diff --git a/lakefile.toml b/lakefile.toml
index 2bca04769..e4a5c654b 100644
--- a/lakefile.toml
+++ b/lakefile.toml
@@ -12,7 +12,7 @@ pp.unicode.fun = true   # universal across all origin projects
 [[require]]
 name = "mathlib"
 git = "https://github.com/leanprover-community/mathlib4.git"
-rev = "e4b72ca0d01c"     # daily steady-state bump 2026-08-25, toolchain v4.34.0-rc2
+rev = "d13f23b723b8a846827a245b89c10fc7d3f11612"     # Zauner compatibility: Lean/Mathlib v4.34.1
 
 # ───────────────────────── shared, refactored-out lemmas ─────────────────────────
 [[lean_lib]]
@@ -85,7 +85,6 @@ globs = ["CebotarevDensity"]
 [lean_lib.leanOptions]
 autoImplicit = false
 relaxedAutoImplicit = false
-maxSynthPendingDepth = 3
 
 # ───────── Belabas–Friedman: effective residue of the Dedekind zeta function ─────────
 # Imports mathlib (and CebotarevDensity for the ζ_K Euler product); built on demand.
diff --git a/lean-toolchain b/lean-toolchain
index b814d987e..ba8ebf2db 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.34.0-rc2
+leanprover/lean4:v4.34.1
diff --git a/projects/Chebotarev/CebotarevDensity/Abelian.lean b/projects/Chebotarev/CebotarevDensity/Abelian.lean
index ceca8bdf6..497e8ecc3 100644
--- a/projects/Chebotarev/CebotarevDensity/Abelian.lean
+++ b/projects/Chebotarev/CebotarevDensity/Abelian.lean
@@ -233,13 +233,13 @@ private theorem prime_dvd_natAbs_discr_cyclotomic_dvd
     (E : Type*) [Field E] [NumberField E] (m : ℕ) [NeZero m] [IsCyclotomicExtension {m} ℚ E]
     {p : ℕ} (hp : p.Prime) (hpd : p ∣ (NumberField.discr E).natAbs) : p ∣ m := by
   by_contra hpm
-  haveI : Fact (Nat.Prime p) := ⟨hp⟩
+  have : Fact (Nat.Prime p) := ⟨hp⟩
   have hpprime : Prime (p : ℤ) := Nat.prime_iff_prime_int.mp hp
   refine absurd (Int.ofNat_dvd_left.mpr hpd) ?_
   rw [NumberField.not_dvd_discr_iff_forall_liesOver E (𝓞 E) hpprime]
   intro P hPmax hlo
-  haveI := hPmax.isPrime
-  haveI := hlo
+  have := hPmax.isPrime
+  have := hlo
   have hspanbot : Ideal.span {(p : ℤ)} ≠ ⊥ := by
     rw [Ne, Ideal.span_singleton_eq_bot]; exact hpprime.ne_zero
   have hPbot : P ≠ ⊥ := Ideal.ne_bot_of_liesOver_of_ne_bot hspanbot P
@@ -263,9 +263,9 @@ private theorem cyclotomicField_finrank_eq
     (Set.mem_singleton m) (NeZero.ne m)
   set K₁ : IntermediateField ℚ M := IntermediateField.adjoin ℚ {ζ} with hK₁def
   set K₂ : IntermediateField ℚ M := (IsScalarTower.toAlgHom ℚ K M).fieldRange with hK₂def
-  haveI hK₁cyc : IsCyclotomicExtension {m} ℚ K₁ :=
+  have hK₁cyc : IsCyclotomicExtension {m} ℚ K₁ :=
     hζ.intermediateField_adjoin_isCyclotomicExtension (K := ℚ)
-  haveI : IsGalois ℚ K₁ := IsCyclotomicExtension.isGalois (S := {m}) (K := ℚ) (L := K₁)
+  have : IsGalois ℚ K₁ := IsCyclotomicExtension.isGalois (S := {m}) (K := ℚ) (L := K₁)
   have hfinK₁ : Module.finrank ℚ K₁ = m.totient :=
     IsCyclotomicExtension.finrank K₁ (Polynomial.cyclotomic.irreducible_rat (NeZero.pos m))
   have hsup : K₁ ⊔ K₂ = ⊤ := by
@@ -317,8 +317,8 @@ private theorem compositum_charProd_bijective
     [IsGalois K L] [IsGalois K M] (m : ℕ) [NeZero m] [IsCyclotomicExtension {m} L M]
     (hcop : ((NumberField.discr L).natAbs).Coprime m) (ζ : M) (hζ : IsPrimitiveRoot ζ m) :
     Function.Bijective ((AlgEquiv.restrictNormalHom L).prod (hζ.autToPow K)) := by
-  haveI : FiniteDimensional K M := inferInstance
-  haveI : IsGalois L M := IsGalois.tower_top_of_isGalois K L M
+  have : FiniteDimensional K M := inferInstance
+  have : IsGalois L M := IsGalois.tower_top_of_isGalois K L M
   set χK : Gal(M/K) →* (ZMod m)ˣ := hζ.autToPow K with hχK
   set Φ : Gal(M/K) →* Gal(L/K) × (ZMod m)ˣ :=
     (AlgEquiv.restrictNormalHom L).prod χK with hΦ
@@ -372,8 +372,8 @@ private theorem autToPow_L_bijective
     [IsGalois K L] [IsGalois K M] (m : ℕ) [NeZero m] [IsCyclotomicExtension {m} L M]
     (hcop : ((NumberField.discr L).natAbs).Coprime m) (ζ : M) (hζ : IsPrimitiveRoot ζ m) :
     Function.Bijective (hζ.autToPow L) := by
-  haveI : FiniteDimensional K M := inferInstance
-  haveI : IsGalois L M := IsGalois.tower_top_of_isGalois K L M
+  have : FiniteDimensional K M := inferInstance
+  have : IsGalois L M := IsGalois.tower_top_of_isGalois K L M
   have hML : Module.finrank L M = m.totient := cyclotomicField_finrank_eq L M m hcop
   have hcardML : Nat.card Gal(M/L) = Nat.card (ZMod m)ˣ := by
     rw [IsGalois.card_aut_eq_finrank L M, hML, Nat.card_eq_fintype_card,
@@ -426,7 +426,7 @@ private theorem compositum_isCyclotomic_over_fixedField
       (IntermediateField.adjoin_le_iff.mpr (Set.singleton_subset_iff.mpr
         (IntermediateField.subset_adjoin K _ hζ.pow_eq_one)))
       (IntermediateField.adjoin_le_iff.mpr fun x hx ↦ by
-        obtain ⟨i, -, rfl⟩ := hζ.eq_pow_of_pow_eq_one (Set.mem_setOf_eq ▸ hx)
+        obtain ⟨i, -, rfl⟩ := hζ.eq_pow_of_pow_eq_one (Set.mem_ofPred_eq ▸ hx)
         exact pow_mem (IntermediateField.subset_adjoin K _ (Set.mem_singleton ζ)) i)
   have hsup : (F ⊔ Kμ).fixingSubgroup = ⊥ := by
     rw [IntermediateField.fixingSubgroup_sup, IntermediateField.fixingSubgroup_fixedField, _hmeet]
@@ -437,8 +437,8 @@ private theorem compositum_isCyclotomic_over_fixedField
     apply IntermediateField.restrictScalars_injective K
     rw [IntermediateField.restrictScalars_adjoin_eq_sup, hadjζ, htop]
     rfl
-  haveI : Algebra.IsIntegral ↥F M := Algebra.IsIntegral.of_finite ↥F M
-  haveI hcyc : IsCyclotomicExtension {m} ↥F (IntermediateField.adjoin (↥F) {ζ}) :=
+  have : Algebra.IsIntegral ↥F M := Algebra.IsIntegral.of_finite ↥F M
+  have hcyc : IsCyclotomicExtension {m} ↥F (IntermediateField.adjoin (↥F) {ζ}) :=
     IsPrimitiveRoot.intermediateField_adjoin_isCyclotomicExtension (K := ↥F) hζ
   rw [htopF] at hcyc
   exact IsCyclotomicExtension.equiv (S := {m}) (A := ↥F) (f := IntermediateField.topEquiv)
@@ -454,7 +454,7 @@ private theorem smul_algebraMap_eq_repl
     [Algebra K L] [Algebra K M] [Algebra L M] [IsScalarTower K L M]
     [IsGalois K L] [IsGalois K M] (σ : Gal(M/K)) (y : 𝓞 L) :
     σ • (algebraMap (𝓞 L) (𝓞 M) y) = algebraMap (𝓞 L) (𝓞 M) ((σ.restrictNormal L) • y) := by
-  haveI : IsScalarTower (𝓞 K) (𝓞 L) (𝓞 M) := inferInstance
+  have : IsScalarTower (𝓞 K) (𝓞 L) (𝓞 M) := inferInstance
   have hbridgeM : ∀ (g : M ≃ₐ[K] M) (x : 𝓞 M), ((g • x : 𝓞 M) : M) = g • (x : M) := fun g x ↦
     by simpa [Algebra.smul_def] using
       (smul_distrib_smul (G := M ≃ₐ[K] M) (R := 𝓞 M) (S := M) g x 1).symm
@@ -479,7 +479,7 @@ private theorem isArithFrobAt_restrictNormal_repl
     [IsGalois K L] [IsGalois K M] (σ : Gal(M/K)) (𝔓 : Ideal (𝓞 M))
     (hσ : IsArithFrobAt (𝓞 K) σ 𝔓) :
     IsArithFrobAt (𝓞 K) (σ.restrictNormal L) (𝔓.under (𝓞 L)) := by
-  haveI : IsScalarTower (𝓞 K) (𝓞 L) (𝓞 M) := inferInstance
+  have : IsScalarTower (𝓞 K) (𝓞 L) (𝓞 M) := inferInstance
   have hunder : (𝔓.under (𝓞 L)).under (𝓞 K) = 𝔓.under (𝓞 K) := Ideal.under_under 𝔓
   intro y
   rw [hunder, Ideal.under, Ideal.mem_comap, map_sub, map_pow,
@@ -495,16 +495,16 @@ private theorem frobeniusClass_proj_isPrime_aux
     (_hfr : frobeniusClass K M 𝔭 = ConjClasses.mk τM) :
     frobeniusClass K L 𝔭 = ConjClasses.mk σ := by
   obtain ⟨𝔓, h𝔓p, h𝔓lo, -⟩ := exists_prime_liesOver K M 𝔭 (UnramifiedIn.ne_bot K M _hunrM)
-  haveI := h𝔓p
-  haveI := h𝔓lo
-  haveI : Finite (𝓞 M ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓
+  have := h𝔓p
+  have := h𝔓lo
+  have : Finite (𝓞 M ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓
     (ne_bot_of_ramificationIdx_eq_one K M (UnramifiedIn.ramificationIdx_eq_one K M _hunrM 𝔓 h𝔓lo))
   set σM : Gal(M/K) := arithFrobAt (𝓞 K) Gal(M/K) 𝔓
   have hMfrobσM : IsArithFrobAt (𝓞 K) σM 𝔓 := IsArithFrobAt.arithFrobAt (𝓞 K) Gal(M/K) 𝔓
   have hconjM : IsConj σM τM := ConjClasses.mk_eq_mk_iff_isConj.mp
     ((frobeniusClass_eq_mk_of_isArithFrobAt K M 𝔭 _hunrM σM 𝔓 hMfrobσM h𝔓lo).symm.trans _hfr)
-  haveI : (𝔓.under (𝓞 L)).IsPrime := Ideal.IsPrime.under (𝓞 L) 𝔓
-  haveI : (𝔓.under (𝓞 L)).LiesOver 𝔭 := ⟨((Ideal.under_under 𝔓).trans h𝔓lo.over.symm).symm⟩
+  have : (𝔓.under (𝓞 L)).IsPrime := Ideal.IsPrime.under (𝓞 L) 𝔓
+  have : (𝔓.under (𝓞 L)).LiesOver 𝔭 := ⟨((Ideal.under_under 𝔓).trans h𝔓lo.over.symm).symm⟩
   rw [frobeniusClass_eq_mk_of_isArithFrobAt K L 𝔭 _hunrL (σM.restrictNormal L) (𝔓.under (𝓞 L))
     (isArithFrobAt_restrictNormal_repl K L M σM 𝔓 hMfrobσM) inferInstance]
   refine ConjClasses.mk_eq_mk_iff_isConj.mpr ?_
@@ -527,12 +527,12 @@ private theorem frobeniusClass_proj
     (_hfr : frobeniusClass K M 𝔭 = ConjClasses.mk τM) :
     frobeniusClass K L 𝔭 = ConjClasses.mk σ := by
   by_cases hp : 𝔭.IsPrime
-  · haveI := hp
+  · have := hp
     exact frobeniusClass_proj_isPrime_aux K L M σ τM _hτM 𝔭 _hunrM _hunrL _hfr
   · have hMjunk : frobeniusClass K M 𝔭 = ConjClasses.mk 1 := by
-      rw [frobeniusClass, dif_neg fun h ↦ hp h.1]
+      rw [frobeniusClass, dite_eq_right fun h ↦ hp h.1]
     have hLjunk : frobeniusClass K L 𝔭 = ConjClasses.mk 1 := by
-      rw [frobeniusClass, dif_neg fun h ↦ hp h.1]
+      rw [frobeniusClass, dite_eq_right fun h ↦ hp h.1]
     have hconj : IsConj (1 : Gal(M/K)) τM :=
       ConjClasses.mk_eq_mk_iff_isConj.mp (hMjunk.symm.trans _hfr)
     have hτM1 : τM = 1 := isConj_one_right.mp hconj
@@ -583,16 +583,16 @@ private theorem isGalois_compositum_base
     [IsScalarTower K L M] [IsCyclotomicExtension {m} L M] : IsGalois K M := by
   obtain ⟨ζ, hζ⟩ : ∃ r : M, IsPrimitiveRoot r m :=
     IsCyclotomicExtension.exists_isPrimitiveRoot (S := {m}) L M (Set.mem_singleton m) (NeZero.ne m)
-  haveI : FiniteDimensional K M := inferInstance
-  haveI hsep : Algebra.IsSeparable K M := inferInstance
+  have : FiniteDimensional K M := inferInstance
+  have hsep : Algebra.IsSeparable K M := inferInstance
   set A : IntermediateField K M := (IsScalarTower.toAlgHom K L M).fieldRange with hA
   set B : IntermediateField K M := IntermediateField.adjoin K {ζ} with hB
-  haveI hAnormal : Normal K A :=
+  have hAnormal : Normal K A :=
     Normal.of_algEquiv (AlgEquiv.ofInjectiveField (IsScalarTower.toAlgHom K L M))
-  haveI hBcyc : IsCyclotomicExtension {m} K B :=
+  have hBcyc : IsCyclotomicExtension {m} K B :=
     hζ.intermediateField_adjoin_isCyclotomicExtension (K := K)
-  haveI hBgal : IsGalois K B := IsCyclotomicExtension.isGalois (S := {m}) (K := K) (L := B)
-  haveI hBnormal : Normal K B := hBgal.to_normal
+  have hBgal : IsGalois K B := IsCyclotomicExtension.isGalois (S := {m}) (K := K) (L := B)
+  have hBnormal : Normal K B := hBgal.to_normal
   have hsup : A ⊔ B = ⊤ := by
     have hζalg : IsAlgebraic K ζ := Algebra.IsAlgebraic.isAlgebraic ζ
     have hsubalg : (IsScalarTower.toAlgHom K L M).range ⊔ Algebra.adjoin K {ζ}
@@ -606,7 +606,7 @@ private theorem isGalois_compositum_base
       IntermediateField.adjoin_simple_toSubalgebra_of_isAlgebraic hζalg,
       AlgHom.fieldRange_toSubalgebra, IntermediateField.top_toSubalgebra]
     exact hsubalg
-  haveI hnormal : Normal K M := by
+  have hnormal : Normal K M := by
     have h := IntermediateField.normal_sup K M A B
     rw [hsup] at h
     exact Normal.of_algEquiv (IntermediateField.topEquiv (F := K) (E := M))
@@ -615,35 +615,38 @@ private theorem isGalois_compositum_base
 /-- **Unramifiedness descends to an intermediate field.** If a prime `𝔭` of `K` is unramified
 in the top field `M` of a tower `K ⊆ L ⊆ M`, it is unramified in `L`: for a maximal prime `𝔮`
 of `𝓞 L` over `𝔭`, pick a prime `𝔓` of `𝓞 M` over `𝔮`; then `e(𝔓/𝔭) = 1` (from the `M`-side
-hypothesis) factors as `e(𝔮/𝔭)·e(𝔓/𝔮)` (`Ideal.ramificationIdx_algebra_tower`), forcing
+hypothesis) factors as `e(𝔮/𝔭)·e(𝔓/𝔮)` (`Ideal.ramificationIdx'_algebra_tower`), forcing
 `e(𝔮/𝔭) = 1`. Supplies C4's `_hunrL : UnramifiedIn K L 𝔭` input from `UnramifiedIn K M 𝔭`. -/
 private theorem unramifiedIn_tower_descend
     (K L M : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Field M] [NumberField M]
     [Algebra K L] [Algebra K M] [Algebra L M] [IsScalarTower K L M] [IsGalois K L] [IsGalois K M]
     (𝔭 : Ideal (𝓞 K)) (hunr : UnramifiedIn K M 𝔭) : UnramifiedIn K L 𝔭 := by
-  haveI : IsScalarTower (𝓞 K) (𝓞 L) (𝓞 M) := inferInstance
+  have : IsScalarTower (𝓞 K) (𝓞 L) (𝓞 M) := inferInstance
   refine ⟨hunr.1, fun 𝔮 h𝔮max h𝔮lo ↦ ?_⟩
-  haveI := h𝔮max
-  haveI := h𝔮lo
-  haveI h𝔮p : 𝔮.IsPrime := h𝔮max.isPrime
+  have := h𝔮max
+  have := h𝔮lo
+  have h𝔮p : 𝔮.IsPrime := h𝔮max.isPrime
   have h𝔮bot : 𝔮 ≠ ⊥ := Ideal.ne_bot_of_liesOver_of_ne_bot hunr.1 𝔮
   obtain ⟨𝔓, _, h𝔓p, h𝔓comap⟩ :=
     Ideal.exists_ideal_over_prime_of_isIntegral (S := 𝓞 M) 𝔮 ⊥ (by simp)
-  haveI := h𝔓p
-  haveI h𝔓lo𝔮 : 𝔓.LiesOver 𝔮 := ⟨h𝔓comap.symm⟩
+  have := h𝔓p
+  have h𝔓lo𝔮 : 𝔓.LiesOver 𝔮 := ⟨h𝔓comap.symm⟩
   have h𝔓bot : 𝔓 ≠ ⊥ := Ideal.ne_bot_of_liesOver_of_ne_bot h𝔮bot 𝔓
-  haveI h𝔓max : 𝔓.IsMaximal := h𝔓p.isMaximal h𝔓bot
+  have h𝔓max : 𝔓.IsMaximal := h𝔓p.isMaximal h𝔓bot
   have h𝔮under : Ideal.under (𝓞 L) 𝔓 = 𝔮 := h𝔓lo𝔮.over.symm
   have h𝔭under : Ideal.under (𝓞 K) 𝔮 = 𝔭 := h𝔮lo.over.symm
-  haveI h𝔓lo𝔭 : 𝔓.LiesOver 𝔭 := ⟨by rw [← h𝔭under, ← h𝔮under, Ideal.under_under]⟩
+  have h𝔓lo𝔭 : 𝔓.LiesOver 𝔭 := ⟨by rw [← h𝔭under, ← h𝔮under, Ideal.under_under]⟩
   have hunderP : Ideal.under (𝓞 K) 𝔓 = 𝔭 := h𝔓lo𝔭.over.symm
   have hP1 : (Ideal.under (𝓞 K) 𝔓).ramificationIdx' 𝔓 = 1 := by
     rw [Ideal.ramificationIdx'_eq_ramificationIdx _ 𝔓 (hunderP ▸ hunr.1)]
     exact Ideal.ramificationIdx_eq_one_iff.mpr (hunr.2 𝔓 h𝔓max h𝔓lo𝔭)
   rw [hunderP] at hP1
-  have htower := Ideal.ramificationIdx_algebra_tower (R := 𝓞 K) (S := 𝓞 L) (T := 𝓞 M)
+  have htower := Ideal.ramificationIdx'_algebra_tower (R := 𝓞 K) (S := 𝓞 L) (T := 𝓞 M)
     (p := 𝔭) (P := 𝔮) (Q := 𝔓) (Ideal.map_ne_bot_of_ne_bot h𝔮bot)
-    (Ideal.map_ne_bot_of_ne_bot hunr.1) (by rw [Ideal.map_le_iff_le_comap, h𝔓comap])
+    (Ideal.map_ne_bot_of_ne_bot hunr.1) (by
+      rw [Ideal.map_le_iff_le_comap]
+      change 𝔮 ≤ 𝔓.under (𝓞 L)
+      rw [h𝔓comap])
   rw [hP1] at htower
   have he𝔮 : 𝔭.ramificationIdx' 𝔮 = 1 := Nat.eq_one_of_mul_eq_one_right htower.symm
   rw [← Ideal.ramificationIdx_eq_one_iff,
@@ -732,15 +735,15 @@ private theorem density_crossing_fibre_aux
         frobeniusClass K M 𝔭 = ConjClasses.mk s}
       ((Nat.card Gal(L/K) * Nat.card ((ZMod m)ˣ) : ℝ)⁻¹) := by
   set F : IntermediateField K M := IntermediateField.fixedField (Subgroup.zpowers s) with hF
-  haveI : IsScalarTower K ↥F M := F.isScalarTower_mid'
-  haveI : IsCyclotomicExtension {m} ↥F M :=
+  have : IsScalarTower K ↥F M := F.isScalarTower_mid'
+  have : IsCyclotomicExtension {m} ↥F M :=
     compositum_isCyclotomic_over_fixedField K L M m s hgate
   set σE : Gal(M/↥F) :=
     IntermediateField.subgroupEquivAlgEquiv (Subgroup.zpowers s) ⟨s, Subgroup.mem_zpowers s⟩
   have hlift := density_lift_through_fixedField_repl K M s F σE
     (by ext x; rfl) rfl (chebotarev_cyclotomic (K := ↥F) (L := M) m hm4 σE)
   have hcarrier : Nat.card (ConjClasses.mk s).carrier = 1 := by
-    letI : CommMonoid Gal(M/K) := IsMulCommutative.instCommMonoid
+    let : CommMonoid Gal(M/K) := IsMulCommutative.instCommMonoid
     have hcar : (ConjClasses.mk s).carrier = {s} := by
       ext a
       rw [ConjClasses.mem_carrier_iff_mk_eq, ConjClasses.mk_eq_mk_iff_isConj,
@@ -782,11 +785,11 @@ private theorem exists_crossing_family_tagged
       (∀ τ, HasDirichletDensity (S τ)
           ((Nat.card Gal(L/K) * Nat.card ((ZMod m)ˣ) : ℝ)⁻¹)) := by
   classical
-  haveI : NeZero m := ⟨by lia⟩
+  have : NeZero m := ⟨by lia⟩
   let M := CyclotomicField m L
-  haveI : IsGalois K M := isGalois_compositum_base K L m M
-  haveI : IsGalois L M := IsGalois.tower_top_of_isGalois K L M
-  haveI : FiniteDimensional K M := inferInstance
+  have : IsGalois K M := isGalois_compositum_base K L m M
+  have : IsGalois L M := IsGalois.tower_top_of_isGalois K L M
+  have : FiniteDimensional K M := inferInstance
   obtain ⟨ζ, hζ⟩ : ∃ r : M, IsPrimitiveRoot r m :=
     IsCyclotomicExtension.exists_isPrimitiveRoot (S := {m}) L M (Set.mem_singleton m) (NeZero.ne m)
   set χK : Gal(M/K) →* (ZMod m)ˣ := hζ.autToPow K with hχK
@@ -796,10 +799,10 @@ private theorem exists_crossing_family_tagged
     MulEquiv.ofBijective _ hΦbij with hequivΦ
   set e2 : Gal(M/L) ≃* (ZMod m)ˣ :=
     MulEquiv.ofBijective (hζ.autToPow L) (autToPow_L_bijective K L M m hcop ζ hζ) with he2
-  haveI : IsMulCommutative Gal(M/L) :=
+  have : IsMulCommutative Gal(M/L) :=
     .of_comm fun a b ↦ e2.injective (by rw [map_mul, map_mul]; exact mul_comm (e2 a) (e2 b))
-  haveI hGcomm : ∀ x y : Gal(L/K), x * y = y * x := fun x y ↦ mul_comm' x y
-  haveI : IsMulCommutative Gal(M/K) :=
+  have hGcomm : ∀ x y : Gal(L/K), x * y = y * x := fun x y ↦ mul_comm' x y
+  have : IsMulCommutative Gal(M/K) :=
     .of_comm fun a b ↦ equivΦ.injective (by
       rw [map_mul, map_mul, Prod.mul_def, Prod.mul_def, hGcomm, mul_comm
         ((equivΦ a).2) ((equivΦ b).2)])
@@ -933,8 +936,8 @@ private theorem factorization_ordCompl_mul_pow (E p v : ℕ) (hp : p.Prime) (hE
   simp only [Finsupp.coe_add, Pi.add_apply, hp.factorization_pow, Finsupp.single_apply,
     Nat.factorization_ordCompl]
   by_cases hq : q = p
-  · subst hq; rw [Finsupp.erase_same, if_pos rfl, if_pos rfl, zero_add]
-  · rw [Finsupp.erase_ne hq, if_neg fun h ↦ hq h.symm, if_neg hq, add_zero]
+  · subst hq; rw [Finsupp.erase_same, ite_eq_left rfl, ite_eq_left rfl, zero_add]
+  · rw [Finsupp.erase_ne hq, ite_eq_right fun h ↦ hq h.symm, ite_eq_right hq, add_zero]
 
 /-- If `d ∣ E` and the `p`-adic valuation of `d` is `≤ v - 1`, then `d` divides the
 "capped" modulus `ordCompl[p] E * p ^ (v - 1)` (which replaces `E`'s `p`-part by
@@ -947,8 +950,8 @@ private theorem dvd_capped (E d p v : ℕ) (hp : p.Prime) (hE : E ≠ 0) (hd : d
   intro q
   rw [factorization_ordCompl_mul_pow E p v hp hE q]
   by_cases hq : q = p
-  · subst hq; rwa [if_pos rfl]
-  · rw [if_neg hq]; exact (Nat.factorization_le_iff_dvd hdne hE).mpr hd q
+  · subst hq; rwa [ite_eq_left rfl]
+  · rw [ite_eq_right hq]; exact (Nat.factorization_le_iff_dvd hdne hE).mpr hd q
 
 /-- The capped modulus `ordCompl[p] E * p ^ (v - 1)` divides `E` when `v - 1 ≤ v_p(E)`. -/
 private theorem M_dvd_E (E p v : ℕ) (hp : p.Prime) (hE : E ≠ 0) (hle : v - 1 ≤ E.factorization p) :
@@ -958,8 +961,8 @@ private theorem M_dvd_E (E p v : ℕ) (hp : p.Prime) (hE : E ≠ 0) (hle : v - 1
   intro q
   rw [factorization_ordCompl_mul_pow E p v hp hE q]
   by_cases hq : q = p
-  · subst hq; rwa [if_pos rfl]
-  · rw [if_neg hq]
+  · subst hq; rwa [ite_eq_left rfl]
+  · rw [ite_eq_right hq]
 
 /-- Factoring out the complementary `p`-power: `E = (ordCompl[p] E * p ^ (v - 1)) *
 p ^ (v_p(E) - (v - 1))`, used to compute `E / M = p ^ (v_p(E) - (v - 1))`. -/
@@ -1341,7 +1344,7 @@ theorem liminf_ratio_ge_inv_card_G
     rw [Nat.coprime_comm, (hmp k).coprime_iff_not_dvd]
     exact fun hdvd ↦ absurd (Nat.le_of_dvd hdBpos hdvd) (Nat.not_le.mpr (hmgt k))
   have hexp : ∀ k, Monoid.exponent (ZMod (m k))ˣ = m k - 1 := fun k ↦ by
-    haveI : Fact (m k).Prime := ⟨hmp k⟩
+    have : Fact (m k).Prime := ⟨hmp k⟩
     rw [IsCyclic.exponent_eq_card, Nat.card_eq_fintype_card, ZMod.card_units_eq_totient,
       Nat.totient_prime (hmp k)]
   have hbound : ∀ k : ℕ,
@@ -1393,7 +1396,7 @@ theorem ratioSum_frobeniusFibres_tendsto_one
   set R : Set (Ideal (𝓞 K)) :=
     {𝔭 : Ideal (𝓞 K) | 𝔭.IsPrime ∧ 𝔭 ≠ ⊥ ∧ ¬ UnramifiedIn K L 𝔭} with hR
   set D : ℝ → ℝ := primeIdealZetaSum (Set.univ : Set (Ideal (𝓞 K))) with hD
-  letI : CommMonoid Gal(L/K) := IsMulCommutative.instCommMonoid
+  let : CommMonoid Gal(L/K) := IsMulCommutative.instCommMonoid
   have hmk_inj : Function.Injective (ConjClasses.mk : Gal(L/K) → ConjClasses Gal(L/K)) :=
     ConjClasses.mk_injective
   have hpd : ((Finset.univ : Finset Gal(L/K)) : Set Gal(L/K)).PairwiseDisjoint S := by
@@ -1514,7 +1517,7 @@ theorem tendsto_inv_card_of_liminf_ge_of_sum_tendsto_one {ι : Type*} [Fintype 
   have hFlimsup : limsup F l = 1 := hsum.limsup_eq
   have hgle : ∀ i, l.IsBoundedUnder (· ≤ ·) (g i) :=
     isBoundedUnder_le_of_isBoundedUnder_le_sum g hFle hbelow
-  haveI : Nonempty ι := ⟨i₀⟩
+  have : Nonempty ι := ⟨i₀⟩
   have hNpos : 0 < N := Fintype.card_pos
   have hNR : (0 : ℝ) < N := by exact_mod_cast hNpos
   set t : Finset ι := Finset.univ.erase i₀ with ht
diff --git a/projects/Chebotarev/CebotarevDensity/AbsoluteDegreeOneDensity.lean b/projects/Chebotarev/CebotarevDensity/AbsoluteDegreeOneDensity.lean
new file mode 100644
index 000000000..a484ff648
--- /dev/null
+++ b/projects/Chebotarev/CebotarevDensity/AbsoluteDegreeOneDensity.lean
@@ -0,0 +1,144 @@
+module
+
+public import CebotarevDensity.Density
+public import CebotarevDensity.ForMathlib.TsumFiberBound
+public import Mathlib.RingTheory.RamificationInertia.Basic
+public import Mathlib.NumberTheory.RamificationInertia.Inertia
+public import Mathlib.Analysis.PSeries
+public import Mathlib.RingTheory.Ideal.Int
+
+/-!
+# Discarding primes of absolute residue degree greater than one
+
+The higher-degree terms are bounded by `[K : ℚ] * ∑ n⁻²`, independently of
+`s > 1`.  They consequently have zero Dirichlet density, even after intersection
+with an arbitrary set of prime ideals.
+-/
+
+@[expose] public section
+
+noncomputable section
+
+open Filter NumberField Topology Set
+
+namespace Chebotarev
+
+variable {K : Type*} [Field K] [NumberField K]
+
+private theorem absNorm_rpow_neg_le_intUnder_sq
+    (P : Ideal (𝓞 K)) [P.IsPrime] (hP : P ≠ ⊥)
+    (hdegree : ¬ P.absNorm.Prime) {s : ℝ} (hs : 1 < s) :
+    (P.absNorm : ℝ) ^ (-s) ≤ ((P.under ℤ).absNorm : ℝ) ^ (-(2 : ℝ)) := by
+  let : NeZero P := ⟨hP⟩
+  have hp : (P.under ℤ).absNorm.Prime := Nat.absNorm_under_prime P
+  have : P.LiesOver (P.under ℤ) := Ideal.over_under (A := ℤ) (P := P)
+  have hpow := (Ideal.absNorm_pow_inertiaDeg (P.under ℤ) P).symm
+  have hdeg : 2 ≤ P.inertiaDeg ℤ := by
+    have hpos : 0 < P.inertiaDeg ℤ := Ideal.inertiaDeg_pos P ℤ
+    have hne : P.inertiaDeg ℤ ≠ 1 := by
+      intro h
+      apply hdegree
+      rw [hpow, h, pow_one]
+      exact hp
+    omega
+  rw [hpow, Nat.cast_pow,
+    ← Real.rpow_natCast ((P.under ℤ).absNorm : ℝ) (P.inertiaDeg ℤ),
+    ← Real.rpow_mul (by positivity)]
+  apply Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast hp.one_lt.le)
+  nlinarith [mul_le_mul (show (2 : ℝ) ≤ (P.inertiaDeg ℤ : ℝ) by exact_mod_cast hdeg)
+    hs.le (by norm_num) (by positivity : (0 : ℝ) ≤ (P.inertiaDeg ℤ : ℝ))]
+
+/-- The non-prime-norm part of any set of prime ideals has a uniform Dirichlet-series bound. -/
+theorem primeIdealZetaSum_nonprimeNorm_le (S : Set (Ideal (𝓞 K))) {s : ℝ} (hs : 1 < s) :
+    primeIdealZetaSum {P | P ∈ S ∧ ¬ P.absNorm.Prime} s ≤
+      (Module.finrank ℚ K : ℝ) * ∑' n : ℕ, (n : ℝ) ^ (-(2 : ℝ)) := by
+  let A := {P : Ideal (𝓞 K) // (P ∈ S ∧ ¬ P.absNorm.Prime) ∧ P.IsPrime ∧ P ≠ ⊥}
+  let B := {p : Ideal ℤ // p.IsPrime ∧ p ≠ ⊥}
+  have hunder : ∀ P : A, (P.1.under ℤ).IsPrime ∧ P.1.under ℤ ≠ ⊥ := by
+    intro P
+    let := P.2.2.1
+    exact ⟨inferInstance, Ideal.IsIntegral.under_ne_bot ℤ P.2.2.2⟩
+  let g : A → B := fun P => ⟨P.1.under ℤ, hunder P⟩
+  have hfib : ∀ p : B, Finite (g ⁻¹' {p} : Set A) ∧
+      Nat.card (g ⁻¹' {p} : Set A) ≤ Module.finrank ℚ K := by
+    intro p
+    let := p.2.1
+    let : p.1.IsMaximal := p.2.1.isMaximal p.2.2
+    have hmem : ∀ P : (g ⁻¹' {p} : Set A), P.1.1.IsPrime ∧ P.1.1.LiesOver p.1 := by
+      intro P
+      let := P.1.2.2.1
+      refine ⟨P.1.2.2.1, ⟨?_⟩⟩
+      exact (congrArg Subtype.val P.2 : P.1.1.under ℤ = p.1) ▸
+        (Ideal.over_under (A := ℤ) (P := P.1.1)).over
+    let f : (g ⁻¹' {p} : Set A) → p.1.primesOver (𝓞 K) := fun P => ⟨P.1.1, hmem P⟩
+    have hfinj : Function.Injective f := by
+      intro P Q h
+      apply Subtype.ext
+      apply Subtype.ext
+      exact congrArg (fun x : p.1.primesOver (𝓞 K) => x.1) h
+    let : Finite (p.1.primesOver (𝓞 K)) :=
+      (IsDedekindDomain.primesOver_finite p.1 _).to_subtype
+    have hcard : Nat.card (p.1.primesOver (𝓞 K)) ≤ Module.finrank ℚ K := by
+      classical
+      let : Fintype (p.1.primesOver (𝓞 K)) := Fintype.ofFinite _
+      calc
+        Nat.card (p.1.primesOver (𝓞 K)) = ∑ _q : p.1.primesOver (𝓞 K), (1 : ℕ) := by
+          simp [Nat.card_eq_fintype_card]
+        _ ≤ ∑ q : p.1.primesOver (𝓞 K), q.1.ramificationIdx ℤ * q.1.inertiaDeg ℤ := by
+          apply Finset.sum_le_sum
+          intro q _
+          have : q.1.IsPrime := q.2.1
+          exact Nat.mul_pos (Ideal.ramificationIdx_pos q.1 ℤ) (Ideal.inertiaDeg_pos q.1 ℤ)
+        _ = Module.finrank ℤ (𝓞 K) := Ideal.sum_ramification_inertia_eq_finrank p.1 (𝓞 K)
+        _ = Module.finrank ℚ K := RingOfIntegers.rank K
+    exact ⟨Finite.of_injective f hfinj, (Nat.card_le_card_of_injective f hfinj).trans hcard⟩
+  have hnorminj : Function.Injective (fun p : B => p.1.absNorm) := by
+    intro p q h
+    change p.1.absNorm = q.1.absNorm at h
+    apply Subtype.ext
+    rw [← Int.ideal_span_absNorm_eq_self p.1, ← Int.ideal_span_absNorm_eq_self q.1, h]
+  have hnat : Summable (fun n : ℕ => (n : ℝ) ^ (-(2 : ℝ))) :=
+    Real.summable_nat_rpow.mpr (by norm_num)
+  have hsum : Summable (fun p : B => (p.1.absNorm : ℝ) ^ (-(2 : ℝ))) :=
+    hnat.comp_injective hnorminj
+  calc
+    primeIdealZetaSum {P | P ∈ S ∧ ¬ P.absNorm.Prime} s ≤
+        (Module.finrank ℚ K : ℝ) * ∑' p : B, (p.1.absNorm : ℝ) ^ (-(2 : ℝ)) := by
+      apply tsum_real_comp_le_card_fibre_mul g _ _ _
+        (summable_prime_absNorm_rpow _ hs) hsum
+        (fun _ => by positivity) (fun _ => by positivity)
+      · intro P
+        let := P.2.2.1
+        exact absNorm_rpow_neg_le_intUnder_sq P.1 P.2.2.2 P.2.1.2 hs
+      · exact fun p => (hfib p).1
+      · exact fun p => (hfib p).2
+    _ ≤ (Module.finrank ℚ K : ℝ) * ∑' n : ℕ, (n : ℝ) ^ (-(2 : ℝ)) := by
+      apply mul_le_mul_of_nonneg_left _ (by positivity)
+      exact hsum.tsum_le_tsum_of_inj _ hnorminj (fun _ _ => by positivity)
+        (fun _ => le_rfl) hnat
+
+/-- Removing prime ideals of absolute residue degree greater than one preserves density. -/
+theorem HasDirichletDensity.primeNorm {S : Set (Ideal (𝓞 K))} {δ : ℝ}
+    (hS : HasDirichletDensity S δ) :
+    HasDirichletDensity {P | P ∈ S ∧ P.absNorm.Prime} δ := by
+  let T : Set (Ideal (𝓞 K)) := {P | P ∈ S ∧ P.absNorm.Prime}
+  let U : Set (Ideal (𝓞 K)) := {P | P ∈ S ∧ ¬ P.absNorm.Prime}
+  have hzero : HasDirichletDensity U 0 :=
+    tendsto_primeIdealZetaSum_div_univ_zero_of_le_const K U _
+      (eventually_nhdsWithin_of_forall fun s hs => primeIdealZetaSum_nonprimeNorm_le S hs)
+  have hdisjoint : Disjoint T U := by
+    rw [Set.disjoint_left]
+    exact fun _ ht hu => hu.2 ht.2
+  have hunion : T ∪ U = S := by
+    ext P
+    simp only [T, U, Set.mem_union, Set.mem_ofPred_eq]
+    tauto
+  have h := hS.sub hzero
+  simp only [sub_zero] at h
+  apply h.congr'
+  filter_upwards [self_mem_nhdsWithin] with s hs
+  have hsum : primeIdealZetaSum S s = primeIdealZetaSum T s + primeIdealZetaSum U s := by
+    rw [← hunion, primeIdealZetaSum_union_of_disjoint hdisjoint hs]
+  rw [hsum, add_div, add_sub_cancel_right]
+
+end Chebotarev
diff --git a/projects/Chebotarev/CebotarevDensity/Cyclotomic.lean b/projects/Chebotarev/CebotarevDensity/Cyclotomic.lean
index c0e83f4b7..7a08eb09f 100644
--- a/projects/Chebotarev/CebotarevDensity/Cyclotomic.lean
+++ b/projects/Chebotarev/CebotarevDensity/Cyclotomic.lean
@@ -205,7 +205,7 @@ theorem character_orthogonality_cyclotomic_eq
       ConjClasses.mk_eq_mk_iff_isConj.mp (hmk.trans _h)
     rw [SemiconjBy, mul_comm' (c : Gal(L/K))] at hc
     rw [mul_right_cancel hc, mul_inv_cancel]
-  rw [sum_galoisCharacter_mul_inv_eq K L σ τ, if_pos heq]
+  rw [sum_galoisCharacter_mul_inv_eq K L σ τ, ite_eq_left heq]
 
