtinLocalizedField
+    (m : ℕ+) (q : Nat.Primes) :=
+  AlgebraicNumberTheory.Valuations.LocalizedCompletion
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q)
+
+@[reducible]
+noncomputable local instance rationalCyclotomicArtinExtensionAlgebra
+    (m : ℕ+) (q : Nat.Primes) :
+    Algebra ℚ
+      (rationalCyclotomicArtinExtension m q).1.Completion :=
+  AbsoluteValue.extensionCompletionAlgebra
+    (K := ℚ) (rationalCyclotomicArtinExtension m q).1
+
+@[reducible]
+noncomputable local instance rationalCyclotomicArtinExtensionSMul
+    (m : ℕ+) (q : Nat.Primes) :
+    SMul ℚ
+      (rationalCyclotomicArtinExtension m q).1.Completion :=
+  (rationalCyclotomicArtinExtensionAlgebra m q).toSMul
+
+@[reducible]
+noncomputable local instance
+    rationalCyclotomicArtinCompletionAlgebra
+    (m : ℕ+) (q : Nat.Primes) :
+    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinExtension m q).1.Completion :=
+  AbsoluteValue.completionAlgebra
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q).1
+    (rationalCyclotomicArtinExtension m q).2
+
+@[reducible]
+noncomputable local instance rationalCyclotomicArtinLocalizedAlgebra
+    (m : ℕ+) (q : Nat.Primes) :
+    Algebra (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  finitePlaceLocalArtinLocalizedAlgebra
+    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
+    (rationalCyclotomicArtinPlace q)
+    (rationalCyclotomicArtinExtension m q)
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedGlobalAlgebra
+    (m : ℕ+) (q : Nat.Primes) :
+    Algebra ℚ (rationalCyclotomicArtinLocalizedField m q) :=
+  LocalClassFieldTheory.localizedCompletionGlobalAlgebra
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q)
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedGlobalSMul
+    (m : ℕ+) (q : Nat.Primes) :
+    SMul ℚ (rationalCyclotomicArtinLocalizedField m q) :=
+  (rationalCyclotomicArtinLocalizedGlobalAlgebra m q).toSMul
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedScalarTower
+    (m : ℕ+) (q : Nat.Primes) :
+    IsScalarTower ℚ
+      (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinLocalizedField m q) := by
+  constructor
+  intro r x y
+  simp only [Algebra.smul_def, map_mul, eq_ratCast,
+    map_ratCast, mul_assoc]
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedFiniteDimensional
+    (m : ℕ+) (q : Nat.Primes) :
+    FiniteDimensional
+      (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  finitePlaceLocalArtinFiniteDimensional
+    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
+    (rationalCyclotomicArtinPlace q)
+    (rationalCyclotomicArtinExtension m q)
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIsAbelianGalois
+    (m : ℕ+) (q : Nat.Primes) :
+    IsAbelianGalois
+      (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  finitePlaceLocalArtinIsAbelianGalois
+    (K := ℚ) (L := rationalCyclotomicArtinLevel m)
+    (rationalCyclotomicArtinPlace q)
+    (rationalCyclotomicArtinExtension m q)
+    (inferInstance :
+      FiniteDimensional ℚ (rationalCyclotomicArtinLevel m))
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIsSeparable
+    (m : ℕ+) (q : Nat.Primes) :
+    Algebra.IsSeparable
+      (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  (rationalCyclotomicArtinLocalizedIsAbelianGalois m q).toIsGalois.to_isSeparable
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIsCyclotomic
+    (m : ℕ+) (q : Nat.Primes) :
+    IsCyclotomicExtension {(m : ℕ)}
+      (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  rationalCyclotomicLevel_localizedCompletion_isCyclotomicExtension
+    m (rationalCyclotomicArtinPlace q)
+
+noncomputable local instance
+    rationalCyclotomicArtinExtensionFiniteDimensional
+    (m : ℕ+) (q : Nat.Primes) :
+    FiniteDimensional
+      (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinExtension m q).1.Completion :=
+  completionModuleFinite
+    (rationalCyclotomicArtinBaseAbv q)
+    (RayClass.adicAbv_isNontrivial
+      (rationalCyclotomicArtinPlace q))
+    (rationalCyclotomicArtinExtension m q)
+
+noncomputable local instance
+    rationalCyclotomicArtinExtensionContinuousSMul
+    (m : ℕ+) (q : Nat.Primes) :
+    ContinuousSMul
+      (rationalCyclotomicArtinBaseAbv q).Completion
+      (rationalCyclotomicArtinExtension m q).1.Completion :=
+  continuousSMul_of_algebraMap _ _
+    (AbsoluteValue.completionMap_isometry
+      (rationalCyclotomicArtinBaseAbv q)
+      (rationalCyclotomicArtinExtension m q).1
+      (rationalCyclotomicArtinExtension m q).2).continuous
+
+noncomputable local instance
+    rationalCyclotomicArtinExtensionLocallyCompact
+    (m : ℕ+) (q : Nat.Primes) :
+    LocallyCompactSpace
+      (rationalCyclotomicArtinExtension m q).1.Completion :=
+  LocallyCompactSpace.of_finiteDimensional_of_complete
+    (rationalCyclotomicArtinBaseAbv q).Completion
+    (rationalCyclotomicArtinExtension m q).1.Completion
+
+private noncomputable def
+    rationalCyclotomicArtinLocalizedEquivCompletion
+    (m : ℕ+) (q : Nat.Primes) :
+    rationalCyclotomicArtinLocalizedField m q ≃ᵢ
+      (rationalCyclotomicArtinExtension m q).1.Completion :=
+  { toEquiv :=
+      (localizedCompletionEquivCompletion
+        (rationalCyclotomicArtinBaseAbv q)
+        (RayClass.adicAbv_isNontrivial
+          (rationalCyclotomicArtinPlace q))
+        (rationalCyclotomicArtinExtension m q)).toEquiv
+    isometry_toFun :=
+      Isometry.of_dist_eq fun _ _ => rfl }
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedLocallyCompact
+    (m : ℕ+) (q : Nat.Primes) :
+    LocallyCompactSpace
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  ((rationalCyclotomicArtinLocalizedEquivCompletion m q).toHomeomorph.locallyCompactSpace_iff).2
+    inferInstance
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIsUltrametricDist
+    (m : ℕ+) (q : Nat.Primes) :
+    IsUltrametricDist
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  localizedCompletionIsUltrametricDist
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q)
+    (HeightOneSpectrum.isNonarchimedean_adicAbv
+      ℚ (rationalCyclotomicArtinPlace q))
+
+noncomputable local instance rationalCyclotomicArtinLocalizedValued
+    (m : ℕ+) (q : Nat.Primes) :
+    Valued (rationalCyclotomicArtinLocalizedField m q) ℝ≥0 :=
+  localizedCompletionFinitePlaceValued
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q)
+    (HeightOneSpectrum.isNonarchimedean_adicAbv
+      ℚ (rationalCyclotomicArtinPlace q))
+
+@[reducible]
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedValuativeRel
+    (m : ℕ+) (q : Nat.Primes) :