 /-- Sharifi 7.2.1 character orthogonality, **non-matching case**:
 when `frobeniusClass K L 𝔭 ≠ ConjClasses.mk σ`, the character sum
@@ -222,7 +222,7 @@ theorem character_orthogonality_cyclotomic_ne
   have hmk : ConjClasses.mk τ = frobeniusClass K L 𝔭 := Quotient.out_eq _
   have hne : σ * τ⁻¹ ≠ 1 := fun hσ ↦
     _h <| hmk.symm.trans (congrArg ConjClasses.mk (mul_inv_eq_one.mp hσ)).symm
-  rw [sum_galoisCharacter_mul_inv_eq K L σ τ, if_neg hne]
+  rw [sum_galoisCharacter_mul_inv_eq K L σ τ, ite_eq_right hne]
 
 /-! ### Complex-analytic core of the cyclotomic χ≠1 bound
 
@@ -615,7 +615,7 @@ private theorem artinLSeries_prime_sum_bounded_of_analytic_extension
   set D : Set ℂ := {s : ℂ | 1 - (Module.finrank ℚ K : ℝ)⁻¹ < s.re} with hD
   have hDopen : IsOpen D := by rw [hD]; exact isOpen_lt continuous_const Complex.continuous_re
   have hmemD : ∀ s : ℝ, 1 ≤ s → (s : ℂ) ∈ D := fun s hs ↦ by
-    rw [hD]; simp only [Set.mem_setOf_eq, Complex.ofReal_re]
+    rw [hD]; simp only [Set.mem_ofPred_eq, Complex.ofReal_re]
     have hfr : 0 < Module.finrank ℚ K := Module.finrank_pos
     have : (0 : ℝ) < (Module.finrank ℚ K : ℝ)⁻¹ := by positivity
     linarith
@@ -796,8 +796,8 @@ private theorem sum_charTwist_mul_twistedPrimeSum_eq
             (Nat.card Gal(L/K) : ℂ) * (Ideal.absNorm 𝔭.1 : ℂ) ^ (-(s : ℂ)) else 0 := fun 𝔭 ↦ by
     have := 𝔭.2.1
     by_cases h : frobeniusClass K L 𝔭.1 = ConjClasses.mk σ
-    · rw [sum_charTwist_eq K L m σ 𝔭.1 𝔭.2.2 h, if_pos h]
-    · rw [sum_charTwist_ne K L m σ 𝔭.1 𝔭.2.2 h, if_neg h, zero_mul]
+    · rw [sum_charTwist_eq K L m σ 𝔭.1 𝔭.2.2 h, ite_eq_left h]
+    · rw [sum_charTwist_ne K L m σ 𝔭.1 𝔭.2.2 h, ite_eq_right h, zero_mul]
   have hfinj : Function.Injective
       (fun 𝔭 : {𝔭 : Ideal (𝓞 K) // 𝔭 ∈ {𝔭 : Ideal (𝓞 K) | 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭 ∧
           frobeniusClass K L 𝔭 = ConjClasses.mk σ} ∧ 𝔭.IsPrime ∧ 𝔭 ≠ ⊥} ↦
@@ -812,11 +812,11 @@ private theorem sum_charTwist_mul_twistedPrimeSum_eq
       (f := fun 𝔭 : {𝔭 : Ideal (𝓞 K) // 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭} ↦
         if frobeniusClass K L 𝔭.1 = ConjClasses.mk σ then (Ideal.absNorm 𝔭.1 : ℝ) ^ (-s) else 0)
       ?_]
-    · exact tsum_congr fun 𝔭 ↦ (if_pos 𝔭.2.1.2.2).symm
+    · exact tsum_congr fun 𝔭 ↦ (ite_eq_left 𝔭.2.1.2.2).symm
     · rintro 𝔭 h𝔭
       have h : frobeniusClass K L 𝔭.1 = ConjClasses.mk σ := by
         by_contra hne
-        exact h𝔭 (if_neg hne)
+        exact h𝔭 (ite_eq_right hne)
       exact ⟨⟨𝔭.1, ⟨𝔭.2.1, 𝔭.2.2, h⟩, 𝔭.2.1, UnramifiedIn.ne_bot K L 𝔭.2.2⟩, rfl⟩
   rw [hinterchange, tsum_congr hcollapse, hfibre, Complex.ofReal_tsum, ← tsum_mul_left]
   refine tsum_congr fun 𝔭 ↦ ?_
diff --git a/projects/Chebotarev/CebotarevDensity/CyclotomicNormResidue.lean b/projects/Chebotarev/CebotarevDensity/CyclotomicNormResidue.lean
index 4cf9574ba..1bcb85b86 100644
--- a/projects/Chebotarev/CebotarevDensity/CyclotomicNormResidue.lean
+++ b/projects/Chebotarev/CebotarevDensity/CyclotomicNormResidue.lean
@@ -56,7 +56,7 @@ theorem cyclotomic_frobenius_acts_as_norm_power
     haveI : Finite (𝓞 L ⧸ 𝔓) := UnramifiedIn.finite_quotient K L hunr 𝔓 hP
     ∀ ζ : L, ζ ∈ primitiveRoots m L →
       arithFrobAt (𝓞 K) Gal(L/K) 𝔓 ζ = ζ ^ Ideal.absNorm 𝔭 := by
-  haveI : Finite (𝓞 L ⧸ 𝔓) := UnramifiedIn.finite_quotient K L hunr 𝔓 hP
+  have : Finite (𝓞 L ⧸ 𝔓) := UnramifiedIn.finite_quotient K L hunr 𝔓 hP
   intro ζ hζmem
   set φ := arithFrobAt (𝓞 K) Gal(L/K) 𝔓
   have hζ : IsPrimitiveRoot ζ m := (mem_primitiveRoots (NeZero.pos m)).mp hζmem
@@ -64,9 +64,8 @@ theorem cyclotomic_frobenius_acts_as_norm_power
   have hzc : (algebraMap (𝓞 L) L) z = ζ := rfl
   have hzpow : z ^ m = 1 := hζ.toInteger_isPrimitiveRoot.pow_eq_one
   set q := Ideal.absNorm 𝔭
-  have h𝔭ne : 𝔭 ≠ ⊥ := UnramifiedIn.ne_bot K L hunr
   have hcopP : (Ideal.absNorm 𝔓).Coprime m := by
-    rw [Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOver 𝔓 𝔭 ‹𝔭.IsPrime› h𝔭ne]
+    rw [← Ideal.absNorm_pow_inertiaDeg 𝔭 𝔓]
     exact Nat.Coprime.pow_left _ hcop
   have hN1 : Ideal.absNorm 𝔓 ≠ 1 := fun h ↦ ‹𝔓.IsPrime›.ne_top (Ideal.absNorm_eq_one_iff.mp h)
   have hmnotmem : (m : 𝓞 L) ∉ 𝔓 := by
@@ -92,7 +91,6 @@ private theorem pow_natModEq_of_pow_eq {S : Type*} [CommRing S] [IsDomain S] {μ
     [NeZero n] (hμ : IsPrimitiveRoot μ n) {a b : ℕ} (h : μ ^ a = μ ^ b) : a ≡ b [MOD n] :=
   hμ.eq_orderOf ▸ (hμ.isOfFinOrder (NeZero.ne n)).pow_eq_pow_iff_modEq.mp h
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **The cyclotomic Frobenius is the norm residue** (multiplicative form). For `L = K(μ_m)`,
 a primitive `m`-th root `ζ` of unity in `L`, and a prime `𝔭` of `K` unramified in `L` with
 `N𝔭` coprime to `m`, the cyclotomic character `IsPrimitiveRoot.autToPow` sends the Frobenius
@@ -105,15 +103,16 @@ theorem autToPow_frobeniusClass_out
     hζ.autToPow K ((frobeniusClass K L 𝔭).out : L ≃ₐ[K] L) =
       ZMod.unitOfCoprime (Ideal.absNorm 𝔭) hcop := by
   obtain ⟨𝔓, h𝔓prime, h𝔓lo, _⟩ := exists_prime_liesOver K L 𝔭 (UnramifiedIn.ne_bot K L hunr)
-  haveI := h𝔓prime
-  haveI := h𝔓lo
-  haveI : Finite (𝓞 L ⧸ 𝔓) := UnramifiedIn.finite_quotient K L hunr 𝔓 h𝔓lo
+  have := h𝔓prime
+  have := h𝔓lo
+  have : Finite (𝓞 L ⧸ 𝔓) := UnramifiedIn.finite_quotient K L hunr 𝔓 h𝔓lo
   set φ : L ≃ₐ[K] L := arithFrobAt (𝓞 K) Gal(L/K) 𝔓
   have hclass : frobeniusClass K L 𝔭 = ConjClasses.mk φ :=
     frobeniusClass_eq_mk_of_isArithFrobAt K L 𝔭 hunr φ 𝔓
       (IsArithFrobAt.arithFrobAt (𝓞 K) Gal(L/K) 𝔓) h𝔓lo
   have hconj : IsConj ((frobeniusClass K L 𝔭).out) φ := by
-    rw [← ConjClasses.mk_eq_mk_iff_isConj, ← hclass, ConjClasses.mk, Quotient.out_eq]
+    apply ConjClasses.mk_eq_mk_iff_isConj.mp
+    exact (Quotient.out_eq (frobeniusClass K L 𝔭)).trans hclass
   rw [isConj_iff_eq.mp ((hζ.autToPow K).map_isConj hconj)]
   have hact : φ ζ = ζ ^ Ideal.absNorm 𝔭 :=
     cyclotomic_frobenius_acts_as_norm_power K L m 𝔭 hunr hcop 𝔓 h𝔓lo ζ
@@ -145,7 +144,7 @@ private theorem smul_algebraMap_eq
     (σ : L ≃ₐ[K] L) (y : 𝓞 F) :
     haveI : IsScalarTower K F L := F.isScalarTower_mid'
     σ • (algebraMap (𝓞 F) (𝓞 L) y) = algebraMap (𝓞 F) (𝓞 L) ((σ.restrictNormal F) • y) := by
-  haveI : IsScalarTower K F L := F.isScalarTower_mid'
+  have : IsScalarTower K F L := F.isScalarTower_mid'
   have hbridgeL : ∀ (g : L ≃ₐ[K] L) (x : 𝓞 L), ((g • x : 𝓞 L) : L) = g • (x : L) := fun g x ↦
     by simpa [Algebra.smul_def] using
       (smul_distrib_smul (G := L ≃ₐ[K] L) (R := 𝓞 L) (S := L) g x 1).symm
@@ -173,7 +172,7 @@ private theorem isArithFrobAt_restrictNormal
     (𝔓 : Ideal (𝓞 L)) (hσ : IsArithFrobAt (𝓞 K) σ 𝔓) :
     haveI : IsScalarTower K F L := F.isScalarTower_mid'
     IsArithFrobAt (𝓞 K) (σ.restrictNormal F) (𝔓.under (𝓞 F)) := by
-  haveI : IsScalarTower K F L := F.isScalarTower_mid'
+  have : IsScalarTower K F L := F.isScalarTower_mid'
   intro y
   rw [Ideal.under_under 𝔓, Ideal.under, Ideal.mem_comap, map_sub, map_pow,
     show (MulSemiringAction.toAlgHom (𝓞 K) (𝓞 F) (σ.restrictNormal F)) y
@@ -188,21 +187,20 @@ private theorem unramifiedIn_intermediateField
     (F : IntermediateField K L) [IsGalois K F]
     (𝔭 : Ideal (𝓞 K)) (hunr : UnramifiedIn K L 𝔭) :
     UnramifiedIn K (↥F) 𝔭 := by
-  haveI : IsScalarTower K F L := F.isScalarTower_mid'
-  haveI : IsScalarTower (𝓞 K) (𝓞 F) (𝓞 L) := inferInstance
+  have : IsScalarTower K F L := F.isScalarTower_mid'
+  have : IsScalarTower (𝓞 K) (𝓞 F) (𝓞 L) := inferInstance
   refine ⟨hunr.1, fun 𝔮 h𝔮max h𝔮lo ↦ ?_⟩
-  haveI := h𝔮lo
-  haveI := h𝔮max.isPrime
+  have := h𝔮lo
+  have := h𝔮max.isPrime
   obtain ⟨𝔓, h𝔓prime, h𝔓lo, -⟩ := exists_prime_liesOver (↥F) L 𝔮
     (Ideal.ne_bot_of_liesOver_of_ne_bot hunr.1 𝔮)
-  haveI := h𝔓prime
-  haveI := h𝔓lo
-  haveI : 𝔓.LiesOver 𝔭 := ⟨by rw [← Ideal.under_under (B := 𝓞 F) 𝔓, h𝔓lo.over.symm, h𝔮lo.over.symm]⟩
-  haveI : Algebra.IsUnramifiedAt (𝓞 K) 𝔓 :=
+  have := h𝔓prime
+  have := h𝔓lo
+  have : 𝔓.LiesOver 𝔭 := ⟨by rw [← Ideal.under_under (B := 𝓞 F) 𝔓, h𝔓lo.over.symm, h𝔮lo.over.symm]⟩
+  have : Algebra.IsUnramifiedAt (𝓞 K) 𝔓 :=
     hunr.2 𝔓 (h𝔓prime.isMaximal (Ideal.ne_bot_of_liesOver_of_ne_bot hunr.1 𝔓)) inferInstance
   exact Algebra.IsUnramifiedAt.of_liesOver (𝓞 K) 𝔮 𝔓
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Step (A): a Frobenius-trivial prime splits completely.** For `F = fixedField H` with `H`
 abelian (hence normal) and a nonzero prime `𝔭` of `𝓞 K` unramified in `L` whose Frobenius
 representative lies in `H`, the `F`-Frobenius class of `𝔭` is trivial: `frobeniusClass K F 𝔭 = [1]`.
@@ -220,19 +218,20 @@ private theorem frobeniusClass_fixedField_eq_one
       IsGalois.of_fixedField_normal_subgroup H
     frobeniusClass K (↥(IntermediateField.fixedField H)) 𝔭 = ConjClasses.mk 1 := by
   set F := IntermediateField.fixedField H with hF
-  haveI : IsScalarTower K F L := F.isScalarTower_mid'
-  haveI : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
-  haveI : NumberField F := NumberField.of_intermediateField F
+  have : IsScalarTower K F L := F.isScalarTower_mid'
+  have : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
+  have : NumberField F := NumberField.of_intermediateField F
   obtain ⟨𝔓, h𝔓p, h𝔓lo, -⟩ := exists_prime_liesOver K L 𝔭 (UnramifiedIn.ne_bot K L hunr)
-  haveI := h𝔓p
-  haveI := h𝔓lo
-  haveI : Finite (𝓞 L ⧸ 𝔓) := UnramifiedIn.finite_quotient K L hunr 𝔓 h𝔓lo
+  have := h𝔓p
+  have := h𝔓lo
+  have : Finite (𝓞 L ⧸ 𝔓) := UnramifiedIn.finite_quotient K L hunr 𝔓 h𝔓lo
   set σ : L ≃ₐ[K] L := arithFrobAt (𝓞 K) Gal(L/K) 𝔓
   have hclass : frobeniusClass K L 𝔭 = ConjClasses.mk σ :=
     frobeniusClass_eq_mk_of_isArithFrobAt K L 𝔭 hunr σ 𝔓
       (IsArithFrobAt.arithFrobAt (𝓞 K) Gal(L/K) 𝔓) h𝔓lo
   have hconj : IsConj σ ((frobeniusClass K L 𝔭).out) := by
-    rw [← ConjClasses.mk_eq_mk_iff_isConj, hclass.symm, ConjClasses.mk, Quotient.out_eq]
+    apply ConjClasses.mk_eq_mk_iff_isConj.mp
+    exact hclass.symm.trans (Quotient.out_eq (frobeniusClass K L 𝔭)).symm
   have hσeq : σ = (frobeniusClass K L 𝔭).out := by
     obtain ⟨c, hc⟩ := hconj
     rw [SemiconjBy, mul_comm' (c : Gal(L/K)) σ] at hc
@@ -244,7 +243,7 @@ private theorem frobeniusClass_fixedField_eq_one
     MonoidHom.mem_ker.mp <| (IntermediateField.restrictNormalHom_ker F).ge hσfix
   have h𝔮lo : (𝔓.under (𝓞 F)).LiesOver 𝔭 :=
     ⟨by rw [← Ideal.under_under (B := 𝓞 F) 𝔓]; exact h𝔓lo.over⟩
-  haveI := h𝔮lo
+  have := h𝔮lo
   have hfrobF : IsArithFrobAt (𝓞 K) (σ.restrictNormal F) (𝔓.under (𝓞 F)) :=
     isArithFrobAt_restrictNormal K L F σ 𝔓 (IsArithFrobAt.arithFrobAt (𝓞 K) Gal(L/K) 𝔓)
   rw [frobeniusClass_eq_mk_of_isArithFrobAt K (↥F) 𝔭 (unramifiedIn_intermediateField K L F 𝔭 hunr)
@@ -267,14 +266,14 @@ private theorem finrank_residue_fixedField_eq_one
     ∀ 𝔮 : Ideal (𝓞 ↥(IntermediateField.fixedField H)), 𝔮.IsPrime → 𝔮.LiesOver 𝔭 →
       Module.finrank (𝓞 K ⧸ 𝔮.under (𝓞 K)) (𝓞 ↥(IntermediateField.fixedField H) ⧸ 𝔮) = 1 := by
   set F := IntermediateField.fixedField H
-  haveI : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
-  haveI : NumberField F := NumberField.of_intermediateField F
+  have : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
+  have : NumberField F := NumberField.of_intermediateField F
   have hunrF : UnramifiedIn K (↥F) 𝔭 := unramifiedIn_intermediateField K L F 𝔭 hunr
   have hfc : frobeniusClass K (↥F) 𝔭 = ConjClasses.mk (1 : Gal(↥F/K)) :=
     frobeniusClass_fixedField_eq_one K L H 𝔭 hunr hmem
   intro 𝔮 h𝔮p h𝔮lo
-  haveI := h𝔮p
-  haveI := h𝔮lo
+  have := h𝔮p
+  have := h𝔮lo
   rw [finrank_residue_eq_orderOf K (↥F) (1 : Gal(↥F/K)) (ConjClasses.mk 1) rfl 𝔭 hunrF hfc 𝔮
     h𝔮lo, orderOf_one]
 
@@ -291,13 +290,13 @@ private theorem card_primesOver_fixedField_eq_finrank
         𝔮.IsPrime ∧ 𝔮.LiesOver 𝔭 ∧ 𝔮 ≠ ⊥} =
       Module.finrank K ↥(IntermediateField.fixedField H) := by
   set F := IntermediateField.fixedField H
-  haveI : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
-  haveI : NumberField F := NumberField.of_intermediateField F
+  have : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
+  have : NumberField F := NumberField.of_intermediateField F
   have hunrF : UnramifiedIn K (↥F) 𝔭 := unramifiedIn_intermediateField K L F 𝔭 hunr
   have hresdeg := finrank_residue_fixedField_eq_one K L H 𝔭 hunr hmem
   obtain ⟨𝔮₀, h𝔮₀p, h𝔮₀lo, -⟩ := exists_prime_liesOver K (↥F) 𝔭 (UnramifiedIn.ne_bot K L hunr)
-  haveI := h𝔮₀p
-  haveI := h𝔮₀lo
+  have := h𝔮₀p
+  have := h𝔮₀lo
   have hcard := card_primesAbove_mul_finrank_eq K (↥F) 𝔭 hunrF 𝔮₀ h𝔮₀lo
   rw [hresdeg 𝔮₀ h𝔮₀p h𝔮₀lo, mul_one] at hcard
   rw [hcard, IsGalois.card_aut_eq_finrank K (↥F)]
@@ -318,18 +317,19 @@ private theorem absNorm_eq_of_liesOver_fixedField
     ∀ 𝔮 : Ideal (𝓞 ↥(IntermediateField.fixedField H)), 𝔮.IsPrime → 𝔮.LiesOver 𝔭 →
       Ideal.absNorm 𝔮 = Ideal.absNorm 𝔭 := by
   set F := IntermediateField.fixedField H
-  haveI : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
-  haveI : NumberField F := NumberField.of_intermediateField F
+  have : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
+  have : NumberField F := NumberField.of_intermediateField F
   have hresdeg := finrank_residue_fixedField_eq_one K L H 𝔭 hunr hmem
   intro 𝔮 h𝔮p h𝔮lo
-  haveI := h𝔮p
-  haveI := h𝔮lo
-  have hinert : (𝔮.under (𝓞 K)).inertiaDeg' 𝔮 = 1 := by
-    rw [Ideal.inertiaDeg'_algebraMap]
+  have := h𝔮p
+  have := h𝔮lo
+  have : 𝔭.IsMaximal := ‹𝔭.IsPrime›.isMaximal (UnramifiedIn.ne_bot K L hunr)
+  have : 𝔮.IsMaximal := Ideal.IsMaximal.of_liesOver_isMaximal 𝔮 𝔭
+  have : (𝔮.under (𝓞 K)).IsMaximal := h𝔮lo.over ▸ ‹𝔭.IsMaximal›
+  have hinert : 𝔮.inertiaDeg (𝓞 K) = 1 := by
+    rw [Ideal.inertiaDeg_eq_of_isMaximal (𝔮.under (𝓞 K)) 𝔮]
     exact hresdeg 𝔮 h𝔮p h𝔮lo
-  have hunder : 𝔮.under (𝓞 K) = 𝔭 := h𝔮lo.over.symm
-  rw [Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOver 𝔮 (𝔮.under (𝓞 K)) inferInstance
-    (hunder ▸ UnramifiedIn.ne_bot K L hunr), hinert, pow_one, hunder]
+  rw [← Ideal.absNorm_pow_inertiaDeg 𝔭 𝔮, hinert, pow_one]
 
 /-! ### Coprime-restricted Frobenii generation
 
@@ -409,8 +409,8 @@ private theorem finite_primesLiesOver_ne_bot (F : IntermediateField K L) [IsGalo
     (𝔭 : Ideal (𝓞 K)) [𝔭.IsMaximal] :
     haveI : NumberField F := NumberField.of_intermediateField F
     Finite {𝔮 : Ideal (𝓞 ↥F) // 𝔮.IsPrime ∧ 𝔮.LiesOver 𝔭 ∧ 𝔮 ≠ ⊥} := by
-  haveI : NumberField F := NumberField.of_intermediateField F
-  haveI : Finite (𝔭.primesOver (𝓞 ↥F)) :=
+  have : NumberField F := NumberField.of_intermediateField F
+  have : Finite (𝔭.primesOver (𝓞 ↥F)) :=
     (IsDedekindDomain.primesOver_finite 𝔭 (𝓞 ↥F)).to_subtype
   refine Finite.of_injective (β := 𝔭.primesOver (𝓞 ↥F)) (fun y ↦ ⟨y.1, y.2.1, y.2.2.1⟩)
     fun a b hab ↦ Subtype.ext ?_
@@ -436,8 +436,8 @@ private theorem primeIdealZetaSum_under_eq_finrank_mul [IsMulCommutative Gal(L/K
       = (Module.finrank K ↥(IntermediateField.fixedField H) : ℝ) * primeIdealZetaSum
           {𝔭 : Ideal (𝓞 K) | 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭 ∧ (Ideal.absNorm 𝔭).Coprime m} s := by
   set F := IntermediateField.fixedField H
-  haveI : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
-  haveI : NumberField F := NumberField.of_intermediateField F
+  have : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
+  have : NumberField F := NumberField.of_intermediateField F
   set U : Set (Ideal (𝓞 K)) :=
     {𝔭 | 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭 ∧ (Ideal.absNorm 𝔭).Coprime m}
   set V : Set (Ideal (𝓞 F)) :=
@@ -446,9 +446,9 @@ private theorem primeIdealZetaSum_under_eq_finrank_mul [IsMulCommutative Gal(L/K
   set IV := {𝔮 : Ideal (𝓞 F) // 𝔮 ∈ V ∧ 𝔮.IsPrime ∧ 𝔮 ≠ ⊥}
   have hφ_mem : ∀ 𝔮 : IV, (𝔮.1.under (𝓞 K)) ∈ U ∧ (𝔮.1.under (𝓞 K)).IsPrime
       ∧ (𝔮.1.under (𝓞 K)) ≠ ⊥ := fun 𝔮 ↦ by
-    haveI := 𝔮.2.2.1
+    have := 𝔮.2.2.1
     exact ⟨⟨inferInstance, 𝔮.2.1.2.1, 𝔮.2.1.2.2⟩, inferInstance,
-      Ideal.IsIntegral.comap_ne_bot (𝓞 K) 𝔮.2.2.2⟩
+      Ideal.IsIntegral.under_ne_bot (𝓞 K) 𝔮.2.2.2⟩
   set φ : IV → IU := fun 𝔮 ↦ ⟨𝔮.1.under (𝓞 K), hφ_mem 𝔮⟩
   set e := Equiv.sigmaFiberEquiv φ
   have hsummSig : Summable (fun p : Σ 𝔭 : IU, {𝔮 : IV // φ 𝔮 = 𝔭} ↦
@@ -458,13 +458,13 @@ private theorem primeIdealZetaSum_under_eq_finrank_mul [IsMulCommutative Gal(L/K
   rw [primeIdealZetaSum_def, ← e.tsum_eq (fun 𝔮 : IV ↦ (Ideal.absNorm (𝔮.1) : ℝ) ^ (-s)),
     hsummSig.tsum_sigma, primeIdealZetaSum_def, ← tsum_mul_left]
   refine tsum_congr (fun 𝔭 ↦ ?_)
-  haveI := 𝔭.2.2.1
+  have := 𝔭.2.2.1
   have hfibeq : {𝔮 : IV // φ 𝔮 = 𝔭} ≃
       {𝔮 : Ideal (𝓞 F) // 𝔮.IsPrime ∧ 𝔮.LiesOver 𝔭.1 ∧ 𝔮 ≠ ⊥} :=
     { toFun := fun x ↦ ⟨x.1.1, x.1.2.2.1, ⟨(Subtype.ext_iff.mp x.2).symm⟩, x.1.2.2.2⟩
       invFun := fun y ↦ ⟨⟨y.1, ⟨y.2.1, by
-          haveI := y.2.1
-          haveI := y.2.2.1
+          have := y.2.1
+          have := y.2.2.1
           exact (y.2.2.1.over ▸ 𝔭.2.1.2 : UnramifiedIn K L (y.1.under (𝓞 K)) ∧
             (Ideal.absNorm (y.1.under (𝓞 K))).Coprime m)⟩, y.2.1, y.2.2.2⟩,
         Subtype.ext (haveI := y.2.2.1; y.2.2.1.over.symm)⟩
@@ -474,10 +474,10 @@ private theorem primeIdealZetaSum_under_eq_finrank_mul [IsMulCommutative Gal(L/K
       = (Ideal.absNorm 𝔭.1 : ℝ) ^ (-s) := fun x ↦ by
     change (Ideal.absNorm x.1.1 : ℝ) ^ (-s) = (Ideal.absNorm 𝔭.1 : ℝ) ^ (-s)
     rw [(hsplit 𝔭.1 𝔭.2.2.1 𝔭.2.1.2.1 𝔭.2.1.2.2).2 x.1.1 x.1.2.2.1 ⟨(Subtype.ext_iff.mp x.2).symm⟩]
-  haveI : 𝔭.1.IsMaximal := 𝔭.2.2.1.isMaximal 𝔭.2.2.2
-  haveI : Finite {𝔮 : Ideal (𝓞 F) // 𝔮.IsPrime ∧ 𝔮.LiesOver 𝔭.1 ∧ 𝔮 ≠ ⊥} :=
+  have : 𝔭.1.IsMaximal := 𝔭.2.2.1.isMaximal 𝔭.2.2.2
+  have : Finite {𝔮 : Ideal (𝓞 F) // 𝔮.IsPrime ∧ 𝔮.LiesOver 𝔭.1 ∧ 𝔮 ≠ ⊥} :=
     finite_primesLiesOver_ne_bot K L F 𝔭.1
-  haveI : Finite {𝔮 : IV // φ 𝔮 = 𝔭} := Finite.of_equiv _ hfibeq.symm
+  have : Finite {𝔮 : IV // φ 𝔮 = 𝔭} := Finite.of_equiv _ hfibeq.symm
   rw [tsum_congr hconst, tsum_const, Nat.card_congr hfibeq,
     (hsplit 𝔭.1 𝔭.2.2.1 𝔭.2.1.2.1 𝔭.2.1.2.2).1, nsmul_eq_mul, mul_comm]
 
@@ -498,8 +498,8 @@ private theorem finrank_mul_unramified_coprime_le_univ
           {𝔭 : Ideal (𝓞 K) | 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭 ∧ (Ideal.absNorm 𝔭).Coprime m} s
       ≤ primeIdealZetaSum (Set.univ : Set (Ideal (𝓞 ↥(IntermediateField.fixedField H)))) s := by
   set F := IntermediateField.fixedField H
-  haveI : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
-  haveI : NumberField F := NumberField.of_intermediateField F
+  have : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
+  have : NumberField F := NumberField.of_intermediateField F
   have hsplit : ∀ 𝔭 : Ideal (𝓞 K), 𝔭.IsPrime → UnramifiedIn K L 𝔭 → (Ideal.absNorm 𝔭).Coprime m →
       Nat.card {𝔮 : Ideal (𝓞 F) // 𝔮.IsPrime ∧ 𝔮.LiesOver 𝔭 ∧ 𝔮 ≠ ⊥} = Module.finrank K ↥F
         ∧ ∀ 𝔮 : Ideal (𝓞 F), 𝔮.IsPrime → 𝔮.LiesOver 𝔭 → Ideal.absNorm 𝔮 = Ideal.absNorm 𝔭 :=
@@ -522,8 +522,8 @@ private theorem finrank_fixedField_le_one_of_forall_frobenius_mem_of_coprime
       (Ideal.absNorm 𝔭).Coprime m → ((frobeniusClass K L 𝔭).out : L ≃ₐ[K] L) ∈ H) :
     Module.finrank K (IntermediateField.fixedField H) ≤ 1 := by
   set F := IntermediateField.fixedField H
-  haveI : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
-  haveI : NumberField F := NumberField.of_intermediateField F
+  have : IsGalois K F := IsGalois.of_fixedField_normal_subgroup H
+  have : NumberField F := NumberField.of_intermediateField F
   set d : ℕ := Module.finrank K ↥F
   rw [← Nat.cast_le (α := ℝ), Nat.cast_one]
   set A : ℝ → ℝ := fun s ↦ primeIdealZetaSum
diff --git a/projects/Chebotarev/CebotarevDensity/FixedFieldDensity.lean b/projects/Chebotarev/CebotarevDensity/FixedFieldDensity.lean
index 7d40bad6a..33f8b2367 100644
--- a/projects/Chebotarev/CebotarevDensity/FixedFieldDensity.lean
+++ b/projects/Chebotarev/CebotarevDensity/FixedFieldDensity.lean
@@ -1,8 +1,9 @@
 module
 
 public import Mathlib.RingTheory.Ideal.Over
-public import Mathlib.NumberTheory.RamificationInertia.Basic
+public import Mathlib.RingTheory.RamificationInertia.Basic
 public import CebotarevDensity.Cyclotomic
+public import CebotarevDensity.ForMathlib.TsumFiberBound
 
 /-!
 # Density transfer through a fixed-field subextension (Sharifi 7.2.2 Step 1)
@@ -65,18 +66,18 @@ theorem frobeniusFibre_card_eq_of_isConj
   simp only [MulAction.toPerm_apply]
   constructor
   · rintro ⟨hp, hP, hne, hfrob⟩
-    haveI := hp
-    haveI := hP
+    have := hp
+    have := hP
     refine ⟨inferInstance, inferInstance, ?_, ?_⟩
     · rw [← Ideal.smul_bot c]
       exact (MulAction.injective c).ne hne
     · exact hc ▸ hfrob.conj c
   · rintro ⟨hp, hP, hne, hfrob⟩
-    haveI := hp
-    haveI := hP
+    have := hp
+    have := hP
     have hsmul : c⁻¹ • (c • 𝔓) = 𝔓 := inv_smul_smul c 𝔓
-    haveI hp' : 𝔓.IsPrime := hsmul ▸ (inferInstance : (c⁻¹ • (c • 𝔓)).IsPrime)
-    haveI hP' : 𝔓.LiesOver 𝔭 := hsmul ▸ (inferInstance : (c⁻¹ • (c • 𝔓)).LiesOver 𝔭)
+    have hp' : 𝔓.IsPrime := hsmul ▸ (inferInstance : (c⁻¹ • (c • 𝔓)).IsPrime)
+    have hP' : 𝔓.LiesOver 𝔭 := hsmul ▸ (inferInstance : (c⁻¹ • (c • 𝔓)).LiesOver 𝔭)
     have hne' : 𝔓 ≠ ⊥ := by
       rw [← hsmul, ← Ideal.smul_bot c⁻¹]
       exact (MulAction.injective c⁻¹).ne hne
@@ -84,7 +85,6 @@ theorem frobeniusFibre_card_eq_of_isConj
     have hconj := hfrob.conj c⁻¹
     rwa [hsmul, ← hc, show c⁻¹ * (c * σ * c⁻¹) * c⁻¹⁻¹ = σ by group] at hconj
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **Balanced fibre count.** If every prime above `𝔭` has Frobenius in the class `C = [σ]`
 and conjugate Frobenius values occur equally often (`hequi`), the total number of primes
 above `𝔭` is `|C|` times the number with `Frob_𝔓 = σ`: partition by the (class-`C`-valued)
@@ -103,21 +103,21 @@ theorem card_primesAbove_eq_card_carrier_mul_frobeniusFibre
         * Nat.card {𝔓 : Ideal (𝓞 L) // ∃ (_ : 𝔓.IsPrime) (_ : 𝔓.LiesOver 𝔭) (_ : 𝔓 ≠ ⊥),
           IsArithFrobAt (𝓞 K) σ 𝔓} := by
   have hpbot : 𝔭 ≠ ⊥ := UnramifiedIn.ne_bot K L hunr
-  haveI : 𝔭.IsMaximal := ‹𝔭.IsPrime›.isMaximal hpbot
-  haveI : Finite (𝔭.primesOver (𝓞 L)) := (IsDedekindDomain.primesOver_finite 𝔭 (𝓞 L)).to_subtype
-  haveI : Finite {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} :=
+  have : 𝔭.IsMaximal := ‹𝔭.IsPrime›.isMaximal hpbot
+  have : Finite (𝔭.primesOver (𝓞 L)) := (IsDedekindDomain.primesOver_finite 𝔭 (𝓞 L)).to_subtype
+  have : Finite {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} :=
     Finite.of_injective
       (fun 𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} ↦
         (⟨𝔓.1, 𝔓.2.1, 𝔓.2.2.1⟩ : 𝔭.primesOver (𝓞 L)))
-      fun _ _ hab ↦ Subtype.ext (by simpa using hab)
-  haveI : Fintype C.carrier := Fintype.ofFinite _
+      fun _ _ hab ↦ Subtype.ext (congrArg (fun q : 𝔭.primesOver (𝓞 L) ↦ q.1) hab)
+  have : Fintype C.carrier := Fintype.ofFinite _
   have hfinP : ∀ (𝔓 : Ideal (𝓞 L)) [𝔓.IsPrime] (hP : 𝔓.LiesOver 𝔭), Finite (𝓞 L ⧸ 𝔓) :=
     fun 𝔓 _ hP ↦ UnramifiedIn.finite_quotient K L hunr 𝔓 hP
   have hmem : ∀ (𝔓 : Ideal (𝓞 L)) [𝔓.IsPrime] (hP : 𝔓.LiesOver 𝔭),
       haveI := hfinP 𝔓 hP
       arithFrobAt (𝓞 K) Gal(L/K) 𝔓 ∈ C.carrier := by
     intro 𝔓 _ hP
-    haveI := hfinP 𝔓 hP
+    have := hfinP 𝔓 hP
     rw [ConjClasses.mem_carrier_iff_mk_eq, ← frobeniusClass_eq_mk_of_isArithFrobAt K L 𝔭 hunr _ 𝔓
       (IsArithFrobAt.arithFrobAt (𝓞 K) Gal(L/K) 𝔓) hP, hCfrob]
   have hconj : ∀ g : C.carrier, IsConj σ g.1 := by
@@ -138,18 +138,18 @@ theorem card_primesAbove_eq_card_carrier_mul_frobeniusFibre
     refine Nat.card_congr ⟨fun x ↦ ⟨x.1.1, x.1.2.1, x.1.2.2.1, x.1.2.2.2, ?_⟩,
       fun x ↦ ⟨⟨x.1, by obtain ⟨hp, hP, hne, _⟩ := x.2; exact ⟨hp, hP, hne⟩⟩, ?_⟩,
       fun _ ↦ rfl, fun _ ↦ rfl⟩
-    · haveI := x.1.2.1
-      haveI := hfinP x.1.1 x.1.2.2.1
+    · have := x.1.2.1
+      have := hfinP x.1.1 x.1.2.2.1
       rw [← Subtype.ext_iff.mp x.2]
       exact IsArithFrobAt.arithFrobAt (𝓞 K) Gal(L/K) x.1.1
     · obtain ⟨hp, hP, hne, hg⟩ := x.2
-      haveI := hp
-      haveI := hP
-      haveI := hfinP x.1 hP
-      haveI : Algebra.IsUnramifiedAt (𝓞 K) x.1 :=
+      have := hp
+      have := hP
+      have := hfinP x.1 hP
+      have : Algebra.IsUnramifiedAt (𝓞 K) x.1 :=
         Ideal.ramificationIdx_eq_one_iff.mp
           ((Ideal.ramificationIdx'_eq_ramificationIdx (x.1.under (𝓞 K)) x.1
-            (Ideal.IsIntegral.comap_ne_bot (𝓞 K) hne)).symm.trans
+            (Ideal.IsIntegral.under_ne_bot (𝓞 K) hne)).symm.trans
             (UnramifiedIn.ramificationIdx_eq_one K L hunr x.1 hP))
       exact Subtype.ext (eq_arithFrobAt_of_isArithFrobAt K L x.1 g.1 hg).symm
   simp_rw [hfib]
@@ -239,11 +239,11 @@ theorem arithFrobAt_restrictScalars_eq (E : IntermediateField K L)
       Ideal.finiteQuotientOfFreeOfNeBot 𝔓 (ne_bot_of_ramificationIdx_eq_one K L hunrK)
     haveI : IsGalois (↥E) L := IsGalois.tower_top_intermediateField E
     (arithFrobAt (𝓞 ↥E) Gal(L/(↥E)) 𝔓).restrictScalars K = arithFrobAt (𝓞 K) Gal(L/K) 𝔓 := by
-  haveI : IsScalarTower K ↥E L := E.isScalarTower_mid'
-  haveI : IsGalois (↥E) L := IsGalois.tower_top_intermediateField E
-  haveI : IsGaloisGroup Gal(L/(↥E)) (↥E) L := IsGaloisGroup.of_isGalois (↥E) L
-  haveI hPbot : 𝔓 ≠ ⊥ := ne_bot_of_ramificationIdx_eq_one K L hunrK
-  haveI : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
+  have : IsScalarTower K ↥E L := E.isScalarTower_mid'
+  have : IsGalois (↥E) L := IsGalois.tower_top_intermediateField E
+  have : IsGaloisGroup Gal(L/(↥E)) (↥E) L := IsGaloisGroup.of_isGalois (↥E) L
+  have hPbot : 𝔓 ≠ ⊥ := ne_bot_of_ramificationIdx_eq_one K L hunrK
+  have : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
   set σE := arithFrobAt (𝓞 ↥E) Gal(L/(↥E)) 𝔓 with hσE
   have hKfrob1 : IsArithFrobAt (𝓞 K) (σE.restrictScalars K) 𝔓 := by
     intro x
@@ -279,28 +279,29 @@ private theorem stabilizer_intermediate_eq_top_of_frobenius
       (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
     MulAction.stabilizer
         Gal(L/↥(IntermediateField.fixedField (Subgroup.zpowers σ))) 𝔓 = ⊤ := by
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
   have hraK := UnramifiedIn.ramificationIdx_eq_one K L hunrK 𝔓 hPK
   have hPbot : 𝔓 ≠ ⊥ := ne_bot_of_ramificationIdx_eq_one K L hraK
   have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := UnramifiedIn.ne_bot K L hunrK
-  haveI : 𝔓.IsMaximal := ‹𝔓.IsPrime›.isMaximal hPbot
-  haveI : (𝔓.under (𝓞 K)).IsMaximal :=
+  have : 𝔓.IsMaximal := ‹𝔓.IsPrime›.isMaximal hPbot
+  have : (𝔓.under (𝓞 K)).IsMaximal :=
     (inferInstance : (𝔓.under (𝓞 K)).IsPrime).isMaximal hpbot
-  haveI : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
-  haveI : Algebra.IsSeparable (𝓞 K ⧸ 𝔓.under (𝓞 K)) (𝓞 L ⧸ 𝔓) := by
-    letI : Field (𝓞 K ⧸ 𝔓.under (𝓞 K)) := Ideal.Quotient.field _
-    letI : Field (𝓞 L ⧸ 𝔓) := Ideal.Quotient.field _
+  have : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
+  have : Algebra.IsSeparable (𝓞 K ⧸ 𝔓.under (𝓞 K)) (𝓞 L ⧸ 𝔓) := by
+    let : Field (𝓞 K ⧸ 𝔓.under (𝓞 K)) := Ideal.Quotient.field _
+    let : Field (𝓞 L ⧸ 𝔓) := Ideal.Quotient.field _
     exact IsGalois.to_isSeparable
   have hmem : σ ∈ MulAction.stabilizer Gal(L/K) 𝔓 := hfrob.mem_stabilizer
-  have hinertK : (𝔓.under (𝓞 K)).inertiaDeg' 𝔓 = orderOf σ := by
-    rw [Ideal.inertiaDeg'_algebraMap, orderOf_eq_finrank_of_isArithFrobAt K L σ 𝔓 hraK hfrob]
+  have hinertK : 𝔓.inertiaDeg (𝓞 K) = orderOf σ := by
+    rw [Ideal.inertiaDeg_eq_of_isMaximal (𝔓.under (𝓞 K)) 𝔓,
+      orderOf_eq_finrank_of_isArithFrobAt K L σ 𝔓 hraK hfrob]
   have hcardstab' : Nat.card (MulAction.stabilizer Gal(L/K) 𝔓) = orderOf σ := by
     rw [Ideal.card_stabilizer_eq (𝔓.under (𝓞 K)) 𝔓,
       Ideal.ramificationIdxIn_eq_ramificationIdx (𝔓.under (𝓞 K)) 𝔓 Gal(L/K),
       ← Ideal.ramificationIdx'_eq_ramificationIdx (𝔓.under (𝓞 K)) 𝔓 hpbot, hraK, one_mul,
       Ideal.inertiaDegIn_eq_inertiaDeg (𝔓.under (𝓞 K)) 𝔓 Gal(L/K),
-      ← Ideal.inertiaDeg'_eq_inertiaDeg (𝔓.under (𝓞 K)) 𝔓, hinertK]
+      hinertK]
   have hstab : Subgroup.zpowers σ = MulAction.stabilizer Gal(L/K) 𝔓 :=
     Subgroup.eq_of_le_of_card_ge (by rwa [Subgroup.zpowers_le])
       (by rw [Nat.card_zpowers, hcardstab'])
@@ -331,34 +332,33 @@ private theorem inertiaDeg_under_E_eq_one_of_frobenius
       (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
     Ideal.ramificationIdx'
         (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) 𝔓 = 1
-      ∧ (𝔓.under (𝓞 K)).inertiaDeg'
-          (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) = 1
+      ∧ (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).inertiaDeg (𝓞 K) = 1
       ∧ Nat.card (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))
             ⧸ 𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))))
           = Nat.card (𝓞 K ⧸ 𝔓.under (𝓞 K)) := by
   set E := IntermediateField.fixedField (Subgroup.zpowers σ) with hE
-  haveI : IsScalarTower K ↥E L := E.isScalarTower_mid'
-  haveI : IsGalois (↥E) L := IsGalois.tower_top_intermediateField _
+  have : IsScalarTower K ↥E L := E.isScalarTower_mid'
+  have : IsGalois (↥E) L := IsGalois.tower_top_intermediateField _
   have hraK := UnramifiedIn.ramificationIdx_eq_one K L hunrK 𝔓 hPK
   have hPbot : 𝔓 ≠ ⊥ := ne_bot_of_ramificationIdx_eq_one K L hraK
   have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := UnramifiedIn.ne_bot K L hunrK
-  haveI : 𝔓.IsMaximal := ‹𝔓.IsPrime›.isMaximal hPbot
-  haveI : (𝔓.under (𝓞 K)).IsMaximal :=
+  have : 𝔓.IsMaximal := ‹𝔓.IsPrime›.isMaximal hPbot
+  have : (𝔓.under (𝓞 K)).IsMaximal :=
     (inferInstance : (𝔓.under (𝓞 K)).IsPrime).isMaximal hpbot
-  haveI : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
-  haveI hPEp : (𝔓.under (𝓞 ↥E)).IsPrime := inferInstance
-  haveI hPK' : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
-  haveI hPEK : (𝔓.under (𝓞 ↥E)).LiesOver (𝔓.under (𝓞 K)) := inferInstance
-  haveI hPPE : 𝔓.LiesOver (𝔓.under (𝓞 ↥E)) := Ideal.over_under (A := 𝓞 ↥E) (P := 𝔓)
-  have hpEbot : 𝔓.under (𝓞 ↥E) ≠ ⊥ := Ideal.IsIntegral.comap_ne_bot (𝓞 ↥E) hPbot
-  haveI : (𝔓.under (𝓞 ↥E)).IsMaximal := hPEp.isMaximal hpEbot
+  have : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
+  have hPEp : (𝔓.under (𝓞 ↥E)).IsPrime := inferInstance
+  have hPK' : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
+  have hPEK : (𝔓.under (𝓞 ↥E)).LiesOver (𝔓.under (𝓞 K)) := inferInstance
+  have hPPE : 𝔓.LiesOver (𝔓.under (𝓞 ↥E)) := Ideal.over_under (A := 𝓞 ↥E) (P := 𝔓)
+  have hpEbot : 𝔓.under (𝓞 ↥E) ≠ ⊥ := Ideal.IsIntegral.under_ne_bot (𝓞 ↥E) hPbot
+  have : (𝔓.under (𝓞 ↥E)).IsMaximal := hPEp.isMaximal hpEbot
   have hraE : Ideal.ramificationIdx' (𝔓.under (𝓞 ↥E)) 𝔓 = 1 := by
     have htower := Ideal.ramificationIdx'_algebra_tower' (𝔓.under (𝓞 K)) (𝔓.under (𝓞 ↥E)) 𝔓
     rw [hraK] at htower
     exact Nat.eq_one_of_mul_eq_one_left htower.symm
-  haveI : Algebra.IsSeparable (𝓞 ↥E ⧸ 𝔓.under (𝓞 ↥E)) (𝓞 L ⧸ 𝔓) := by
-    letI : Field (𝓞 ↥E ⧸ 𝔓.under (𝓞 ↥E)) := Ideal.Quotient.field _
-    letI : Field (𝓞 L ⧸ 𝔓) := Ideal.Quotient.field _
+  have : Algebra.IsSeparable (𝓞 ↥E ⧸ 𝔓.under (𝓞 ↥E)) (𝓞 L ⧸ 𝔓) := by
+    let : Field (𝓞 ↥E ⧸ 𝔓.under (𝓞 ↥E)) := Ideal.Quotient.field _
+    let : Field (𝓞 L ⧸ 𝔓) := Ideal.Quotient.field _
     exact IsGalois.to_isSeparable
   have hstabE : MulAction.stabilizer Gal(L/(↥E)) 𝔓 = ⊤ :=
     stabilizer_intermediate_eq_top_of_frobenius σ 𝔓 hunrK hPK hfrob horderE
@@ -369,26 +369,25 @@ private theorem inertiaDeg_under_E_eq_one_of_frobenius
     Ideal.ramificationIdxIn_eq_ramificationIdx (𝔓.under (𝓞 ↥E)) 𝔓 Gal(L/(↥E)),
     ← Ideal.ramificationIdx'_eq_ramificationIdx (𝔓.under (𝓞 ↥E)) 𝔓 hpEbot, hraE, one_mul,
     Ideal.inertiaDegIn_eq_inertiaDeg (𝔓.under (𝓞 ↥E)) 𝔓 Gal(L/(↥E)),
-    ← Ideal.inertiaDeg'_eq_inertiaDeg (𝔓.under (𝓞 ↥E)) 𝔓,
-    Ideal.inertiaDeg'_algebraMap] at hcardE
-  have hinertTower : (𝔓.under (𝓞 K)).inertiaDeg' 𝔓
-      = (𝔓.under (𝓞 K)).inertiaDeg' (𝔓.under (𝓞 ↥E))
-        * (𝔓.under (𝓞 ↥E)).inertiaDeg' 𝔓 :=
-    Ideal.inertiaDeg'_algebra_tower (𝔓.under (𝓞 K)) (𝔓.under (𝓞 ↥E)) 𝔓
-  have hinertK : (𝔓.under (𝓞 K)).inertiaDeg' 𝔓 = orderOf σ := by
-    rw [Ideal.inertiaDeg'_algebraMap, orderOf_eq_finrank_of_isArithFrobAt K L σ 𝔓 hraK hfrob]
-  have hfPE : (𝔓.under (𝓞 ↥E)).inertiaDeg' 𝔓 = orderOf σ := by
-    rw [Ideal.inertiaDeg'_algebraMap, ← hcardE, horderE]
+    Ideal.inertiaDeg_eq_of_isMaximal (𝔓.under (𝓞 ↥E)) 𝔓] at hcardE
+  have hinertTower : 𝔓.inertiaDeg (𝓞 K)
+      = (𝔓.under (𝓞 ↥E)).inertiaDeg (𝓞 K)
+        * 𝔓.inertiaDeg (𝓞 ↥E) :=
+    Ideal.inertiaDeg_tower (R := 𝓞 K) (𝔓.under (𝓞 ↥E)) 𝔓
+  have hinertK : 𝔓.inertiaDeg (𝓞 K) = orderOf σ := by
+    rw [Ideal.inertiaDeg_eq_of_isMaximal (𝔓.under (𝓞 K)) 𝔓,
+      orderOf_eq_finrank_of_isArithFrobAt K L σ 𝔓 hraK hfrob]
+  have hfPE : 𝔓.inertiaDeg (𝓞 ↥E) = orderOf σ := by
+    rw [Ideal.inertiaDeg_eq_of_isMaximal (𝔓.under (𝓞 ↥E)) 𝔓, ← hcardE, horderE]
   have hpos : 0 < orderOf σ := orderOf_pos_iff.mpr (isOfFinOrder_of_finite σ)
-  have hinertPK : (𝔓.under (𝓞 K)).inertiaDeg' (𝔓.under (𝓞 ↥E)) = 1 := by
+  have hinertPK : (𝔓.under (𝓞 ↥E)).inertiaDeg (𝓞 K) = 1 := by
     rw [hinertK, hfPE] at hinertTower
     exact Nat.eq_of_mul_eq_mul_right hpos (by rw [one_mul]; exact hinertTower.symm)
   refine ⟨hraE, hinertPK, ?_⟩
   have hnormP : Nat.card (𝓞 ↥E ⧸ 𝔓.under (𝓞 ↥E))
-      = Nat.card (𝓞 K ⧸ 𝔓.under (𝓞 K)) ^ (𝔓.under (𝓞 K)).inertiaDeg' (𝔓.under (𝓞 ↥E)) := by
+      = Nat.card (𝓞 K ⧸ 𝔓.under (𝓞 K)) ^ (𝔓.under (𝓞 ↥E)).inertiaDeg (𝓞 K) := by
     simpa [Submodule.cardQuot_apply, Ideal.absNorm_apply] using
-      Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOver (𝔓.under (𝓞 ↥E)) (𝔓.under (𝓞 K))
-        inferInstance hpbot
+      (Ideal.absNorm_pow_inertiaDeg (𝔓.under (𝓞 K)) (𝔓.under (𝓞 ↥E))).symm
   rw [hnormP, hinertPK, pow_one]
 
 open scoped Pointwise in
@@ -405,15 +404,15 @@ private theorem eq_of_liesOver_under_E_of_frobenius
         (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
       𝔔.LiesOver (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))))) :
     𝔔 = 𝔓 := by
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
+  have : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
     IsGalois.tower_top_intermediateField _
-  haveI : IsGaloisGroup Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ))))
+  have : IsGaloisGroup Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ))))
       (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L := IsGaloisGroup.of_isGalois _ L
   set E := IntermediateField.fixedField (Subgroup.zpowers σ) with hE
-  haveI := hQ
-  haveI : 𝔓.LiesOver (𝔓.under (𝓞 ↥E)) := Ideal.over_under (A := 𝓞 ↥E) (P := 𝔓)
+  have := hQ
+  have : 𝔓.LiesOver (𝔓.under (𝓞 ↥E)) := Ideal.over_under (A := 𝓞 ↥E) (P := 𝔓)
   have hstabE : MulAction.stabilizer Gal(L/(↥E)) 𝔓 = ⊤ :=
     stabilizer_intermediate_eq_top_of_frobenius σ 𝔓 hunrK hPK hfrob horderE
   obtain ⟨τ, hτ⟩ := Ideal.exists_smul_eq_of_isGaloisGroup (𝔓.under (𝓞 ↥E)) 𝔓 𝔔 Gal(L/(↥E))
@@ -446,18 +445,18 @@ private theorem arithFrobAt_E_eq_of_isArithFrobAt
       IsGalois.tower_top_intermediateField _
     arithFrobAt (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))
       Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) 𝔓 = σE := by
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
+  have : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
     IsGalois.tower_top_intermediateField _
   have hraK : Ideal.ramificationIdx' (𝔓.under (𝓞 K)) 𝔓 = 1 :=
     UnramifiedIn.ramificationIdx_eq_one K L hunrK 𝔓 hPK
   have hPbot : 𝔓 ≠ ⊥ := ne_bot_of_ramificationIdx_eq_one K L hraK
-  haveI : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
-  haveI : Algebra.IsUnramifiedAt (𝓞 K) 𝔓 :=
+  have : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
+  have : Algebra.IsUnramifiedAt (𝓞 K) 𝔓 :=
     Ideal.ramificationIdx_eq_one_iff.mp
       ((Ideal.ramificationIdx'_eq_ramificationIdx (𝔓.under (𝓞 K)) 𝔓
-        (Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPbot)).symm.trans hraK)
+        (Ideal.IsIntegral.under_ne_bot (𝓞 K) hPbot)).symm.trans hraK)
   have hbridge := arithFrobAt_restrictScalars_eq
     (IntermediateField.fixedField (Subgroup.zpowers σ)) 𝔓 hraK hraE hnorm
   rw [(eq_arithFrobAt_of_isArithFrobAt K L 𝔓 σ hfrob).symm] at hbridge
@@ -480,49 +479,47 @@ private theorem exists_arithFrobAt_over_fibrePrime
     (hPunr : UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P)
     (hPfrob : frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
       = ConjClasses.mk σE)
-    (hPdeg : (P.under (𝓞 K)).inertiaDeg' P = 1) (hPbot : P ≠ ⊥) :
+    (hPdeg : P.inertiaDeg (𝓞 K) = 1) (hPbot : P ≠ ⊥) :
     ∃ (𝔓 : Ideal (𝓞 L)) (_ : 𝔓.IsPrime) (_ : 𝔓.LiesOver P) (_ : 𝔓 ≠ ⊥),
       𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) = P ∧
         IsArithFrobAt (𝓞 K) σ 𝔓 := by
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
+  have : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
     IsGalois.tower_top_intermediateField _
   obtain ⟨𝔓, h𝔓p, h𝔓lo, h𝔓bot⟩ := exists_prime_liesOver
     (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L P hPbot
-  haveI := h𝔓p
-  haveI := h𝔓lo
+  have := h𝔓p
+  have := h𝔓lo
   have hPeq : 𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) = P :=
     h𝔓lo.over.symm
-  haveI hPPE : 𝔓.LiesOver (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
+  have hPPE : 𝔓.LiesOver (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
     Ideal.over_under (A := 𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) (P := 𝔓)
   have hunderK : 𝔓.under (𝓞 K) = P.under (𝓞 K) := by
     rw [← Ideal.under_under (B := 𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) 𝔓, hPeq]
   have hunrK : UnramifiedIn K L (𝔓.under (𝓞 K)) := hunderK ▸ hunrP
-  haveI : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
-  have hinertPK1 : (𝔓.under (𝓞 K)).inertiaDeg'
-      (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) = 1 := by
-    rw [hPeq, hunderK]
+  have : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
+  have hinertPK1 : (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).inertiaDeg (𝓞 K) = 1 := by
+    rw [hPeq]
     exact hPdeg
   have hraE : Ideal.ramificationIdx'
       (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) 𝔓 = 1 :=
     (Ideal.ramificationIdx'_eq_ramificationIdx
         (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) 𝔓
-        (Ideal.IsIntegral.comap_ne_bot _ h𝔓bot)).trans
+        (Ideal.IsIntegral.under_ne_bot _ h𝔓bot)).trans
       (Ideal.ramificationIdx_eq_one_iff.mpr
         (hPunr.2 𝔓 (h𝔓p.isMaximal h𝔓bot) (hPeq ▸ hPPE)))
   have hnorm : Nat.card (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))
         ⧸ 𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))))
       = Nat.card (𝓞 K ⧸ 𝔓.under (𝓞 K)) := by
-    have hnP := Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOver
-      (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) (𝔓.under (𝓞 K))
-      inferInstance (UnramifiedIn.ne_bot K L hunrK)
+    have hnP := (Ideal.absNorm_pow_inertiaDeg (𝔓.under (𝓞 K))
+      (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))))).symm
     simp only [Submodule.cardQuot_apply, Ideal.absNorm_apply] at hnP ⊢
     rw [hnP, hinertPK1, pow_one]
-  haveI : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 h𝔓bot
+  have : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 h𝔓bot
   have hfrEeqσE : arithFrobAt (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))
       Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) 𝔓 = σE := by
-    letI : CommMonoid Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
+    let : CommMonoid Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
       IsMulCommutative.instCommMonoid
     have hcl := frobeniusClass_eq_mk_of_isArithFrobAt
       (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L P hPunr _ 𝔓
@@ -560,40 +557,40 @@ private theorem under_E_mem_fibre_of_isArithFrobAt
         {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) |
           P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P ∧
           frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
-            = ConjClasses.mk σE ∧ (P.under (𝓞 K)).inertiaDeg' P = 1}
+            = ConjClasses.mk σE ∧ P.inertiaDeg (𝓞 K) = 1}
       ∧ (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).LiesOver 𝔭
       ∧ (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) ≠ ⊥ := by
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
+  have : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
     IsGalois.tower_top_intermediateField _
-  haveI := hP
+  have := hP
   have hunderK : 𝔓.under (𝓞 K) = 𝔭 := hP.over.symm
   have hunrK : UnramifiedIn K L (𝔓.under (𝓞 K)) := hunderK ▸ hunr
-  haveI : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
-  haveI : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
+  have : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
+  have : Finite (𝓞 L ⧸ 𝔓) := Ideal.finiteQuotientOfFreeOfNeBot 𝔓 hPbot
   obtain ⟨hraE, hinPK, hnorm⟩ :=
     inertiaDeg_under_E_eq_one_of_frobenius σ 𝔓 hunrK inferInstance hfrob horderE
   have hfrE := arithFrobAt_E_eq_of_isArithFrobAt σ σE hσE 𝔓 hunrK inferInstance hfrob horderE
     hraE hnorm
   have hunram : UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L
       (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) := by
-    refine ⟨Ideal.IsIntegral.comap_ne_bot _ hPbot, fun 𝔔 h𝔔max h𝔔lo ↦ ?_⟩
-    haveI := h𝔔max.isPrime
+    refine ⟨Ideal.IsIntegral.under_ne_bot _ hPbot, fun 𝔔 h𝔔max h𝔔lo ↦ ?_⟩
+    have := h𝔔max.isPrime
     have h𝔔eq : 𝔔 = 𝔓 := eq_of_liesOver_under_E_of_frobenius σ 𝔓 hunrK inferInstance hfrob
       horderE 𝔔 h𝔔lo
     subst h𝔔eq
     exact Ideal.ramificationIdx_eq_one_iff.mp
       ((Ideal.ramificationIdx'_eq_ramificationIdx
         (𝔔.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) 𝔔
-        (Ideal.IsIntegral.comap_ne_bot _ hPbot)).symm.trans hraE)
-  haveI hPEK : (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).LiesOver 𝔭 := by
-    haveI : (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).LiesOver
+        (Ideal.IsIntegral.under_ne_bot _ hPbot)).symm.trans hraE)
+  have hPEK : (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).LiesOver 𝔭 := by
+    have : (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).LiesOver
         (𝔓.under (𝓞 K)) := inferInstance
     rwa [hunderK] at this
-  haveI hPPE : 𝔓.LiesOver (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
+  have hPPE : 𝔓.LiesOver (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
     Ideal.over_under (A := 𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) (P := 𝔓)
-  refine ⟨⟨inferInstance, hunram, ?_, ?_⟩, hPEK, Ideal.IsIntegral.comap_ne_bot _ hPbot⟩
+  refine ⟨⟨inferInstance, hunram, ?_, ?_⟩, hPEK, Ideal.IsIntegral.under_ne_bot _ hPbot⟩
   · rw [frobeniusClass_eq_mk_of_isArithFrobAt
       (↥(IntermediateField.fixedField (Subgroup.zpowers σ)))
       L (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) hunram _ 𝔓
@@ -601,9 +598,7 @@ private theorem under_E_mem_fibre_of_isArithFrobAt
         Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) 𝔓) hPPE,
       ConjClasses.mk_eq_mk_iff_isConj]
     exact isConj_iff.mpr ⟨1, by simp [hfrE]⟩
-  · rw [show (𝔓.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))).under (𝓞 K)
-        = 𝔓.under (𝓞 K) from Ideal.under_under 𝔓]
-    exact hinPK
+  · exact hinPK
 
 /-- **Fibre bijection: degree-one `E`-primes with Frobenius `σ_E` ↔ `L`-primes with
 Frobenius `σ`** (Sharifi 7.2.2 p. 143). For a prime `𝔭` of `𝓞 K` unramified in `L` with
@@ -628,12 +623,12 @@ private theorem card_fibre_E_eq_card_fibre_L
         P ∈ {P | P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
               ∧ frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
                 = ConjClasses.mk σE
-              ∧ (P.under (𝓞 K)).inertiaDeg' P = 1} ∧ P.LiesOver 𝔭 ∧ P ≠ ⊥}
+              ∧ P.inertiaDeg (𝓞 K) = 1} ∧ P.LiesOver 𝔭 ∧ P ≠ ⊥}
       = Nat.card {𝔓 : Ideal (𝓞 L) // ∃ (_ : 𝔓.IsPrime) (_ : 𝔓.LiesOver 𝔭) (_ : 𝔓 ≠ ⊥),
           IsArithFrobAt (𝓞 K) σ 𝔓} := by
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
+  have : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
     IsGalois.tower_top_intermediateField _
   refine (Nat.card_congr (Equiv.ofBijective
       (fun 𝔓 ↦ ⟨𝔓.1.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))),
@@ -641,37 +636,37 @@ private theorem card_fibre_E_eq_card_fibre_L
            exact under_E_mem_fibre_of_isArithFrobAt σ σE hσE horderE 𝔭 hunr 𝔓.1 hP hPbot hfrob⟩)
       ⟨?_, ?_⟩)).symm
   · rintro ⟨𝔓₁, h𝔓₁, hP₁, hP₁bot, hfrob₁⟩ ⟨𝔓₂, h𝔓₂, hP₂, hP₂bot, hfrob₂⟩ hΦ
-    haveI := h𝔓₁
-    haveI := h𝔓₂
-    haveI := hP₁
-    haveI := hP₂
+    have := h𝔓₁
+    have := h𝔓₂
+    have := hP₁
+    have := hP₂
     have hunderK₁ : 𝔓₁.under (𝓞 K) = 𝔭 := hP₁.over.symm
     have hunrK₁ : UnramifiedIn K L (𝔓₁.under (𝓞 K)) := hunderK₁ ▸ hunr
-    haveI : 𝔓₁.LiesOver (𝔓₁.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓₁)
+    have : 𝔓₁.LiesOver (𝔓₁.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓₁)
     have hΦ' : 𝔓₂.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))
         = 𝔓₁.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) :=
       congrArg Subtype.val hΦ |>.symm
-    haveI hP₂lo : 𝔓₂.LiesOver
+    have hP₂lo : 𝔓₂.LiesOver
         (𝔓₁.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) := by
-      haveI : 𝔓₂.LiesOver
+      have : 𝔓₂.LiesOver
         (𝔓₂.under (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
         Ideal.over_under (A := 𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) (P := 𝔓₂)
       rwa [hΦ'] at this
     exact Subtype.ext (eq_of_liesOver_under_E_of_frobenius σ 𝔓₁ hunrK₁ inferInstance hfrob₁
       horderE 𝔓₂ hP₂lo).symm
   · rintro ⟨P, ⟨hPp, hPunr, hPfrob, hPdeg⟩, hPlo, hPbot⟩
-    haveI := hPp
-    haveI := hPlo
+    have := hPp
+    have := hPlo
     have hunrP : UnramifiedIn K L (P.under (𝓞 K)) := hPlo.over.symm ▸ hunr
     obtain ⟨𝔓, h𝔓p, h𝔓lo, h𝔓bot, hPeq, hfrobK⟩ :=
       exists_arithFrobAt_over_fibrePrime σ σE hσE P hunrP hPunr hPfrob hPdeg hPbot
-    haveI := h𝔓p
-    haveI := h𝔓lo
+    have := h𝔓p
+    have := h𝔓lo
     have hunderK : 𝔓.under (𝓞 K) = 𝔭 := by
       rw [← Ideal.under_under (B := 𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) 𝔓,
         hPeq]
       exact hPlo.over.symm
-    haveI hPK𝔓 : 𝔓.LiesOver 𝔭 := hunderK ▸ Ideal.over_under (A := 𝓞 K) (P := 𝔓)
+    have hPK𝔓 : 𝔓.LiesOver 𝔭 := hunderK ▸ Ideal.over_under (A := 𝓞 K) (P := 𝔓)
     exact ⟨⟨𝔓, h𝔓p, hPK𝔓, h𝔓bot, hfrobK⟩, Subtype.ext hPeq⟩
 
 /-- **A degree-one fibre prime has `K`-Frobenius class `[σ]`** (Sharifi 7.2.2 p. 143). If `P`
@@ -690,18 +685,18 @@ private theorem frobeniusClass_under_eq_of_mem_fibre
     (hPunr : UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P)
     (hPfrob : frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
       = ConjClasses.mk σE)
-    (hPdeg : (P.under (𝓞 K)).inertiaDeg' P = 1) (hPbot : P ≠ ⊥) :
+    (hPdeg : P.inertiaDeg (𝓞 K) = 1) (hPbot : P ≠ ⊥) :
     frobeniusClass K L (P.under (𝓞 K)) = ConjClasses.mk σ := by
-  haveI : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
+  have : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
     IsGalois.tower_top_intermediateField _
   obtain ⟨𝔓, h𝔓p, h𝔓lo, h𝔓bot, hPeq, hfrobK⟩ :=
     exists_arithFrobAt_over_fibrePrime σ σE hσE P hunrP hPunr hPfrob hPdeg hPbot
-  haveI := h𝔓p
-  haveI := h𝔓lo
+  have := h𝔓p
+  have := h𝔓lo
   have hunderK : P.under (𝓞 K) = 𝔓.under (𝓞 K) := by
     rw [← hPeq, Ideal.under_under]
   have hunrK : UnramifiedIn K L (𝔓.under (𝓞 K)) := hunderK ▸ hunrP
-  haveI : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
+  have : 𝔓.LiesOver (𝔓.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓)
   rw [hunderK,
     frobeniusClass_eq_mk_of_isArithFrobAt K L (𝔓.under (𝓞 K)) hunrK _ 𝔓 hfrobK inferInstance]
 
@@ -715,8 +710,8 @@ private theorem tsum_comp_eq_card_fibre_smul {β γ : Type*} (g : β → γ) (h
     ∑' b, h (g b) = c * ∑' y, h y := by
   rw [← (hsumm.hasSum.tsum_fiberwise g).tsum_eq, ← tsum_mul_left]
   refine tsum_congr fun y ↦ ?_
-  haveI := hfin y
-  letI := Fintype.ofFinite (g ⁻¹' {y} : Set β)
+  have := hfin y
+  let := Fintype.ofFinite (g ⁻¹' {y} : Set β)
   rw [tsum_congr fun b : (g ⁻¹' {y} : Set β) ↦ congrArg h b.2, tsum_fintype, Finset.sum_const,
     Finset.card_univ, ← Nat.card_eq_fintype_card, nsmul_eq_mul, hcard, mul_comm]
 
@@ -738,13 +733,12 @@ private theorem card_fibre_T1_over_prime
         Nat.card {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) //
           P ∈ {P | P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
                 ∧ frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
-                  = ConjClasses.mk σE ∧ (P.under (𝓞 K)).inertiaDeg' P = 1}
+                  = ConjClasses.mk σE ∧ P.inertiaDeg (𝓞 K) = 1}
             ∧ P.LiesOver 𝔭 ∧ P ≠ ⊥}
       = Nat.card Gal(L/K) := by
   rw [card_fibre_E_eq_card_fibre_L σ σE hσE horderE 𝔭 hunr𝔭 hfrob𝔭, mul_comm, ← mul_assoc]
   exact count_primes_above_with_frobenius_eq_sigma K L σ (ConjClasses.mk σ) rfl 𝔭 hunr𝔭 hfrob𝔭
 
-set_option backward.isDefEq.respectTransparency false in
 /-- **LEAF A: the degree-one part of `T` carries the main term** (Sharifi 7.2.2 p. 143). For
 `1 < s`, the partial Dirichlet sum over the set `T₁` of degree-one (over `K`) primes `P` of
 `𝓞 E` above an unramified-in-`L` prime, with `Frob^E_P = [σ_E]`, equals `|G|/(f·|C|)` times the
@@ -767,26 +761,26 @@ private theorem primeIdealZetaSum_fibre_eq_smul
         {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) |
           P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P ∧
           frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
-            = ConjClasses.mk σE ∧ (P.under (𝓞 K)).inertiaDeg' P = 1 ∧
+            = ConjClasses.mk σE ∧ P.inertiaDeg (𝓞 K) = 1 ∧
           UnramifiedIn K L (P.under (𝓞 K))} s
       = ((Nat.card Gal(L/K) : ℝ) / (orderOf σ * Nat.card (ConjClasses.mk σ).carrier))
         * primeIdealZetaSum {𝔭 : Ideal (𝓞 K) | 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭 ∧
             frobeniusClass K L 𝔭 = ConjClasses.mk σ} s := by
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
+  have : IsGalois (↥(IntermediateField.fixedField (Subgroup.zpowers σ))) L :=
     IsGalois.tower_top_intermediateField _
   set Sset := {𝔭 : Ideal (𝓞 K) | 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭 ∧
     frobeniusClass K L 𝔭 = ConjClasses.mk σ} with hSset
   set T₁set := {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) |
     P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P ∧
     frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P = ConjClasses.mk σE ∧
-    (P.under (𝓞 K)).inertiaDeg' P = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))} with hT₁set
+    P.inertiaDeg (𝓞 K) = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))} with hT₁set
   set S' := {𝔭 : Ideal (𝓞 K) // 𝔭 ∈ Sset ∧ 𝔭.IsPrime ∧ 𝔭 ≠ ⊥} with hS'
   have hgmem : ∀ P : {P // P ∈ T₁set ∧ P.IsPrime ∧ P ≠ ⊥},
       P.1.under (𝓞 K) ∈ Sset ∧ (P.1.under (𝓞 K)).IsPrime ∧ P.1.under (𝓞 K) ≠ ⊥ := by
     rintro ⟨P, ⟨hPp, hPunr, hPfrob, hPdeg, hunrP⟩, _, hPbot⟩
-    haveI := hPp
+    have := hPp
     refine ⟨⟨inferInstance, hunrP, ?_⟩, inferInstance, UnramifiedIn.ne_bot K L hunrP⟩
     exact frobeniusClass_under_eq_of_mem_fibre σ σE hσE horderE P hunrP hPunr hPfrob hPdeg hPbot
   set g : {P // P ∈ T₁set ∧ P.IsPrime ∧ P ≠ ⊥} → S' :=
@@ -794,28 +788,28 @@ private theorem primeIdealZetaSum_fibre_eq_smul
   have hnormeq : ∀ P : {P // P ∈ T₁set ∧ P.IsPrime ∧ P ≠ ⊥},
       (Ideal.absNorm P.1 : ℝ) = (Ideal.absNorm (P.1.under (𝓞 K)) : ℝ) := by
     rintro ⟨P, ⟨hPp, _, _, hPdeg, _⟩, _, hPbot⟩
-    haveI := hPp
-    have hpbot : P.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPbot
-    haveI : P.LiesOver (P.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := P)
-    have hpow := Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOver P (P.under (𝓞 K)) inferInstance hpbot
+    have := hPp
+    have hpbot : P.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.under_ne_bot (𝓞 K) hPbot
+    have : P.LiesOver (P.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := P)
+    have hpow := (Ideal.absNorm_pow_inertiaDeg (P.under (𝓞 K)) P).symm
     rw [hPdeg, pow_one] at hpow
     rw [hpow]
   have hcardfib : ∀ 𝔭 : S', (orderOf σ * Nat.card (ConjClasses.mk σ).carrier) *
       Nat.card {P : {P // P ∈ T₁set ∧ P.IsPrime ∧ P ≠ ⊥} // g P = 𝔭} = Nat.card Gal(L/K) := by
     intro 𝔭
     obtain ⟨hp𝔭, hunr𝔭, hfrob𝔭⟩ := 𝔭.2.1
-    haveI := hp𝔭
+    have := hp𝔭
     have hreindex : Nat.card {P : {P // P ∈ T₁set ∧ P.IsPrime ∧ P ≠ ⊥} // g P = 𝔭}
         = Nat.card {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) //
             P ∈ {P | P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ))
                   L P ∧ frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
-                  = ConjClasses.mk σE ∧ (P.under (𝓞 K)).inertiaDeg' P = 1}
+                  = ConjClasses.mk σE ∧ P.inertiaDeg (𝓞 K) = 1}
               ∧ P.LiesOver 𝔭.1 ∧ P ≠ ⊥} := by
       refine Nat.card_congr ⟨fun x ↦ ⟨x.1.1, ?_, ?_, x.1.2.2.2⟩,
         fun y ↦ ⟨⟨y.1, ?_, y.2.1.1, y.2.2.2⟩, ?_⟩, fun _ ↦ rfl, fun _ ↦ rfl⟩
       · exact ⟨x.1.2.1.1, x.1.2.1.2.1, x.1.2.1.2.2.1, x.1.2.1.2.2.2.1⟩
       · exact ⟨(congrArg Subtype.val x.2).symm ▸ (Ideal.over_under (A := 𝓞 K) (P := x.1.1)).over⟩
-      · haveI := y.2.1.1
+      · have := y.2.1.1
         have hunderK : y.1.under (𝓞 K) = 𝔭.1 := (y.2.2.1).over.symm
         exact ⟨y.2.1.1, y.2.1.2.1, y.2.1.2.2.1, y.2.1.2.2.2, by rw [hunderK]; exact hunr𝔭⟩
       · exact Subtype.ext (y.2.2.1).over.symm
@@ -823,18 +817,20 @@ private theorem primeIdealZetaSum_fibre_eq_smul
     exact card_fibre_T1_over_prime σ σE hσE horderE 𝔭.1 hunr𝔭 hfrob𝔭
   have hfibfin : ∀ 𝔭 : S', Finite {P : {P // P ∈ T₁set ∧ P.IsPrime ∧ P ≠ ⊥} // g P = 𝔭} := by
     intro 𝔭
-    haveI := 𝔭.2.2.1
-    haveI : 𝔭.1.IsMaximal := 𝔭.2.2.1.isMaximal 𝔭.2.2.2
-    haveI : Finite (𝔭.1.primesOver
+    have := 𝔭.2.2.1
+    have : 𝔭.1.IsMaximal := 𝔭.2.2.1.isMaximal 𝔭.2.2.2
+    have : Finite (𝔭.1.primesOver
         (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
       (IsDedekindDomain.primesOver_finite 𝔭.1 _).to_subtype
     refine Finite.of_injective (β := 𝔭.1.primesOver
         (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))))
       (fun P ↦ ⟨P.1.1, P.1.2.2.1, ?_⟩) ?_
-    · haveI := P.1.2.2.1
+    · have := P.1.2.2.1
       exact ⟨(congrArg Subtype.val P.2).symm ▸ (Ideal.over_under (A := 𝓞 K) (P := P.1.1)).over⟩
     · rintro ⟨⟨P, hP⟩, hgP⟩ ⟨⟨Q, hQ⟩, hgQ⟩ hPQ
-      simpa using hPQ
+      exact Subtype.ext (Subtype.ext (congrArg
+        (fun q : 𝔭.1.primesOver
+          (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) ↦ q.1) hPQ))
   have hordC_pos : (0 : ℝ) < orderOf σ * Nat.card (ConjClasses.mk σ).carrier := by
     have h₁ : 0 < orderOf σ := orderOf_pos_iff.mpr (isOfFinOrder_of_finite σ)
     have : Nonempty (ConjClasses.mk σ).carrier := ⟨⟨σ, ConjClasses.mem_carrier_mk⟩⟩
@@ -871,83 +867,55 @@ most `[E:K]` primes of `𝓞 E` above it, by `Ideal.card_primesOverFinset_le_fin
 over `B` is bounded by the finite cardinality of `B`. Both are constants in `s`, so
 `Σ_{T₂} s ≤ C` for all `s > 1`, whence `Σ_{T₂}/Σ_univ^E → 0` since `Σ_univ^E → ∞`. -/
 