+    ValuativeRel (rationalCyclotomicArtinLocalizedField m q) :=
+  localizedCompletionFinitePlaceValuativeRel
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q)
+    (HeightOneSpectrum.isNonarchimedean_adicAbv
+      ℚ (rationalCyclotomicArtinPlace q))
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedValuationCompatible
+    (m : ℕ+) (q : Nat.Primes) :
+    (Valued.v : Valuation
+      (rationalCyclotomicArtinLocalizedField m q) ℝ≥0).Compatible :=
+  Valuation.Compatible.ofValuation _
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedValuationHasExtension
+    (m : ℕ+) (q : Nat.Primes) :
+    Valuation.HasExtension
+      (ValuativeRel.valuation
+        (rationalCyclotomicArtinBaseAbv q).Completion)
+      (ValuativeRel.valuation
+        (rationalCyclotomicArtinLocalizedField m q)) :=
+  localizedCompletionValuationHasExtension
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q)
+    (HeightOneSpectrum.isNonarchimedean_adicAbv
+      ℚ (rationalCyclotomicArtinPlace q))
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedValuationIsNontrivial
+    (m : ℕ+) (q : Nat.Primes) :
+    (ValuativeRel.valuation
+      (rationalCyclotomicArtinLocalizedField m q)).IsNontrivial :=
+  Valuation.IsNontrivial.of_hasExtension
+    (ValuativeRel.valuation
+      (rationalCyclotomicArtinBaseAbv q).Completion)
+    (ValuativeRel.valuation
+      (rationalCyclotomicArtinLocalizedField m q))
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
+    (m : ℕ+) (q : Nat.Primes) :
+    ValuativeRel.IsNontrivial
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  (ValuativeRel.isNontrivial_iff_isNontrivial
+    (ValuativeRel.valuation
+      (rationalCyclotomicArtinLocalizedField m q))).2 inferInstance
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIsValuativeTopology
+    (m : ℕ+) (q : Nat.Primes) :
+    IsValuativeTopology
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  isValuativeTopology_of_valued_ofValuation
+    (rationalCyclotomicArtinLocalizedField m q) ℝ≥0
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
+    (m : ℕ+) (q : Nat.Primes) :
+    IsNonarchimedeanLocalField
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  { toIsValuativeTopology := inferInstance
+    toLocallyCompactSpace := inferInstance
+    toIsNontrivial := inferInstance }
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIntegerAlgebra
+    (m : ℕ+) (q : Nat.Primes) :
+    Algebra
+      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  Algebra.ofSubsemiring
+    𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIsIntegralClosure
+    (m : ℕ+) (q : Nat.Primes) :
+    IsIntegralClosure
+      𝒪[rationalCyclotomicArtinLocalizedField m q]
+      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
+      (rationalCyclotomicArtinLocalizedField m q) :=
+  localizedCompletionIsIntegralClosureWithExtension
+    (rationalCyclotomicArtinBaseAbv q)
+    (rationalCyclotomicArtinExtension m q)
+    (RayClass.adicAbv_isNontrivial
+      (rationalCyclotomicArtinPlace q))
+    (HeightOneSpectrum.isNonarchimedean_adicAbv
+      ℚ (rationalCyclotomicArtinPlace q))
+
+noncomputable local instance
+    rationalCyclotomicArtinLocalizedIntegerModuleFinite
+    (m : ℕ+) (q : Nat.Primes) :
+    Module.Finite
+      𝒪[(rationalCyclotomicArtinBaseAbv q).Completion]
+      𝒪[rationalCyclotomicArtinLocalizedField m q] :=
+  integerRing_moduleFinite_of_isIntegralClosure
+    (rationalCyclotomicArtinBaseAbv q).Completion
+    (rationalCyclotomicArtinLocalizedField m q)
+
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinTransport.lean b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinTransport.lean
new file mode 100644
index 0000000..666b029
--- /dev/null
+++ b/Lean4/ClassFieldTheory/GlobalClassFieldTheory/Reciprocity/RationalCyclotomicFinitePlaceArtinTransport.lean
@@ -0,0 +1,375 @@
+/-
+Copyright (c) 2026 Naganori Yamaguchi (https://github.com/n-yamaguchi-0729). All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Naganori Yamaguchi (assisted by OpenAI Codex)
+-/
+
+import ClassFieldTheory.GlobalClassFieldTheory.Reciprocity.RationalCyclotomicFinitePlaceArtinPadic
+
+
+set_option autoImplicit false
+
+open scoped Classical NNReal NumberField ValuativeRel
+open NumberField IsDedekindDomain
+open AlgebraicNumberTheory.Valuations
+open HilbertRamification
+open LocalClassFieldTheory
+open LocalFieldTheory
+open LocalFieldTheory.DiscreteValuationField
+open LocalFieldTheory.DiscreteValuationField.Examples.Qp
+open LubinTate
+
+noncomputable section
+
+namespace GlobalClassFieldTheory
+namespace Reciprocity
+
+section RationalCyclotomicPrincipalPrime
+
+attribute [local instance]
+  rationalFinitePlaceCompletionRatAlgebra
+  primeFact
+  levelNeZero
+  rationalCyclotomicLevelFiniteDimensional
+  rationalCyclotomicLevelIsAbelianGalois
+  rationalFinitePlaceBaseNontriviallyNormedField
+  rationalFinitePlaceBaseLocallyCompactSpace
+  rationalFinitePlaceBaseIsUltrametricDist
+  rationalFinitePlaceBaseValued
+  rationalFinitePlaceBaseValuativeRel
+  rationalFinitePlaceBaseValuationIsNontrivial
+  rationalFinitePlaceBaseValuationCompatible
+  rationalFinitePlaceBaseValuativeRelIsNontrivial
+  rationalFinitePlaceBaseIsValuativeTopology
+  rationalFinitePlaceBaseIsNonarchimedeanLocalField
+  rationalCyclotomicArtinExtensionAlgebra
+  rationalCyclotomicArtinExtensionSMul
+  rationalCyclotomicArtinCompletionAlgebra
+  rationalCyclotomicArtinLocalizedAlgebra
+  rationalCyclotomicArtinLocalizedGlobalAlgebra
+  rationalCyclotomicArtinLocalizedGlobalSMul
+  rationalCyclotomicArtinLocalizedScalarTower
+  rationalCyclotomicArtinLocalizedFiniteDimensional
+  rationalCyclotomicArtinLocalizedIsAbelianGalois
+  rationalCyclotomicArtinLocalizedIsSeparable
+  rationalCyclotomicArtinLocalizedIsCyclotomic
+  rationalCyclotomicArtinExtensionFiniteDimensional
+  rationalCyclotomicArtinExtensionContinuousSMul
+  rationalCyclotomicArtinExtensionLocallyCompact
+  rationalCyclotomicArtinLocalizedLocallyCompact
+  rationalCyclotomicArtinLocalizedIsUltrametricDist
+  rationalCyclotomicArtinLocalizedValued
+  rationalCyclotomicArtinLocalizedValuativeRel
+  rationalCyclotomicArtinLocalizedValuationCompatible
+  rationalCyclotomicArtinLocalizedValuationHasExtension
+  rationalCyclotomicArtinLocalizedValuationIsNontrivial
+  rationalCyclotomicArtinLocalizedValuativeRelIsNontrivial
+  rationalCyclotomicArtinLocalizedIsValuativeTopology
+  rationalCyclotomicArtinLocalizedIsNonarchimedeanLocalField
+  rationalCyclotomicArtinLocalizedIntegerAlgebra
+  rationalCyclotomicArtinLocalizedIsIntegralClosure