-/-- **Fibre-counting bound for `ℝ≥0∞`-valued sums.** If every fibre `g ⁻¹' {y}` is finite with at
-most `d` elements, then `Σ_b f(g b) ≤ d · Σ_y f y`: group `b` by its image `g b`, on each fibre
-the summand is the constant `f y`, and the fibre has `≤ d` terms. -/
-private theorem tsum_comp_le_card_fibre_mul {β γ : Type*} (g : β → γ) (f : γ → ℝ≥0∞) (d : ℕ)
-    (hfin : ∀ y, Finite (g ⁻¹' {y} : Set β)) (hfib : ∀ y, Nat.card (g ⁻¹' {y} : Set β) ≤ d) :
-    ∑' b, f (g b) ≤ (d : ℝ≥0∞) * ∑' y, f y := by
-  rw [← ENNReal.tsum_fiberwise (fun b ↦ f (g b)) g, ← ENNReal.tsum_mul_left]
-  refine ENNReal.tsum_le_tsum (fun y ↦ ?_)
-  rw [tsum_congr (fun b : (g ⁻¹' {y} : Set β) ↦ by rw [b.2])]
-  haveI := hfin y
-  letI := Fintype.ofFinite (g ⁻¹' {y} : Set β)
-  rw [tsum_fintype, Finset.sum_const, Finset.card_univ, ← Nat.card_eq_fintype_card, nsmul_eq_mul]
-  gcongr
-  exact_mod_cast hfib y
-
-/-- **Fibre-counting bound for real-valued sums.** The `ℝ`-valued companion of
-`tsum_comp_le_card_fibre_mul`: for nonnegative summable `FA, FK` with `FA b ≤ FK (g b)` and every
-fibre `g ⁻¹' {y}` finite of size `≤ d`, the sum `Σ_b FA b` is at most `d · Σ_y FK y`. -/
-private theorem tsum_real_comp_le_card_fibre_mul {β γ : Type*} (g : β → γ) (FA : β → ℝ)
-    (FK : γ → ℝ) (d : ℕ) (hsummA : Summable FA) (hsummK : Summable FK) (hnonnegA : ∀ b, 0 ≤ FA b)
-    (hnonnegK : ∀ y, 0 ≤ FK y) (hterm : ∀ b, FA b ≤ FK (g b))
-    (hfin : ∀ y, Finite (g ⁻¹' {y} : Set β)) (hfib : ∀ y, Nat.card (g ⁻¹' {y} : Set β) ≤ d) :
-    ∑' b, FA b ≤ (d : ℝ) * ∑' y, FK y := by
-  have hchain : ∑' b, ENNReal.ofReal (FA b)
-      ≤ (d : ℝ≥0∞) * ∑' y, ENNReal.ofReal (FK y) :=
-    calc ∑' b, ENNReal.ofReal (FA b) ≤ ∑' b, ENNReal.ofReal (FK (g b)) :=
-          ENNReal.tsum_le_tsum fun b ↦ ENNReal.ofReal_le_ofReal (hterm b)
-      _ ≤ (d : ℝ≥0∞) * ∑' y, ENNReal.ofReal (FK y) :=
-          tsum_comp_le_card_fibre_mul g (ENNReal.ofReal <| FK ·) d hfin hfib
-  rw [← ENNReal.ofReal_tsum_of_nonneg hnonnegA hsummA,
-    ← ENNReal.ofReal_tsum_of_nonneg hnonnegK hsummK] at hchain
-  rw [← ENNReal.toReal_ofReal (tsum_nonneg hnonnegA),
-    ← ENNReal.toReal_ofReal (mul_nonneg (Nat.cast_nonneg d) (tsum_nonneg hnonnegK)),
-    ENNReal.ofReal_mul (Nat.cast_nonneg d), ENNReal.ofReal_natCast]
-  exact ENNReal.toReal_mono (ENNReal.mul_ne_top (by simp) ENNReal.ofReal_ne_top) hchain
-
 omit [IsGalois K L] in
 /-- For a degree-`≥ 2` prime `P` of `𝓞 E` over `𝔭 = P ∩ 𝓞 K`, the Dirichlet term is dominated by
 the square term of `𝔭`: `N P^{-s} ≤ N𝔭^{-2}` for `1 < s`. Here `N P = N𝔭^{f}` with `f ≥ 2` and
 `N𝔭 ≥ 2`, so the exponent `f·s ≥ 2` dominates. -/
 private theorem absNorm_rpow_neg_le_under_sq (σ : Gal(L/K))
     (P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) [P.IsPrime]
-    (hPb : P ≠ ⊥) {s : ℝ} (hs : 1 < s) (hdeg : 2 ≤ (P.under (𝓞 K)).inertiaDeg' P) :
+    (hPb : P ≠ ⊥) {s : ℝ} (hs : 1 < s) (hdeg : 2 ≤ P.inertiaDeg (𝓞 K)) :
     (Ideal.absNorm P : ℝ) ^ (-s) ≤ (Ideal.absNorm (P.under (𝓞 K)) : ℝ) ^ (-(2 : ℝ)) := by
   have hppr : (P.under (𝓞 K)).IsPrime := inferInstance
-  have hpbot : P.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPb
-  haveI : P.LiesOver (P.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := P)
-  have hpow := Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOver P (P.under (𝓞 K)) hppr hpbot
+  have hpbot : P.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.under_ne_bot (𝓞 K) hPb
+  have : P.LiesOver (P.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := P)
+  have hpow := (Ideal.absNorm_pow_inertiaDeg (P.under (𝓞 K)) P).symm
   have hn2 : 2 ≤ Ideal.absNorm (P.under (𝓞 K)) := by
     have h0 : Ideal.absNorm (P.under (𝓞 K)) ≠ 0 := Ideal.absNorm_eq_zero_iff.not.mpr hpbot
     have h1 : Ideal.absNorm (P.under (𝓞 K)) ≠ 1 := Ideal.absNorm_eq_one_iff.not.mpr hppr.ne_top
     omega
   rw [hpow, Nat.cast_pow,
-    ← Real.rpow_natCast (Ideal.absNorm (P.under (𝓞 K)) : ℝ) ((P.under (𝓞 K)).inertiaDeg' P),
+    ← Real.rpow_natCast (Ideal.absNorm (P.under (𝓞 K)) : ℝ) (P.inertiaDeg (𝓞 K)),
     ← Real.rpow_mul (by positivity)]
   refine Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast Nat.one_le_of_lt hn2) ?_
-  nlinarith [mul_le_mul (show (2 : ℝ) ≤ ((P.under (𝓞 K)).inertiaDeg' P : ℝ) by exact_mod_cast hdeg)
-    hs.le (by norm_num) (by positivity : (0 : ℝ) ≤ ((P.under (𝓞 K)).inertiaDeg' P : ℝ))]
+  nlinarith [mul_le_mul (show (2 : ℝ) ≤ (P.inertiaDeg (𝓞 K) : ℝ) by exact_mod_cast hdeg)
+    hs.le (by norm_num) (by positivity : (0 : ℝ) ≤ (P.inertiaDeg (𝓞 K) : ℝ))]
 
 omit [IsGalois K L] in
 /-- The number of primes of `𝓞 E` over a fixed maximal prime `𝔭` of `𝓞 K` is at most `[E : K]`,
 a `Nat.card` repackaging of `Ideal.card_primesOverFinset_le_finrank`. -/
 private theorem card_primesOver_le_finrank (σ : Gal(L/K))
     [NoZeroSMulDivisors (𝓞 K) (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))]
-    (𝔭 : Ideal (𝓞 K)) [𝔭.IsMaximal] (h𝔭 : 𝔭 ≠ ⊥) :
+    (𝔭 : Ideal (𝓞 K)) [𝔭.IsMaximal] (_h𝔭 : 𝔭 ≠ ⊥) :
     Nat.card {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) //
         P.IsPrime ∧ P.LiesOver 𝔭}
       ≤ Module.finrank K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) := by
-  letI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
-    (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  rw [show {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) //
-        P.IsPrime ∧ P.LiesOver 𝔭}
-      = ↥(𝔭.primesOver (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) from rfl,
-    Nat.card_coe_set_eq, ← IsDedekindDomain.coe_primesOverFinset h𝔭, Set.ncard_coe_finset]
-  exact Ideal.card_primesOverFinset_le_finrank (R := 𝓞 K)
-    (S := 𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))
-    (K := K) (L := ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) h𝔭
+  classical
+  let E := IntermediateField.fixedField (Subgroup.zpowers σ)
+  let : IsScalarTower K ↥E L := E.isScalarTower_mid'
+  let : Finite (𝔭.primesOver (𝓞 ↥E)) :=
+    (IsDedekindDomain.primesOver_finite 𝔭 (𝓞 ↥E)).to_subtype
+  let : Fintype (𝔭.primesOver (𝓞 ↥E)) := Fintype.ofFinite _
+  change Nat.card (𝔭.primesOver (𝓞 ↥E)) ≤ Module.finrank K ↥E
+  calc
+    Nat.card (𝔭.primesOver (𝓞 ↥E)) = ∑ _q : 𝔭.primesOver (𝓞 ↥E), (1 : ℕ) := by
+      simp [Nat.card_eq_fintype_card]
+    _ ≤ ∑ q : 𝔭.primesOver (𝓞 ↥E), q.1.ramificationIdx (𝓞 K) * q.1.inertiaDeg (𝓞 K) := by
+      apply Finset.sum_le_sum
+      intro q _
+      have : q.1.IsPrime := q.2.1
+      exact Nat.mul_pos (Ideal.ramificationIdx_pos q.1 (𝓞 K)) (Ideal.inertiaDeg_pos q.1 (𝓞 K))
+    _ = Module.finrank (𝓞 K) (𝓞 ↥E) := Ideal.sum_ramification_inertia_eq_finrank 𝔭 (𝓞 ↥E)
+    _ = Module.finrank K ↥E := (IsFractionRing.finrank_eq (𝓞 K) K (𝓞 ↥E) ↥E).symm
 
 /-- **The degree-`≥ 2` part of `T₂` is bounded by a constant.** For `1 < s`, the partial sum over
 the set `A` of primes `P` of `𝓞 E` whose underlying `K`-prime is unramified in `L` but of inertia
@@ -958,13 +926,13 @@ degree `≥ 2` is bounded by `[E:K]·Σ_𝔭 N𝔭^{-2}`. Indeed `N P = N𝔭^{f
 private theorem primeIdealZetaSum_degTwo_le (σ : Gal(L/K)) {s : ℝ}
     (hs : 1 < s) (Aset : Set (Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ)))))
     (hA : Aset = {P | P.IsPrime ∧ P ≠ ⊥ ∧
-      UnramifiedIn K L (P.under (𝓞 K)) ∧ 2 ≤ (P.under (𝓞 K)).inertiaDeg' P}) :
+      UnramifiedIn K L (P.under (𝓞 K)) ∧ 2 ≤ P.inertiaDeg (𝓞 K)}) :
     primeIdealZetaSum Aset s
       ≤ (Module.finrank K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) : ℝ)
         * primeIdealZetaSum (univ : Set (Ideal (𝓞 K))) 2 := by
-  letI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  let : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : NoZeroSMulDivisors (𝓞 K)
+  have : NoZeroSMulDivisors (𝓞 K)
       (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) :=
     ⟨fun {c x} h ↦ by
       rw [Algebra.smul_def, mul_eq_zero] at h
@@ -974,24 +942,24 @@ private theorem primeIdealZetaSum_degTwo_le (σ : Gal(L/K)) {s : ℝ}
   set KP := {𝔭 : Ideal (𝓞 K) // 𝔭 ∈ (univ : Set (Ideal (𝓞 K))) ∧ 𝔭.IsPrime ∧ 𝔭 ≠ ⊥} with hKP
   have hunder : ∀ P : AP, (P.1.under (𝓞 K)).IsPrime ∧ P.1.under (𝓞 K) ≠ ⊥ := by
     rintro ⟨P, hPA, hPp, hPb⟩
-    haveI := hPp
-    exact ⟨inferInstance, Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPb⟩
+    have := hPp
+    exact ⟨inferInstance, Ideal.IsIntegral.under_ne_bot (𝓞 K) hPb⟩
   set g : AP → KP := fun P ↦ ⟨P.1.under (𝓞 K), mem_univ _, (hunder P).1, (hunder P).2⟩ with hg
   have hterm : ∀ P : AP, (Ideal.absNorm P.1 : ℝ) ^ (-s)
       ≤ (Ideal.absNorm (g P).1 : ℝ) ^ (-(2 : ℝ)) := by
     rintro ⟨P, hPA, hPp, hPb⟩
-    haveI := hPp
+    have := hPp
     rw [hA] at hPA
     exact absNorm_rpow_neg_le_under_sq σ P hPb hs hPA.2.2.2
   have hinj : ∀ 𝔭 : KP, Finite (g ⁻¹' {𝔭} : Set AP) ∧
       Nat.card (g ⁻¹' {𝔭} : Set AP)
         ≤ Module.finrank K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) := by
     intro 𝔭
-    haveI := 𝔭.2.2.1
-    haveI : 𝔭.1.IsMaximal := 𝔭.2.2.1.isMaximal 𝔭.2.2.2
+    have := 𝔭.2.2.1
+    have : 𝔭.1.IsMaximal := 𝔭.2.2.1.isMaximal 𝔭.2.2.2
     have hmem : ∀ P : (g ⁻¹' {𝔭} : Set AP), P.1.1.IsPrime ∧ P.1.1.LiesOver 𝔭.1 := by
       rintro ⟨⟨P, hPA, hPp, hPb⟩, hgP⟩
-      haveI := hPp
+      have := hPp
       exact ⟨hPp, ⟨(congrArg Subtype.val hgP : P.under (𝓞 K) = 𝔭.1) ▸
         (Ideal.over_under (A := 𝓞 K) (P := P)).over⟩⟩
     set hmap : (g ⁻¹' {𝔭} : Set AP) → {P : Ideal (𝓞 ↥(IntermediateField.fixedField
@@ -999,7 +967,7 @@ private theorem primeIdealZetaSum_degTwo_le (σ : Gal(L/K)) {s : ℝ}
     have hmapinj : Function.Injective hmap := by
       rintro ⟨⟨P, hP⟩, hgP⟩ ⟨⟨Q, hQ⟩, hgQ⟩ hPQ
       exact Subtype.ext (Subtype.ext (by simpa only [hhmap, Subtype.mk.injEq] using hPQ))
-    haveI : Finite {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) //
+    have : Finite {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) //
         P.IsPrime ∧ P.LiesOver 𝔭.1} :=
       (IsDedekindDomain.primesOver_finite 𝔭.1 _).to_subtype
     exact ⟨Finite.of_injective hmap hmapinj, (Nat.card_le_card_of_injective hmap hmapinj).trans
@@ -1021,12 +989,12 @@ private theorem ramifiedBelow_finite (σ : Gal(L/K))
     (𝔭.primesOver (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))))) ?_)
   · rw [hB]
     rintro P ⟨hPp, hPb, hPnu⟩
-    haveI := hPp
-    exact Set.mem_biUnion ⟨inferInstance, Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPb, hPnu⟩
+    have := hPp
+    exact Set.mem_biUnion ⟨inferInstance, Ideal.IsIntegral.under_ne_bot (𝓞 K) hPb, hPnu⟩
       ⟨hPp, Ideal.over_under (A := 𝓞 K) (P := P)⟩
   · rintro 𝔭 ⟨hp, hb, -⟩
-    haveI := hp
-    haveI : 𝔭.IsMaximal := hp.isMaximal hb
+    have := hp
+    have : 𝔭.IsMaximal := hp.isMaximal hb
     exact IsDedekindDomain.primesOver_finite 𝔭 _
 
 /-- **LEAF B: the degree-`≥ 2` part of `T` vanishes in the density ratio** (Sharifi 7.2.2
@@ -1051,32 +1019,32 @@ private theorem primeIdealZetaSum_T2_div_univ_tendsto_zero
           P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P ∧
           frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
             = ConjClasses.mk σE} |
-        (P.under (𝓞 K)).inertiaDeg' P = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))}) :
+        P.inertiaDeg (𝓞 K) = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))}) :
     Tendsto (fun s : ℝ ↦ primeIdealZetaSum T₂set s
       / primeIdealZetaSum (univ : Set (Ideal (𝓞 ↥(IntermediateField.fixedField
         (Subgroup.zpowers σ))))) s) (𝓝[>] 1) (𝓝 0) := by
-  haveI : IsGalois ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsGalois ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     IsGalois.tower_top_intermediateField _
   set Aset := {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) |
     P.IsPrime ∧ P ≠ ⊥ ∧ UnramifiedIn K L (P.under (𝓞 K)) ∧
-    2 ≤ (P.under (𝓞 K)).inertiaDeg' P} with hAdef
+    2 ≤ P.inertiaDeg (𝓞 K)} with hAdef
   set Bset := {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) |
     P.IsPrime ∧ P ≠ ⊥ ∧ ¬ UnramifiedIn K L (P.under (𝓞 K))} with hBdef
   have hsub : T₂set ⊆ Aset ∪ Bset := by
     rw [hT₂]
     rintro P ⟨⟨hPp, hPunr, hPfr⟩, hPnotT1⟩
-    haveI := hPp
-    simp only [Set.mem_setOf_eq, not_and] at hPnotT1
+    have := hPp
+    simp only [Set.mem_ofPred_eq, not_and] at hPnotT1
     have hPb : P ≠ ⊥ := UnramifiedIn.ne_bot _ L hPunr
     by_cases hunrK : UnramifiedIn K L (P.under (𝓞 K))
     · refine Or.inl ⟨hPp, hPb, hunrK, ?_⟩
-      have hdegne : (P.under (𝓞 K)).inertiaDeg' P ≠ 1 := fun hdeg1 ↦
+      have hdegne : P.inertiaDeg (𝓞 K) ≠ 1 := fun hdeg1 ↦
         hPnotT1 ⟨hPp, hPunr, hPfr⟩ hdeg1 hunrK
       have hppr : (P.under (𝓞 K)).IsPrime := inferInstance
-      haveI : (P.under (𝓞 K)).IsMaximal :=
-        hppr.isMaximal (Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPb)
-      haveI : P.LiesOver (P.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := P)
-      have hpos : 0 < (P.under (𝓞 K)).inertiaDeg' P := Ideal.inertiaDeg_pos' _ _
+      have : (P.under (𝓞 K)).IsMaximal :=
+        hppr.isMaximal (Ideal.IsIntegral.under_ne_bot (𝓞 K) hPb)
+      have : P.LiesOver (P.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := P)
+      have hpos : 0 < P.inertiaDeg (𝓞 K) := Ideal.inertiaDeg_pos P (𝓞 K)
       omega
     · exact Or.inr ⟨hPp, hPb, hunrK⟩
   have hdisj : Disjoint Aset Bset := by
@@ -1149,9 +1117,9 @@ theorem density_lift_through_fixedField
         frobeniusClass K L 𝔭 = ConjClasses.mk σ}
       ((Nat.card (ConjClasses.mk σ).carrier : ℝ) / Nat.card Gal(L/K)) := by
   subst _hEfix
-  haveI : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
+  have : IsScalarTower K ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L :=
     (IntermediateField.fixedField (Subgroup.zpowers σ)).isScalarTower_mid'
-  haveI : IsMulCommutative Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
+  have : IsMulCommutative Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
     isMulCommutative_galGroup_fixedField σ
   have horderE' :
       orderOf σ = Nat.card Gal(L/(↥(IntermediateField.fixedField (Subgroup.zpowers σ)))) :=
@@ -1160,7 +1128,7 @@ theorem density_lift_through_fixedField
     P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P ∧
     frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P
       = ConjClasses.mk σE} with hTset
-  set T₁set := {P ∈ Tset | (P.under (𝓞 K)).inertiaDeg' P = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))}
+  set T₁set := {P ∈ Tset | P.inertiaDeg (𝓞 K) = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))}
     with hT₁set
   set T₂set := Tset \ T₁set with hT₂set
   have hT₁sub : T₁set ⊆ Tset := fun x hx ↦ hx.1
@@ -1212,10 +1180,10 @@ theorem density_lift_through_fixedField
   have hT₁flat : T₁set = {P : Ideal (𝓞 ↥(IntermediateField.fixedField (Subgroup.zpowers σ))) |
       P.IsPrime ∧ UnramifiedIn ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P ∧
       frobeniusClass ↥(IntermediateField.fixedField (Subgroup.zpowers σ)) L P = ConjClasses.mk σE ∧
-      (P.under (𝓞 K)).inertiaDeg' P = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))} := by
+      P.inertiaDeg (𝓞 K) = 1 ∧ UnramifiedIn K L (P.under (𝓞 K))} := by
     rw [hT₁set, hTset]
     ext P
-    simp only [Set.mem_setOf_eq]
+    simp only [Set.mem_ofPred_eq]
     tauto
   have hleafA := primeIdealZetaSum_fibre_eq_smul σ σE hσE horderE' hs
   rw [← hT₁flat] at hleafA
diff --git a/projects/Chebotarev/CebotarevDensity/ForMathlib/CharacterOrthogonality.lean b/projects/Chebotarev/CebotarevDensity/ForMathlib/CharacterOrthogonality.lean
index a82fed3f7..a4a9a2d72 100644
--- a/projects/Chebotarev/CebotarevDensity/ForMathlib/CharacterOrthogonality.lean
+++ b/projects/Chebotarev/CebotarevDensity/ForMathlib/CharacterOrthogonality.lean
@@ -78,7 +78,7 @@ theorem card_mul_eq_sum_of_sum_char_mul_eq_zero {G : Type*} [CommGroup G] [Finty
     intro s
     by_cases hs : s = u
     · subst hs; simp
-    · rw [if_neg hs]
+    · rw [ite_eq_right hs]
       exact sum_char_apply_eq_zero_of_ne_one fun h ↦ hs (inv_mul_eq_one.mp h).symm
   calc (Fintype.card (G →* Rˣ) : R) * f u
       = ∑ s : G, (if s = u then (Fintype.card (G →* Rˣ) : R) else 0) * f s := by
diff --git a/projects/Chebotarev/CebotarevDensity/ForMathlib/IdealCongruenceCount.lean b/projects/Chebotarev/CebotarevDensity/ForMathlib/IdealCongruenceCount.lean
index 96f816150..4499927f7 100644
--- a/projects/Chebotarev/CebotarevDensity/ForMathlib/IdealCongruenceCount.lean
+++ b/projects/Chebotarev/CebotarevDensity/ForMathlib/IdealCongruenceCount.lean
@@ -331,7 +331,7 @@ theorem exists_card_coset_inter_smul_sub_volume_mul_rpow_le
   set Ts : (ι → ℝ) →L[ℝ] (ι → ℝ) := (T.symm.toContinuousLinearEquiv : (ι → ℝ) →L[ℝ] (ι → ℝ))
     with hTs
   have hTslip : LipschitzWith ‖Ts‖₊ (T.symm : (ι → ℝ) → (ι → ℝ)) := by
-    simpa [hTs] using Ts.lipschitz
+    simpa [hTs] using Ts.lipschitzWith
   have hD'bdd : Bornology.IsBounded D' := hTslip.isBounded_image hbdd
   have hD'meas : MeasurableSet D' :=
     (T.symm.toContinuousLinearEquiv.toHomeomorph.toMeasurableEquiv).measurableSet_image.mpr hmeas
@@ -434,10 +434,10 @@ private theorem norm_eq_prod_real_emb_mul_prod_complex {K : Type*} [Field K] [Nu
           star_involutive.eq_iff]
       rw [hfilter]
       by_cases hw : IsReal w
-      · rw [dif_pos hw, ComplexEmbedding.isReal_iff.mp (isReal_iff.mp hw),
+      · rw [dite_eq_left hw, ComplexEmbedding.isReal_iff.mp (isReal_iff.mp hw),
           Finset.insert_eq_self.mpr (Finset.mem_singleton_self _), Finset.prod_singleton,
           embedding_of_isReal_apply hw]
-      · rw [dif_neg hw, Finset.prod_pair]
+      · rw [dite_eq_right hw, Finset.prod_pair]
         · rw [ComplexEmbedding.conjugate_coe_eq, Complex.mul_conj]
           norm_cast
           rw [Complex.normSq_eq_norm_sq]
@@ -452,8 +452,8 @@ private theorem norm_eq_prod_real_emb_mul_prod_complex {K : Type*} [Field K] [Nu
     simp_rw [hperplace]
     rw [prod_eq_prod_mul_prod]
     congr 1
-    · rw [Finset.prod_congr rfl (fun w _ ↦ by rw [dif_pos w.2]), Complex.ofReal_prod]
-    · rw [Finset.prod_congr rfl (fun w _ ↦ by rw [dif_neg (not_isReal_iff_isComplex.mpr w.2)]),
+    · rw [Finset.prod_congr rfl (fun w _ ↦ by rw [dite_eq_left w.2]), Complex.ofReal_prod]
+    · rw [Finset.prod_congr rfl (fun w _ ↦ by rw [dite_eq_right (not_isReal_iff_isComplex.mpr w.2)]),
         Complex.ofReal_prod]
   exact_mod_cast hcc
 
@@ -520,7 +520,7 @@ open Ideal in
 /-- **Class split of the residue count.** The number of nonzero integral ideals of norm `≤ N`
 with norm residue `a (mod c)` is the sum over the (finite) class group of the per-class counts.
 The class group is a `Fintype`; finiteness of each fibre follows from
-`Ideal.finite_setOf_absNorm_le₀`. -/
+`Ideal.finite_setOfPred_absNorm_le₀`. -/
 private theorem card_norm_le_residue_eq_sum_class {K : Type*} [Field K] [NumberField K]
     (c : ℕ) [NeZero c] (a : ZMod c) (N : ℕ) :
     Nat.card {I : (Ideal (𝓞 K))⁰ // Ideal.absNorm (I : Ideal (𝓞 K)) ≤ N ∧
@@ -530,7 +530,7 @@ private theorem card_norm_le_residue_eq_sum_class {K : Type*} [Field K] [NumberF
           ((Ideal.absNorm (I : Ideal (𝓞 K)) : ZMod c)) = a) ∧ ClassGroup.mk0 I = C} := by
   classical
   have hbase : Finite {I : (Ideal (𝓞 K))⁰ // Ideal.absNorm (I : Ideal (𝓞 K)) ≤ N} :=
-    Ideal.finite_setOf_absNorm_le₀ N
+    Ideal.finite_setOfPred_absNorm_le₀ N
   have hfin : Finite {I : (Ideal (𝓞 K))⁰ // Ideal.absNorm (I : Ideal (𝓞 K)) ≤ N ∧
         ((Ideal.absNorm (I : Ideal (𝓞 K)) : ZMod c)) = a} :=
     Finite.of_injective (fun I ↦ (⟨I.1, I.2.1⟩ :
@@ -660,7 +660,7 @@ private theorem cone_normLe_eq_smul_normLeOne {K : Type*} [Field K] [NumberField
   have ht0 : (0 : ℝ) < t := lt_of_lt_of_le one_pos ht
   have htne : t ≠ 0 := ht0.ne'
   ext x
-  simp only [Set.mem_setOf_eq, Set.mem_smul_set, normLeOne, Set.mem_inter_iff, Set.mem_setOf_eq]
+  simp only [Set.mem_ofPred_eq, Set.mem_smul_set, normLeOne, Set.mem_inter_iff, Set.mem_ofPred_eq]
   constructor
   · rintro ⟨hcone, hnorm⟩
     refine ⟨t⁻¹ • x, ⟨(smul_mem_iff_mem (inv_ne_zero htne)).mpr hcone, ?_⟩, ?_⟩
@@ -736,8 +736,8 @@ private theorem exists_lipschitz_cube_cover_hyperplane_slab {ι : Type*} [Fintyp
     rw [dist_pi_le_iff (by positivity)]
     intro i
     by_cases hij : i = j
-    · simp only [hφ, dif_pos hij, dist_self]; positivity
-    · simp only [hφ, dif_neg hij]
+    · simp only [hφ, dite_eq_left hij, dist_self]; positivity
+    · simp only [hφ, dite_eq_right hij]
       have hreorg : (2 * R) * c (σ.symm ⟨i, hij⟩) - R - ((2 * R) * c' (σ.symm ⟨i, hij⟩) - R)
           = (2 * R) * (c (σ.symm ⟨i, hij⟩) - c' (σ.symm ⟨i, hij⟩)) := by ring
       rw [Real.dist_eq, hreorg, abs_mul, abs_of_nonneg (by positivity : (0 : ℝ) ≤ 2 * R),
@@ -752,8 +752,8 @@ private theorem exists_lipschitz_cube_cover_hyperplane_slab {ι : Type*} [Fintyp
       refine ⟨0, ⟨le_refl _, zero_le_one⟩, ?_⟩
       ext i; simp only [hφ]
       by_cases hij : i = j
-      · rw [dif_pos hij, hx0]; rfl
-      · rw [dif_neg hij, hx0]; simp [← hR0]
+      · rw [dite_eq_left hij, hx0]; rfl
+      · rw [dite_eq_right hij, hx0]; simp [← hR0]
     · refine ⟨fun k ↦ (x (σ k) + R) / (2 * R), ⟨?_, ?_⟩, ?_⟩
       · intro k; simp only [Pi.zero_apply]
         rw [le_div_iff₀ (by positivity)]; have := (abs_le.mp (hxbd (σ k))).1; linarith
@@ -761,8 +761,8 @@ private theorem exists_lipschitz_cube_cover_hyperplane_slab {ι : Type*} [Fintyp
         rw [div_le_one (by positivity)]; have := (abs_le.mp (hxbd (σ k))).2; linarith
       · ext i
         by_cases hij : i = j
-        · rw [hφ]; simp only; rw [dif_pos hij, hij]; exact hxj.symm
-        · rw [hφ]; simp only [dif_neg hij, Equiv.apply_symm_apply]; field_simp; ring
+        · rw [hφ]; simp only; rw [dite_eq_left hij, hij]; exact hxj.symm
+        · rw [hφ]; simp only [dite_eq_right hij, Equiv.apply_symm_apply]; field_simp; ring
 
 /-- **Union of two Lipschitz cube covers.** If `A` and `B` are each covered by finitely many
 `Lipschitz`-images of `[0,1]^(card ι - 1)`, so is `A ∪ B` (concatenate the families, take the max
@@ -822,13 +822,13 @@ private theorem frontier_signOrthant_subset {ι κ : Type*} [Finite κ] (g : κ
   set O : Set (ι → ℝ) := {y | (∀ k ∈ s, y (g k) ≤ 0) ∧ (∀ k ∉ s, 0 ≤ y (g k))} with hO
   set Os : Set (ι → ℝ) := {y | (∀ k ∈ s, y (g k) < 0) ∧ (∀ k ∉ s, 0 < y (g k))} with hOs
   have hOclosed : IsClosed O := by
-    simp only [hO, Set.setOf_and, Set.setOf_forall]
+    simp only [hO, Set.ofPred_and, Set.ofPred_forall]
     exact (isClosed_iInter fun k ↦ isClosed_iInter fun _ ↦
         isClosed_le (continuous_apply (g k)) continuous_const).inter
       (isClosed_iInter fun k ↦ isClosed_iInter fun _ ↦
         isClosed_le continuous_const (continuous_apply (g k)))
   have hOsopen : IsOpen Os := by
-    simp only [hOs, Set.setOf_and, Set.setOf_forall]
+    simp only [hOs, Set.ofPred_and, Set.ofPred_forall]
     exact (isOpen_iInter_of_finite fun k ↦ isOpen_iInter_of_finite fun _ ↦
         isOpen_lt (continuous_apply (g k)) continuous_const).inter
       (isOpen_iInter_of_finite fun k ↦ isOpen_iInter_of_finite fun _ ↦
@@ -840,7 +840,7 @@ private theorem frontier_signOrthant_subset {ι κ : Type*} [Finite κ] (g : κ
     rw [frontier_eq_closure_inter_closure] at hy
     rw [interior_eq_compl_closure_compl]; exact fun hh ↦ hh hy.2
   by_contra hcon
-  simp only [Set.mem_iUnion, Set.mem_setOf_eq, not_exists] at hcon
+  simp only [Set.mem_iUnion, Set.mem_ofPred_eq, not_exists] at hcon
   exact hyni (mem_interior.mpr ⟨Os, hsub, hOsopen,
     ⟨fun k hk ↦ lt_of_le_of_ne (hyO.1 k hk) (hcon k),
      fun k hk ↦ lt_of_le_of_ne (hyO.2 k hk) (Ne.symm (hcon k))⟩⟩)
@@ -985,14 +985,14 @@ private theorem exists_card_cell_sub_mul_rpow_le_explicit {ι : Type*} [Fintype
               * t ^ (Fintype.card ι)|
         ≤ C * t ^ (Fintype.card ι - 1 : ℕ) := by
   classical
-  haveI : Fintype κ := Fintype.ofFinite κ
+  have : Fintype κ := Fintype.ofFinite κ
   set T' : (ι → ℝ) ≃ₗ[ℝ] (ι → ℝ) := (LinearEquiv.smulOfNeZero ℝ (ι → ℝ) (m : ℝ) hm).trans T
   set Ds : Set (ι → ℝ) :=
     D₀ ∩ {y : ι → ℝ | (∀ k ∈ s, y (g k) ≤ 0) ∧ (∀ k ∉ s, 0 ≤ y (g k))}
   have hDsbdd : Bornology.IsBounded Ds := hbdd.subset Set.inter_subset_left
   have hOclosed : IsClosed {y : ι → ℝ | (∀ k ∈ s, y (g k) ≤ 0) ∧ (∀ k ∉ s, 0 ≤ y (g k))} := by
     classical
-    rw [setOf_and]
+    rw [ofPred_and]
     refine IsClosed.inter ?_ ?_
     · have h : {y : ι → ℝ | ∀ k ∈ s, y (g k) ≤ 0} = ⋂ k ∈ s, {y : ι → ℝ | y (g k) ≤ 0} := by
         ext y; simp
@@ -1514,7 +1514,7 @@ private theorem exists_card_residue_fibre_sub_mul_rpow_le_explicit {K : Type*} [
       ((Φ.toHomeomorph.toMeasurableEquiv).measurableSet_image.mpr (measurableSet_normLeOne K))
       hcov (Sum.inl : {w : InfinitePlace K // IsReal w} → index K) s
     refine ⟨cellC, fun t ht ↦ ?_⟩
-    rw [if_pos ⟨a₀, horth₀, hcos₀, hres₀⟩]
+    rw [ite_eq_left ⟨a₀, horth₀, hcos₀, hres₀⟩]
     have hfibre : Nat.card {a : idealSet K J //
         (mixedEmbedding.norm (a : mixedSpace K) ≤ t ^ Module.finrank ℚ K ∧
           ((intNorm (idealSetEquiv K J a).val : ZMod m) = (b : ZMod m))) ∧
@@ -1538,7 +1538,7 @@ private theorem exists_card_residue_fibre_sub_mul_rpow_le_explicit {K : Type*} [
     rw [hpow1, hpow2]
     exact hcell (T (fun i ↦ ((k i).val : ℝ))) t ht
   · refine ⟨0, fun t ht ↦ ?_⟩
-    rw [if_neg hQ]
+    rw [ite_eq_right hQ]
     have hempty : IsEmpty {a : idealSet K J //
         (mixedEmbedding.norm (a : mixedSpace K) ≤ t ^ Module.finrank ℚ K ∧
           ((intNorm (idealSetEquiv K J a).val : ZMod m) = (b : ZMod m))) ∧
@@ -1590,14 +1590,14 @@ private theorem exists_card_residue_fibre_sub_mul_rpow_le {K : Type*} [Field K]
 open Ideal NumberField.mixedEmbedding NumberField.mixedEmbedding.fundamentalCone Units in
 /-- **Finiteness of bounded-norm cone points.** The cone points of `idealSet K J` of norm `≤ s`
 form a finite set: they inject (via `integerSetEquiv ∘ idealSetMap`) into the product of the
-finite set of integral ideals of norm `≤ ⌊s⌋` (`Ideal.finite_setOf_absNorm_le₀`) with the finite
+finite set of integral ideals of norm `≤ ⌊s⌋` (`Ideal.finite_setOfPred_absNorm_le₀`) with the finite
 torsion group. -/
 private theorem finite_idealSet_norm_le {K : Type*} [Field K] [NumberField K]
     (J : (Ideal (𝓞 K))⁰) (s : ℝ) :
     Finite {a : idealSet K J // mixedEmbedding.norm (a : mixedSpace K) ≤ s} := by
   classical
   have : Finite {I : (Ideal (𝓞 K))⁰ // Ideal.absNorm (I : Ideal (𝓞 K)) ≤ ⌊s⌋₊} :=
-    (Ideal.finite_setOf_absNorm_le₀ ⌊s⌋₊).to_subtype
+    (Ideal.finite_setOfPred_absNorm_le₀ ⌊s⌋₊).to_subtype
   refine Finite.of_injective (β := {I : (Ideal (𝓞 K))⁰ // Ideal.absNorm (I : Ideal (𝓞 K)) ≤ ⌊s⌋₊} ×
       torsion K) (fun a ↦ ⟨⟨(integerSetEquiv K (idealSetMap K J a.1)).1.1, ?_⟩,
     (integerSetEquiv K (idealSetMap K J a.1)).2⟩) ?_
@@ -1838,7 +1838,7 @@ private theorem exists_card_norm_le_residue_class_eq_sub_mul_rpow_le
   classical
   obtain ⟨J, hJ⟩ := ClassGroup.mk0_surjective C⁻¹
   have hNJ : 0 < Ideal.absNorm (J : Ideal (𝓞 K)) := absNorm_pos_of_nonZeroDivisors J
-  haveI : NeZero (c * Ideal.absNorm (J : Ideal (𝓞 K))) :=
+  have : NeZero (c * Ideal.absNorm (J : Ideal (𝓞 K))) :=
     ⟨Nat.mul_ne_zero (NeZero.ne c) hNJ.ne'⟩
   obtain ⟨κ, C', hκ⟩ := exists_card_dvd_principal_residue_eq_sub_mul_rpow_le
     (c * Ideal.absNorm (J : Ideal (𝓞 K))) (a.val * Ideal.absNorm (J : Ideal (𝓞 K))) J
@@ -2403,7 +2403,7 @@ private theorem crt_single_coset {ι : Type*} [Finite ι] (L L' : Submodule ℤ
       = {z | ∃ x ∈ M, z = m • x} := by
     intro M
     ext z
-    simp only [Set.mem_smul_set, SetLike.mem_coe, Set.mem_setOf_eq]
+    simp only [Set.mem_smul_set, SetLike.mem_coe, Set.mem_ofPred_eq]
     exact ⟨fun ⟨x, hx, h⟩ ↦ ⟨x, hx, by rw [← h, Nat.cast_smul_eq_nsmul]⟩,
       fun ⟨x, hx, h⟩ ↦ ⟨x, hx, by rw [h, Nat.cast_smul_eq_nsmul]⟩⟩
   have hbij := hcop.nsmul_right_bijective
@@ -2429,7 +2429,7 @@ private theorem crt_single_coset {ι : Type*} [Finite ι] (L L' : Submodule ℤ
     rw [QuotientAddGroup.eq_zero_iff, AddSubgroup.mem_addSubgroupOf] at hq0
     simpa using hq0
   ext a
-  simp only [Set.mem_setOf_eq, hmsmul, Set.mem_vadd_set, Set.mem_setOf_eq, vadd_eq_add]
+  simp only [Set.mem_ofPred_eq, hmsmul, Set.mem_vadd_set, Set.mem_ofPred_eq, vadd_eq_add]
   constructor
   · rintro ⟨haL', w, ⟨x, hxL, rfl⟩, hweq⟩
     rw [hξeq] at hweq
@@ -2578,7 +2578,7 @@ private theorem exists_card_fibre_dvd_eq_card_cell {K : Type*} [Field K] [Number
   have hrel : L'.toAddSubgroup.relIndex L.toAddSubgroup = Ideal.absNorm (𝔟 : Ideal (𝓞 K)) :=
     relIndex_chart_eq_absNorm J 𝔟 T T' hT hT'
   have hNB : 0 < Ideal.absNorm (𝔟 : Ideal (𝓞 K)) := absNorm_pos_of_nonZeroDivisors 𝔟
-  haveI hfin : Finite (L.toAddSubgroup ⧸ L'.toAddSubgroup.addSubgroupOf L.toAddSubgroup) := by
+  have hfin : Finite (L.toAddSubgroup ⧸ L'.toAddSubgroup.addSubgroupOf L.toAddSubgroup) := by
     rw [← AddSubgroup.index_ne_zero_iff_finite]
     rw [show (L'.toAddSubgroup.addSubgroupOf L.toAddSubgroup).index
         = L'.toAddSubgroup.relIndex L.toAddSubgroup from rfl, hrel]
@@ -2779,7 +2779,7 @@ private theorem exists_card_fibre_dvd_residue_sub_mul_rpow_le {K : Type*} [Field
       ((fun i ↦ (round ((T.symm (Φ (a : mixedSpace K))) i) : ZMod m)) = k) ∧
       ((intNorm (idealSetEquiv K J a).val : ZMod m) = (b : ZMod m))
   · obtain ⟨a₀, horth₀, hcos₀, hres₀⟩ := hQ
-    rw [if_pos ⟨a₀, horth₀, hcos₀, hres₀⟩]
+    rw [ite_eq_left ⟨a₀, horth₀, hcos₀, hres₀⟩]
     have hdrop : Nat.card {a : idealSet K (𝔟 * J) //
         (mixedEmbedding.norm (a : mixedSpace K) ≤ t ^ d ∧
           ((intNorm (idealSetEquiv K (𝔟 * J) a).val : ZMod m) = (b : ZMod m))) ∧
@@ -2806,7 +2806,7 @@ private theorem exists_card_fibre_dvd_residue_sub_mul_rpow_le {K : Type*} [Field
     have hcell'' := hcell' ξ' t ht
     rw [smul_chart_lattice_eq T' m hm, ← hOs, hdetratio, hcard] at hcell''
     exact hcell''.trans (by gcongr; exact le_abs_self _)
-  · rw [if_neg hQ, zero_div, zero_mul, sub_zero]
+  · rw [ite_eq_right hQ, zero_div, zero_mul, sub_zero]
     have : IsEmpty {a : idealSet K (𝔟 * J) //
         (mixedEmbedding.norm (a : mixedSpace K) ≤ t ^ d ∧
           ((intNorm (idealSetEquiv K (𝔟 * J) a).val : ZMod m) = (b : ZMod m))) ∧
@@ -3011,7 +3011,7 @@ private theorem cardNormLeResidueClassDvd_div_density {K : Type*} [Field K] [Num
   have hNJ : 0 < NJ := absNorm_pos_of_nonZeroDivisors J
   have hNBc : NB.Coprime c := by rw [hNBdef, ZMod.isUnit_iff_coprime] at hu; exact hu
   have hcop : NB.Coprime (c * NJ) := Nat.Coprime.mul_right hNBc (hNJdef ▸ hJcop.symm)
-  haveI : NeZero (c * NJ) := ⟨Nat.mul_ne_zero (NeZero.ne c) hNJ.ne'⟩
+  have : NeZero (c * NJ) := ⟨Nat.mul_ne_zero (NeZero.ne c) hNJ.ne'⟩
   have hm : ((c * NJ : ℕ) : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (NeZero.ne (c * NJ))
   obtain ⟨κ, C', hJcone, h𝔟Jcone⟩ :=
     exists_card_idealSet_residue_real_le_dvd (c * NJ) hm (y.val * NJ) J 𝔟 hcop
diff --git a/projects/Chebotarev/CebotarevDensity/ForMathlib/LatticePointCount.lean b/projects/Chebotarev/CebotarevDensity/ForMathlib/LatticePointCount.lean
index d1ed572f5..ad3fc2802 100644
--- a/projects/Chebotarev/CebotarevDensity/ForMathlib/LatticePointCount.lean
+++ b/projects/Chebotarev/CebotarevDensity/ForMathlib/LatticePointCount.lean
@@ -296,7 +296,7 @@ theorem abs_card_inter_sub_volume_mul_pow_le {s : Set (ι → ℝ)}
   set Tag : Set (ι → ℤ) := {ν | tag n ν ∈ s} with hTag
   have himg : index n '' (s ∩ (n : ℝ)⁻¹ • span ℤ (Set.range (Pi.basisFun ℝ ι))) = Tag := by
     ext ν
-    simp only [hTag, Set.mem_image, Set.mem_inter_iff, Set.mem_setOf_eq]
+    simp only [hTag, Set.mem_image, Set.mem_inter_iff, Set.mem_ofPred_eq]
     constructor
     · rintro ⟨x, ⟨hxs, hxL⟩, rfl⟩
       rwa [tag_index_eq_self_of_mem_smul_span n hxL]
diff --git a/projects/Chebotarev/CebotarevDensity/ForMathlib/NormLeOneLipschitz.lean b/projects/Chebotarev/CebotarevDensity/ForMathlib/NormLeOneLipschitz.lean
index 8241f43b0..e8d648364 100644
--- a/projects/Chebotarev/CebotarevDensity/ForMathlib/NormLeOneLipschitz.lean
+++ b/projects/Chebotarev/CebotarevDensity/ForMathlib/NormLeOneLipschitz.lean
@@ -152,19 +152,19 @@ open scoped Classical in
 theorem contDiff_faceMapZero : ContDiff ℝ 1 (faceMapZero K) := by
   refine (contDiff_expMapBasis K).comp (contDiff_pi.mpr fun w ↦ ?_)
   by_cases hw : w = w₀
-  · simpa only [dif_pos hw] using contDiff_const
-  · simpa only [dif_neg hw] using contDiff_apply ℝ ℝ _
+  · simpa only [dite_eq_left hw] using contDiff_const
+  · simpa only [dite_eq_right hw] using contDiff_apply ℝ ℝ _
 
 open scoped Classical in
 theorem contDiff_faceMapSide (i : {w : InfinitePlace K // w ≠ w₀}) (a : ℝ) :
     ContDiff ℝ 1 (faceMapSide K i a) := by
   refine (contDiff_apply ℝ ℝ i).smul ((contDiff_expMapBasis K).comp (contDiff_pi.mpr fun w ↦ ?_))
   by_cases hw : w = w₀
-  · simpa only [dif_pos hw] using contDiff_const
-  · simp only [dif_neg hw]
+  · simpa only [dite_eq_left hw] using contDiff_const
+  · simp only [dite_eq_right hw]
     by_cases hi : (⟨w, hw⟩ : {w // w ≠ w₀}) = i
-    · simpa only [if_pos hi] using contDiff_const
-    · simpa only [if_neg hi] using contDiff_apply ℝ ℝ _
+    · simpa only [ite_eq_left hi] using contDiff_const
+    · simpa only [ite_eq_right hi] using contDiff_apply ℝ ℝ _
 
 /-- **Abstract topological core.** For an open injective map `f` and a set `s`, if the closure of
 `f '' s` is contained in `f '' closure s` together with one extra point `p`, then the frontier of
@@ -222,12 +222,12 @@ private theorem expMapBasis_mem_iUnion_faceMapSide
         fun w' ↦ if w' = w₀ then 0 else y w' := by
       funext w'
       by_cases hw'₀ : w' = w₀
-      · simp only [dif_pos hw'₀, if_pos hw'₀]
-      · simp only [dif_neg hw'₀, if_neg hw'₀]
+      · simp only [dite_eq_left hw'₀, ite_eq_left hw'₀]
+      · simp only [dite_eq_right hw'₀, ite_eq_right hw'₀]
         by_cases hw'w : (⟨w', hw'₀⟩ : {w // w ≠ w₀}) = i
         · obtain rfl : w' = w := by rw [hi, Subtype.mk_eq_mk] at hw'w; exact hw'w
-          simp only [if_pos hw'w]
-        · simp only [hc, if_neg hw'w]
+          simp only [ite_eq_left hw'w]
+        · simp only [hc, ite_eq_right hw'w]
     rw [faceMapSide, expMapBasis_apply'' y, hci, hfun]
   refine Set.mem_iUnion.mpr ⟨i, Set.mem_iUnion₂.mpr ⟨y w, ?_, ⟨c, hcmem, hkey⟩⟩⟩
   rcases ha with h | h <;> simp [h]
@@ -252,16 +252,16 @@ theorem image_boundary_subset_faces :
   have hIcc : ∀ v : InfinitePlace K, v ≠ w₀ → y v ∈ Icc (0 : ℝ) 1 := fun v hv ↦ by
     simpa [hv] using hyc v
   by_cases hwe : w = w₀
-  · rw [if_pos hwe, Set.mem_Iio, not_lt, hwe] at hw
+  · rw [ite_eq_left hwe, Set.mem_Iio, not_lt, hwe] at hw
     have hy0 : y w₀ = 0 := le_antisymm hw₀ hw
     refine Or.inl ⟨fun i ↦ y i.1, ⟨fun i ↦ (hIcc i.1 i.2).1, fun i ↦ (hIcc i.1 i.2).2⟩, ?_⟩
     rw [faceMapZero]
     congr 1
     funext v
     by_cases hv : v = w₀
-    · rw [dif_pos hv, hv, hy0]
-    · simp only [dif_neg hv]
-  · rw [if_neg hwe] at hw
+    · rw [dite_eq_left hv, hv, hy0]
+    · simp only [dite_eq_right hv]
+  · rw [ite_eq_right hwe] at hw
     have h1 := hIcc w hwe
     rw [Set.mem_Ioo, not_and_or, not_lt, not_lt] at hw
     refine Or.inr (expMapBasis_mem_iUnion_faceMapSide K hwe hw₀ hIcc ?_)
@@ -598,8 +598,8 @@ theorem mem_iUnion_image_liftToMixed_of_eq
   refine Prod.ext (funext fun w ↦ ?_) (funext fun w ↦ ?_)
   · simp only [hε, hmodreal w]
     by_cases hpos : 0 ≤ x.1 w
-    · rw [decide_eq_true hpos, if_pos rfl, one_mul, abs_of_nonneg hpos]
-    · rw [decide_eq_false hpos, if_neg (by simp), abs_of_neg (not_le.mp hpos)]
+    · rw [decide_eq_true hpos, ite_eq_left rfl, one_mul, abs_of_nonneg hpos]
+    · rw [decide_eq_false hpos, ite_eq_right (by simp), abs_of_neg (not_le.mp hpos)]
       ring
   · simp only [hcph w, hmodcplx w]
     exact hθeq w
@@ -675,7 +675,7 @@ theorem normLeOne_frontier_lipschitz_cover_index :
   refine ⟨m, ‖(Φ : mixedSpace K →L[ℝ] (index K → ℝ))‖₊ * (M * 1),
     fun j c ↦ Φ (φ j (fun a ↦ c (g.symm a))),
     fun j ↦ (Φ : mixedSpace K →L[ℝ] (index K → ℝ)).lipschitzWith.comp ((hφ j).comp
-      (IsometryEquiv.piCongrLeft' (Y := fun _ ↦ ℝ) g).isometry.lipschitz), ?_⟩
+      (IsometryEquiv.piCongrLeft' (Y := fun _ ↦ ℝ) g).isometry.lipschitzWith), ?_⟩
   rw [← Φ.coe_toHomeomorph, ← Φ.toHomeomorph.image_frontier]
   refine (Set.image_mono hcov).trans ?_
   rw [Set.image_iUnion]
diff --git a/projects/Chebotarev/CebotarevDensity/ForMathlib/TsumFiberBound.lean b/projects/Chebotarev/CebotarevDensity/ForMathlib/TsumFiberBound.lean
new file mode 100644
index 000000000..02e8c54a0
--- /dev/null
+++ b/projects/Chebotarev/CebotarevDensity/ForMathlib/TsumFiberBound.lean
@@ -0,0 +1,51 @@
+module
+
+public import Mathlib.Topology.Algebra.InfiniteSum.Real
+public import Mathlib.Topology.Instances.ENNReal.Lemmas
+
+/-! Fibre-counting bounds for nonnegative sums. -/
+
+@[expose] public section
+
+open Set
+open scoped ENNReal
+
+namespace Chebotarev
+
+/-- **Fibre-counting bound for `ℝ≥0∞`-valued sums.** If every fibre `g ⁻¹' {y}` is finite with at
+most `d` elements, then `Σ_b f(g b) ≤ d · Σ_y f y`: group `b` by its image `g b`, on each fibre
+the summand is the constant `f y`, and the fibre has `≤ d` terms. -/
+private theorem tsum_comp_le_card_fibre_mul {β γ : Type*} (g : β → γ) (f : γ → ℝ≥0∞) (d : ℕ)
+    (hfin : ∀ y, Finite (g ⁻¹' {y} : Set β)) (hfib : ∀ y, Nat.card (g ⁻¹' {y} : Set β) ≤ d) :
+    ∑' b, f (g b) ≤ (d : ℝ≥0∞) * ∑' y, f y := by
+  rw [← ENNReal.tsum_fiberwise (fun b ↦ f (g b)) g, ← ENNReal.tsum_mul_left]
+  refine ENNReal.tsum_le_tsum (fun y ↦ ?_)
+  rw [tsum_congr (fun b : (g ⁻¹' {y} : Set β) ↦ by rw [b.2])]
+  have := hfin y
+  let := Fintype.ofFinite (g ⁻¹' {y} : Set β)
+  rw [tsum_fintype, Finset.sum_const, Finset.card_univ, ← Nat.card_eq_fintype_card, nsmul_eq_mul]
+  gcongr
+  exact_mod_cast hfib y
+
+/-- **Fibre-counting bound for real-valued sums.** The `ℝ`-valued companion of
+`tsum_comp_le_card_fibre_mul`: for nonnegative summable `FA, FK` with `FA b ≤ FK (g b)` and every
+fibre `g ⁻¹' {y}` finite of size `≤ d`, the sum `Σ_b FA b` is at most `d · Σ_y FK y`. -/
+theorem tsum_real_comp_le_card_fibre_mul {β γ : Type*} (g : β → γ) (FA : β → ℝ)
+    (FK : γ → ℝ) (d : ℕ) (hsummA : Summable FA) (hsummK : Summable FK) (hnonnegA : ∀ b, 0 ≤ FA b)
+    (hnonnegK : ∀ y, 0 ≤ FK y) (hterm : ∀ b, FA b ≤ FK (g b))
+    (hfin : ∀ y, Finite (g ⁻¹' {y} : Set β)) (hfib : ∀ y, Nat.card (g ⁻¹' {y} : Set β) ≤ d) :
+    ∑' b, FA b ≤ (d : ℝ) * ∑' y, FK y := by
+  have hchain : ∑' b, ENNReal.ofReal (FA b)
+      ≤ (d : ℝ≥0∞) * ∑' y, ENNReal.ofReal (FK y) :=
+    calc ∑' b, ENNReal.ofReal (FA b) ≤ ∑' b, ENNReal.ofReal (FK (g b)) :=
+          ENNReal.tsum_le_tsum fun b ↦ ENNReal.ofReal_le_ofReal (hterm b)
+      _ ≤ (d : ℝ≥0∞) * ∑' y, ENNReal.ofReal (FK y) :=
+          tsum_comp_le_card_fibre_mul g (ENNReal.ofReal <| FK ·) d hfin hfib
+  rw [← ENNReal.ofReal_tsum_of_nonneg hnonnegA hsummA,
+    ← ENNReal.ofReal_tsum_of_nonneg hnonnegK hsummK] at hchain
+  rw [← ENNReal.toReal_ofReal (tsum_nonneg hnonnegA),
+    ← ENNReal.toReal_ofReal (mul_nonneg (Nat.cast_nonneg d) (tsum_nonneg hnonnegK)),
+    ENNReal.ofReal_mul (Nat.cast_nonneg d), ENNReal.ofReal_natCast]
+  exact ENNReal.toReal_mono (ENNReal.mul_ne_top (by simp) ENNReal.ofReal_ne_top) hchain
+
+end Chebotarev
diff --git a/projects/Chebotarev/CebotarevDensity/Frobenius.lean b/projects/Chebotarev/CebotarevDensity/Frobenius.lean
index a5556b12b..de813c534 100644
--- a/projects/Chebotarev/CebotarevDensity/Frobenius.lean
+++ b/projects/Chebotarev/CebotarevDensity/Frobenius.lean
@@ -102,7 +102,7 @@ theorem UnramifiedIn.ramificationIdx_eq_one [IsGalois K L]
     (hP : 𝔓.LiesOver 𝔭) : Ideal.ramificationIdx' (𝔓.under (𝓞 K)) 𝔓 = 1 := by
   have := hP
   have h𝔓 : 𝔓 ≠ ⊥ := Ideal.ne_bot_of_liesOver_of_ne_bot hunr.1 𝔓
-  have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.comap_ne_bot (𝓞 K) h𝔓
+  have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.under_ne_bot (𝓞 K) h𝔓
   have : Algebra.IsUnramifiedAt (𝓞 K) 𝔓 := hunr.2 𝔓 (‹𝔓.IsPrime›.isMaximal h𝔓) hP
   rw [Ideal.ramificationIdx'_eq_ramificationIdx (𝔓.under (𝓞 K)) 𝔓 hpbot]
   exact Ideal.ramificationIdx_eq_one_of_isUnramifiedAt
@@ -120,7 +120,7 @@ theorem inertiaGroup_trivial_of_unramified [IsGalois K L]
     (hunr : Ideal.ramificationIdx' (𝔓.under (𝓞 K)) 𝔓 = 1) :
     Ideal.inertia Gal(L/K) 𝔓 = ⊥ := by
   have hPbot : 𝔓 ≠ ⊥ := ne_bot_of_ramificationIdx_eq_one K L hunr
-  have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPbot
+  have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.under_ne_bot (𝓞 K) hPbot
   have : 𝔓.IsMaximal := ‹𝔓.IsPrime›.isMaximal hPbot
   have : (𝔓.under (𝓞 K)).IsMaximal :=
     (inferInstance : (𝔓.under (𝓞 K)).IsPrime).isMaximal hpbot
@@ -129,7 +129,7 @@ theorem inertiaGroup_trivial_of_unramified [IsGalois K L]
     let : Field (𝓞 K ⧸ 𝔓.under (𝓞 K)) := Ideal.Quotient.field _
     let : Field (𝓞 L ⧸ 𝔓) := Ideal.Quotient.field _
     exact IsGalois.to_isSeparable
-  haveI : Finite (𝓞 K ⧸ 𝔓.under (𝓞 K)) := Ideal.finiteQuotientOfFreeOfNeBot _ hpbot
+  have : Finite (𝓞 K ⧸ 𝔓.under (𝓞 K)) := Ideal.finiteQuotientOfFreeOfNeBot _ hpbot
   have hcard :
       Nat.card (Ideal.inertia Gal(L/K) 𝔓) =
         Ideal.ramificationIdx' (𝔓.under (𝓞 K)) 𝔓 := by
@@ -206,7 +206,7 @@ theorem frobeniusClass_eq_mk_of_isArithFrobAt [IsGalois K L]
     (σ : Gal(L/K)) (𝔓 : Ideal (𝓞 L)) [𝔓.IsPrime] (hσ : IsArithFrobAt (𝓞 K) σ 𝔓)
     (hP : 𝔓.LiesOver 𝔭) :
     frobeniusClass K L 𝔭 = ConjClasses.mk σ := by
-  rw [frobeniusClass, dif_pos ⟨‹𝔭.IsPrime›, hunr⟩]
+  rw [frobeniusClass, dite_eq_left ⟨‹𝔭.IsPrime›, hunr⟩]
   exact (exists_frobeniusClass K L 𝔭 hunr).choose_spec σ 𝔓 hσ hP
 
 /-- The order of an arithmetic Frobenius at an unramified prime is its residue degree. -/
@@ -218,7 +218,7 @@ theorem orderOf_eq_finrank_of_isArithFrobAt
     (hσ : IsArithFrobAt (𝓞 K) σ 𝔓) :
     orderOf σ = Module.finrank (𝓞 K ⧸ 𝔓.under (𝓞 K)) (𝓞 L ⧸ 𝔓) := by
   have hPbot : 𝔓 ≠ ⊥ := ne_bot_of_ramificationIdx_eq_one K L h
-  have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.comap_ne_bot (𝓞 K) hPbot
+  have hpbot : 𝔓.under (𝓞 K) ≠ ⊥ := Ideal.IsIntegral.under_ne_bot (𝓞 K) hPbot
   have : 𝔓.IsMaximal := ‹𝔓.IsPrime›.isMaximal hPbot
   have : (𝔓.under (𝓞 K)).IsMaximal :=
     (inferInstance : (𝔓.under (𝓞 K)).IsPrime).isMaximal hpbot
@@ -287,14 +287,13 @@ theorem card_primesAbove_mul_finrank_eq
     let : Field (𝓞 K ⧸ 𝔓₀.under (𝓞 K)) := Ideal.Quotient.field _
     let : Field (𝓞 L ⧸ 𝔓₀) := Ideal.Quotient.field _
     exact IsGalois.to_isSeparable
-  haveI : Finite (𝓞 K ⧸ 𝔓₀.under (𝓞 K)) :=
+  have : Finite (𝓞 K ⧸ 𝔓₀.under (𝓞 K)) :=
     Ideal.finiteQuotientOfFreeOfNeBot _ hp_under_bot
   have H :=
     Ideal.ncard_primesOver_mul_card_inertia_mul_finrank
       (G := Gal(L/K)) (𝔓₀.under (𝓞 K)) 𝔓₀
   rw [inertiaGroup_trivial_of_unramified K L 𝔓₀ he, Subgroup.card_bot, mul_one,
-      ← Ideal.inertiaDeg'_eq_inertiaDeg (𝔓₀.under (𝓞 K)) 𝔓₀,
-      Ideal.inertiaDeg'_algebraMap (𝔓₀.under (𝓞 K)) 𝔓₀] at H
+      Ideal.inertiaDeg_eq_of_isMaximal (𝔓₀.under (𝓞 K)) 𝔓₀] at H
   have hset : (𝔓₀.under (𝓞 K)).primesOver (𝓞 L)
       = {𝔓 : Ideal (𝓞 L) | 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} := by
     ext 𝔓
@@ -427,7 +426,7 @@ theorem finite_primes_natCast_mem (p : ℕ) (hp : p ≠ 0) :
   have hfin := Ideal.finite_factors (R := 𝓞 K) hspan
   apply Set.Finite.ofFinset (hfin.toFinset.image (·.asIdeal))
   intro 𝔭
-  simp only [Set.Finite.mem_toFinset, Finset.mem_image, Set.mem_setOf_eq]
+  simp only [Set.Finite.mem_toFinset, Finset.mem_image, Set.mem_ofPred_eq]
   constructor
   · rintro ⟨v, hv, rfl⟩
     exact
diff --git a/projects/Chebotarev/CebotarevDensity/Main.lean b/projects/Chebotarev/CebotarevDensity/Main.lean
index 678f99731..5cb630f0a 100644
--- a/projects/Chebotarev/CebotarevDensity/Main.lean
+++ b/projects/Chebotarev/CebotarevDensity/Main.lean
@@ -90,7 +90,7 @@ so `|C| = 1`. -/
 theorem ConjClasses_carrier_card_eq_one_of_comm
     {G : Type*} [Monoid G] [IsMulCommutative G] [Finite G] (g : G) :
     Nat.card (ConjClasses.mk g).carrier = 1 := by
-  letI : CommMonoid G := IsMulCommutative.instCommMonoid
+  let : CommMonoid G := IsMulCommutative.instCommMonoid
   have h : (ConjClasses.mk g).carrier = {g} := by
     ext a
     simp [ConjClasses.mem_carrier_iff_mk_eq, ConjClasses.mk_eq_mk_iff_isConj, isConj_iff_eq]
@@ -260,7 +260,7 @@ private theorem ratPrime_eq_span (𝔭 : Ideal (𝓞 ℚ)) (hp : 𝔭.IsPrime) (
 `p`, via the same `𝓞 ℚ ≃+* ℤ` transport (`comap` of the prime `span {(p : ℤ)}`). -/
 private theorem span_nat_isPrime {p : ℕ} (hpp : p.Prime) :
     (Ideal.span {(p : 𝓞 ℚ)}).IsPrime := by
-  haveI : (Ideal.span {(p : ℤ)}).IsPrime :=
+  have : (Ideal.span {(p : ℤ)}).IsPrime :=
     (Ideal.span_singleton_prime (by exact_mod_cast hpp.ne_zero)).mpr
       (Nat.prime_iff_prime_int.mp hpp)
   rw [ratSpan_eq_comap_intSpan]
@@ -278,8 +278,8 @@ private theorem unramifiedIn_cyclotomic_of_coprime {K : Type*} [Field K] [Number
     UnramifiedIn K L 𝔭 := by
   classical
   refine ⟨h𝔭, fun 𝔓 h𝔓max h𝔓lo ↦ ?_⟩
-  haveI := h𝔓lo
-  haveI : 𝔓.IsPrime := h𝔓max.isPrime
+  have := h𝔓lo
+  have : 𝔓.IsPrime := h𝔓max.isPrime
   rw [← not_dvd_differentIdeal_iff (A := 𝓞 K) (B := 𝓞 L)]
   intro hdvd
   obtain ⟨ζ, hζ⟩ := IsCyclotomicExtension.exists_isPrimitiveRoot K L
@@ -337,7 +337,7 @@ private theorem frobeniusClass_eq_iff_residue
     (𝔭 : Ideal (𝓞 ℚ)) [𝔭.IsPrime] (hunr : UnramifiedIn ℚ L 𝔭)
     (hcop : (Ideal.absNorm 𝔭).Coprime n) :
     frobeniusClass ℚ L 𝔭 = ConjClasses.mk σ ↔ (Ideal.absNorm 𝔭 : ZMod n) = a := by
-  letI : CommMonoid (L ≃ₐ[ℚ] L) := IsMulCommutative.instCommMonoid
+  let : CommMonoid (L ≃ₐ[ℚ] L) := IsMulCommutative.instCommMonoid
   have hdict := autToPow_frobeniusClass_out ℚ L n hζ 𝔭 hunr hcop
   rw [show frobeniusClass ℚ L 𝔭 = ConjClasses.mk (frobeniusClass ℚ L 𝔭).out from
     (Quotient.out_eq _).symm, ConjClasses.mk_eq_mk_iff_isConj, isConj_iff_eq]
@@ -389,7 +389,7 @@ private theorem dirichlet_AP_fibre_diff_image_subset_bad
       (fun p : ℕ ↦ Ideal.span {(p : 𝓞 ℚ)}) '' {p : ℕ | p.Prime ∧ (p : ZMod n) = a} ⊆
       (fun q : ℕ ↦ Ideal.span {(q : 𝓞 ℚ)}) '' {q : ℕ | q.Prime ∧ q ∣ n} := by
   rintro 𝔭 ⟨⟨hpr, hunr, hfrob⟩, hnotI⟩
-  haveI := hpr
+  have := hpr
   obtain ⟨q, hqp, hqeq⟩ := ratPrime_eq_span 𝔭 hpr (UnramifiedIn.ne_bot ℚ L hunr)
   have hnorm : Ideal.absNorm 𝔭 = q := by rw [hqeq, absNorm_span_nat]
   by_cases hcop : (Ideal.absNorm 𝔭).Coprime n
@@ -414,7 +414,7 @@ private theorem dirichlet_AP_image_diff_fibre_subset_bad
       (fun q : ℕ ↦ Ideal.span {(q : 𝓞 ℚ)}) '' {q : ℕ | q.Prime ∧ q ∣ n} := by
   rintro 𝔭 ⟨⟨p, ⟨hpp, hpa⟩, rfl⟩, hnotF⟩
   have hprime : (Ideal.span {(p : 𝓞 ℚ)}).IsPrime := span_nat_isPrime hpp
-  haveI := hprime
+  have := hprime
   have hp0 : (p : 𝓞 ℚ) ≠ 0 := Nat.cast_ne_zero.mpr hpp.ne_zero
   by_cases hdvd : p ∣ n
   · exact ⟨p, ⟨hpp, hdvd⟩, rfl⟩
@@ -438,11 +438,11 @@ private theorem dirichlet_AP_main (n : ℕ) (hn4 : n % 4 ≠ 2) (hn : 1 ≤ n)
       ((Nat.totient n : ℝ)⁻¹) := by
   have : NeZero n := ⟨by lia⟩
   -- The module system does not auto-synthesise `NeZero (n : ℚ)` for `CyclotomicField n ℚ` here.
-  haveI : NeZero ((n : ℕ) : ℚ) := ⟨by exact_mod_cast (show n ≠ 0 by lia)⟩
+  have : NeZero ((n : ℕ) : ℚ) := ⟨by exact_mod_cast (show n ≠ 0 by lia)⟩
   set L := CyclotomicField n ℚ
-  haveI : IsCyclotomicExtension {n} ℚ L := CyclotomicField.isCyclotomicExtension n ℚ
-  haveI : IsGalois ℚ L := IsCyclotomicExtension.isGalois {n} ℚ L
-  haveI : IsMulCommutative (L ≃ₐ[ℚ] L) :=
+  have : IsCyclotomicExtension {n} ℚ L := CyclotomicField.isCyclotomicExtension n ℚ
+  have : IsGalois ℚ L := IsCyclotomicExtension.isGalois {n} ℚ L
+  have : IsMulCommutative (L ≃ₐ[ℚ] L) :=
     IsCyclotomicExtension.isMulCommutative (S := {n}) ℚ L
   have hirr : Irreducible (Polynomial.cyclotomic n ℚ) :=
     Polynomial.cyclotomic.irreducible_rat (by lia)
diff --git a/projects/Chebotarev/CebotarevDensity/NumberFieldEulerProduct.lean b/projects/Chebotarev/CebotarevDensity/NumberFieldEulerProduct.lean
index 753a2f1ee..28b3b5b8b 100644
--- a/projects/Chebotarev/CebotarevDensity/NumberFieldEulerProduct.lean
+++ b/projects/Chebotarev/CebotarevDensity/NumberFieldEulerProduct.lean
@@ -2,7 +2,7 @@ module
 
 public import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
 public import Mathlib.Analysis.Normed.Ring.InfiniteSum
-public import Mathlib.Data.Finite.Vector
+public import Mathlib.Data.Fintype.Vector
 public import Mathlib.Data.Finsupp.Multiset
 public import Mathlib.Data.Sym.Card
 public import Mathlib.NumberTheory.NumberField.Cyclotomic.Basic
@@ -249,7 +249,7 @@ lemma sum_idealNormMultiplicity_isBigO :
   classical
   have h_finite : ∀ (b : ℕ), {I : NonzeroIdeal L | Ideal.absNorm I.1 = b}.Finite := fun b ↦
     Set.Finite.preimage (f := fun I : NonzeroIdeal L ↦ I.1) (fun _ _ _ _ ↦ Subtype.ext)
-      (Ideal.finite_setOf_absNorm_eq (S := 𝓞 L) b)
+      (Ideal.finite_setOfPred_absNorm_eq (S := 𝓞 L) b)
   have h_sum_card : ∀ n : ℕ, ∑ k ∈ Finset.Icc 1 n, idealNormMultiplicity L k =
       Nat.card {I : NonzeroIdeal L // Ideal.absNorm I.1 ≤ n} := fun n ↦ by
     have key := Finset.card_preimage_eq_sum_card_image_eq (f := fun I : NonzeroIdeal L ↦
@@ -257,7 +257,7 @@ lemma sum_idealNormMultiplicity_isBigO :
     rw [show ((fun I : NonzeroIdeal L ↦ Ideal.absNorm I.1) ⁻¹' ↑(Finset.Icc 1 n)) =
         {I : NonzeroIdeal L | Ideal.absNorm I.1 ≤ n} by
       ext ⟨I, hI⟩
-      simp only [Set.mem_preimage, Finset.coe_Icc, Set.mem_Icc, Set.mem_setOf_eq]
+      simp only [Set.mem_preimage, Finset.coe_Icc, Set.mem_Icc, Set.mem_ofPred_eq]
       exact ⟨fun h ↦ h.2, fun h ↦
         ⟨Nat.one_le_iff_ne_zero.mpr (mt Ideal.absNorm_eq_zero_iff.mp hI), h⟩⟩] at key
     exact key.symm
@@ -501,7 +501,7 @@ eventually captured, so the partial sums tend to `∑_𝔞 N𝔞^{-s} = ζ_K(s)`
 private instance instFiniteAbsNormFiber (n : ℕ) :
     Finite {I : NonzeroIdeal L // Ideal.absNorm I.1 = n} :=
   Set.Finite.to_subtype <| Set.Finite.of_finite_image (f := fun I : NonzeroIdeal L ↦ I.1)
-    ((Ideal.finite_setOf_absNorm_eq (S := 𝓞 L) n).subset (by rintro _ ⟨⟨I, _⟩, rfl, rfl⟩; rfl))
+    ((Ideal.finite_setOfPred_absNorm_eq (S := 𝓞 L) n).subset (by rintro _ ⟨⟨I, _⟩, rfl, rfl⟩; rfl))
     (fun _ _ _ _ ↦ Subtype.ext)
 
 private lemma tsum_absNormFiber {M : Type*} [AddCommGroup M] [TopologicalSpace M] [T2Space M]
@@ -583,14 +583,14 @@ private theorem factorization_prod_primePow_eq
   rw [factorization_prod_primePow_apply L S e 𝔮]
   simp_rw [factorization_primePow_apply L]
   by_cases h𝔮 : 𝔮 ∈ S
-  · rw [if_pos h𝔮, Finset.sum_eq_single (⟨𝔮, h𝔮⟩ : {x // x ∈ S})]
-    · rw [if_pos rfl]
+  · rw [ite_eq_left h𝔮, Finset.sum_eq_single (⟨𝔮, h𝔮⟩ : {x // x ∈ S})]
+    · rw [ite_eq_left rfl]
     · rintro b _ hb
-      rw [if_neg (fun h ↦ hb (Subtype.ext h.symm))]
+      rw [ite_eq_right (fun h ↦ hb (Subtype.ext h.symm))]
     · exact fun h ↦ absurd (Finset.mem_attach S _) h
-  · rw [if_neg h𝔮, Finset.sum_eq_zero]
+  · rw [ite_eq_right h𝔮, Finset.sum_eq_zero]
     rintro ⟨b, hb⟩ -
-    rw [if_neg (fun h : 𝔮 = b ↦ h𝔮 (h.symm ▸ hb))]
+    rw [ite_eq_right (fun h : 𝔮 = b ↦ h𝔮 (h.symm ▸ hb))]
 
 open UniqueFactorizationMonoid in
 /-- The normalized prime factors of `𝔞` as a `Finset` of nonzero prime ideals. -/
@@ -624,9 +624,9 @@ private theorem factorization_idealOfExp_eq
   classical
   have hprod : (∏ 𝔭 ∈ S.attach, 𝔭.1.1 ^ (fun q ↦ if h : q ∈ S then f ⟨q, h⟩ else 0) 𝔭.1)
       = ∏ 𝔭 ∈ S.attach, 𝔭.1.1 ^ f 𝔭 :=
-    Finset.prod_congr rfl fun 𝔭 _ ↦ by simp only [dif_pos 𝔭.2]
+    Finset.prod_congr rfl fun 𝔭 _ ↦ by simp only [dite_eq_left 𝔭.2]
   rw [← hprod, factorization_prod_primePow_eq L S
-    (fun q ↦ if h : q ∈ S then f ⟨q, h⟩ else 0) 𝔮.1, if_pos 𝔮.2, dif_pos 𝔮.2]
+    (fun q ↦ if h : q ∈ S then f ⟨q, h⟩ else 0) 𝔮.1, ite_eq_left 𝔮.2, dite_eq_left 𝔮.2]
 
 open UniqueFactorizationMonoid in
 private theorem prod_primePow_count_eq (S : Finset {𝔭 : Ideal (𝓞 L) // 𝔭.IsPrime ∧ 𝔭 ≠ ⊥})
diff --git a/projects/Chebotarev/CebotarevDensity/ZetaProduct.lean b/projects/Chebotarev/CebotarevDensity/ZetaProduct.lean
index 356695e9a..c67aa3c90 100644
--- a/projects/Chebotarev/CebotarevDensity/ZetaProduct.lean
+++ b/projects/Chebotarev/CebotarevDensity/ZetaProduct.lean
@@ -172,11 +172,11 @@ private theorem norm_galoisCharacterOnIdeal_le_one
     (χ : galoisCharacter K L) (𝔞 : Ideal (𝓞 K)) :
     ‖galoisCharacterOnIdeal K L χ 𝔞‖ ≤ 1 := by
   rw [galoisCharacterOnIdeal, norm_prod]
-  refine Finset.prod_le_one (fun i _ ↦ norm_nonneg _) (fun 𝔭 _ ↦ ?_)
+  refine Finset.prod_le_one₀ (fun i _ ↦ norm_nonneg _) (fun 𝔭 _ ↦ ?_)
   rw [norm_pow]
   by_cases h : UnramifiedIn K L 𝔭
-  · rw [if_pos h, norm_galoisCharacter_out, one_pow]
-  · rw [if_neg h, norm_zero]
+  · rw [ite_eq_left h, norm_galoisCharacter_out, one_pow]
+  · rw [ite_eq_right h, norm_zero]
     exact zero_pow_le_one _
 
 /-- Sharifi 7.1.18 (p. 141): Euler product for an abelian Galois
@@ -212,18 +212,18 @@ theorem exists_artinLSeries_eulerProduct_abelian
   have hsupp : Function.mulSupport f ⊆ Set.range g := by
     intro 𝔭 hmem
     simp only [Function.mem_mulSupport, hf] at hmem
-    haveI := 𝔭.2.1
+    have := 𝔭.2.1
     have hunr : UnramifiedIn K L 𝔭.1 := by
       by_contra hnr
       apply hmem
-      rw [hw, galoisCharacterOnIdeal_apply_prime K L χ 𝔭.1 𝔭.2.2, if_neg hnr, zero_mul, sub_zero,
+      rw [hw, galoisCharacterOnIdeal_apply_prime K L χ 𝔭.1 𝔭.2.2, ite_eq_right hnr, zero_mul, sub_zero,
         inv_one]
     exact ⟨⟨𝔭.1, 𝔭.2.1, hunr⟩, rfl⟩
   rw [← hg_inj.tprod_eq hsupp]
   refine tprod_congr fun 𝔭 ↦ ?_
   simp only [hf, hg, hw]
-  haveI := 𝔭.2.1
-  rw [galoisCharacterOnIdeal_apply_prime K L χ 𝔭.1 𝔭.2.2.ne_bot, if_pos 𝔭.2.2]
+  have := 𝔭.2.1
+  rw [galoisCharacterOnIdeal_apply_prime K L χ 𝔭.1 𝔭.2.2.ne_bot, ite_eq_left 𝔭.2.2]
 
 /-! ### Sub-lemmas for `dedekindZeta_local_factor_eq_product_artin_local`
 
@@ -350,7 +350,6 @@ private theorem cpow_neg_absNorm_eq_pow {a b : ℕ} (f : ℕ) (s : ℂ)
     (h : b = a ^ f) : ((b : ℂ)) ^ (-s) = ((a : ℂ) ^ (-s)) ^ f := by
   rw [h, Nat.cast_pow, ← Complex.natCast_cpow_natCast_mul, Complex.cpow_nat_mul]
 