+  rationalCyclotomicArtinLocalizedIntegerModuleFinite
+  rationalCyclotomicPrincipalPrimeLevelIsCyclotomicExtension
+  rationalCyclotomicArtinLocalizedPadicAlgebra
+  rationalCyclotomicArtinLocalizedPadicScalarTower
+  rationalCyclotomicPrincipalPrimePadicLevelIsAbelianGalois
+  rationalPrimeFactorCompletionPadicAlgebra
+  rationalCyclotomicPrincipalPrimePadicLevelFiniteDimensional
+
+/-- The chosen localized global cyclotomic level, transported over the
+completion equivalence, is the standard multiplicative Lubin--Tate level. -/
+noncomputable def rationalCyclotomicLocalizedCompletionPadicAlgEquiv
+    (p : Nat.Primes) (n : ℕ) :
+    rationalCyclotomicArtinLocalizedField
+        (rationalCyclotomicPrincipalPrimeModulus p n) p ≃ₐ[ℚ_[p.1]]
+      RationalCyclotomicPrincipalPrimePadicLevel p n := by
+  letI : IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ_[p.1]
+      (rationalCyclotomicArtinLocalizedField
+        (rationalCyclotomicPrincipalPrimeModulus p n) p) :=
+    rationalCyclotomicPrincipalPrimeLocalizedLevel_isCyclotomicExtension p n
+  letI : IsCyclotomicExtension {p.1 ^ (n + 1)} ℚ_[p.1]
+      (RationalCyclotomicPrincipalPrimePadicLevel p n) :=
+    padicMultiplicativeLevel_isCyclotomicExtension p.1 n
+  exact
+    IsCyclotomicExtension.algEquiv
+      {p.1 ^ (n + 1)} ℚ_[p.1]
+      (rationalCyclotomicArtinLocalizedField
+        (rationalCyclotomicPrincipalPrimeModulus p n) p)
+      (RationalCyclotomicPrincipalPrimePadicLevel p n)
+
+/-! ## The ramified principal finite-place factor -/
+
+theorem
+    rationalCyclotomicPrincipalPrime_galEquivZMod_eq_of_action
+    (p : Nat.Primes) (n : ℕ)
+    (sigma : Gal(
+      rationalCyclotomicPrincipalPrimeLevel
+        (rationalCyclotomicPrincipalPrimeModulus p n) / ℚ))
+    (a : (ZMod (p.1 ^ (n + 1)))ˣ)
+    (haction :
+      sigma (rationalCyclotomicLevelPrimitiveRoot
+          (rationalCyclotomicPrincipalPrimeModulus p n)) =
+        rationalCyclotomicLevelPrimitiveRoot
+            (rationalCyclotomicPrincipalPrimeModulus p n) ^ a.val.val) :
+    IsCyclotomicExtension.Rat.galEquivZMod
+        (p.1 ^ (n + 1))
+        (rationalCyclotomicPrincipalPrimeLevel
+          (rationalCyclotomicPrincipalPrimeModulus p n))
+        (hK :=
+          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension
+            (rationalCyclotomicPrincipalPrimeModulus p n)) sigma =
+      a := by
+  let m := rationalCyclotomicPrincipalPrimeModulus p n
+  let L := rationalCyclotomicPrincipalPrimeLevel m
+  let zeta : L := rationalCyclotomicLevelPrimitiveRoot m
+  have hzeta : IsPrimitiveRoot zeta (p.1 ^ (n + 1)) := by
+    change
+      IsPrimitiveRoot
+        (rationalCyclotomicLevelPrimitiveRoot m) (m : ℕ)
+    exact rationalCyclotomicLevelPrimitiveRoot_isPrimitiveRoot m
+  change
+    IsCyclotomicExtension.Rat.galEquivZMod
+        (p.1 ^ (n + 1)) L
+        (hK :=
+          KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension m)
+        sigma = a
+  let c :=
+    IsCyclotomicExtension.Rat.galEquivZMod
+      (p.1 ^ (n + 1)) L
+      (hK :=
+        KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension m)
+      sigma
+  have hc :
+      sigma zeta = zeta ^ c.val.val :=
+    IsCyclotomicExtension.Rat.galEquivZMod_apply_of_pow_eq
+      (hK :=
+        KummerTheory.rationalCyclotomicLevel_isCyclotomicExtension m)
+      (p.1 ^ (n + 1)) L sigma hzeta.pow_eq_one
+  have hpowers :
+      zeta ^ c.val.val = zeta ^ a.val.val :=
+    hc.symm.trans haction
+  rw [(hzeta.isOfFinOrder m.ne_zero).pow_inj_mod,
+    ← hzeta.eq_orderOf,
+    ← ZMod.natCast_eq_natCast_iff'] at hpowers
+  change
+      (c.val.val : ZMod (p.1 ^ (n + 1))) =
+        (a.val.val : ZMod (p.1 ^ (n + 1))) at hpowers
+  have hValues : c.val = a.val := by
+    calc
+      c.val = (c.val.val : ZMod (p.1 ^ (n + 1))) :=
+        (ZMod.natCast_zmod_val c.val).symm
+      _ = (a.val.val : ZMod (p.1 ^ (n + 1))) := hpowers
+      _ = a.val := ZMod.natCast_zmod_val a.val
+  change c = a
+  apply Units.ext
+  exact hValues
+
+/-- The chosen finite-place Artin map factors through any extension identified
+with the chosen one. -/
+theorem chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+    (v : HeightOneSpectrum (𝓞 K))
+    (w : AbsoluteValueExtension
+      (HeightOneSpectrum.adicAbv K v) L)
+    (hw : chosenFinitePlaceExtension (L := L) v = w)
+    (x : (v.adicCompletion K)ˣ) :
+    chosenFinitePlaceArtinMonoidHom
+        (K := K) (L := L) v x =
+      finitePlaceLocalToGlobalMonoidHom
+        (K := K) (L := L) v w
+        (finitePlaceLocalArtinMonoidHom
+          (K := K) (L := L) v w x) := by
+  subst w
+  exact
+    congrArg
+      (fun f : (v.adicCompletion K)ˣ →* (L ≃ₐ[K] L) => f x)
+      (finitePlaceArtinMonoidHomOfExtension_factor
+        (K := K) (L := L) v
+        (chosenFinitePlaceExtension (L := L) v))
+
+private theorem
+    chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq_at
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+    (v : HeightOneSpectrum (𝓞 K))
+    (w : AbsoluteValueExtension
+      (HeightOneSpectrum.adicAbv K v) L)
+    (hw : chosenFinitePlaceExtension (L := L) v = w)
+    (x : (v.adicCompletion K)ˣ) (z : L) :
+    chosenFinitePlaceArtinMonoidHom
+        (K := K) (L := L) v x z =
+      finitePlaceLocalToGlobalMonoidHom
+        (K := K) (L := L) v w
+        (finitePlaceLocalArtinMonoidHom
+          (K := K) (L := L) v w x) z := by
+  exact
+    congrArg (fun sigma : Gal(L / K) => sigma z)
+      (chosenFinitePlaceArtinMonoidHom_apply_factor_of_extension_eq
+        (K := K) (L := L) v w hw x)
+
+theorem finitePlaceLocalToGlobalMonoidHom_apply_pow_of_localized_action
+    {K L : Type}
+    [Field K] [NumberField K]
+    [Field L] [Algebra K L]
+    [FiniteDimensional K L] [IsAbelianGalois K L]
+    (v : HeightOneSpectrum (𝓞 K))
+    (w : AbsoluteValueExtension
+      (HeightOneSpectrum.adicAbv K v) L)
+    (sigma :
+      let vK := HeightOneSpectrum.adicAbv K v
+      let E := LocalizedCompletion vK w
+      letI : Algebra vK.Completion E :=
+        finitePlaceLocalArtinLocalizedAlgebra v w
+      Gal(E / vK.Completion))
+    (z : L)
+    (zLocal :
+      let vK := HeightOneSpectrum.adicAbv K v
+      LocalizedCompletion vK w)
+    (e : ℕ)
+    (hLocalization :
+      let vK := HeightOneSpectrum.adicAbv K v
+      let E := LocalizedCompletion vK w
+      letI : Algebra vK.Completion E :=
+        finitePlaceLocalArtinLocalizedAlgebra v w
+      let eLoc : L →+* E :=
+        AbsoluteValue.toAlgebraicLocalization vK w.1 w.2
+      eLoc z = zLocal)
+    (hlocal : sigma zLocal = zLocal ^ e) :