-set_option backward.isDefEq.respectTransparency false in
 /-- Sharifi 7.1.16 (p. 141) local step: the local Euler factor at an
 unramified prime `𝔭` of `K` factors as a product over characters.
 Source quote (paraphrased identity): the local factor
@@ -365,12 +364,12 @@ theorem dedekindZeta_local_factor_eq_product_artin_local
         (1 - (χ (frobeniusClass K L 𝔭).out : ℂ) * (Ideal.absNorm 𝔭 : ℂ) ^ (-s))⁻¹ := by
   classical
   open scoped IsMulCommutative in
-  letI : CommGroup Gal(L/K) := inferInstance
+  let : CommGroup Gal(L/K) := inferInstance
   set σ : Gal(L/K) := (frobeniusClass K L 𝔭).out
   set Y : ℂ := (Ideal.absNorm 𝔭 : ℂ) ^ (-s) with hY
   set f : ℕ := orderOf σ with hf
-  haveI : Fintype Gal(L/K) := Fintype.ofFinite _
-  haveI : Fintype (Gal(L/K) →* ℂˣ) := Fintype.ofFinite _
+  have : Fintype Gal(L/K) := Fintype.ofFinite _
+  have : Fintype (Gal(L/K) →* ℂˣ) := Fintype.ofFinite _
   have hfpos : 0 < f := hf ▸ orderOf_pos_iff.mpr (isOfFinOrder_of_finite σ)
   have hcount : Nat.card {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥}
       = Nat.card Gal(L/K) / f := by
@@ -383,29 +382,29 @@ theorem dedekindZeta_local_factor_eq_product_artin_local
       = ((1 - Y ^ f) ^ (Nat.card Gal(L/K) / f))⁻¹ := by
     rw [tprod_fintype, Finset.prod_inv_distrib, prod_galoisCharacter_one_sub σ Y, hf]
   have hpbot : 𝔭 ≠ ⊥ := UnramifiedIn.ne_bot K L _hunr
-  haveI : 𝔭.IsMaximal := ‹𝔭.IsPrime›.isMaximal hpbot
-  haveI : Finite (𝔭.primesOver (𝓞 L)) := (IsDedekindDomain.primesOver_finite 𝔭 (𝓞 L)).to_subtype
-  haveI : Finite {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} :=
+  have : 𝔭.IsMaximal := ‹𝔭.IsPrime›.isMaximal hpbot
+  have : Finite (𝔭.primesOver (𝓞 L)) := (IsDedekindDomain.primesOver_finite 𝔭 (𝓞 L)).to_subtype
+  have : Finite {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} :=
     Finite.of_injective
       (fun 𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} ↦
         (⟨𝔓.1, 𝔓.2.1, 𝔓.2.2.1⟩ : 𝔭.primesOver (𝓞 L)))
-      fun _ _ hab ↦ Subtype.ext (by simpa using hab)
-  haveI : Fintype {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} := Fintype.ofFinite _
+      fun _ _ hab ↦ Subtype.ext (congrArg (fun q : 𝔭.primesOver (𝓞 L) ↦ q.1) hab)
+  have : Fintype {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥} := Fintype.ofFinite _
   have hterm : ∀ 𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver 𝔭 ∧ 𝔓 ≠ ⊥},
       (1 - (Ideal.absNorm 𝔓.1 : ℂ) ^ (-s))⁻¹ = (1 - Y ^ f)⁻¹ := by
     intro 𝔓
-    haveI := 𝔓.2.1
-    haveI hlo : 𝔓.1.LiesOver 𝔭 := 𝔓.2.2.1
-    have hdeg : (𝔓.1.under (𝓞 K)).inertiaDeg' 𝔓.1 = f := by
-      rw [Ideal.inertiaDeg'_algebraMap, hf]
+    have := 𝔓.2.1
+    have hlo : 𝔓.1.LiesOver 𝔭 := 𝔓.2.2.1
+    have : 𝔓.1.IsMaximal := 𝔓.2.1.isMaximal 𝔓.2.2.2
+    have : (𝔓.1.under (𝓞 K)).IsMaximal := hlo.over ▸ ‹𝔭.IsMaximal›
+    have hdeg : 𝔓.1.inertiaDeg (𝓞 K) = f := by
+      rw [Ideal.inertiaDeg_eq_of_isMaximal (𝔓.1.under (𝓞 K)) 𝔓.1, hf]
       exact finrank_residue_eq_orderOf K L σ (frobeniusClass K L 𝔭) (Quotient.out_eq _)
         𝔭 _hunr rfl 𝔓.1 hlo
-    haveI : 𝔓.1.LiesOver (𝔓.1.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓.1)
-    have hpubot : 𝔓.1.under (𝓞 K) ≠ ⊥ := hlo.over ▸ hpbot
-    haveI : (𝔓.1.under (𝓞 K)).IsPrime := hlo.over ▸ ‹𝔭.IsPrime›
+    have : 𝔓.1.LiesOver (𝔓.1.under (𝓞 K)) := Ideal.over_under (A := 𝓞 K) (P := 𝔓.1)
+    have : (𝔓.1.under (𝓞 K)).IsPrime := hlo.over ▸ ‹𝔭.IsPrime›
     have hnorm : Ideal.absNorm 𝔓.1 = Ideal.absNorm 𝔭 ^ f := by
-      rw [Ideal.absNorm_eq_pow_inertiaDeg'_of_liesOver 𝔓.1 (𝔓.1.under (𝓞 K)) inferInstance hpubot,
-        hdeg, ← hlo.over]
+      rw [← Ideal.absNorm_pow_inertiaDeg 𝔭 𝔓.1, hdeg]
     rw [cpow_neg_absNorm_eq_pow f s hnorm, hY]
   rw [tprod_congr hterm, tprod_fintype, Finset.prod_const, Finset.card_univ,
     ← Nat.card_eq_fintype_card, hcount, hRHS, Nat.card_eq_fintype_card, inv_pow]
@@ -460,7 +459,7 @@ open Classical in
     (K L : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] [IsGalois K L]
     [IsMulCommutative Gal(L/K)] (𝔭 : Ideal (𝓞 K)) [𝔭.IsPrime] (h𝔭 : 𝔭 ≠ ⊥) :
     frobeniusIdeal K L 𝔭 = (frobeniusClass K L 𝔭).out := by
-  letI : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
+  let : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
   rw [frobeniusIdeal, UniqueFactorizationMonoid.normalizedFactors_irreducible
     (Ideal.prime_of_isPrime h𝔭 ‹_›).irreducible, normalize_eq, Multiset.map_singleton,
     Multiset.prod_singleton]
@@ -470,7 +469,7 @@ theorem frobeniusIdeal_mul
     (K L : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] [IsGalois K L]
     [IsMulCommutative Gal(L/K)] {𝔞 𝔟 : Ideal (𝓞 K)} (h𝔞 : 𝔞 ≠ ⊥) (h𝔟 : 𝔟 ≠ ⊥) :
     frobeniusIdeal K L (𝔞 * 𝔟) = frobeniusIdeal K L 𝔞 * frobeniusIdeal K L 𝔟 := by
-  letI : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
+  let : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
   rw [frobeniusIdeal, frobeniusIdeal, frobeniusIdeal,
     UniqueFactorizationMonoid.normalizedFactors_mul h𝔞 h𝔟, Multiset.map_add, Multiset.prod_add]
 
@@ -479,7 +478,7 @@ theorem frobeniusIdeal_mul
     (K L : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] [IsGalois K L]
     [IsMulCommutative Gal(L/K)] :
     frobeniusIdeal K L ⊤ = 1 := by
-  letI : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
+  let : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
   rw [frobeniusIdeal, ← Ideal.one_eq_top, UniqueFactorizationMonoid.normalizedFactors_one,
     Multiset.map_zero, Multiset.prod_zero]
 
@@ -495,7 +494,7 @@ theorem galoisCharacterOnIdeal_eq_char_frobeniusIdeal
     [IsCyclotomicExtension {m} K L] (χ : galoisCharacter K L) {𝔞 : Ideal (𝓞 K)}
     (hU : ∀ 𝔭 ∈ UniqueFactorizationMonoid.normalizedFactors 𝔞, UnramifiedIn K L 𝔭) :
     galoisCharacterOnIdeal K L χ 𝔞 = (χ (frobeniusIdeal K L 𝔞) : ℂ) := by
-  letI : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
+  let : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
   have hfrob : (χ (frobeniusIdeal K L 𝔞) : ℂ) =
       ((UniqueFactorizationMonoid.normalizedFactors 𝔞).map
         (fun 𝔭 ↦ (χ (frobeniusClass K L 𝔭).out : ℂ))).prod := by
@@ -504,7 +503,7 @@ theorem galoisCharacterOnIdeal_eq_char_frobeniusIdeal
     rfl
   rw [galoisCharacterOnIdeal_eq_map_prod, hfrob]
   refine congrArg Multiset.prod (Multiset.map_congr rfl fun 𝔭 h𝔭 ↦ ?_)
-  rw [if_pos (hU 𝔭 h𝔭)]
+  rw [ite_eq_left (hU 𝔭 h𝔭)]
 
 open Classical in
 /-- If the ideal character `χ(𝔞)` is nonzero then every prime factor of `𝔞` is unramified in `L`
@@ -516,7 +515,7 @@ private theorem unramifiedIn_of_mem_normalizedFactors_of_galoisCharacterOnIdeal_
   by_contra hnr
   refine h ?_
   rw [galoisCharacterOnIdeal_eq_map_prod]
-  exact Multiset.prod_eq_zero (Multiset.mem_map.mpr ⟨𝔭, h𝔭, if_neg hnr⟩)
+  exact Multiset.prod_eq_zero (Multiset.mem_map.mpr ⟨𝔭, h𝔭, ite_eq_right hnr⟩)
 
 open Classical in
 /-- **Helper 1a (cardinality form) — value-fibre = unramified-supported Frobenius-value-fibre.** For
@@ -555,8 +554,8 @@ theorem charFibre_mem_range {G : Type*} [CommGroup G] [Finite G] (χ : G →* 
     (hζ : ζ ^ orderOf χ = 1) :
     ∃ g : G, χ g = ζ := by
   classical
-  haveI : NeZero (orderOf χ) := ⟨(orderOf_pos_iff.mpr (isOfFinOrder_of_finite χ)).ne'⟩
-  haveI : Finite (MonoidHom.range χ) :=
+  have : NeZero (orderOf χ) := ⟨(orderOf_pos_iff.mpr (isOfFinOrder_of_finite χ)).ne'⟩
+  have : Finite (MonoidHom.range χ) :=
     Finite.of_surjective χ.rangeRestrict χ.rangeRestrict_surjective
   have hpow : ∀ g : G, (χ g) ^ orderOf χ = 1 := fun g ↦ by
     rw [← MonoidHom.pow_apply, pow_orderOf_eq_one, MonoidHom.one_apply]
@@ -620,8 +619,8 @@ private theorem unramifiedIn_of_coprime_absNorm
     UnramifiedIn K L 𝔭 := by
   classical
   refine ⟨h𝔭, fun 𝔓 h𝔓max h𝔓lo ↦ ?_⟩
-  haveI := h𝔓lo
-  haveI : 𝔓.IsPrime := h𝔓max.isPrime
+  have := h𝔓lo
+  have : 𝔓.IsPrime := h𝔓max.isPrime
   rw [← not_dvd_differentIdeal_iff (A := 𝓞 K) (B := 𝓞 L)]
   intro hdvd
   obtain ⟨ζ, hζ⟩ := IsCyclotomicExtension.exists_isPrimitiveRoot K L
@@ -697,7 +696,7 @@ private theorem autToPow_frobeniusIdeal
       intro hpa hcop
       have hp' : p ≠ ⊥ := hp.ne_zero
       have ha' : a ≠ ⊥ := ha
-      haveI : p.IsPrime := Ideal.isPrime_of_prime hp
+      have : p.IsPrime := Ideal.isPrime_of_prime hp
       have hsplit : Ideal.absNorm (p * a) = Ideal.absNorm p * Ideal.absNorm a :=
         map_mul Ideal.absNorm p a
       have hcp : (Ideal.absNorm p).Coprime m :=
@@ -885,7 +884,7 @@ private theorem card_fibre_eq_card_good_fibre
     intro 𝔭 h𝔭
     rw [normalizedFactors_mul h0 h𝔟, Multiset.mem_add] at h𝔭
     rcases h𝔭 with h𝔭 | h𝔭
-    · haveI : 𝔭.IsPrime := Ideal.isPrime_of_prime (prime_of_normalized_factor _ h𝔭)
+    · have : 𝔭.IsPrime := Ideal.isPrime_of_prime (prime_of_normalized_factor _ h𝔭)
       exact unramifiedIn_of_coprime_absNorm K L m 𝔭
         (prime_of_normalized_factor _ h𝔭).ne_zero
         (coprime_absNorm_of_mem_factors_of_coprime K m hcop h𝔭)
@@ -921,18 +920,18 @@ omit [NumberField L] [FiniteDimensional K L] [IsMulCommutative Gal(L/K)] [NeZero
 /-- The bad-supported ideals of norm `≤ N` form a finite set: they are a subset of the (finitely
 many) ideals of norm `≤ N`. -/
 private theorem finite_isBadPart (N : ℕ) : {𝔟 : Ideal (𝓞 K) | IsBadPart K L m N 𝔟}.Finite :=
-  (Ideal.finite_setOf_absNorm_le (S := 𝓞 K) N).subset fun _ h𝔟 ↦ h𝔟.2.2
+  (Ideal.finite_setOfPred_absNorm_le (S := 𝓞 K) N).subset fun _ h𝔟 ↦ h𝔟.2.2
 
 open UniqueFactorizationMonoid in
 /-- The L2 fibre subtype at `g` is finite (subset of all ideals of norm `≤ N`). -/
 private instance finite_L2 (g : Gal(L/K)) (N : ℕ) :
     Finite {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ N ∧
       (∀ 𝔭 ∈ normalizedFactors 𝔞, UnramifiedIn K L 𝔭) ∧ frobeniusIdeal K L 𝔞 = g} := by
-  haveI : Finite {I : Ideal (𝓞 K) // Ideal.absNorm I ≤ N} :=
-    (Ideal.finite_setOf_absNorm_le (S := 𝓞 K) N).to_subtype
+  have : Finite {I : Ideal (𝓞 K) // Ideal.absNorm I ≤ N} :=
+    (Ideal.finite_setOfPred_absNorm_le (S := 𝓞 K) N).to_subtype
   exact Finite.of_injective (β := {I : Ideal (𝓞 K) // Ideal.absNorm I ≤ N})
     (fun 𝔞 ↦ ⟨𝔞.1, 𝔞.2.2.1⟩)
-    (fun _ _ hab ↦ Subtype.ext (by simpa using hab))
+    (fun _ _ hab ↦ Subtype.ext (congrArg (fun q : {I : Ideal (𝓞 K) // Ideal.absNorm I ≤ N} ↦ q.1) hab))
 
 omit [FiniteDimensional K L] [NeZero m] [IsCyclotomicExtension {m} K L] in
 open UniqueFactorizationMonoid in
@@ -1054,7 +1053,7 @@ private theorem autToPow_range_le_realizedResidues
   have hHtop : H = ⊤ := by
     refine subgroup_eq_top_of_forall_frobenius_mem_of_coprime K L m H
       (fun 𝔭 h𝔭p h𝔭ne h𝔭unr h𝔭cop ↦ ?_)
-    haveI := h𝔭p
+    have := h𝔭p
     rw [hH, Subgroup.mem_comap, autToPow_frobeniusClass_out K L m hζ 𝔭 h𝔭unr h𝔭cop]
     exact ⟨⟨𝔭, mem_nonZeroDivisors_of_ne_zero h𝔭ne⟩, by rw [ZMod.coe_unitOfCoprime]⟩
   intro a ha
@@ -1203,7 +1202,7 @@ private theorem sum_rpow_le_euler_prod (K : Type*) [Field K] [NumberField K]
         rw [Finset.prod_sum P (fun _ ↦ Finset.range (Kn + 1))
           (fun 𝔭 k ↦ (((Ideal.absNorm 𝔭 : ℝ)) ^ e) ^ k)]
     _ ≤ ∏ 𝔭 ∈ P, (1 - ((Ideal.absNorm 𝔭 : ℝ)) ^ e)⁻¹ := by
-        refine Finset.prod_le_prod
+        refine Finset.prod_le_prod₀
           (fun 𝔭 h𝔭 ↦ Finset.sum_nonneg fun k _ ↦ pow_nonneg (hx0 𝔭 h𝔭) k) (fun 𝔭 h𝔭 ↦ ?_)
         have h1x : 0 < 1 - ((Ideal.absNorm 𝔭 : ℝ)) ^ e := by have := hxlt 𝔭 h𝔭; linarith
         have hkey := geom_sum_mul (((Ideal.absNorm 𝔭 : ℝ)) ^ e) (Kn + 1)
@@ -1658,7 +1657,7 @@ private theorem coprime_absNorm_of_unramified_of_finrank_eq_one
   by_contra hncop
   obtain ⟨p, hpm, hpmem𝔭⟩ := exists_primeFactor_natCast_mem_of_not_coprime K m 𝔭 h𝔭 hncop
   have hp : p.Prime := (Nat.mem_primeFactors.mp hpm).1
-  haveI : Fact p.Prime := ⟨hp⟩
+  have : Fact p.Prime := ⟨hp⟩
   have hpdvd : p ∣ m := Nat.dvd_of_mem_primeFactors hpm
   have hm0 : m ≠ 0 := (Nat.mem_primeFactors.mp hpm).2.2
   set v := m.factorization p with hv
@@ -1684,11 +1683,11 @@ private theorem coprime_absNorm_of_unramified_of_finrank_eq_one
   have hφ2 : 2 ≤ p ^ k * (p - 1) := two_le_pow_mul_pred hp hbad
   have hspan𝔭 : Ideal.span {(p : 𝓞 K)} = 𝔭 :=
     span_singleton_natCast_eq_of_finrank_eq_one K hd1 p hp 𝔭 hpmem𝔭
-  haveI : 𝔭.IsMaximal := ‹𝔭.IsPrime›.isMaximal h𝔭
+  have : 𝔭.IsMaximal := ‹𝔭.IsPrime›.isMaximal h𝔭
   obtain ⟨𝔓, h𝔓max, h𝔓lo⟩ :=
     Ideal.exists_maximal_ideal_liesOver_of_isIntegral (R := 𝓞 K) (S := 𝓞 L) 𝔭
-  haveI : 𝔓.IsPrime := h𝔓max.isPrime
-  haveI := h𝔓lo
+  have : 𝔓.IsPrime := h𝔓max.isPrime
+  have := h𝔓lo
   have hnotdvd : ¬ 𝔓 ∣ differentIdeal (𝓞 K) (𝓞 L) := by
     rw [not_dvd_differentIdeal_iff (A := 𝓞 K) (B := 𝓞 L)]
     exact hunr.2 𝔓 h𝔓max h𝔓lo
@@ -1747,7 +1746,7 @@ private theorem card_fibre_bound_eq_one {ζ : L} (hζ : IsPrimitiveRoot ζ m)
         rw [Ideal.one_eq_top] at this; exact htop this
       obtain ⟨𝔭, h𝔭⟩ := Multiset.exists_mem_of_ne_zero hfac0
       have hprime := prime_of_normalized_factor 𝔭 h𝔭
-      haveI : 𝔭.IsPrime := Ideal.isPrime_of_prime hprime
+      have : 𝔭.IsPrime := Ideal.isPrime_of_prime hprime
       exact (hfac 𝔭 h𝔭).2 (coprime_absNorm_of_unramified_of_finrank_eq_one K L m hd1 𝔭
         hprime.ne_zero (hfac 𝔭 h𝔭).1 hm)
   rw [card_L2_eq_sum_residue K L m hζ g N, hbadtop, Finset.sum_singleton,
@@ -1819,9 +1818,9 @@ private theorem card_unramifiedSupported_frobeniusValueFibre_eq_sum
             (∀ 𝔭 ∈ UniqueFactorizationMonoid.normalizedFactors 𝔞, UnramifiedIn K L 𝔭) ∧
               frobeniusIdeal K L 𝔞 = g.1} := by
   classical
-  haveI hfinN : Finite {𝔞 : Ideal (𝓞 K) // Ideal.absNorm 𝔞 ≤ N} :=
-    (Ideal.finite_setOf_absNorm_le (S := 𝓞 K) N).to_subtype
-  haveI hfin : ∀ g : {g : Gal(L/K) // (χ g : ℂ) = ζ},
+  have hfinN : Finite {𝔞 : Ideal (𝓞 K) // Ideal.absNorm 𝔞 ≤ N} :=
+    (Ideal.finite_setOfPred_absNorm_le (S := 𝓞 K) N).to_subtype
+  have hfin : ∀ g : {g : Gal(L/K) // (χ g : ℂ) = ζ},
       Finite {𝔞 : Ideal (𝓞 K) //
         𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ N ∧
           (∀ 𝔭 ∈ UniqueFactorizationMonoid.normalizedFactors 𝔞, UnramifiedIn K L 𝔭) ∧
@@ -1916,7 +1915,7 @@ theorem exists_card_galoisCharacterOnIdeal_eq_const_mul_add_pow
       rw [← hζuval]
       exact ⟨fun h ↦ Units.ext h, fun h ↦ congrArg Units.val h⟩
     rw [heq]
-    letI : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
+    let : CommGroup Gal(L/K) := { mul_comm := mul_comm' }
     exact card_charFibre_eq_card_ker χ ζu hζun
   have hcardℝ : (Fintype.card {g : Gal(L/K) // (χ g : ℂ) = ζ} : ℝ) = (κ₀ : ℝ) := by
     rw [← Nat.card_eq_fintype_card, hSκ₀]
@@ -1985,7 +1984,7 @@ private theorem galoisCharacterOnIdeal_mem_insert_zero_nthRootsFinset
   · push Not at hU
     obtain ⟨𝔭, h𝔭, hram⟩ := hU
     rw [galoisCharacterOnIdeal_eq_map_prod,
-      Multiset.prod_eq_zero (Multiset.mem_map.mpr ⟨𝔭, h𝔭, if_neg hram⟩)]
+      Multiset.prod_eq_zero (Multiset.mem_map.mpr ⟨𝔭, h𝔭, ite_eq_right hram⟩)]
     exact Finset.mem_insert_self _ _
 
 private theorem sum_galoisCharacterOnIdeal_eq_sum_card_sub_mul
@@ -2044,9 +2043,9 @@ many) ideals of norm `≤ N`. -/
 private theorem finite_nonzeroIdeal_absNorm_le (K : Type*) [Field K] [NumberField K] (N : ℕ) :
     Finite {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ N} :=
   haveI : Finite {𝔞 : Ideal (𝓞 K) // Ideal.absNorm 𝔞 ≤ N} :=
-    (Ideal.finite_setOf_absNorm_le (S := 𝓞 K) N).to_subtype
+    (Ideal.finite_setOfPred_absNorm_le (S := 𝓞 K) N).to_subtype
   Finite.of_injective (fun a ↦ (⟨a.1, a.2.2⟩ : {𝔞 : Ideal (𝓞 K) // Ideal.absNorm 𝔞 ≤ N}))
-    fun _ _ hab ↦ Subtype.ext (by simpa using hab)
+    fun _ _ hab ↦ Subtype.ext (congrArg (fun q : {I : Ideal (𝓞 K) // Ideal.absNorm I ≤ N} ↦ q.1) hab)
 
 /-- Sharifi 7.1.19 step 1 (p. 142): geometry-of-numbers bound. The
 partial-sum character sum `Σ_{N𝔞≤N} χ(𝔞)` (with `χ(𝔞) = galoisCharacterOnIdeal K L χ 𝔞` the
@@ -2067,7 +2066,7 @@ theorem character_sum_geometry_of_numbers_bound
   refine ⟨(orderOf χ : ℝ) * C', fun N ↦ ?_⟩
   have hC' : 0 ≤ C' := (abs_nonneg _).trans (by simpa using hcount 1 (one_pow _) 1 le_rfl)
   rcases Nat.eq_zero_or_pos N with rfl | hN1
-  · haveI : IsEmpty {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ 0} :=
+  · have : IsEmpty {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ 0} :=
       ⟨fun 𝔞 ↦ 𝔞.2.1 (Ideal.absNorm_eq_zero_iff.mp (Nat.le_zero.mp 𝔞.2.2))⟩
     rw [tsum_empty, norm_zero]
     positivity
@@ -2079,8 +2078,8 @@ theorem character_sum_geometry_of_numbers_bound
   set R : Finset ℂ := Polynomial.nthRootsFinset (orderOf χ) (1 : ℂ) with hR
   have hmemR : ∀ {z : ℂ}, z ∈ R ↔ z ^ orderOf χ = 1 := fun {z} ↦
     Polynomial.mem_nthRootsFinset (Nat.pos_of_ne_zero hord0) 1
-  haveI := finite_nonzeroIdeal_absNorm_le K N
-  haveI := Fintype.ofFinite {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ N}
+  have := finite_nonzeroIdeal_absNorm_le K N
+  have := Fintype.ofFinite {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ N}
   rw [tsum_fintype, sum_galoisCharacterOnIdeal_eq_sum_card_sub_mul K L m χ hord2 C₀ N]
   calc ‖∑ v ∈ R, (((Nat.card {𝔞 : Ideal (𝓞 K) //
         𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ N ∧ galoisCharacterOnIdeal K L χ 𝔞 = v} : ℝ)
@@ -2111,7 +2110,7 @@ private theorem finite_nonzeroIdeal_absNorm_eq
     (K : Type*) [Field K] [NumberField K] (n : ℕ) :
     Finite {𝔞 : NonzeroIdeal K // Ideal.absNorm 𝔞.1 = n} :=
   Set.Finite.to_subtype <| Set.Finite.of_finite_image (f := fun I : NonzeroIdeal K ↦ I.1)
-    ((Ideal.finite_setOf_absNorm_eq (S := 𝓞 K) n).subset (by rintro _ ⟨⟨I, _⟩, rfl, rfl⟩; rfl))
+    ((Ideal.finite_setOfPred_absNorm_eq (S := 𝓞 K) n).subset (by rintro _ ⟨⟨I, _⟩, rfl, rfl⟩; rfl))
     (fun _ _ _ _ ↦ Subtype.ext)
 
 /-- The `0`-th coefficient vanishes: no nonzero ideal has norm `0`, so the fibre is empty. -/
@@ -2128,8 +2127,8 @@ private theorem norm_galoisCharacterCoeff_le
     (K L : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] [IsGalois K L]
     (χ : galoisCharacter K L) (n : ℕ) :
     ‖galoisCharacterCoeff K L χ n‖ ≤ (idealNormMultiplicity K n : ℝ) := by
-  haveI := finite_nonzeroIdeal_absNorm_eq K n
-  haveI := Fintype.ofFinite {𝔞 : NonzeroIdeal K // Ideal.absNorm 𝔞.1 = n}
+  have := finite_nonzeroIdeal_absNorm_eq K n
+  have := Fintype.ofFinite {𝔞 : NonzeroIdeal K // Ideal.absNorm 𝔞.1 = n}
   calc ‖galoisCharacterCoeff K L χ n‖
       ≤ ∑' 𝔞 : {𝔞 : NonzeroIdeal K // Ideal.absNorm 𝔞.1 = n},
           ‖galoisCharacterOnIdeal K L χ 𝔞.1.1‖ :=
@@ -2154,8 +2153,8 @@ private theorem sum_galoisCharacterCoeff_eq_tsum_absNorm_le
       ∑' 𝔞 : {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ n},
         galoisCharacterOnIdeal K L χ 𝔞.1 := by
   classical
-  haveI := finite_nonzeroIdeal_absNorm_le K n
-  haveI := Fintype.ofFinite {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ n}
+  have := finite_nonzeroIdeal_absNorm_le K n
+  have := Fintype.ofFinite {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ n}
   rw [tsum_fintype, ← Finset.sum_fiberwise_of_maps_to (t := Finset.Icc 1 n)
       (g := fun 𝔞 : {𝔞 : Ideal (𝓞 K) // 𝔞 ≠ ⊥ ∧ Ideal.absNorm 𝔞 ≤ n} ↦ Ideal.absNorm 𝔞.1)
       (fun 𝔞 _ ↦ Finset.mem_Icc.mpr
@@ -2163,8 +2162,8 @@ private theorem sum_galoisCharacterCoeff_eq_tsum_absNorm_le
       (fun 𝔞 ↦ galoisCharacterOnIdeal K L χ 𝔞.1)]
   refine Finset.sum_congr rfl fun k hk ↦ ?_
   rw [galoisCharacterCoeff, ← Finset.sum_subtype_eq_sum_filter, Finset.subtype_univ]
-  haveI := finite_nonzeroIdeal_absNorm_eq K k
-  haveI := Fintype.ofFinite {𝔞 : NonzeroIdeal K // Ideal.absNorm 𝔞.1 = k}
+  have := finite_nonzeroIdeal_absNorm_eq K k
+  have := Fintype.ofFinite {𝔞 : NonzeroIdeal K // Ideal.absNorm 𝔞.1 = k}
   rw [tsum_fintype]
   exact Fintype.sum_equiv
     { toFun := fun ⟨⟨𝔞, h𝔞ne⟩, hnorm⟩ ↦
@@ -2260,8 +2259,8 @@ private theorem setIntegral_Ioi_one_mul_cpow_eq_mellin (S : ℝ → ℂ) (hS : 
       ae_iff.mpr (by simp : volume {x : ℝ | ¬x ≠ 1} = 0)] with t ht _
     rw [indicator_apply]
     by_cases h1 : t ∈ Ioi (1 : ℝ)
-    · rw [if_pos h1]
-    · rw [if_neg h1, hS t (lt_of_le_of_ne (not_lt.mp (by simpa using h1)) ht), smul_zero]
+    · rw [ite_eq_left h1]
+    · rw [ite_eq_right h1, hS t (lt_of_le_of_ne (not_lt.mp (by simpa using h1)) ht), smul_zero]
 
 open Filter Topology Set MeasureTheory Asymptotics in
 /-- Sharifi 7.1.19 step 1b (p. 142) — analytic extension of `L(χ,·)`.
@@ -2467,7 +2466,7 @@ private theorem norm_one_sub_inv_sub_one_le {y : ℂ} (hy : ‖y‖ ≤ 1 / 2) :
 
 /-- A nonzero prime ideal `𝔭` of a number ring has `2 ≤ N𝔭`: its norm is neither `0` (only `⊥` has
 norm `0`) nor `1` (only `⊤` has norm `1`). -/
-private theorem two_le_absNorm {R : Type*} [CommRing R] [IsDedekindDomain R]
+private theorem two_le_absNorm {R : Type*} [CommRing R] [IsDedekindDomain R] [Infinite R]
     [Module.Free ℤ R] [Module.Finite ℤ R] {𝔭 : Ideal R} (hp : 𝔭.IsPrime) (hb : 𝔭 ≠ ⊥) :
     2 ≤ Ideal.absNorm 𝔭 := by
   have hne0 : Ideal.absNorm 𝔭 ≠ 0 := fun h ↦ hb (Ideal.absNorm_eq_zero_iff.mp h)
@@ -2476,7 +2475,7 @@ private theorem two_le_absNorm {R : Type*} [CommRing R] [IsDedekindDomain R]
 
 /-- For a nonzero prime `𝔭` of a number ring and `Re s > 1`, `‖N𝔭^{-s}‖ ≤ 1/2` (since `N𝔭 ≥ 2`,
 `Re s > 1`). The bound that lets the Euler factors enter `norm_one_sub_inv_sub_one_le`. -/
-private theorem norm_absNorm_cpow_neg_le_half {R : Type*} [CommRing R] [IsDedekindDomain R]
+private theorem norm_absNorm_cpow_neg_le_half {R : Type*} [CommRing R] [IsDedekindDomain R] [Infinite R]
     [Module.Free ℤ R] [Module.Finite ℤ R] {s : ℂ} (hs : 1 < s.re)
     (𝔭 : {𝔭 : Ideal R // 𝔭.IsPrime ∧ 𝔭 ≠ ⊥}) :
     ‖(Ideal.absNorm 𝔭.1 : ℂ) ^ (-s)‖ ≤ 1 / 2 := by
@@ -2581,7 +2580,7 @@ private def underUP
     (K L : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] [IsGalois K L]
     (𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧ UnramifiedIn K L (𝔓.under (𝓞 K))}) :
     {𝔭 : Ideal (𝓞 K) // 𝔭.IsPrime ∧ UnramifiedIn K L 𝔭} :=
-  ⟨𝔓.1.under (𝓞 K), by haveI := 𝔓.2.1; exact inferInstance, 𝔓.2.2.2⟩
+  ⟨𝔓.1.under (𝓞 K), by have := 𝔓.2.1; exact inferInstance, 𝔓.2.2.2⟩
 
 @[simp] private theorem underUP_val
     (K L : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] [IsGalois K L]
@@ -2600,8 +2599,8 @@ private def fiberUnderEquiv
   toFun 𝔓 := ⟨𝔓.1.1, 𝔓.1.2.1, ⟨by
     have h := congrArg Subtype.val 𝔓.2; rw [underUP_val] at h; rw [← h]⟩, 𝔓.1.2.2.1⟩
   invFun 𝔔 := ⟨⟨𝔔.1, 𝔔.2.1, 𝔔.2.2.2, by
-      haveI := 𝔔.2.1; haveI := 𝔔.2.2.1; rw [← 𝔔.2.2.1.over]; exact c.2.2⟩, by
-    haveI := 𝔔.2.1; haveI := 𝔔.2.2.1
+      have := 𝔔.2.1; have := 𝔔.2.2.1; rw [← 𝔔.2.2.1.over]; exact c.2.2⟩, by
+    have := 𝔔.2.1; have := 𝔔.2.2.1
     exact Subtype.ext (by rw [underUP_val]; exact 𝔔.2.2.1.over.symm)⟩
   left_inv 𝔓 := by ext; rfl
   right_inv 𝔔 := by ext; rfl
@@ -2632,7 +2631,6 @@ private def ramifiedFlattenEquiv
   left_inv _ := rfl
   right_inv _ := rfl
 
-set_option backward.isDefEq.respectTransparency false in
 /-- The unramified part of the prime-ideal Euler product equals `∏_χ L_χ`. Regroup the unramified
 `L`-primes fibrewise over the `K`-prime below them (`Equiv.sigmaFiberEquiv` +
 `Multipliable.tprod_sigma`); each fibre product is `∏_χ (1 - χ(σ_𝔭) N𝔭^{-s})^{-1}`
@@ -2672,16 +2670,16 @@ private theorem tprod_unramified_eq_prod_artinDirichletSeries
       HasProd (fun 𝔓 : {𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧
           UnramifiedIn K L (𝔓.under (𝓞 K))} // underUP K L 𝔓 = c} ↦ F 𝔓.1.1) (G c) := by
     intro c
-    haveI : c.1.IsPrime := c.2.1
-    haveI : c.1.IsMaximal := c.2.1.isMaximal (UnramifiedIn.ne_bot K L c.2.2)
-    haveI : Finite (c.1.primesOver (𝓞 L)) :=
+    have : c.1.IsPrime := c.2.1
+    have : c.1.IsMaximal := c.2.1.isMaximal (UnramifiedIn.ne_bot K L c.2.2)
+    have : Finite (c.1.primesOver (𝓞 L)) :=
       (IsDedekindDomain.primesOver_finite c.1 (𝓞 L)).to_subtype
-    haveI : Finite {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver c.1 ∧ 𝔓 ≠ ⊥} :=
+    have : Finite {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver c.1 ∧ 𝔓 ≠ ⊥} :=
       Finite.of_injective
         (fun 𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓.LiesOver c.1 ∧ 𝔓 ≠ ⊥} ↦
           (⟨𝔓.1, 𝔓.2.1, 𝔓.2.2.1⟩ : c.1.primesOver (𝓞 L)))
-        (fun _ _ hab ↦ Subtype.ext (by simpa using hab))
-    haveI : Finite {𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧
+        (fun _ _ hab ↦ Subtype.ext (congrArg (fun q : c.1.primesOver (𝓞 L) ↦ q.1) hab))
+    have : Finite {𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧
         UnramifiedIn K L (𝔓.under (𝓞 K))} // underUP K L 𝔓 = c} :=
       Finite.of_equiv _ (fiberUnderEquiv K L c).symm
     have hval : (∏' 𝔓 : {𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧
@@ -2786,12 +2784,12 @@ private instance finite_ramifiedAbove
     (K L : Type*) [Field K] [NumberField K] [Field L] [NumberField L] [Algebra K L] [IsGalois K L] :
     Finite {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧ ¬ UnramifiedIn K L (𝔓.under (𝓞 K))} := by
   classical
-  haveI : Finite {𝔭 : Ideal (𝓞 K) // 𝔭.IsPrime ∧ 𝔭 ≠ ⊥ ∧ ¬ UnramifiedIn K L 𝔭} :=
+  have : Finite {𝔭 : Ideal (𝓞 K) // 𝔭.IsPrime ∧ 𝔭 ≠ ⊥ ∧ ¬ UnramifiedIn K L 𝔭} :=
     (finite_ramifiedIn K L).to_subtype
-  haveI : ∀ 𝔭 : {𝔭 : Ideal (𝓞 K) // 𝔭.IsPrime ∧ 𝔭 ≠ ⊥ ∧ ¬ UnramifiedIn K L 𝔭},
+  have : ∀ 𝔭 : {𝔭 : Ideal (𝓞 K) // 𝔭.IsPrime ∧ 𝔭 ≠ ⊥ ∧ ¬ UnramifiedIn K L 𝔭},
       Finite (𝔭.1.primesOver (𝓞 L)) := fun 𝔭 ↦ by
-    haveI : 𝔭.1.IsPrime := 𝔭.2.1
-    haveI : 𝔭.1.IsMaximal := 𝔭.2.1.isMaximal 𝔭.2.2.1
+    have : 𝔭.1.IsPrime := 𝔭.2.1
+    have : 𝔭.1.IsMaximal := 𝔭.2.1.isMaximal 𝔭.2.2.1
     exact (IsDedekindDomain.primesOver_finite 𝔭.1 (𝓞 L)).to_subtype
   refine Finite.of_injective
     (fun 𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧ ¬ UnramifiedIn K L (𝔓.under (𝓞 K))} ↦
@@ -2834,7 +2832,7 @@ private theorem log_norm_ramified_factor_bounded
           ¬ UnramifiedIn K L (𝔓.under (𝓞 K))}, (1 - (Ideal.absNorm 𝔓.1 : ℂ) ^ (-(s : ℂ)))⁻¹‖| ≤
         C := by
   classical
-  haveI : Fintype {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧
+  have : Fintype {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧
       ¬ UnramifiedIn K L (𝔓.under (𝓞 K))} := Fintype.ofFinite _
   set R : ℝ → ℂ := fun s ↦ ∏ 𝔓 : {𝔓 : Ideal (𝓞 L) // 𝔓.IsPrime ∧ 𝔓 ≠ ⊥ ∧
       ¬ UnramifiedIn K L (𝔓.under (𝓞 K))}, (1 - (Ideal.absNorm 𝔓.1 : ℂ) ^ (-(s : ℂ)))⁻¹ with hR
@@ -2932,7 +2930,7 @@ private theorem analyticAt_one_of_analyticOn_finrankDomain (K : Type*) [Field K]
     have hdpos : (0 : ℝ) < (Module.finrank ℚ K : ℝ)⁻¹ := by
       have : 0 < Module.finrank ℚ K := Module.finrank_pos
       positivity
-    simp only [Set.mem_setOf_eq, Complex.one_re]; linarith
+    simp only [Set.mem_ofPred_eq, Complex.one_re]; linarith
   exact hLf.analyticAt ((isOpen_lt continuous_const Complex.continuous_re).mem_nhds hmem)
 
 open Filter Topology Set in
@@ -3034,8 +3032,8 @@ private theorem sum_ite_pole_zero_cancel
       (if χ' = 1 then a else if χ' = χ then -a else 0) =
         (if χ' = 1 then a else 0) + (if χ' = χ then -a else 0) := fun χ' ↦ by
     by_cases h1 : χ' = 1
-    · rw [if_pos h1, if_pos h1, if_neg (h1 ▸ Ne.symm hχ), add_zero]
-    · rw [if_neg h1, if_neg h1]; by_cases hc : χ' = χ <;> simp [hc]
+    · rw [ite_eq_left h1, ite_eq_left h1, ite_eq_right (h1 ▸ Ne.symm hχ), add_zero]
+    · rw [ite_eq_right h1, ite_eq_right h1]; by_cases hc : χ' = χ <;> simp [hc]
   rw [Finset.sum_congr rfl fun χ' _ ↦ hsplit χ', Finset.sum_add_distrib,
     Finset.sum_ite_eq' Finset.univ (1 : galoisCharacter K L), Finset.sum_ite_eq' Finset.univ χ]
   simp
@@ -3061,14 +3059,14 @@ private theorem log_norm_artinDirichletSeries_le_pole_zero_ite
   by_cases h1 : χ' = 1
   · subst h1
     obtain ⟨C1, hC1⟩ := log_norm_artinDirichletSeries_one_le K L
-    exact ⟨C1, by filter_upwards [hC1] with s hs; rwa [if_pos rfl]⟩
+    exact ⟨C1, by filter_upwards [hC1] with s hs; rwa [ite_eq_left rfl]⟩
   · by_cases hc : χ' = χ
     · subst hc
-      exact ⟨Cχ, by filter_upwards [hCχ] with s hs; rwa [if_neg h1, if_pos rfl]⟩
+      exact ⟨Cχ, by filter_upwards [hCχ] with s hs; rwa [ite_eq_right h1, ite_eq_left rfl]⟩
     · obtain ⟨C, hC⟩ := artinDirichletSeries_norm_le_of_ne_one K L m hm χ' h1
       refine ⟨Real.log (max C 1), ?_⟩
       filter_upwards [hC] with s hs
-      simp only [if_neg h1, if_neg hc, zero_add]
+      simp only [ite_eq_right h1, ite_eq_right hc, zero_add]
       have hmax1 : (1 : ℝ) ≤ max C 1 := le_max_right _ _
       rcases le_total ‖artinDirichletSeries K L χ' (s : ℂ)‖ 0 with h0 | h0
       · have hz : ‖artinDirichletSeries K L χ' (s : ℂ)‖ = 0 := le_antisymm h0 (norm_nonneg _)
diff --git a/projects/DedekindResidue/DedekindResidue/AuxiliaryFunction.lean b/projects/DedekindResidue/DedekindResidue/AuxiliaryFunction.lean
index 8e22b1de6..f9545f056 100644
--- a/projects/DedekindResidue/DedekindResidue/AuxiliaryFunction.lean
+++ b/projects/DedekindResidue/DedekindResidue/AuxiliaryFunction.lean
@@ -44,7 +44,7 @@ noncomputable def auxF (s : ℂ) (X t : ℝ) : ℂ :=
 
 /-- On the plateau `|t| ≤ log X`, `F_{s,X}(t) = 1` — eq. (11). -/
 theorem auxF_of_le (s : ℂ) (X : ℝ) {t : ℝ} (h : |t| ≤ Real.log X) : auxF s X t = 1 := by
-  unfold auxF; exact if_pos h
+  unfold auxF; exact ite_eq_left h
 
 /-- `F_{s,X}(0) = 1` whenever `X ≥ 1` (so `log X ≥ 0`, putting `0` on the plateau). -/
 theorem auxF_zero (s : ℂ) {X : ℝ} (hX : 1 ≤ X) : auxF s X 0 = 1 :=
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/AnalyticControl.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/AnalyticControl.lean
index 5c23f3c90..8c56d23e6 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/AnalyticControl.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/AnalyticControl.lean
@@ -228,7 +228,7 @@ theorem exists_re_norm_dedekindZeta_ge_half :
     · simp
     rcases eq_or_ne n 0 with rfl | hn0
     · simp
-    rw [if_neg hn1, LSeries.norm_term_eq, LSeries.norm_term_eq, if_neg hn0, if_neg hn0,
+    rw [ite_eq_right hn1, LSeries.norm_term_eq, LSeries.norm_term_eq, ite_eq_right hn0, ite_eq_right hn0,
       show ((2:ℂ)).re = (2:ℝ) by norm_num]
     have hn2 : (2:ℝ) ≤ (n:ℝ) := by exact_mod_cast (by omega : 2 ≤ n)
     have hnpos : (0:ℝ) < n := by linarith
@@ -259,7 +259,7 @@ theorem exists_re_norm_dedekindZeta_ge_half :
     refine Summable.of_nonneg_of_le (fun n => norm_nonneg _) (fun n => ?_) hsummS.norm
     rcases eq_or_ne n 1 with rfl | hn1
     · simp
-    · rw [if_neg hn1]
+    · rw [ite_eq_right hn1]
   have hcompar : Summable (fun n : ℕ => (1/2:ℝ)^m * ‖LSeries.term f 2 n‖) :=
     (hsum2.norm).mul_left _
   have hRbound : ‖∑' n : ℕ, ite (n = 1) 0 (LSeries.term f s n)‖ ≤ T * (1/2)^m := by
@@ -463,7 +463,7 @@ theorem norm_dedekindZeta_le_of_two_le_re {s : ℂ} (hs : 2 ≤ s.re) :
     intro n
     rcases eq_or_ne n 0 with rfl | hn0
     · simp
-    rw [LSeries.norm_term_eq, LSeries.norm_term_eq, if_neg hn0, if_neg hn0,
+    rw [LSeries.norm_term_eq, LSeries.norm_term_eq, ite_eq_right hn0, ite_eq_right hn0,
       show ((2:ℂ)).re = (2:ℝ) by norm_num]
     have hn1 : (1:ℝ) ≤ (n:ℝ) := by exact_mod_cast Nat.one_le_iff_ne_zero.mpr hn0
     gcongr
@@ -602,11 +602,10 @@ theorem one_add_abs_im_le_two_norm_sub_four {z : ℂ} (_h1 : -1 ≤ z.re) (h2 :
     simpa using h
   linarith
 
-/-- The comparator sine-exponent: `n_K = r₁ + 2r₂`. -/
+
 noncomputable def gammaExponent : ℕ := nrRealPlaces K + 2 * nrComplexPlaces K
 
-/-- Right boundary bound for the comparator
-`G = H⁴·sin(πz)^{n_K}/(z-4)^{4n_K+8}` on `Re z = 2`. -/
+
 theorem comparator_bound_right :
     ∃ M : ℝ, 0 < M ∧ ∀ t : ℝ,
       ‖completedDedekindZetaEntire K (((2:ℝ) : ℂ) + (t : ℂ) * Complex.I)‖^4
@@ -715,8 +714,7 @@ theorem comparator_bound_right :
           + (B * 25)^4 * Real.exp ((n:ℝ) * (π * 2)) / 2^(4*n+8) :=
           le_add_of_nonneg_left (by positivity)
 
-/-- Left boundary bound for the comparator on `Re z = -1`, by reflecting to the right
-boundary through the functional equation. -/
+
 theorem comparator_bound_left :
     ∃ M : ℝ, 0 < M ∧ ∀ t : ℝ,
       ‖completedDedekindZetaEntire K (((-1:ℝ) : ℂ) + (t : ℂ) * Complex.I)‖^4
@@ -794,10 +792,7 @@ theorem comparator_bound_left :
           (by positivity) (pow_le_pow_left₀ (norm_nonneg _) hden _)
     _ ≤ M := hM (-t)
 
-/-- Sub-double-exponential growth of the AC-A3 comparator
-`completedDedekindZetaEntire K ^ 4 * sin (π z) ^ n / (z - 4) ^ (4 n + 8)` on the strip
-`-1 ≤ Re ≤ 2`, from the polynomial strip bound `‖H‖ ≤ B (1 + ‖z‖)²`. Supplies the
-growth hypothesis of the Phragmén–Lindelöf step in `comparator_bound_strip`. -/
+
 private theorem comparator_isBigO_growth {B : ℝ} (hB0 : 0 ≤ B)
     (hB : ∀ w : ℂ, -1 ≤ w.re → w.re ≤ 2 →
         ‖completedDedekindZetaEntire K w‖ ≤ B * (1 + ‖w‖) ^ 2) :
@@ -904,8 +899,7 @@ private theorem comparator_isBigO_growth {B : ℝ} (hB0 : 0 ≤ B)
     nlinarith
   exact mul_le_mul_of_nonneg_left hexp_le (by positivity)
 
-/-- **The comparator is bounded on the whole strip** `-1 ≤ Re z ≤ 2`
-(Phragmén–Lindelöf between the two boundary bounds). -/
+
 theorem comparator_bound_strip :
     ∃ M : ℝ, 0 < M ∧ ∀ z : ℂ, -1 ≤ z.re → z.re ≤ 2 →
       ‖completedDedekindZetaEntire K z^4 * Complex.sin (π * z)^(gammaExponent K)
@@ -925,7 +919,7 @@ theorem comparator_bound_strip :
     simp [sub_eq_zero] at this
     rw [this] at hw
     norm_num at hw
-  -- differentiability up to the boundary
+
   have hfd : DiffContOnCl ℂ f (Complex.re ⁻¹' Set.Ioo (-1 : ℝ) 2) := by
     have hcl : closure (Complex.re ⁻¹' Set.Ioo (-1 : ℝ) 2)
         ⊆ Complex.re ⁻¹' Set.Icc (-1 : ℝ) 2 :=
@@ -946,7 +940,7 @@ theorem comparator_bound_strip :
         fun z => Real.exp (Bg * Real.exp (c * |z.im|)) := by
     simp only [hf, hn]
     exact comparator_isBigO_growth K hB0 hB
-  -- boundary bounds in PL form
+
   have hle_a : ∀ w : ℂ, w.re = -1 → ‖f w‖ ≤ M₁ + M₂ := by
     intro w hw
     have hweq : w = ((-1:ℝ) : ℂ) + (w.im : ℂ) * Complex.I := by
@@ -995,7 +989,7 @@ theorem exists_H_strip_decay :
   set t : ℝ := z.im with hts
   set C₀ : ℝ := (M * 5^(4*n+8) * 3^n) ^ ((1:ℝ)/4) + 1 with hC₀
   have hC₀0 : 0 < C₀ := by positivity
-  -- the comparator bound, cleared of the normalizer
+
   have hcomp := hM z h1 h2
   rw [norm_div, norm_mul, norm_pow, norm_pow, norm_pow] at hcomp
   -- sine lower bound
@@ -1041,7 +1035,7 @@ theorem exists_H_strip_decay :
             rw [abs_sub_comm, abs_of_pos (by linarith : (0:ℝ) < 4 - z.re)]
             linarith
           nlinarith [abs_nonneg t]
-  -- solve the comparator inequality for ‖H‖⁴
+
   have hsinpos : (0:ℝ) < ‖Complex.sin (π * z)‖ := by
     have h3 : (0:ℝ) < Real.exp (π * |t|) / 3 := by positivity
     linarith
@@ -1656,8 +1650,7 @@ theorem exists_H_upper_right (σ₁ : ℝ) (hσ₁ : 2 ≤ σ₁) :
         rw [pow_add]
         ring
 
-/-- Sphere-sup bound for the Jensen ball: at center `A+iT` with radius `A+1`, every
-point of the closed ball satisfies the decaying envelope bound (in terms of `|T|`). -/
+
 theorem exists_H_ball_sup (A : ℝ) (hA : 2 ≤ A) :
     ∃ C : ℝ, 0 < C ∧ ∃ P : ℕ, ∀ T : ℝ, A + 5 ≤ |T| →
       ∀ z ∈ Metric.closedBall ((A : ℂ) + (T : ℂ) * Complex.I) (A + 1),
@@ -1743,10 +1736,7 @@ theorem exists_H_ball_sup (A : ℝ) (hA : 2 ≤ A) :
         rw [mul_pow]
         ring
 
-/-- **AC-A4: per-height zero counting.** There are an abscissa `A` and a constant `C`
-such that for all heights `|T| ≥ A+5`, the number of zeros of `H` (with multiplicity)
-in the closed ball of radius `√(A²+1)` around `A+iT` — a ball containing the critical
-slab `0 ≤ Re ≤ 1`, `|Im - T| ≤ 1` — is at most `C·log(2+|T|)`. -/
+
 theorem exists_ball_zero_count :
     ∃ A : ℝ, 2 ≤ A ∧ ∃ C : ℝ, 0 < C ∧ ∀ T : ℝ, A + 5 ≤ |T| →
       ((∑ᶠ u, (MeromorphicOn.divisor (completedDedekindZetaEntire K)
@@ -1858,7 +1848,7 @@ theorem exists_ball_zero_count :
       meromorphicOrderAt (fun _ : ℂ => (completedDedekindZetaEntire K c)⁻¹) z ≠ ⊤ := by
     intro z _
     classical
-    rw [meromorphicOrderAt_const, if_neg (inv_ne_zero hHcne)]
+    rw [meromorphicOrderAt_const, ite_eq_right (inv_ne_zero hHcne)]
     exact WithTop.zero_ne_top
   have hcmem : c ∈ Metric.closedBall c |Real.sqrt (A^2+1)| := by
     rw [habs_r]
@@ -2079,10 +2069,7 @@ theorem exists_H_ball_factorization (c : ℂ) {R : ℝ}
   congr 1
   rw [Function.FactorizedRational.finprod_eq_fun hfinsupp]
 
-/-- Sphere-sup bound for the Landau ball: at center `A+iT` with the larger radius
-`A+3`, every point of the closed ball satisfies the decaying envelope bound (in
-terms of `|T|`). Left of the critical strip the functional equation reflects the
-bound from the right half-plane. -/
+
 theorem exists_H_ball_sup_big (A : ℝ) (hA : 2 ≤ A) :
     ∃ C : ℝ, 0 < C ∧ ∃ P : ℕ, ∀ T : ℝ, A + 5 ≤ |T| →
       ∀ z ∈ Metric.closedBall ((A : ℂ) + (T : ℂ) * Complex.I) (A + 3),
@@ -2186,10 +2173,7 @@ theorem exists_H_ball_sup_big (A : ℝ) (hA : 2 ≤ A) :
         rw [mul_pow]
         ring
 
-/-- **Per-height zero counting at the Landau radius.** Given the abscissa `A` with the
-envelope-matched center lower bound, the number of zeros of `H` (with multiplicity) in
-the closed ball of radius `A+2` around `A+iT` is at most `C·log(2+|T|)`. Parametric in
-`A` and the center bound so that downstream users can share one `A`. -/
+
 theorem exists_ball_zero_count_big (A : ℝ) (hA2 : 2 ≤ A) (cL : ℝ) (hcL : 0 < cL)
     (hlow : ∀ t : ℝ, 2 ≤ |t| →
       cL * Real.exp (-(((gammaExponent K : ℕ) : ℝ) * (π * |t|)) / 4)
@@ -2298,7 +2282,7 @@ theorem exists_ball_zero_count_big (A : ℝ) (hA2 : 2 ≤ A) (cL : ℝ) (hcL : 0
       meromorphicOrderAt (fun _ : ℂ => (completedDedekindZetaEntire K c)⁻¹) z ≠ ⊤ := by
     intro z _
     classical
-    rw [meromorphicOrderAt_const, if_neg (inv_ne_zero hHcne)]
+    rw [meromorphicOrderAt_const, ite_eq_right (inv_ne_zero hHcne)]
     exact WithTop.zero_ne_top
   have hcmem : c ∈ Metric.closedBall c |A + 2| := by
     rw [habs_r]
@@ -2482,11 +2466,7 @@ theorem exists_H_two_radius_factorization (c : ℂ) {r₁ r₂ : ℝ} (h12 : r
     rw [hfac z hz, hsplit z]
     ring
 
-/-- **Landau's logarithmic-derivative bound** (A5-ii-c, generic form): if `h` is
-holomorphic and zero-free on `ball c r` with `‖h‖ ≤ mS` there and `mL ≤ ‖h c‖` at the
-center, then `‖h'/h‖ ≤ 32·r·(log(mS/mL) + 1)` on the concentric closed ball of radius
-`r - 3/4`. Chain: holomorphic logarithm on the convex ball, Borel–Carathéodory from the
-`Re`-bound `log(mS/mL)`, then the Schwarz derivative estimate on quarter-balls. -/
+
 theorem norm_logDeriv_le_of_norm_le {h : ℂ → ℂ} {c : ℂ} {r : ℝ} (hr : 3/4 < r)
     (hd : DifferentiableOn ℂ h (Metric.ball c r))
     (h0 : ∀ z ∈ Metric.ball c r, h z ≠ 0)
@@ -2535,7 +2515,7 @@ theorem norm_logDeriv_le_of_norm_le {h : ℂ → ℂ} {c : ℂ} {r : ℝ} (hr :
     intro w hw
     refine Complex.borelCaratheodory_zero hM0 hGd ?_ hr0 hw ?_
     · intro x hx
-      simp only [Set.mem_setOf_eq]
+      simp only [Set.mem_ofPred_eq]
       refine hreG (c + x) ?_
       rw [Metric.mem_ball] at hx ⊢
       simpa [dist_eq_norm] using hx
@@ -2630,7 +2610,7 @@ private theorem pow_sum_toNat_le_norm_prod_sub {F : Finset ℂ} {D : ℂ → ℤ
     (hzu : ∀ u ∈ F, b ≤ ‖z - u‖) :
     b ^ (∑ u ∈ F, (D u).toNat) ≤ ‖∏ u ∈ F, (z - u) ^ D u‖ := by
   rw [norm_prod_sub_zpow hD, ← Finset.prod_pow_eq_pow_sum]
-  exact Finset.prod_le_prod (fun u _ => by positivity)
+  exact Finset.prod_le_prod₀ (fun u _ => by positivity)
     (fun u hu => pow_le_pow_left₀ hb (hzu u hu) _)
 
 /-- Upper bound for the norm of a monic-type product: if every factor distance `‖z - u‖`
@@ -2640,13 +2620,10 @@ private theorem norm_prod_sub_le_pow {F : Finset ℂ} {D : ℂ → ℤ}
     (hzu : ∀ u ∈ F, ‖z - u‖ ≤ B) :
     ‖∏ u ∈ F, (z - u) ^ D u‖ ≤ B ^ (∑ u ∈ F, (D u).toNat) := by
   rw [norm_prod_sub_zpow hD, ← Finset.prod_pow_eq_pow_sum]
-  exact Finset.prod_le_prod (fun u _ => by positivity)
+  exact Finset.prod_le_prod₀ (fun u _ => by positivity)
     (fun u hu => pow_le_pow_left₀ (norm_nonneg _) (hzu u hu) _)
 
-/-- **Divisor monotonicity from an open ball to its closed ball.** For an everywhere-analytic
-`f`, the (nonnegative) zero-count divisor over `ball c r` sums to at most the divisor over
-`closedBall c r`, since the local orders agree and the open-ball support is contained in the
-closed-ball support. -/
+
 private theorem finsum_divisor_ball_le_closedBall {f : ℂ → ℂ}
     (hf : ∀ z : ℂ, AnalyticAt ℂ f z) (c : ℂ) (r : ℝ) :
     (∑ᶠ u, (MeromorphicOn.divisor f (Metric.ball c r)) u)
@@ -2692,9 +2669,7 @@ private theorem finsum_divisor_ball_le_closedBall {f : ℂ → ℂ}
           exact hu (hfin₁.mem_toFinset.mpr (Function.mem_support.mpr h0)))
     _ ≤ ∑ u ∈ hfincl.toFinset, Dcl u := Finset.sum_le_sum (fun u _ => hpt u)
 
-/-- **Maximum-modulus sup bound on a closed ball from a bound on its sphere.** If `f` is
-analytic on the open ball `ball c R` and `‖f‖ ≤ C` on the sphere `sphere c ρ` (with
-`0 < ρ < R`), then `‖f‖ ≤ C` on the whole closed ball `closedBall c ρ`. -/
+
 private theorem norm_le_of_frontier_ball {f : ℂ → ℂ} {c : ℂ} {ρ R : ℝ}
     (hρ : 0 < ρ) (hρR : ρ < R)
     (hanal : AnalyticOnNhd ℂ f (Metric.ball c R))
@@ -2891,7 +2866,7 @@ theorem exists_H_landau_cofactor :
       _ ≤ ((∑ᶠ u, (MeromorphicOn.divisor (completedDedekindZetaEntire K)
             (Metric.closedBall c (A+2))) u : ℤ) : ℝ) := by exact_mod_cast hDnatZ_le
       _ ≤ Cc * Real.log (2 + |T|) := hclosed
-  -- export the open-ball zero count
+
   have hcount_open : ((∑ᶠ u, D₁ u : ℤ) : ℝ)
       ≤ (32 * (A+2)
         * (|Real.log (cB/cL)| + (P+Pn) + Cc * Real.log (2*(A+2)) + 1) + Cc)
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/Existence.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/Existence.lean
index d305719d4..3dc187db1 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/Existence.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/Existence.lean
@@ -48,7 +48,7 @@ theorem prod_place_gamma (σ : ℝ) :
       rw [Finset.prod_const, Finset.card_univ]]
     refine Finset.prod_congr rfl (fun w _ => ?_)
     rw [show (mult (w : InfinitePlace K) : ℝ) = 1 by
-      rw [mult, if_pos w.2]
+      rw [mult, ite_eq_left w.2]
       norm_num]
     norm_num
   · rw [show (π ^ (-(2 * σ)) * Real.Gamma (2 * σ)) ^ (nrComplexPlaces K)
@@ -60,7 +60,7 @@ theorem prod_place_gamma (σ : ℝ) :
         (fun w : InfinitePlace K => not_isReal_iff_isComplex))).trans rfl).symm]
     refine Finset.prod_congr rfl (fun w _ => ?_)
     rw [show (mult (w : InfinitePlace K) : ℝ) = 2 by
-      rw [mult, if_neg w.2]
+      rw [mult, ite_eq_right w.2]
       norm_num]
 
 open scoped Classical in
@@ -231,7 +231,7 @@ theorem analyticOnNhd_completedDedekindZeta :
     intro h
     have : z = 0 := by linear_combination (2:ℂ) * h
     rw [this] at hz
-    norm_num [Set.mem_setOf_eq] at hz
+    norm_num [Set.mem_ofPred_eq] at hz
   have hzk : z / 2 ≠ (((heckeFEPair K).k : ℝ) : ℂ) := by
     intro h
     rw [show (heckeFEPair K).k = 1/2 from rfl] at h
@@ -239,7 +239,7 @@ theorem analyticOnNhd_completedDedekindZeta :
       push_cast at h
       linear_combination (2:ℂ) * h
     rw [this] at hz
-    norm_num [Set.mem_setOf_eq] at hz
+    norm_num [Set.mem_ofPred_eq] at hz
   have hd := (heckeFEPair K).differentiableAt_Λ (Or.inl hz2) (Or.inl hzk)
   have hhalf : DifferentiableAt ℂ (fun w : ℂ => w / 2) z := by fun_prop
   exact DifferentiableAt.comp (𝕜 := ℂ) (g := (heckeFEPair K).Λ)
@@ -358,7 +358,7 @@ theorem completedDedekindZeta_eq_of_one_lt_re {s : ℂ} (hs : 1 < s.re) :
     exact (LSeries_hasDerivAt hz').differentiableAt
   have hF₂ := analyticOnNhd_completedZetaPrefactor_mul_dedekindZeta K hζat
   have h2mem : (2:ℂ) ∈ {z : ℂ | 1 < z.re} := by
-    norm_num [Set.mem_setOf_eq]
+    norm_num [Set.mem_ofPred_eq]
   have hfreq := frequently_completedDedekindZeta_eq K
   have hEq := hF₁.eqOn_of_preconnected_of_frequently_eq hF₂ hUconn h2mem hfreq
   exact hEq hs
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/FunctionalEquation.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/FunctionalEquation.lean
index e9cae2b15..8df65b3b9 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/FunctionalEquation.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/FunctionalEquation.lean
@@ -82,10 +82,10 @@ theorem IsCompletedDedekindZeta.eqOn {K : Type*} [Field K] [NumberField K]
       intro s hs
       have hs0' : s ≠ 0 := by
         rintro rfl
-        norm_num [Set.mem_setOf_eq, Complex.zero_re] at hs
+        norm_num [Set.mem_ofPred_eq, Complex.zero_re] at hs
       have hs1' : s ≠ 1 := by
         rintro rfl
-        norm_num [Set.mem_setOf_eq, Complex.one_re] at hs
+        norm_num [Set.mem_ofPred_eq, Complex.one_re] at hs
       rw [hH₁eq s hs0' hs1', hH₂eq s hs0' hs1', h₁.1 s hs, h₂.1 s hs]
     have h2mem : (2 : ℂ) ∈ {s : ℂ | 1 < s.re} := by norm_num [Complex.ofReal_re]
     have := AnalyticOnNhd.eqOn_of_preconnected_of_eventuallyEq
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/HeckeTheta.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/HeckeTheta.lean
index 372944eb4..ba69d2efb 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/HeckeTheta.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/HeckeTheta.lean
@@ -8,23 +8,6 @@ module
 public import Mathlib
 public import DedekindResidue.CompletedZeta.IdealLattice
 
-/-!
-# The multivariable Hecke theta of a fractional ideal  (SP1-AGE-3)
-
-Hecke's functional-equation argument for a number field of unit rank `> 0` uses the
-**multivariable** theta of an ideal lattice, with one weight per infinite place, averaged
-over a fundamental domain of the unit action. This file builds the per-place weight
-machinery and the theta function itself:
-
-* `placeWeights c` — expand per-place weights to the `index K` coordinates (equal on the
-  `(re, im)` pair of a complex place — the shape preserved by the unit action, since complex
-  multiplication mixes the pair but preserves `re² + im²`);
-* `sum_placeWeights_embeddingCoords_sq` — `∑ᵢ c_i·ζ(x)ᵢ² = ∑_w c_w·w(x)²`: the weighted
-  square-sum of embedding coordinates is the place-absolute-value form.
-
-The unit equivariance `Θ(c·w(ε)², L_I) = Θ(c, L_I)` and the box-averaged `g_I` follow
-(SP1-AGE-3 continuation), then the Mellin definition of `Λ_K` (AGE-4).
--/
 
 namespace DedekindResidue
 
@@ -36,8 +19,7 @@ open scoped nonZeroDivisors Real
 
 variable (K : Type*) [Field K] [NumberField K]
 
-/-- The `ℝ`-cast field degree `[K : ℚ]` is positive — the packaged `Module.finrank_pos` used
-for the degree denominators throughout the Hecke-weight normalisation. -/
+
 theorem finrank_pos_real : (0 : ℝ) < (Module.finrank ℚ K : ℝ) := by
   have := Module.finrank_pos (R := ℚ) (M := K)
   positivity
@@ -198,10 +180,10 @@ theorem placeWeights_dualPlaceWeights (c : InfinitePlace K → ℝ) (i : index K
     placeWeights K (dualPlaceWeights K c) i
       = (placeWeights K c i)⁻¹ * (dualityWeights K i) ^ 2 := by
   rcases i with w | ⟨w, j⟩
-  · simp only [placeWeights, Sum.elim_inl, dualPlaceWeights, dualityWeights, if_pos w.2]
+  · simp only [placeWeights, Sum.elim_inl, dualPlaceWeights, dualityWeights, ite_eq_left w.2]
     norm_num
   · simp only [placeWeights, Sum.elim_inr, dualPlaceWeights, dualityWeights]
-    rw [if_neg (by rw [not_isReal_iff_isComplex]; exact w.2)]
+    rw [ite_eq_right (by rw [not_isReal_iff_isComplex]; exact w.2)]
     fin_cases j <;> norm_num <;> ring
 
 open scoped Classical in
@@ -237,7 +219,6 @@ theorem heckeTheta_inversion (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
   ring
 
 
-
 open scoped Classical in
 /-- Extend hyperplane coordinates `u : logSpace K` (indexed by `w ≠ w₀`) to all infinite
 places by the trace-zero condition: the `w₀`-component is `-∑_{w ≠ w₀} u_w`. On the image of
@@ -268,10 +249,8 @@ theorem fullLog_add (u v : logSpace K) :
   · rfl
 
 open scoped Classical in
-/-- **The Hecke weight family**: `c(t,u)_w = t^{1/n}·exp(2·(trace-zero extension of u)_w / mult w)`.
-Pinned by two requirements: `∏_w c_w^{mult w} = t` (the norm ray, giving `N(𝔞)^{-s}` under the
-Mellin transform) and equivariance under `u ↦ u + logEmbedding ε` matching
-`heckeTheta_unit_mul` (making the theta integrand periodic modulo the unit lattice). -/
+
+
 noncomputable def heckeWeights (t : ℝ) (u : logSpace K) : InfinitePlace K → ℝ := fun w =>
   t ^ ((1 : ℝ) / (Module.finrank ℚ K)) * Real.exp (2 * fullLog K u w / mult w)
 
@@ -322,11 +301,11 @@ theorem sum_fullLog (u : logSpace K) : ∑ w : InfinitePlace K, fullLog K u w =
   have h1 : fullLog K u (w₀ : InfinitePlace K)
       = -∑ w' : {w : InfinitePlace K // w ≠ (w₀ : InfinitePlace K)}, u w' := by
     rw [fullLog]
-    exact dif_pos rfl
+    exact dite_eq_left rfl
   have h2 : ∀ w : {w : InfinitePlace K // w ≠ (w₀ : InfinitePlace K)},
       fullLog K u (w : InfinitePlace K) = u w := by
     intro w
-    rw [fullLog, dif_neg w.2]
+    rw [fullLog, dite_eq_right w.2]
   rw [h1, Finset.sum_congr rfl (fun w _ => h2 w)]
   ring
 
@@ -404,7 +383,7 @@ theorem prod_placeWeights (c : InfinitePlace K → ℝ) :
   congr 1
   · refine Finset.prod_congr rfl (fun w _ => ?_)
     simp only [placeWeights, Sum.elim_inl]
-    rw [mult, if_pos w.2, pow_one]
+    rw [mult, ite_eq_left w.2, pow_one]
   · have hre : (∏ i : {w : InfinitePlace K // ¬ IsReal w}, c ↑i ^ mult (i : InfinitePlace K))
         = ∏ w : {w : InfinitePlace K // IsComplex w}, c ↑w ^ mult (w : InfinitePlace K) :=
       Fintype.prod_equiv (Equiv.subtypeEquivRight
@@ -413,7 +392,7 @@ theorem prod_placeWeights (c : InfinitePlace K → ℝ) :
     refine Finset.prod_congr rfl (fun w _ => ?_)
     rw [Fin.prod_univ_two]
     simp only [placeWeights, Sum.elim_inr]
-    rw [mult, if_neg (by rw [not_isReal_iff_isComplex]; exact w.2)]
+    rw [mult, ite_eq_right (by rw [not_isReal_iff_isComplex]; exact w.2)]
     ring
 
 open scoped Classical in
@@ -471,7 +450,7 @@ theorem fullLog_dualShift (w : InfinitePlace K) :
           (mult (w : InfinitePlace K) : ℝ) / 2 *
             (if IsReal (w : InfinitePlace K) then 0 else Real.log 4)) = 0 := by
         refine Finset.sum_eq_zero (fun w _ => ?_)
-        rw [if_pos w.2, mul_zero]
+        rw [ite_eq_left w.2, mul_zero]
       have hc : (∑ w : {w : InfinitePlace K // ¬ IsReal w},
           (mult (w : InfinitePlace K) : ℝ) / 2 *
             (if IsReal (w : InfinitePlace K) then 0 else Real.log 4))
@@ -480,7 +459,7 @@ theorem fullLog_dualShift (w : InfinitePlace K) :
             (mult (w : InfinitePlace K) : ℝ) / 2 *
               (if IsReal (w : InfinitePlace K) then 0 else Real.log 4) = Real.log 4 := by
           intro w
-          rw [if_neg w.2, mult, if_neg w.2]
+          rw [ite_eq_right w.2, mult, ite_eq_right w.2]
           norm_num
         rw [Finset.sum_congr rfl (fun w _ => hterm w), Finset.sum_const, nsmul_eq_mul]
         congr 1
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/MellinAgreement.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/MellinAgreement.lean
index c1ddb8fb9..cb30568d9 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/MellinAgreement.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/MellinAgreement.lean
@@ -341,7 +341,7 @@ theorem lintegral_eq_tsum_box_shift (f : logSpace K → ℝ≥0∞) :
   classical
   set B := (basisUnitLattice K).ofZLatticeBasis ℝ with hB
   have hFD := ZSpan.isAddFundamentalDomain B volume
-  haveI : VAddInvariantMeasure (Submodule.span ℤ (Set.range ⇑B)) (logSpace K) volume :=
+  have : VAddInvariantMeasure (Submodule.span ℤ (Set.range ⇑B)) (logSpace K) volume :=
     inferInstanceAs
       (VAddInvariantMeasure (Submodule.span ℤ (Set.range ⇑B)).toAddSubgroup
         (logSpace K) volume)
@@ -453,11 +453,11 @@ theorem tsum_ite_eq_tsum_coneUnfold_ennreal (J : (Ideal (𝓞 K))⁰)
     have hgpr : g ((euclidMixedEquiv K).symm
         ((∏ i, fundSystem K i ^ (pr.2 i)) • ((pr.1 : mixedSpace K)))) ≠ 0 := by
       rw [hvw]
-      rwa [if_neg hvne] at hv
+      rwa [ite_eq_right hvne] at hv
     refine ⟨⟨pr, hgpr⟩, ?_⟩
     exact Subtype.ext hvw
   · rintro ⟨p, hp⟩
-    rw [if_neg (hne p)]
+    rw [ite_eq_right (hne p)]
 
 open scoped Classical in
 /-- The canonical algebraic preimage of a cone point. -/
@@ -577,9 +577,9 @@ theorem lintegral_box_theta_tail (J : (Ideal (𝓞 K))⁰) {t : ℝ} (ht : 0 < t
   rw [lintegral_tsum (fun v => ?_)]
   swap
   · by_cases hv : v = 0
-    · simp only [if_pos hv]
+    · simp only [ite_eq_left hv]
       exact aemeasurable_const
-    · simp only [if_neg hv]
+    · simp only [ite_eq_right hv]
       exact (ENNReal.continuous_ofReal.comp (continuous_gaussTerm K t _)).aemeasurable
   -- Step 3: the per-v box integral is `ite (v = 0) 0 (G ↑v)`
   have hstep3 : ∀ v : idealZLattice K (FractionalIdeal.mk0 K J),
@@ -589,12 +589,12 @@ theorem lintegral_box_theta_tail (J : (Ideal (𝓞 K))⁰) {t : ℝ} (ht : 0 < t
         = if v = 0 then 0 else G (v : EuclideanSpace ℝ (index K)) := by
     intro v
     by_cases hv : v = 0
-    · rw [if_pos hv]
+    · rw [ite_eq_left hv]
       rw [setLIntegral_congr_fun (ZSpan.fundamentalDomain_measurableSet _)
-        (fun u _ => if_pos hv), lintegral_zero]
-    · rw [if_neg hv, hG]
+        (fun u _ => ite_eq_left hv), lintegral_zero]
+    · rw [ite_eq_right hv, hG]
       exact setLIntegral_congr_fun (ZSpan.fundamentalDomain_measurableSet _)
-        (fun u _ => if_neg hv)
+        (fun u _ => ite_eq_right hv)
   rw [tsum_congr hstep3]
   -- Step 4: cone reindex (on the G-valued family)
   rw [tsum_ite_eq_tsum_coneUnfold_ennreal K J G]
@@ -692,7 +692,7 @@ theorem heckeLogMap_injective : Function.Injective (heckeLogMap K) := by
       field_simp at this
       linarith
     rw [fullLog] at hfl
-    rw [dif_neg w'.2] at hfl
+    rw [dite_eq_right w'.2] at hfl
     simpa using hfl
   rw [hτ, hu]
   rfl
@@ -906,7 +906,7 @@ theorem tsum_idealSet_norm_rpow (J : (Ideal (𝓞 K))⁰) (σ : ℝ) :
       (∑' _ζ : torsion K, ENNReal.ofReal ((((n : ℝ)) ^ 2) ^ (-σ)))
         = (torsionOrder K) * ENNReal.ofReal ((((n : ℝ)) ^ 2) ^ (-σ)) := by
     intro I
-    letI := Fintype.ofFinite (torsion K)
+    let := Fintype.ofFinite (torsion K)
     rw [tsum_fintype, Finset.sum_const, nsmul_eq_mul, Units.torsionOrder,
       Nat.card_eq_fintype_card]
     norm_cast
@@ -1142,8 +1142,8 @@ theorem sum_mult_heckeLogCLE (p : ℝ × logSpace K) :
   field_simp
 
 open scoped Classical in
-/-- **The universal Mellin constant in closed form**: the change of variables along the
-Hecke substitution factorises the master integral into per-place Gamma integrals. -/
+
+
 theorem lintegral_exp_heckeLog (σ : ℝ) (hσ : 0 < σ) :
     ∫⁻ p : ℝ × logSpace K, ENNReal.ofReal (Real.exp (σ * p.1
         - π * ∑ w : InfinitePlace K, Real.exp ((heckeLogCLE K p) w)))
@@ -1367,7 +1367,7 @@ theorem setLIntegral_box_swap (f : logSpace K → ℝ≥0∞)
   have h2 := ZSpan.isAddFundamentalDomain ((basisUnitLattice K).ofZLatticeBasis ℝ) volume
   rw [(Module.Free.chooseBasis ℤ (unitLattice K)).ofZLatticeBasis_span ℝ] at h1
   rw [(basisUnitLattice K).ofZLatticeBasis_span ℝ] at h2
-  haveI : VAddInvariantMeasure (unitLattice K) (logSpace K) volume :=
+  have : VAddInvariantMeasure (unitLattice K) (logSpace K) volume :=
     inferInstanceAs (VAddInvariantMeasure (unitLattice K).toAddSubgroup (logSpace K) volume)
   refine h1.setLIntegral_eq h2 (f := f) (fun l x => ?_)
   rw [Submodule.vadd_def, vadd_eq_add]
@@ -1774,8 +1774,8 @@ theorem sum_count_eq_card_le (n : ℕ) :
     ∑ k ∈ Finset.Icc 1 n,
       (Nat.card {I : Ideal (𝓞 K) // Ideal.absNorm I = k} : ℝ)
       = (Nat.card {I : (Ideal (𝓞 K))⁰ // Ideal.absNorm (I : Ideal (𝓞 K)) ≤ n} : ℝ) := by
-  haveI : ∀ k : (Finset.Icc 1 n), Finite {I : Ideal (𝓞 K) // Ideal.absNorm I = (k : ℕ)} :=
-    fun k => (Ideal.finite_setOf_absNorm_eq (S := 𝓞 K) (k : ℕ)).to_subtype
+  have : ∀ k : (Finset.Icc 1 n), Finite {I : Ideal (𝓞 K) // Ideal.absNorm I = (k : ℕ)} :=
+    fun k => (Ideal.finite_setOfPred_absNorm_eq (S := 𝓞 K) (k : ℕ)).to_subtype
   rw [show (Nat.card {I : (Ideal (𝓞 K))⁰ // Ideal.absNorm (I : Ideal (𝓞 K)) ≤ n})
     = Nat.card (Σ k : (Finset.Icc 1 n),
         {I : Ideal (𝓞 K) // Ideal.absNorm I = (k : ℕ)}) from
@@ -1885,18 +1885,18 @@ theorem summable_ideal_norm_rpow {s : ℝ} (hs : 1 < s) :
       exact h1.congr (fun n => (h2 n))
     refine h3.congr (fun n => ?_)
     by_cases hn : n = 0
-    · rw [if_pos hn, hn]
+    · rw [ite_eq_left hn, hn]
       rw [Nat.cast_zero, Real.zero_rpow (by linarith), mul_zero]
-    · rw [if_neg hn, Real.rpow_neg (Nat.cast_nonneg n), div_eq_mul_inv]
+    · rw [ite_eq_right hn, Real.rpow_neg (Nat.cast_nonneg n), div_eq_mul_inv]
   -- transfer to the ideal-indexed sum by fibering over the norm
   have hnn : ∀ p : (Σ n : ℕ, {b : (Ideal (𝓞 K))⁰ //
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n}),
       0 ≤ ((Ideal.absNorm ((p.2 : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) : ℝ)) ^ (-s) :=
     fun p => Real.rpow_nonneg (Nat.cast_nonneg _) _
-  haveI hfibfin : ∀ n : ℕ, Finite {b : (Ideal (𝓞 K))⁰ //
+  have hfibfin : ∀ n : ℕ, Finite {b : (Ideal (𝓞 K))⁰ //
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n} := fun n => by
-    haveI : Finite {I : Ideal (𝓞 K) // Ideal.absNorm I = n} :=
-      (Ideal.finite_setOf_absNorm_eq (S := 𝓞 K) n).to_subtype
+    have : Finite {I : Ideal (𝓞 K) // Ideal.absNorm I = n} :=
+      (Ideal.finite_setOfPred_absNorm_eq (S := 𝓞 K) n).to_subtype
     exact Finite.of_injective _ (normFiber_forget_injective K n)
   rw [← Equiv.summable_iff (Equiv.sigmaFiberEquiv (fun b : (Ideal (𝓞 K))⁰ =>
     Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K))))]
@@ -1909,7 +1909,7 @@ theorem summable_ideal_norm_rpow {s : ℝ} (hs : 1 < s) :
         = ((n : ℝ)) ^ (-s) := by
     intro b
     rw [b.2]
-  haveI : Fintype {b : (Ideal (𝓞 K))⁰ //
+  have : Fintype {b : (Ideal (𝓞 K))⁰ //
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n} := Fintype.ofFinite _
   rw [show (∑' b : {b : (Ideal (𝓞 K))⁰ //
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n},
@@ -1961,10 +1961,10 @@ theorem dedekindZeta_real_eq {s : ℝ} (hs : 1 < s) :
   rw [tsum_congr h2, ← Complex.ofReal_tsum]
   congr 1
   -- fiber-glue the real n-sum into the ideal-type sum
-  haveI hfibfin : ∀ n : ℕ, Finite {b : (Ideal (𝓞 K))⁰ //
+  have hfibfin : ∀ n : ℕ, Finite {b : (Ideal (𝓞 K))⁰ //
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n} := fun n => by
-    haveI : Finite {I : Ideal (𝓞 K) // Ideal.absNorm I = n} :=
-      (Ideal.finite_setOf_absNorm_eq (S := 𝓞 K) n).to_subtype
+    have : Finite {I : Ideal (𝓞 K) // Ideal.absNorm I = n} :=
+      (Ideal.finite_setOfPred_absNorm_eq (S := 𝓞 K) n).to_subtype
     exact Finite.of_injective _ (normFiber_forget_injective K n)
   have hsum := summable_ideal_norm_rpow K hs
   have hsigma : Summable (fun p : (Σ n : ℕ, {b : (Ideal (𝓞 K))⁰ //
@@ -1986,7 +1986,7 @@ theorem dedekindZeta_real_eq {s : ℝ} (hs : 1 < s) :
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n},
       ((Ideal.absNorm ((b.val : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) : ℝ)) ^ (-s)
         = ((n : ℝ)) ^ (-s) := fun b => by rw [b.2]
-  haveI : Fintype {b : (Ideal (𝓞 K))⁰ //
+  have : Fintype {b : (Ideal (𝓞 K))⁰ //
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n} := Fintype.ofFinite _
   rw [show (∑' b : {b : (Ideal (𝓞 K))⁰ //
       Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) = n},
@@ -1997,8 +1997,8 @@ theorem dedekindZeta_real_eq {s : ℝ} (hs : 1 < s) :
   rw [tsum_fintype, Finset.sum_const, nsmul_eq_mul, Finset.card_univ]
   by_cases hn : n = 0
   · subst hn
-    rw [if_pos rfl, Nat.cast_zero, Real.zero_rpow (by linarith), mul_zero]
-  · rw [if_neg hn]
+    rw [ite_eq_left rfl, Nat.cast_zero, Real.zero_rpow (by linarith), mul_zero]
+  · rw [ite_eq_right hn]
     congr 2
     rw [← Nat.card_eq_fintype_card]
     exact Nat.card_congr (normFiberEquiv K n hn).symm
@@ -2014,8 +2014,7 @@ theorem heckeF_dev_nonneg {t : ℝ} (ht : 0 < t) :
   rw [hFsum]
   exact Finset.sum_nonneg (fun C _ => heckeGClass_dev_nonneg K C ht)
 