+    finitePlaceLocalToGlobalMonoidHom
+        (K := K) (L := L) v w sigma z = z ^ e := by
+  let vK := HeightOneSpectrum.adicAbv K v
+  let hvK : vK.IsNontrivial := RayClass.adicAbv_isNontrivial v
+  let E := LocalizedCompletion vK w
+  let : Algebra vK.Completion E :=
+    finitePlaceLocalArtinLocalizedAlgebra v w
+  let eD : absoluteValueDecompositionGroup K w.1 ≃* Gal(E / vK.Completion) :=
+    decompositionGroupEquivAlgebraicLocalizationAut vK hvK w
+  let eLoc : L →+* E :=
+    AbsoluteValue.toAlgebraicLocalization vK w.1 w.2
+  let delta : absoluteValueDecompositionGroup K w.1 := eD.symm sigma
+  have hDecomposition : eD delta = sigma := eD.apply_symm_apply sigma
+  apply eLoc.injective
+  calc
+    eLoc (finitePlaceLocalToGlobalMonoidHom
+        (K := K) (L := L) v w sigma z) =
+        eD delta (eLoc z) :=
+      (localizationRamificationGroups_decompositionGroupEquiv_toLocalization
+        vK hvK w delta z).symm
+    _ = sigma (eLoc z) :=
+      congrArg (fun tau : Gal(E / vK.Completion) => tau (eLoc z))
+        hDecomposition
+    _ = sigma zLocal :=
+      congrArg (fun y : E => sigma y) hLocalization
+    _ = zLocal ^ e := hlocal
+    _ = (eLoc z) ^ e :=
+      congrArg (fun y : E => y ^ e) hLocalization.symm
+    _ = eLoc (z ^ e) := (map_pow eLoc z e).symm
+
+theorem map_primitiveRoot_eq_pow_of_eq_pow
+    {M : Type} [CommRing M] [IsDomain M]
+    (f : M →* M) (zeta rho : M) (order exponent : ℕ)
+    [NeZero order]
+    (hzeta : IsPrimitiveRoot zeta order)
+    (hrho : IsPrimitiveRoot rho order)
+    (hf : f zeta = zeta ^ exponent) :
+    f rho = rho ^ exponent := by
+  obtain ⟨j, -, hj⟩ :=
+    hzeta.eq_pow_of_pow_eq_one hrho.pow_eq_one
+  calc
+    f rho = f (zeta ^ j) := congrArg f hj.symm
+    _ = (f zeta) ^ j := map_pow f zeta j
+    _ = (zeta ^ exponent) ^ j := congrArg (fun z => z ^ j) hf
+    _ = zeta ^ (exponent * j) := (pow_mul zeta exponent j).symm
+    _ = zeta ^ (j * exponent) :=
+      congrArg (fun e : ℕ => zeta ^ e) (Nat.mul_comm exponent j)
+    _ = (zeta ^ j) ^ exponent := pow_mul zeta j exponent
+    _ = rho ^ exponent := congrArg (fun z => z ^ exponent) hj
+
+theorem finitePlaceLocalArtinMonoidHom_apply_semilinear
+    {K L K' L' : Type}
+    [Field K] [NumberField K]
+    [Field L] [Algebra K L]
+    [hKLfinite : FiniteDimensional K L] [IsAbelianGalois K L]
+    [Field K'] [ValuativeRel K'] [TopologicalSpace K']
+    [IsNonarchimedeanLocalField K']
+    [Field L'] [Algebra K' L']
+    [FiniteDimensional K' L'] [IsAbelianGalois K' L']
+    (v : HeightOneSpectrum (𝓞 K))
+    (w : AbsoluteValueExtension
+      (HeightOneSpectrum.adicAbv K v) L)
+    (eK : (HeightOneSpectrum.adicAbv K v).Completion ≃+* K')
+    (eL : LocalizedCompletion
+      (HeightOneSpectrum.adicAbv K v) w ≃+* L')
+    (hcomm : ∀ y : (HeightOneSpectrum.adicAbv K v).Completion,
+      eL (@algebraMap
+        (HeightOneSpectrum.adicAbv K v).Completion
+        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
+        _ _ (finitePlaceLocalArtinLocalizedAlgebra v w) y) =
+        algebraMap K' L' (eK y))
+    (hExt : SemilinearValuationCompatible
+      (HeightOneSpectrum.adicAbv K v).Completion K' eK)
+    (x : (v.adicCompletion K)ˣ)
+    (z : LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w) :
+    eL (finitePlaceLocalArtinMonoidHom
+        (K := K) (L := L) v w x z) =
+      LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
+        (Units.map eK.toMonoidHom
+          (finitePlaceLocalArtinInput v x)) (eL z) := by
+  calc
+    eL (finitePlaceLocalArtinMonoidHom
+        (K := K) (L := L) v w x z) =
+      eL ((@LocalClassFieldTheory.abelianLocalArtinMonoidHom
+        (HeightOneSpectrum.adicAbv K v).Completion
+        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w)
+        (inferInstance : Field
+          (HeightOneSpectrum.adicAbv K v).Completion)
+        (inferInstance : Field
+          (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w))
+        (finitePlaceLocalArtinLocalizedAlgebra v w)
+        (finitePlaceLocalArtinCompletionValuativeRel v)
+        (inferInstance : TopologicalSpace
+          (HeightOneSpectrum.adicAbv K v).Completion)
+        (finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v)
+        (finitePlaceLocalArtinFiniteDimensional v w)
+        (finitePlaceLocalArtinIsAbelianGalois v w hKLfinite)
+        (finitePlaceLocalArtinInput v x)) z) :=
+      congrArg eL
+        (finitePlaceLocalArtinMonoidHom_apply_normalized_at
+          (K := K) (L := L) v w x z)
+    _ = LocalClassFieldTheory.abelianLocalArtinMonoidHom K' L'
+          (Units.map eK.toMonoidHom
+            (finitePlaceLocalArtinInput v x)) (eL z) :=
+      @abelianLocalArtinMonoidHom_semilinear_action
+        (HeightOneSpectrum.adicAbv K v).Completion K'
+        (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w) L'
+        (inferInstance : Field
+          (HeightOneSpectrum.adicAbv K v).Completion)
+        (finitePlaceLocalArtinCompletionValuativeRel v)
+        (inferInstance : TopologicalSpace
+          (HeightOneSpectrum.adicAbv K v).Completion)
+        (finitePlaceLocalArtinCompletionIsNonarchimedeanLocalField v)
+        (inferInstance : Field K')
+        (inferInstance : ValuativeRel K')
+        (inferInstance : TopologicalSpace K')
+        (inferInstance : IsNonarchimedeanLocalField K')
+        (inferInstance : Field
+          (LocalizedCompletion (HeightOneSpectrum.adicAbv K v) w))
+        (inferInstance : Field L')
+        (finitePlaceLocalArtinLocalizedAlgebra v w)
+        (inferInstance : Algebra K' L')
+        (finitePlaceLocalArtinFiniteDimensional v w)
+        (finitePlaceLocalArtinIsAbelianGalois v w hKLfinite)
+        (inferInstance : FiniteDimensional K' L')
+        (inferInstance : IsAbelianGalois K' L')
+        eK eL hcomm hExt (finitePlaceLocalArtinInput v x) z
+
+
+end RationalCyclotomicPrincipalPrime
+
+end Reciprocity
+end GlobalClassFieldTheory
diff --git a/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FiniteResidueFinrankTransfer.lean b/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FiniteResidueFinrankTransfer.lean
index 1ff2c21..3705c23 100644
--- a/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FiniteResidueFinrankTransfer.lean
+++ b/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FiniteResidueFinrankTransfer.lean
@@ -9,6 +9,7 @@ import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.FiniteResi
 import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.ResidueExtension
 import ValuedFieldTheory.Valuation.UniqueRing
 
+
 set_option autoImplicit false
 
 namespace LocalClassFieldTheory
@@ -16,15 +17,6 @@ open CyclicCohomology RamificationTheory ClassFormation
 
 open LocalFieldTheory ValuationTheory
 
-/-!
-# Finite local reciprocity: comparison of the two finite residue-field models
-
-The residue-action exact sequence presents the residue field of a
-finite fixed field intrinsically, inside the selected residue algebraic
-closure.  The norm formula uses the literal residue field of the spectral
-valuation on that fixed field.  This file compares those presentations by
-the uniqueness of the finite extension valuation.
--/
 
 noncomputable section
 
@@ -53,9 +45,27 @@ noncomputable def valuationSubringEquivDecompositionFieldOfEqTop
   let eFZ : F ≃ₐ[F] Z :=
     (IntermediateField.botEquiv F Omega).symm.trans
       (IntermediateField.equivOfEq hZ.symm)
-  refine
-    { toFun := fun x => ⟨eFZ (x : F), ?_⟩
-      invFun := fun z => ⟨eFZ.symm (z : Z), ?_⟩
+  exact
+    { toFun := fun x => ⟨eFZ (x : F), by
+        change ((eFZ x : Z) : Omega) ∈ A
+        have he : ((eFZ x : Z) : Omega) =
+            algebraMap F Omega (x : F) := by
+          rfl
+        rw [he]
+        have hx : (x : F) ∈ A.comap (algebraMap F Omega) := by
+          rw [hC]
+          exact x.property
+        exact hx⟩
+      invFun := fun z => ⟨eFZ.symm (z : Z), by
+        have hz : eFZ.symm (z : Z) ∈ A.comap (algebraMap F Omega) := by
+          change algebraMap F Omega (eFZ.symm (z : Z)) ∈ A
+          have he : algebraMap F Omega (eFZ.symm (z : Z)) =
+              ((z : Z) : Omega) := by
+            exact congrArg Subtype.val (eFZ.apply_symm_apply (z : Z))
+          rw [he]
+          exact z.property
+        rw [hC] at hz
+        exact hz⟩
       left_inv := fun x => by
         apply Subtype.ext
         exact eFZ.symm_apply_apply (x : F)
@@ -68,24 +78,6 @@ noncomputable def valuationSubringEquivDecompositionFieldOfEqTop
       map_mul' := fun x y => by
         apply Subtype.ext
         exact map_mul eFZ (x : F) (y : F) }
-  · change ((eFZ x : Z) : Omega) ∈ A
-    have he : ((eFZ x : Z) : Omega) =
-        algebraMap F Omega (x : F) := by
-      rfl
-    rw [he]
-    have hx : (x : F) ∈ A.comap (algebraMap F Omega) := by
-      rw [hC]
-      exact x.property
-    exact hx
-  · have hz : eFZ.symm (z : Z) ∈ A.comap (algebraMap F Omega) := by
-      change algebraMap F Omega (eFZ.symm (z : Z)) ∈ A
-      have he : algebraMap F Omega (eFZ.symm (z : Z)) =
-          ((z : Z) : Omega) := by
-        exact congrArg Subtype.val (eFZ.apply_symm_apply (z : Z))
-      rw [he]
-      exact z.property
-    rw [hC] at hz
-    exact hz
 
 /-- The corresponding equivalence between literal and intrinsic residue
 fields. -/
diff --git a/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FixedFieldNormResidueNaturality.lean b/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FixedFieldNormResidueNaturality.lean
index 08eb9ea..dd20dc0 100644
--- a/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FixedFieldNormResidueNaturality.lean
+++ b/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/FixedFieldNormResidueNaturality.lean
@@ -10,15 +10,9 @@ import ClassFieldTheory.LocalClassFieldTheory.Finite.LocalReciprocity.LocalHense
 import ClassFieldTheory.AbstractClassFieldTheory.Reciprocity.Main
 import Mathlib.FieldTheory.Galois.Notation
 
-set_option autoImplicit false
 
-/-!
-# Naturality of fixed-field norm-residue symbols
+set_option autoImplicit false
 
-This file transports norm-restriction and transfer-inclusion naturality
-from the closed-subgroup class formation to actual fixed fields in a single
-Galois ambient field.
--/
 
 noncomputable section
 
@@ -521,7 +515,12 @@ theorem upperAbsoluteFinite
 /-- The fixed-field norm-residue symbol for the lower horizontal extension,
 induced by the canonical local class formation. -/
 noncomputable def lowerNormResidueSymbol
-    (T : LocalFixedFieldNormRestrictionSquare k) := by
+    (T : LocalFixedFieldNormRestrictionSquare k) :
+    Additive (abstractFixedField k (SeparableClosure k) T.lowerBase)ˣ →+
+      Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          T.lowerTop_le_lowerBase /
+          abstractFixedField k (SeparableClosure k) T.lowerBase)) := by
   letI := T.lowerNormal
   letI := T.lowerFinite
   letI := T.lowerAbsoluteFinite
@@ -534,7 +533,12 @@ noncomputable def lowerNormResidueSymbol
 /-- The fixed-field norm-residue symbol for the upper horizontal extension,
 induced by the canonical local class formation. -/
 noncomputable def upperNormResidueSymbol
-    (T : LocalFixedFieldNormRestrictionSquare k) := by
+    (T : LocalFixedFieldNormRestrictionSquare k) :
+    Additive (abstractFixedField k (SeparableClosure k) T.upperBase)ˣ →+
+      Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          T.upperTop_le_upperBase /
+          abstractFixedField k (SeparableClosure k) T.upperBase)) := by
   letI := T.upperNormal
   letI := T.upperFinite
   letI := upperAbsoluteFinite T
@@ -552,7 +556,15 @@ noncomputable def normUnits
 
 /-- Restriction between the abelianized actual relative Galois groups. -/
 noncomputable def abelianizedRestriction
-    (T : LocalFixedFieldNormRestrictionSquare k) := by
+    (T : LocalFixedFieldNormRestrictionSquare k) :
+    Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          T.upperTop_le_upperBase /
+          abstractFixedField k (SeparableClosure k) T.upperBase)) →+
+      Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          T.lowerTop_le_lowerBase /
+          abstractFixedField k (SeparableClosure k) T.lowerBase)) := by
   letI := T.lowerNormal
   letI := T.upperNormal
   exact
@@ -681,7 +693,12 @@ theorem intermediateAbsoluteFinite
 /-- The fixed-field norm-residue symbol for the total extension, induced by
 the canonical local class formation. -/
 noncomputable def baseNormResidueSymbol
-    (T : LocalFixedFieldTransferTower k) := by
+    (T : LocalFixedFieldTransferTower k) :
+    Additive (abstractFixedField k (SeparableClosure k) T.base)ˣ →+
+      Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          (T.top_le_intermediate.trans T.intermediate_le_base) /
+          abstractFixedField k (SeparableClosure k) T.base)) := by
   letI := T.totalNormal
   letI := T.totalFinite
   letI := T.baseAbsoluteFinite
@@ -695,7 +712,12 @@ noncomputable def baseNormResidueSymbol
 /-- The fixed-field norm-residue symbol after changing the base to the
 intermediate fixed field, induced by the canonical local class formation. -/
 noncomputable def intermediateNormResidueSymbol
-    (T : LocalFixedFieldTransferTower k) := by
+    (T : LocalFixedFieldTransferTower k) :
+    Additive (abstractFixedField k (SeparableClosure k) T.intermediate)ˣ →+
+      Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          T.top_le_intermediate /
+          abstractFixedField k (SeparableClosure k) T.intermediate)) := by
   letI := intermediateNormal T
   letI := intermediateFinite T
   letI := intermediateAbsoluteFinite T
@@ -713,7 +735,15 @@ noncomputable def unitsInclusion
 
 /-- Transfer between the abelianized actual relative Galois groups. -/
 noncomputable def abelianizedTransfer
-    (T : LocalFixedFieldTransferTower k) := by
+    (T : LocalFixedFieldTransferTower k) :
+    Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          (T.top_le_intermediate.trans T.intermediate_le_base) /
+          abstractFixedField k (SeparableClosure k) T.base)) →+
+      Additive (Abelianization
+        Gal(abstractRelativeFixedField k (SeparableClosure k)
+          T.top_le_intermediate /
+          abstractFixedField k (SeparableClosure k) T.intermediate)) := by
   letI := T.totalNormal
   letI := T.totalFinite
   exact
diff --git a/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/LocalResidueDatum.lean b/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/LocalResidueDatum.lean
index bc7a1de..9da0120 100644
--- a/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/LocalResidueDatum.lean
+++ b/Lean4/ClassFieldTheory/LocalClassFieldTheory/Finite/LocalReciprocity/LocalResidueDatum.lean
@@ -9,6 +9,7 @@ import ValuedFieldTheory.Valuation.DiscreteValuationField.FiniteIntegralClosure
 import ValuedFieldTheory.Ramification.GaloisValuation.AbsoluteGalois.FiniteExtensionCorrespondence
 import ValuedFieldTheory.LocalField.NonarchimedeanLocalField.FiniteExtensionCompleteDVF
 
+
 set_option autoImplicit false
 
 namespace LocalClassFieldTheory
@@ -17,24 +18,6 @@ open ClassFormation
 
 open ValuationTheory RamificationTheory LocalFieldTheory
 
-/-!
-# Finite local reciprocity: the local residue degree datum
-
-For a nonarchimedean local field `K`, this file makes the choices implicit in
-the construction explicit.  The canonical local valuation is packaged as a complete
-DVF, Chevalley's theorem chooses an extension to `AlgebraicClosure K`, and
-that valuation is pulled back to `SeparableClosure K`.  Finite-separable
-uniqueness shows that its decomposition subgroup is the whole Galois group.
-The residue field of the decomposition field is then identified with the
-finite residue field of `K`.
-
-The remaining step is topological: the reduction action is shown continuous
-for the two Krull topologies and is composed with the intrinsic finite-field
-degree map from `ResidueAlgebraicClosureDegree`.  The selected residue field
-is algebraically closed because its extension to the residue of
-`AlgebraicClosure K` is purely inseparable and the selected residue field is
-perfect over the finite base residue field.
--/
 
 noncomputable section
 
@@ -187,9 +170,23 @@ private noncomputable def localBaseValuationSubringEquivDecompositionField :
   let eKZ : K ≃ₐ[K] Z :=
     (IntermediateField.botEquiv K (SeparableClosure K)).symm.trans
       (IntermediateField.equivOfEq hZ.symm)
-  refine
-    { toFun := fun x => ⟨eKZ (x : K), ?_⟩
-      invFun := fun z => ⟨eKZ.symm (z : Z), ?_⟩
+  exact
+    { toFun := fun x => ⟨eKZ (x : K), by
+        change ((eKZ x : Z) : SeparableClosure K) ∈ A
+        have he : ((eKZ x : Z) : SeparableClosure K) =
+            algebraMap K (SeparableClosure K) (x : K) := by
+          rfl
+        rw [he]
+        exact (localSeparableValuationSubring_pullback K (x : K)).2 x.property⟩
+      invFun := fun z => ⟨eKZ.symm (z : Z), by
+        change eKZ.symm (z : Z) ∈
+          (localCompleteDVF K).valuation.valuationSubring
+        apply (localSeparableValuationSubring_pullback K (eKZ.symm (z : Z))).1
+        have he : algebraMap K (SeparableClosure K) (eKZ.symm (z : Z)) =
+            ((z : Z) : SeparableClosure K) := by
+          exact congrArg Subtype.val (eKZ.apply_symm_apply (z : Z))
+        rw [he]
+        exact z.property⟩
       left_inv := fun x => by
         apply Subtype.ext
         exact eKZ.symm_apply_apply (x : K)
@@ -202,20 +199,6 @@ private noncomputable def localBaseValuationSubringEquivDecompositionField :
       map_mul' := fun x y => by
         apply Subtype.ext
         exact map_mul eKZ (x : K) (y : K) }
-  · change ((eKZ x : Z) : SeparableClosure K) ∈ A
-    have he : ((eKZ x : Z) : SeparableClosure K) =
-        algebraMap K (SeparableClosure K) (x : K) := by
-      rfl
-    rw [he]
-    exact (localSeparableValuationSubring_pullback K (x : K)).2 x.property
-  · change eKZ.symm (z : Z) ∈
-      (localCompleteDVF K).valuation.valuationSubring
-    apply (localSeparableValuationSubring_pullback K (eKZ.symm (z : Z))).1
-    have he : algebraMap K (SeparableClosure K) (eKZ.symm (z : Z)) =
-        ((z : Z) : SeparableClosure K) := by
-      exact congrArg Subtype.val (eKZ.apply_symm_apply (z : Z))
-    rw [he]
-    exact z.property
 
 /-- The residue field in the residue-action exact sequence is canonically the finite residue
 field of the original local field. -/
diff --git a/Lean4/ClassFieldTheory/LubinTate/Padic/CompletedChangedUniformizerFixedField.lean b/Lean4/ClassFieldTheory/LubinTate/Padic/CompletedChangedUniformizerFixedField.lean
index c747c98..aa28793 100644
--- a/Lean4/ClassFieldTheory/LubinTate/Padic/CompletedChangedUniformizerFixedField.lean
+++ b/Lean4/ClassFieldTheory/LubinTate/Padic/CompletedChangedUniformizerFixedField.lean
@@ -9,6 +9,7 @@ import ClassFieldTheory.LubinTate.Padic.CompletedChangedUniformizerPrimitive
 import ClassFieldTheory.LubinTate.Padic.CompletedChangedUniformizerThetaFixed
 import ClassFieldTheory.LubinTate.Padic.CompletedUnramifiedFrobeniusFixed
 
+
 set_option autoImplicit false
 
 /-!
@@ -42,7 +43,8 @@ noncomputable def padicCompletedChangedUniformizerFrobeniusAlgEquiv
     (u : (padicLocalField p).valuationSubringˣ) (n : ℕ) :
     padicCompletedLevelField p n ≃ₐ[ℚ_[p]]
       padicCompletedLevelField p n :=
-  AlgEquiv.ofRingEquiv (by
+  AlgEquiv.ofRingEquiv
+    (f := padicCompletedUnitFrobeniusLiftEquiv p n u⁻¹) (by
     intro b
     change
       padicCompletedUnitFrobeniusLiftEquiv p n u⁻¹
diff --git a/Lean4/GaloisCohomology/Cyclic/Herbrand/HerbrandLowDegree/Basic.lean b/Lean4/GaloisCohomology/Cyclic/Herbrand/HerbrandLowDegree/Basic.lean
index ea6356d..eb19ced 100644
--- a/Lean4/GaloisCohomology/Cyclic/Herbrand/HerbrandLowDegree/Basic.lean
+++ b/Lean4/GaloisCohomology/Cyclic/Herbrand/HerbrandLowDegree/Basic.lean
@@ -14,6 +14,7 @@ import Mathlib.GroupTheory.OrderOfElement
 import Mathlib.GroupTheory.QuotientGroup.Basic
 import Mathlib.GroupTheory.SpecificGroups.Cyclic
 
+
 set_option autoImplicit false
 
 namespace CyclicCohomology
@@ -34,8 +35,7 @@ namespace Herbrand
 
 universe uG uA uB uC
 
-/-- Herbrand-quotient theory: the finite-group norm `N_G a = ∏ g, g • a`
-for a multiplicative `G`-module. -/
+
 def tateNorm (G : Type uG) (A : Type uA) [Group G] [Fintype G] [CommGroup A]
     [MulDistribMulAction G A] (a : A) : A :=
   ∏ g : G, g • a
@@ -92,9 +92,8 @@ theorem fixed_of_forall_mem_zpowers_of_sigmaMinusOne_eq_one
 omit [Group G] [Fintype G] [CommGroup A] [CommGroup B] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- First-isomorphism cardinality source for finite exactness calculations:
-the cardinality of the source of a group homomorphism is the product of the
-cardinalities of its kernel and range. -/
+
+
 theorem monoidHom_card_eq_card_ker_mul_card_range
     {X : Type uA} {Y : Type uB} [Group X] [Group Y] [Finite X] (f : X →* Y) :
     Nat.card X = Nat.card (MonoidHom.ker f) * Nat.card (MonoidHom.range f) := by
@@ -128,9 +127,8 @@ theorem card_subgroupOf_eq_card
 omit [Fintype G] [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- The multiplicative induced module from the trivial subgroup, represented
-as functions on `G` with the right-regular action. This is the concrete model
-used in the low-degree Herbrand comparison. -/
+
+
 @[reducible] def rightRegularFunctionMulDistribMulAction :
     MulDistribMulAction G (G → B) where
   smul g f := fun x => f (x * g)
@@ -255,7 +253,7 @@ def tateNormSubgroup (G : Type uG) (A : Type uA) [Group G] [Fintype G] [CommGrou
     [MulDistribMulAction G A] : Subgroup A :=
   MonoidHom.range (tateNormHom (G := G) (A := A))
 
-/-- Herbrand-quotient theory: the norm kernel `{a | N_G a = 1}`. -/
+
 def normKernelSubgroup (G : Type uG) (A : Type uA) [Group G] [Fintype G] [CommGroup A]
     [MulDistribMulAction G A] : Subgroup A :=
   MonoidHom.ker (tateNormHom (G := G) (A := A))
@@ -267,7 +265,7 @@ def augmentationSubgroup (G : Type uG) (A : Type uA) [Group G] [CommGroup A]
 
 omit [Fintype G] [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- If `σ` generates `G`, the kernel of `a ↦ a^(σ-1)` is the fixed subgroup. -/
+
 theorem sigmaMinusOneHom_ker_eq_fixedSubgroup
     (σ : G) (hgen : ∀ g : G, g ∈ Subgroup.zpowers σ) :
     MonoidHom.ker (sigmaMinusOneHom (G := G) (A := A) σ) = fixedSubgroup G A := by
@@ -282,8 +280,8 @@ theorem sigmaMinusOneHom_ker_eq_fixedSubgroup
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- Finite-module source: the norm homomorphism factors the cardinality of
-`A` into norm-kernel and norm-image cardinalities. -/
+
+
 theorem card_eq_card_normKernelSubgroup_mul_card_tateNormSubgroup [Finite A] :
     Nat.card A = Nat.card (normKernelSubgroup G A) * Nat.card (tateNormSubgroup G A) := by
   simpa [normKernelSubgroup, tateNormSubgroup] using
@@ -291,9 +289,8 @@ theorem card_eq_card_normKernelSubgroup_mul_card_tateNormSubgroup [Finite A] :
 
 omit [Fintype G] [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- Finite-module source: for a cyclic generator `σ`, the coboundary
-homomorphism factors the cardinality of `A` into fixed and augmentation
-cardinalities. -/
+
+
 theorem card_eq_card_fixedSubgroup_mul_card_augmentationSubgroup
     (σ : G) (hgen : ∀ g : G, g ∈ Subgroup.zpowers σ) [Finite A] :
     Nat.card A = Nat.card (fixedSubgroup G A) * Nat.card (augmentationSubgroup G A σ) := by
@@ -304,8 +301,8 @@ theorem card_eq_card_fixedSubgroup_mul_card_augmentationSubgroup
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- Finite-module cardinality balance behind `h(G,A)=1`: for a finite cyclic
-group action, the product `#A^G · #I_G A` equals `#ker N_G · #N_G A`. -/
+
+
 theorem herbrand_finite_module_cardinality_balance
     (σ : G) (hgen : ∀ g : G, g ∈ Subgroup.zpowers σ) [Finite A] :
     Nat.card (fixedSubgroup G A) * Nat.card (augmentationSubgroup G A σ) =
@@ -380,7 +377,7 @@ def HerbrandH0.mk : fixedSubgroup G A →* HerbrandH0 G A := by
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- Canonical projection from the norm kernel to `H⁻¹(G,A)`. -/
+
 def HerbrandHMinusOne.mk (σ : G) :
     normKernelSubgroup G A →* HerbrandHMinusOne G A σ := by
   unfold HerbrandHMinusOne
@@ -422,7 +419,7 @@ theorem HerbrandH0.mk_surjective :
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- Every degree-minus-one Herbrand class has a norm-kernel representative. -/
+
 theorem HerbrandHMinusOne.mk_surjective (σ : G) :
     Function.Surjective (HerbrandHMinusOne.mk (G := G) (A := A) σ) := by
   intro q
@@ -458,8 +455,8 @@ theorem HerbrandH0.mk_eq_one_iff (a : fixedSubgroup G A) :
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- A norm-kernel representative gives the trivial degree-minus-one class
-exactly when it is an augmentation. -/
+
+
 @[simp]
 theorem HerbrandHMinusOne.mk_eq_one_iff (σ : G)
     (a : normKernelSubgroup G A) :
@@ -573,8 +570,8 @@ protected theorem HerbrandH0.inductionOn
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- Eliminate a degree-minus-one Herbrand class through an arbitrary
-norm-kernel representative and the canonical class map. -/
+
+
 protected theorem HerbrandHMinusOne.inductionOn (σ : G)
     {motive : HerbrandHMinusOne G A σ → Prop}
     (q : HerbrandHMinusOne G A σ)
@@ -649,7 +646,7 @@ def HerbrandH0.lift {M : Type*} [Group M]
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- Descend a homomorphism on the norm kernel through `H⁻¹(G,A)`. -/
+
 def HerbrandHMinusOne.lift {M : Type*} [Group M] (σ : G)
     (f : normKernelSubgroup G A →* M)
     (h : (augmentationSubgroup G A σ).subgroupOf (normKernelSubgroup G A) ≤
@@ -703,7 +700,7 @@ noncomputable instance herbrandHMinusOneFiniteOfFinite (σ : G) [Finite A] :
 omit [Fintype G] [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- Fixed subgroup of the right-regular multiplicative induced module. -/
+
 abbrev rightRegularFunctionFixedSubgroup : Subgroup (G → B) := by
   letI := rightRegularFunctionMulDistribMulAction (G := G) (B := B)
   exact fixedSubgroup G (G → B)
@@ -711,7 +708,7 @@ abbrev rightRegularFunctionFixedSubgroup : Subgroup (G → B) := by
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- Norm image subgroup of the right-regular multiplicative induced module. -/
+
 abbrev rightRegularFunctionTateNormSubgroup : Subgroup (G → B) := by
   letI := rightRegularFunctionMulDistribMulAction (G := G) (B := B)
   exact tateNormSubgroup G (G → B)
@@ -719,8 +716,8 @@ abbrev rightRegularFunctionTateNormSubgroup : Subgroup (G → B) := by
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- `H⁰` of the right-regular multiplicative induced module. -/
-def rightRegularFunctionHerbrandH0 := by
+
+def rightRegularFunctionHerbrandH0 : Type (max uG uB) := by
   letI := rightRegularFunctionMulDistribMulAction (G := G) (B := B)
   exact HerbrandH0 G (G → B)
 
@@ -737,7 +734,7 @@ noncomputable instance rightRegularFunctionHerbrandH0CommGroup :
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- Norm-kernel subgroup of the right-regular multiplicative induced module. -/
+
 abbrev rightRegularFunctionNormKernelSubgroup : Subgroup (G → B) := by
   letI := rightRegularFunctionMulDistribMulAction (G := G) (B := B)
   exact normKernelSubgroup G (G → B)
@@ -745,7 +742,7 @@ abbrev rightRegularFunctionNormKernelSubgroup : Subgroup (G → B) := by
 omit [Fintype G] [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- Augmentation subgroup of the right-regular multiplicative induced module. -/
+
 abbrev rightRegularFunctionAugmentationSubgroup (σ : G) : Subgroup (G → B) := by
   letI := rightRegularFunctionMulDistribMulAction (G := G) (B := B)
   exact augmentationSubgroup G (G → B) σ
@@ -753,8 +750,8 @@ abbrev rightRegularFunctionAugmentationSubgroup (σ : G) : Subgroup (G → B) :=
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- `H^{-1}` of the right-regular multiplicative induced module. -/
-def rightRegularFunctionHerbrandHMinusOne (σ : G) := by
+
+def rightRegularFunctionHerbrandHMinusOne (σ : G) : Type (max uG uB) := by
   letI := rightRegularFunctionMulDistribMulAction (G := G) (B := B)
   exact HerbrandHMinusOne G (G → B) σ
 
@@ -791,7 +788,7 @@ theorem tateNorm_sigmaMinusOne_eq_one (σ : G) (a : A) :
     _ = 1 := by
       rw [hshift, Finset.prod_inv_distrib, mul_inv_cancel]
 
-/-- The augmentation image lies in the norm kernel. -/
+
 theorem augmentationSubgroup_le_normKernelSubgroup (σ : G) :
     augmentationSubgroup G A σ ≤ normKernelSubgroup G A := by
   intro a ha
@@ -858,9 +855,8 @@ theorem herbrandH0_subsingleton_iff_fixed_le_tateNormSubgroup :
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- A quotient-level criterion for vanishing of `H^{-1}`: if every norm-kernel
-element is an augmentation element, then the low-degree Herbrand `H^{-1}`
-quotient is a subsingleton. -/
+
+
 theorem herbrandHMinusOne_subsingleton_of_normKernel_le_augmentationSubgroup
     (σ : G) (h : normKernelSubgroup G A ≤ augmentationSubgroup G A σ) :
     Subsingleton (HerbrandHMinusOne G A σ) := by
@@ -879,8 +875,7 @@ theorem herbrandHMinusOne_subsingleton_of_normKernel_le_augmentationSubgroup
     exact h a.2
   rw [hq, hr]
 
-/-- Degree-minus-one Herbrand cohomology is trivial exactly when every
-norm-kernel element is an augmentation element. -/
+
 theorem herbrandHMinusOne_subsingleton_iff_normKernel_le_augmentationSubgroup
     (σ : G) :
     Subsingleton (HerbrandHMinusOne G A σ) ↔
@@ -986,8 +981,8 @@ theorem herbrandHMinusOne_subsingleton_of_mulEquiv
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- Membership in the right-regular norm kernel is the single product
-condition `∏ g, f g = 1`. -/
+
+
 theorem rightRegularFunction_mem_normKernelSubgroup_iff (f : G → B) :
     f ∈ rightRegularFunctionNormKernelSubgroup (G := G) (B := B) ↔
       (∏ g : G, f g) = 1 := by
@@ -1031,9 +1026,8 @@ theorem rightRegularFunction_mem_normKernelSubgroup_iff (f : G → B) :
 omit [Fintype G] [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- The partial primitive along the cyclic powers of `σ`.  This is the
-candidate used to solve `b (x * σ) * (b x)⁻¹ = f x` on the right-regular
-induced module. -/
+
+
 def rightRegularCyclicPartialProduct (σ : G) (f : G → B) (i : ℕ) : B :=
   (Finset.range i).prod (fun k => f (σ ^ k))
 
@@ -1266,8 +1260,8 @@ theorem rightRegularCyclicPrimitive_mul_sigma_mul_inv_of_partialProduct_eq_one
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- A right-regular norm-kernel element has product `1` along one full cyclic
-enumeration by a generator. -/
+
+
 theorem rightRegularFunction_prod_powers_eq_one_of_mem_normKernel
     (σ : G) (hgen : ∀ g : G, g ∈ Subgroup.zpowers σ)
     {f : G → B}
@@ -1325,8 +1319,8 @@ theorem rightRegularFunction_mem_augmentationSubgroup_iff
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- Concrete `H^{-1}` source for the right-regular multiplicative induced
-module: every norm-kernel element is an augmentation element. -/
+
+
 theorem rightRegularFunction_normKernel_le_augmentationSubgroup
     (σ : G) (hgen : ∀ g : G, g ∈ Subgroup.zpowers σ) :
     rightRegularFunctionNormKernelSubgroup (G := G) (B := B) ≤
@@ -1348,8 +1342,8 @@ theorem rightRegularFunction_normKernel_le_augmentationSubgroup
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- The `H^{-1}` quotient of the right-regular multiplicative induced module
-is trivial for a cyclic group generated by `σ`. -/
+
+
 theorem rightRegularFunction_herbrandHMinusOne_subsingleton
     (σ : G) (hgen : ∀ g : G, g ∈ Subgroup.zpowers σ) :
     Subsingleton (rightRegularFunctionHerbrandHMinusOne (G := G) (B := B) σ) := by
@@ -1362,8 +1356,8 @@ theorem rightRegularFunction_herbrandHMinusOne_subsingleton
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- Concrete `H⁰` source for induced modules: for the right-regular
-multiplicative induced module `G → B`, every fixed point is a norm. -/
+
+
 theorem rightRegularFunction_fixed_le_tateNormSubgroup :
     rightRegularFunctionFixedSubgroup (G := G) (B := B) ≤
       rightRegularFunctionTateNormSubgroup (G := G) (B := B) := by
@@ -1420,8 +1414,8 @@ theorem rightRegularFunction_fixed_le_tateNormSubgroup :
 omit [CommGroup A] [CommGroup C]
     [MulDistribMulAction G A] [MulDistribMulAction G B]
     [MulDistribMulAction G C] in
-/-- The `H⁰` quotient of the right-regular multiplicative induced module is
-trivial. This is the degree-zero half of the low-degree Herbrand comparison. -/
+
+
 theorem rightRegularFunction_herbrandH0_subsingleton :
     Subsingleton (rightRegularFunctionHerbrandH0 (G := G) (B := B)) := by
   let := rightRegularFunctionMulDistribMulAction (G := G) (B := B)
@@ -1677,8 +1671,8 @@ theorem card_normKernelSubgroup_eq_card_herbrandHMinusOne_mul_card_augmentationS
 
 omit [CommGroup B] [CommGroup C]
     [MulDistribMulAction G B] [MulDistribMulAction G C] in
-/-- The two low-degree Herbrand quotients of a finite cyclic module have equal
-cardinality, the finite-module source for Herbrand-quotient multiplicativity. -/
+
+
 theorem herbrand_finite_module_card_eq
     (σ : G) (hgen : ∀ g : G, g ∈ Subgroup.zpowers σ) [Finite A] :
     Nat.card (HerbrandH0 G A) = Nat.card (HerbrandHMinusOne G A σ) := by
diff --git a/Lean4/ValuedFieldTheory/LocalField/Padic/UnitDecomposition.lean b/Lean4/ValuedFieldTheory/LocalField/Padic/UnitDecomposition.lean
index c3ea784..4f582bb 100644
--- a/Lean4/ValuedFieldTheory/LocalField/Padic/UnitDecomposition.lean
+++ b/Lean4/ValuedFieldTheory/LocalField/Padic/UnitDecomposition.lean
@@ -12,14 +12,9 @@ import Mathlib.NumberTheory.Padics.ValuativeRel
 import Mathlib.NumberTheory.Padics.ProperSpace
 import Mathlib.GroupTheory.Torsion
 
-set_option autoImplicit false
 
-/-!
-# Unit decomposition of the p-adic integers
+set_option autoImplicit false
 
-This file constructs the reusable topological decomposition of
-`ℤ_[p]ˣ` into its finite factor and its principal `p`-adic factor.
--/
 
 open scoped Topology
 
@@ -457,7 +452,7 @@ noncomputable def padicPrincipalData
     rfl
   have htop : directTopology = standardTopology :=
     hdirect.trans (padicPrincipalUnitDirectTopology_eq_standard p)
-  let P := fun T : TopologicalSpace U => by
+  let P : TopologicalSpace U → Type := fun T => by
     letI : TopologicalSpace U := T
     exact Σ a : ℕ,
       Multiplicative
diff --git a/lakefile.toml b/lakefile.toml
index ed5e179..5e8b720 100644
--- a/lakefile.toml
+++ b/lakefile.toml
@@ -10,8 +10,8 @@ relaxedAutoImplicit = false
 
 [[require]]
 name = "mathlib"
-scope = "leanprover-community"
-rev = "5ed2965256430c3649e86755f9576b54eca72435"
+git = "https://github.com/leanprover-community/mathlib4.git"
+rev = "d13f23b723b8a846827a245b89c10fc7d3f11612"
 
 [[lean_lib]]
 name = "ClassFieldTheory.All"
diff --git a/lean-toolchain b/lean-toolchain
index 12359f9..ba8ebf2 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.34.0
+leanprover/lean4:v4.34.1