-/-- Positivity of the Gamma-product factor `∏ w, π^(-mult w·σ)·Γ(mult w·σ)` of the closed
-Hecke Λ-form (each factor is positive for `σ > 0`). -/
+
 theorem prod_pi_rpow_mul_Gamma_nonneg {σ : ℝ} (hσ0 : 0 < σ) :
     0 ≤ ∏ w : InfinitePlace K,
       π ^ (-((mult w : ℝ) * σ)) * Real.Gamma ((mult w : ℝ) * σ) := by
@@ -2026,8 +2025,7 @@ theorem prod_pi_rpow_mul_Gamma_nonneg {σ : ℝ} (hσ0 : 0 < σ) :
     positivity
   positivity
 
-/-- Nonnegativity of the closed real Λ-form (a `heckeBeta` power times the nonnegative
-Gamma-product times a nonnegative ideal-norm `tsum`). -/
+
 theorem heckeClosedLambdaForm_nonneg {σ : ℝ} (hσ0 : 0 < σ) :
     (0:ℝ) ≤ (heckeBeta K) ^ (-σ)
       * ((((heckeJacobian K : ℝ≥0)) : ℝ)
@@ -2044,8 +2042,8 @@ theorem heckeClosedLambdaForm_nonneg {σ : ℝ} (hσ0 : 0 < σ) :
   positivity
 
 open scoped Classical in
-/-- The `ENNReal.ofReal` of the Mellin integral of the Hecke theta deviation equals the
-`ofReal` of the closed Γ–ζ form; the ENNReal core of `heckeFEPair_Λ_real`. -/
+
+
 theorem ennreal_ofReal_integral_heckeF_dev_eq {σ : ℝ} (hσ0 : 0 < σ) (h2σ : 1 < 2 * σ)
     (hri : IntegrableOn (fun t : ℝ => t ^ (σ - 1) * (heckeF K t - heckeFConst K))
       (Set.Ioi (0:ℝ)) volume)
@@ -2066,7 +2064,7 @@ theorem ennreal_ofReal_integral_heckeF_dev_eq {σ : ℝ} (hσ0 : 0 < σ) (h2σ :
     fun t ht => ENNReal.ofReal_mul (Real.rpow_nonneg (le_of_lt ht) _)
   rw [setLIntegral_congr_fun measurableSet_Ioi hsplit,
     lintegral_mellin_heckeF_dev K hσ0]
-  -- identify the closed ENNReal form with ofReal of the real one
+
   have hzsum : (∑' b : (Ideal (𝓞 K))⁰,
       ENNReal.ofReal ((((Ideal.absNorm ((b : (Ideal (𝓞 K))⁰) : Ideal (𝓞 K)) : ℝ)) ^ 2)
         ^ (-σ)))
@@ -2094,8 +2092,8 @@ theorem ennreal_ofReal_integral_heckeF_dev_eq {σ : ℝ} (hσ0 : 0 < σ) (h2σ :
     ← ENNReal.ofReal_mul (by positivity)]
 
 open scoped Classical in
-/-- **The Λ-value identification (e-vi)**: at real `σ > 1/2` the abstract completed
-function of the Hecke theta pair is the closed Γ–ζ form. -/
+
+
 theorem heckeFEPair_Λ_real {σ : ℝ} (hσ : 1/2 < σ) :
     (heckeFEPair K).Λ (σ : ℂ)
       = (((heckeBeta K) ^ (-σ)
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/PoissonLattice.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/PoissonLattice.lean
index db7f899ca..d8a411dcb 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/PoissonLattice.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/PoissonLattice.lean
@@ -9,22 +9,6 @@ public import Mathlib
 public import DedekindResidue.CompletedZeta.DualLattice
 public import DedekindResidue.CompletedZeta.PoissonSummation
 
-/-!
-# Poisson summation over a general `ℤ`-lattice  (SP1-AGP, leaf P.3)
-
-Transport of the `ℤ^ι` Poisson formula (`tsum_eq_tsum_fourier_zpoint`) to an arbitrary
-`ℤ`-lattice `L ⊂ EuclideanSpace ℝ ι`: conjugate by the linear equivalence sending the standard
-lattice to `L`. The Fourier side transforms by the **`GL` change-of-variables law**
-(`fourier_comp_linearEquiv`, new to this development: mathlib only has the isometry case
-`Real.fourier_comp_linearIsometry`), the dual lattice appears via `dualZLattice_eq_span`
-(P.1), and the covolume factor via `ZLattice.covolume_eq_det_mul_measureReal`.
-
-## Main results (this file)
-* `DedekindResidue.fourier_comp_linearEquiv` —
-  `𝓕(g ∘ T) w = |det T|⁻¹ • 𝓕 g ((T⁻¹)^* w)` for `T ∈ GL(EuclideanSpace ℝ ι)`.
-* `DedekindResidue.tsum_eq_tsum_fourier_zlattice` — **Poisson summation over a `ℤ`-lattice**:
-  `∑'_{v ∈ L} g(v) = covol(L)⁻¹ • ∑'_{w ∈ L♯} 𝓕g(w)`.
--/
 
 namespace DedekindResidue
 
@@ -160,7 +144,7 @@ theorem adjoint_symm_latticeEquiv_zpoint [DecidableEq ι]
         innerₗ_nondegenerate c i j
       rw [← innerₗ_apply_apply]
       exact this
-    simp only [this, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq, Finset.mem_univ, if_true]
+    simp only [this, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq, Finset.mem_univ, ite_true]
   rw [hLHS, hRHS]
 
 /-- **P3d.** The determinant of the lattice-basis change of variables is (up to sign) the
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/PoissonSummation.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/PoissonSummation.lean
index d37431765b6cd844993a19f34fa0da37c8e6c6ad..9e1c836c4e548e89a5aebceddd3da2d4fbff7de7 100644
GIT binary patch
delta 148
zcmbQ$!F2Q!<Aj}?<2hFGY~CoE$jp|X7EqL$vRPd60~3@znMdX}jQK|95sb+r6FFH&
zt_j9{BV)Juqnrsl3rPFqy_r&*?bU+VH!JDoF>m%Xs$<$LWp-SeBP72dKPNxEa`VGL
USD5*mwL>O}g0*eln;9zw0CzSwivR!s

diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaEstimates.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaEstimates.lean
index c9a2e644c..4c7e21d05 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaEstimates.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaEstimates.lean
@@ -91,12 +91,12 @@ theorem tsum_ite_gaussian_tail (L : Submodule ℤ (EuclideanSpace ℝ ι))
   rw [← smul_eq_mul, ← tsum_const_smul'' (Real.exp (-π * (a - a₀) * δ ^ 2))]
   refine Summable.tsum_le_tsum (fun v => ?_) ?_ (hsum₀.const_smul _)
   · by_cases hv : v = 0
-    · rw [if_pos hv]
+    · rw [ite_eq_left hv]
       have : (0:ℝ) < Real.exp (-π * (a - a₀) * δ ^ 2)
           * Real.exp (-π * ∑ i, a₀ * ((v : EuclideanSpace ℝ ι) i) ^ 2) := by
         positivity
       exact le_of_lt (by simpa [smul_eq_mul] using this)
-    · rw [if_neg hv, smul_eq_mul, ← Real.exp_add]
+    · rw [ite_eq_right hv, smul_eq_mul, ← Real.exp_add]
       refine Real.exp_le_exp.mpr ?_
       have hnorm : δ ^ 2 ≤ ∑ i, ((v : EuclideanSpace ℝ ι) i) ^ 2 := by
         have h1 : δ ^ 2 ≤ ‖(v : EuclideanSpace ℝ ι)‖ ^ 2 :=
diff --git a/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaLattice.lean b/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaLattice.lean
index dab1b9d9e..efc5f57f2 100644
--- a/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaLattice.lean
+++ b/projects/DedekindResidue/DedekindResidue/CompletedZeta/ThetaLattice.lean
@@ -10,30 +10,6 @@ public import DedekindResidue.CompletedZeta.DualLattice
 public import DedekindResidue.CompletedZeta.PoissonSummation
 public import DedekindResidue.CompletedZeta.PoissonLattice
 
-/-!
-# The lattice Gaussian theta function and its transformation law  (SP1-AGΘ)
-
-For a `ℤ`-lattice `L ⊂ EuclideanSpace ℝ ι` the **theta function**
-`Θ_L(t) := ∑_{v ∈ L} e^{-π t ‖v‖²}` (`t > 0`) satisfies the **inversion law**
-
-`Θ_L(t) = covol(L)⁻¹ · t^{-n/2} · Θ_{L♯}(1/t)`,
-
-the standard lattice theta transformation (Neukirch, *Algebraic Number Theory*, VII §3;
-the 1-D case is mathlib's `jacobiTheta` functional equation). It is the Gaussian
-instantiation of the lattice Poisson formula `tsum_eq_tsum_fourier_zlattice` (P.3): mathlib's
-`fourier_gaussian_innerProductSpace` at `b = πt` gives `𝓕 g_t(w) = t^{-n/2} e^{-π‖w‖²/t}`,
-and `summable_gaussian_zlattice` discharges the convergence hypotheses.
-
-This is the analytic engine of the Hecke functional-equation route (SP1-AGE → SP1-FE):
-applied to the ideal lattices of a number field it produces the completed `Λ_K`.
-
-## Main definitions / results (this file)
-* `DedekindResidue.thetaLattice L t` — `Θ_L(t) = ∑'_{v ∈ L} exp(-π t ‖v‖²)` (real-valued).
-* `thetaLattice_transform` — **the inversion law** `Θ_L(t) = covol(L)⁻¹·t^{-n/2}·Θ_{L♯}(1/t)`
-  for `t > 0`, fully proven from the lattice Poisson formula + mathlib's Gaussian Fourier
-  transform (`fourier_gaussianCM`), with both convergence hypotheses discharged
-  (`summable_norm_restrict_gaussianCM`, `summable_fourier_gaussianCM`).
--/
 
 namespace DedekindResidue
 
@@ -72,12 +48,12 @@ theorem norm_gaussianCM_apply (t : ℝ) (x : EuclideanSpace ℝ ι) :
     push_cast; ring
   rw [this, Complex.ofReal_re]
 
-/-- Sub-level sets of a lattice are finite (lattice ∩ closed ball in a proper space). -/
+
 theorem finite_norm_le_zlattice (L : Submodule ℤ (EuclideanSpace ℝ ι))
     [DiscreteTopology L] (R : ℝ) :
     {v : L | ‖(v : EuclideanSpace ℝ ι)‖ ≤ R}.Finite := by
   have hcl : IsClosed (L : Set (EuclideanSpace ℝ ι)) := by
-    rw [← Submodule.coe_toAddSubgroup]; exact AddSubgroup.isClosed_of_discrete
+    rw [← Submodule.coe_toAddSubgroup]; exact AddSubgroup.isClosed_of_discreteTopology
   have hfb : (Metric.closedBall 0 R ∩ (L : Set (EuclideanSpace ℝ ι))).Finite :=
     Metric.finite_isBounded_inter_isClosed DiscreteTopology.isDiscrete
       Metric.isBounded_closedBall hcl
@@ -107,7 +83,7 @@ theorem summable_norm_restrict_gaussianCM (L : Submodule ℤ (EuclideanSpace ℝ
   · exact (summable_gaussian_zlattice L (by positivity)).mul_left _
   · rw [Filter.eventually_cofinite]
     refine (finite_norm_le_zlattice L R').subset (fun v hv => ?_)
-    simp only [Set.mem_setOf_eq] at hv ⊢
+    simp only [Set.mem_ofPred_eq] at hv ⊢
     by_contra hnotle
     rw [not_le] at hnotle
     refine hv ?_
diff --git a/projects/DedekindResidue/DedekindResidue/ExplicitFormula/FourierJordan.lean b/projects/DedekindResidue/DedekindResidue/ExplicitFormula/FourierJordan.lean
index 5171d0c7f..e5f838c02 100644
--- a/projects/DedekindResidue/DedekindResidue/ExplicitFormula/FourierJordan.lean
+++ b/projects/DedekindResidue/DedekindResidue/ExplicitFormula/FourierJordan.lean
@@ -10,7 +10,7 @@ public import Mathlib.Analysis.Fourier.RiemannLebesgueLemma
 public import DedekindResidue.ExplicitFormula.PhiTransform
 
 /-!
-# Fourier–Jordan inversion: the Dirichlet integral (SP2-FJ-c)
+# Fourier–Jordan inversion: the Dirichlet integral
 
 The classical Dirichlet integral `∫₀^∞ (sin x)/x dx = π/2`, phrased as convergence of the
 truncated integrals `∫₀^b (sin x)/x dx → π/2` as `b → ∞`. This is the analytic engine of
@@ -18,7 +18,7 @@ Jordan's pointwise Fourier-inversion criterion for functions of bounded variatio
 Poitou's proof of the explicit formula (Poitou, *Sur les petits discriminants*, Séminaire
 DPP 1976/77, exposé 6, p. 6-03) to invert the transform `Φ` on the prime side.
 
-Not available in mathlib. We follow the Frullani/Fubini route:
+The Frullani/Fubini argument uses
 `1/x = ∫₀^∞ e^{-xy} dy`, so
 `∫₀^b (sin x)/x dx = ∫₀^∞ (1 - e^{-yb}(cos y + y sin b))/(1+y²) dy`
 by Fubini and the closed form of the damped sine integral; dominated convergence then gives
@@ -181,7 +181,7 @@ theorem integral_sinc_eq {b : ℝ} (hb : 0 < b) :
               rw [mul_comm x y]; ring)]
         exact integral_exp_neg_mul_sin y b
 
-/-- **The Dirichlet integral** (SP2-FJ-c): `∫₀^∞ sin x/x dx = π/2`, as a limit of the
+/-- **The Dirichlet integral**: `∫₀^∞ sin x/x dx = π/2`, as a limit of the
 truncated integrals. Fubini against `1/x = ∫ e^{-xy} dy` plus dominated convergence. -/
 theorem tendsto_integral_sinc_atTop :
     Filter.Tendsto (fun b : ℝ => ∫ x in (0:ℝ)..b, Real.sin x / x)
@@ -490,7 +490,7 @@ theorem integral_fourier_window_collapse {H : ℝ → ℂ} (hH : Integrable H) {
   refine MeasureTheory.integral_congr_ae ?_
   have h0 : ∀ᵐ (u : ℝ), u ≠ 0 := by
     rw [MeasureTheory.ae_iff]
-    simp only [ne_eq, not_not, Set.setOf_eq_eq_singleton]
+    simp only [ne_eq, not_not, Set.ofPred_eq_eq_singleton]
     exact MeasureTheory.measure_singleton 0
   filter_upwards [h0] with u hu
   rw [MeasureTheory.integral_const_mul]
@@ -740,7 +740,7 @@ theorem abs_integral_monotone_kernel_le {g K : ℝ → ℝ} (hg : Monotone g) {
   have hcount : Set.Countable {x : ℝ | g x ≠ Function.rightLim g x} := by
     refine Set.Countable.mono ?_ hg.countable_not_continuousAt
     intro x hx
-    simp only [Set.mem_setOf_eq] at hx ⊢
+    simp only [Set.mem_ofPred_eq] at hx ⊢
     intro hcont
     exact hx (hcont.continuousWithinAt.rightLim_eq).symm
   have hnull : volume {x : ℝ | g x ≠ Function.rightLim g x} = 0 :=
@@ -1023,8 +1023,8 @@ theorem integral_dirichlet_split {H : ℝ → ℂ} (hH : Integrable H) {δ : ℝ
       funext u
       simp only [Set.indicator_apply, Set.mem_Ico, Set.mem_Ioc]
       by_cases hu : -δ ≤ -u ∧ -u < 0
-      · rw [if_pos hu, if_pos ⟨by linarith [hu.2], by linarith [hu.1]⟩]
-      · rw [if_neg hu, if_neg (fun h => hu ⟨by linarith [h.2], by linarith [h.1]⟩)]
+      · rw [ite_eq_left hu, ite_eq_left ⟨by linarith [hu.2], by linarith [hu.1]⟩]
+      · rw [ite_eq_right hu, ite_eq_right (fun h => hu ⟨by linarith [h.2], by linarith [h.1]⟩)]
     rw [h1, h2, h3, MeasureTheory.integral_indicator measurableSet_Ioc]
   have hK_even : ∀ u : ℝ, f (-u) = H (-u) * ((2 * Real.sin (T * u) / u : ℝ) : ℂ) := by
     intro u
@@ -1122,8 +1122,7 @@ theorem tendsto_integral_dirichlet_plateau {δ : ℝ} (hδ : 0 < δ) :
   push_cast
   ring
 
-/-- **Jordan's Fourier inversion criterion, Dirichlet-kernel form** (SP2-FJ-a/g; Poitou
-p. 6-03): for integrable `H` whose real and imaginary parts have bounded variation, with
+/-- **Jordan's Fourier inversion criterion, Dirichlet-kernel form**: for integrable `H` whose real and imaginary parts have bounded variation, with
 one-sided limits `Hp` (from the right) and `Hm` (from the left) at `0`,
 
 `∫_ℝ H(u) · 2sin(Tu)/u du  →  π(Hp + Hm)`  as `T → ∞`.
@@ -1394,7 +1393,7 @@ theorem tendsto_integral_dirichlet_jordan {H : ℝ → ℂ} (hH : Integrable H)
           (add_lt_add_of_lt_of_lt hfarT hplat_piece) (hRright.trans hδR.le)) (hRleft.trans hδL.le)
     _ = ε := by ring
 
-/-- **Jordan's Fourier inversion, symmetric-window form** (SP2-FJ-a; Poitou p. 6-03): for
+/-- **Jordan's Fourier inversion, symmetric-window form**: for
 integrable `H` of bounded variation with one-sided limits `Hp`, `Hm` at `0`,
 
 `∫_{-T}^{T} (∫_ℝ H(u) e^{itu} du) dt  →  π(Hp + Hm)`  as `T → ∞`.
diff --git a/projects/DedekindResidue/DedekindResidue/ExplicitFormula/GammaSide.lean b/projects/DedekindResidue/DedekindResidue/ExplicitFormula/GammaSide.lean
index 1f1506cfc..ebf01aa5c 100644
--- a/projects/DedekindResidue/DedekindResidue/ExplicitFormula/GammaSide.lean
+++ b/projects/DedekindResidue/DedekindResidue/ExplicitFormula/GammaSide.lean
@@ -10,35 +10,13 @@ public import DedekindResidue.ExplicitFormula.RectangleContour
 public import DedekindResidue.Lemma2
 
 /-!
-# The archimedean side: digamma integrals (SP2-Γψ)
+Gauss's integral representation of the digamma function supplies the
+archimedean part of Poitou's explicit formula. It follows from the
+logarithmic derivative of Euler's gamma integral, the Frullani representation
+of the logarithm and Fubini's theorem. Taking digamma differences cancels
+the counterterms and gives the kernel used on vertical lines.
 
-Poitou's computation of the archimedean part of the explicit formula (Poitou, *Sur les
-petits discriminants*, exposé 6, pp. 6-03–6-06) rests on **Gauss's integral formula**
-(his eq. (5))
-
-`ψ(z) = -∫₀^∞ (e^{-xz}/(1-e^{-x}) - e^{-x}/x) dx`,   `Re z > 0`,
-
-and its consequences for `Re ψ` on vertical lines. Mathlib defines `Complex.digamma`
-(as `logDeriv Gamma`) but has no integral representation (explicit TODO in
-`Mathlib.Analysis.SpecialFunctions.Gamma.Digamma`), so this file builds it from the
-Euler integral: `Γ'(z) = ∫₀^∞ t^{z-1}e^{-t} log t dt` (mathlib's
-`hasDerivAt_GammaIntegral`), the Frullani representation of `log`, and Fubini — the
-classical Gauss derivation.
-
-## Main results
-
-- `integral_frullani_log` : `∫₀^∞ (e^{-x} - e^{-tx})/x dx = log t` for `t > 0`.
-- `integrableOn_rpow_mul_exp_neg_mul_abs_log` : the log-weighted Euler integrand is
-  integrable.
-- `abs_frullani_kernel_le` : exponential domination of the Frullani kernel.
-- `integral_gauss_inner` : the inner evaluation
-  `∫₀^∞ t^{z-1}e^{-t}(e^{-x}-e^{-tx})/x dt = Γ(z)(e^{-x} - (1+x)^{-z})/x`.
-- `digamma_eq_integral_gauss_one` : **Gauss's first form**
-  `ψ(z) = ∫₀^∞ (e^{-x} - (1+x)^{-z})/x dx` for `Re z > 0`.
-- `integrableOn_gauss_one_integrand` : integrability of that integrand.
-- `digamma_sub_digamma_eq_integral` : **Poitou's difference form**
-  `ψ(w) - ψ(σ) = ∫₀^∞ (e^{-σu} - e^{-wu})/(1-e^{-u}) du` (`x = e^u - 1`), the form his
-  eq. (5) is used in — both counterterms cancel in the difference.
+Reference: Poitou, *Sur les petits discriminants*, exposé 6, pp. 6-03–6-06.
 -/
 
 @[expose] public section
@@ -1216,7 +1194,7 @@ theorem hasDerivAt_gammaFT {F : ℝ → ℂ} (hF : Integrable F)
     refine h0.congr ?_
     refine (MeasureTheory.ae_restrict_iff' measurableSet_Ioc).mpr ?_
     filter_upwards [MeasureTheory.ae_iff.mpr (by
-      simp only [ne_eq, not_not, Set.setOf_eq_eq_singleton]
+      simp only [ne_eq, not_not, Set.ofPred_eq_eq_singleton]
       exact MeasureTheory.measure_singleton (0:ℝ) : volume {x : ℝ | ¬ x ≠ 0} = 0)]
       with x hx _
     have hxne : (x:ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hx
@@ -1364,7 +1342,7 @@ theorem hasDerivAt_gammaFT {F : ℝ → ℂ} (hF : Integrable F)
       refine MeasureTheory.integral_congr_ae ?_
       refine ((MeasureTheory.ae_restrict_iff' measurableSet_Ioc).mpr ?_)
       filter_upwards [MeasureTheory.ae_iff.mpr (by
-        simp only [ne_eq, not_not, Set.setOf_eq_eq_singleton]
+        simp only [ne_eq, not_not, Set.ofPred_eq_eq_singleton]
         exact MeasureTheory.measure_singleton (0:ℝ) : volume {x : ℝ | ¬ x ≠ 0} = 0)]
         with x hx _
       have hxne : (x:ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hx
@@ -1518,7 +1496,7 @@ theorem hasDerivAt_rhoFT {k : ℝ → ℂ} (hk : Integrable k) (t : ℝ) :
     rw [h0, Real.exp_zero, mul_one]
   · exact hk.norm
   · filter_upwards [MeasureTheory.ae_iff.mpr (by
-      simp only [ne_eq, not_not, Set.setOf_eq_eq_singleton]
+      simp only [ne_eq, not_not, Set.ofPred_eq_eq_singleton]
       exact MeasureTheory.measure_singleton (0:ℝ) : volume {x : ℝ | ¬ x ≠ 0} = 0)]
       with x hx
     intro τ _
@@ -2479,7 +2457,8 @@ theorem tendsto_eLpNorm_indicator_truncation {g : ℝ → ℂ}
       = (∫⁻ x, ((Set.Ioc (-(n:ℝ)) (n:ℝ))ᶜ.indicator (fun y => ‖g y‖ₑ ^ (2:ℝ)) x))
         ^ (1/(2:ℝ)) := by
     intro n
-    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num) (by norm_num)]
+    rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num) (by norm_num)
+      ((hg.aestronglyMeasurable.indicator measurableSet_Ioc).sub hg.aestronglyMeasurable)]
     rw [show ((2:ℝ≥0∞).toReal) = (2:ℝ) by norm_num]
     congr 1
     refine lintegral_congr (fun x => ?_)
@@ -2507,7 +2486,8 @@ theorem tendsto_eLpNorm_indicator_truncation {g : ℝ → ℂ}
       · rw [Set.indicator_of_notMem hx]
         exact bot_le
     · have := hg.eLpNorm_lt_top
-      rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num) (by norm_num),
+      rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num) (by norm_num)
+          hg.aestronglyMeasurable,
         show ((2:ℝ≥0∞).toReal) = (2:ℝ) by norm_num] at this
       have h1 : (∫⁻ x, ‖g x‖ₑ ^ (2:ℝ)) < ⊤ := by
         by_contra hcon
@@ -2615,11 +2595,7 @@ theorem gammaFT_ae_eq_fourierL2 {F : ℝ → ℂ} (hF : Integrable F)
     exact tendsto_eLpNorm_indicator_truncation hFdiv2
   have hmeas : TendstoInMeasure volume (fun n : ℕ => 𝓕 (fun x : ℝ => hn n x)) atTop
       ((𝓕 (hFdiv2.toLp h) : Lp ℂ 2 (volume : Measure ℝ)) : ℝ → ℂ) := by
-    refine tendstoInMeasure_of_tendsto_eLpNorm (p := 2) (by norm_num) ?_ ?_ heLp
-    · intro n
-      refine Continuous.aestronglyMeasurable ?_
-      exact VectorFourier.fourierIntegral_continuous (by fun_prop) (by fun_prop) (hn1 n)
-    · exact (MeasureTheory.Lp.memLp _).aestronglyMeasurable
+    exact MeasureTheory.tendstoInMeasure_of_tendsto_eLpNorm (p := 2) (by norm_num) heLp
   obtain ⟨ns, hns_mono, hns_ae⟩ := hmeas.exists_seq_tendsto_ae
   have hqmp : Measure.QuasiMeasurePreserving (fun t : ℝ => -t/(2*π))
       volume volume := by
@@ -2637,7 +2613,7 @@ theorem gammaFT_ae_eq_fourierL2 {F : ℝ → ℂ} (hF : Integrable F)
   have hae_t := hqmp.ae hns_ae
   have hne : ∀ᵐ t : ℝ, t ≠ 0 := by
     rw [MeasureTheory.ae_iff]
-    simp only [ne_eq, not_not, Set.setOf_eq_eq_singleton]
+    simp only [ne_eq, not_not, Set.ofPred_eq_eq_singleton]
     exact MeasureTheory.measure_singleton 0
   filter_upwards [hae_t, hne] with t hlim ht
   have hpt : Tendsto (fun i : ℕ => 𝓕 (fun x : ℝ => hn (ns i) x) (-t/(2*π)))
@@ -4275,7 +4251,7 @@ theorem differentiableAt_logDeriv_gammaFactor (K : Type*) [Field K] [NumberField
   have hU : IsOpen {w : ℂ | 0 < w.re} := isOpen_lt continuous_const Complex.continuous_re
   have hdiff : DifferentiableOn ℂ (gammaFactor K) {w : ℂ | 0 < w.re} := by
     intro w hw
-    rw [Set.mem_setOf_eq] at hw
+    rw [Set.mem_ofPred_eq] at hw
     exact (((differentiableAt_Gammaℝ_of_re_pos hw).pow _).mul
       ((differentiableAt_Gammaℂ_of_re_pos hw).pow _)).differentiableWithinAt
   have han : AnalyticOnNhd ℂ (gammaFactor K) {w : ℂ | 0 < w.re} :=
diff --git a/projects/DedekindResidue/DedekindResidue/ExplicitFormula/PrimeSide.lean b/projects/DedekindResidue/DedekindResidue/ExplicitFormula/PrimeSide.lean
index 257b1ee8f..bfe4ae128 100644
--- a/projects/DedekindResidue/DedekindResidue/ExplicitFormula/PrimeSide.lean
+++ b/projects/DedekindResidue/DedekindResidue/ExplicitFormula/PrimeSide.lean
@@ -557,11 +557,11 @@ instance countable_ideal_ringOfIntegers (K : Type*) [Field K] [NumberField K] :
     Countable (Ideal (𝓞 K)) := by
   have h0 : (Set.univ : Set (Ideal (𝓞 K))) = ⋃ n : ℕ, {I | Ideal.absNorm I = n} := by
     ext I
-    simp only [Set.mem_univ, Set.mem_iUnion, Set.mem_setOf_eq, true_iff]
+    simp only [Set.mem_univ, Set.mem_iUnion, Set.mem_ofPred_eq, true_iff]
     exact ⟨Ideal.absNorm I, rfl⟩
   have h1 : (Set.univ : Set (Ideal (𝓞 K))).Countable := by
     rw [h0]
-    exact Set.countable_iUnion (fun n => (Ideal.finite_setOf_absNorm_eq n).countable)
+    exact Set.countable_iUnion (fun n => (Ideal.finite_setOfPred_absNorm_eq n).countable)
   exact Set.countable_univ_iff.mp h1
 
 /-- A pointwise `tsum` of integrable functions with summable `L¹` norms is
diff --git a/projects/DedekindResidue/DedekindResidue/ExplicitFormula/ZeroCapture.lean b/projects/DedekindResidue/DedekindResidue/ExplicitFormula/ZeroCapture.lean
index 7d31587b8f2e079d6c96c21bf6d5ff6fd91172a5..74884c4ad50c792dfd89f70ce83086df94199c96 100644
GIT binary patch
delta 407
zcmY+9u}T9$5Qd3~2&au$3NnRQg&>HHDI?lxV;jO`cP_WMH+$|*5|T!IhU=o0jh4Y!
zi4S69?F0A%?h#M}+bsV#|9tcM(0H3Q=B{>o$$$*5AO=rLkO!8UAQ^#jd0|ON!zpRH
zHmqSvf{+ebiE#`mDsesl#+MY{>7f$P6qGl8!vio9;)Hhr#N-qqV8<h56S68sMmsqZ
zqkbD(aFtQA0*#F@g#ad&L?=&Z!Q%w5??wcUqw6>$Uw*Fb?qM{npjc3?1+tVA3;Kw&
zF7N6)2T`IZ?4(*08<z)|DOTqsVHhJ5J*9l9H5VWCi`t^S{&buO_2{N&DV6f;<b8Ab
q^=6RC;<-KB_@AG5_E++v^WED1n}<v}*4=WxbyPn3?h4unhxb1T(V3_K

diff --git a/projects/DedekindResidue/DedekindResidue/Lemma2.lean b/projects/DedekindResidue/DedekindResidue/Lemma2.lean
index 8ff22340f..27b966470 100644
--- a/projects/DedekindResidue/DedekindResidue/Lemma2.lean
+++ b/projects/DedekindResidue/DedekindResidue/Lemma2.lean
@@ -9,17 +9,6 @@ public import Mathlib
 public import DedekindResidue.AuxiliaryFunction
 public import DedekindResidue.CompletedZeta.Normalisation
 
-/-!
-# Lemma 2: the Fourier transform of the auxiliary function  (T003)
-
-Belabas–Friedman Lemma 2 (eq. 8): the paper-convention Fourier transform (eq. 2,
-`F̂(γ) = ∫ F(t)e^{itγ}dt`) of the test function `F_{s,X}` in closed form. This file builds
-it bottom-up: the evenness reduction to a cosine transform, the plateau contribution
-`2 sin(γT)/γ`, the exponential-tail integration by parts (paper eq. 7), and the assembly.
-
-The main statement is for `γ ≠ 0`; the `sin(γT)/γ` term forces a separate `γ = 0`
-companion (the pointwise limit), faithful to the paper's implicit convention.
--/
 
 namespace DedekindResidue
 
@@ -123,8 +112,7 @@ theorem integrable_exp_neg_mul_abs {b : ℝ} (hb : 0 < b) :
       rw [abs_of_pos ht]) measurableSet_Ioi
   simpa only [Set.Iic_union_Ioi, integrableOn_univ] using hneg.union hpos
 
-/-- The paper Fourier kernel against `F_{s,X}` is integrable on the line
-(`Re s > 1/2`, `X ≥ 1`). -/
+
 theorem integrable_auxF_kernel (s : ℂ) {X : ℝ} (hX : 1 ≤ X) (hs : 1/2 < s.re) (γ : ℝ) :
     Integrable (fun t : ℝ => auxF s X t * Complex.exp (Complex.I * t * γ)) := by
   refine Integrable.mono'
@@ -186,9 +174,7 @@ theorem hasDerivAt_gAux_deriv (h : ℂ) {t : ℝ} (ht : 0 < t) :
   field_simp
   ring
 
-/-- Master decay bound: a continuous multiplier `φ` bounded on `(T, ∞)` times the
-kernel `e^{-ht}/t` (with `Re h > 0`, `T > 0`) is integrable on `(T, ∞)`, by comparison
-with `(C/T)·e^{-Re h · t}`. -/
+
 theorem integrableOn_bounded_mul_exp_div {h : ℂ} (hh : 0 < h.re) {T : ℝ} (hT : 0 < T)
     {φ : ℝ → ℂ} (hφc : ContinuousOn φ (Set.Ioi T)) {C : ℝ}
     (hφb : ∀ t ∈ Set.Ioi T, ‖φ t‖ ≤ C) :
@@ -218,7 +204,7 @@ theorem integrableOn_bounded_mul_exp_div {h : ℂ} (hh : 0 < h.re) {T : ℝ} (hT
         mul_le_mul (hφb t ht) hle (by positivity) hC0
     _ = C / T * Real.exp (-h.re * t) := by ring
 
-/-- The kernel `e^{-ht}/t` times an eventually-bounded multiplier tends to `0` at `∞`. -/
+
 theorem tendsto_exp_div_mul_atTop {h : ℂ} (hh : 0 < h.re) {v : ℝ → ℂ} {C : ℝ}
     (hv : ∀ᶠ t : ℝ in Filter.atTop, ‖v t‖ ≤ C) :
     Filter.Tendsto (fun t : ℝ => Complex.exp (-h * t) / t * v t) Filter.atTop (nhds 0) := by
@@ -343,7 +329,7 @@ theorem integral_Ioi_gAux_ibp₁ (h : ℂ) (hh : 0 < h.re) {T γ : ℝ} (hT : 0
     gcongr
     exact abs_le.mpr ⟨Real.neg_one_le_sin _, Real.sin_le_one _⟩
 
-/-- Integrability of the kernel against a bounded continuous real multiplier. -/
+
 theorem integrableOn_exp_div_mul_real {h : ℂ} (hh : 0 < h.re) {T : ℝ} (hT : 0 < T)
     {ψ : ℝ → ℝ} (hψc : Continuous ψ) {C : ℝ} (hψb : ∀ t, |ψ t| ≤ C) :
     IntegrableOn (fun t : ℝ => Complex.exp (-h * t) / t * ((ψ t : ℝ) : ℂ))
@@ -412,9 +398,7 @@ theorem integrableOn_gAux_deriv2_mul_real {h : ℂ} (hh : 0 < h.re) {T : ℝ} (h
     rw [norm_pow]
     linarith
 
-/-- Second integration by parts for the Fourier tail: with `g(t) = e^{-ht}/t`,
-`∫_T^∞ (g″(t)·(-cos(tγ)/γ²) + g′(t)·sin(tγ)/γ) dt =
-(h+1/T)·g(T)·cos(Tγ)/γ²`, stated with the boundary value in raw `-(u(T)·v(T))` form. -/
+
 theorem integral_Ioi_gAux_ibp₂ (h : ℂ) (hh : 0 < h.re) {T γ : ℝ} (hT : 0 < T)
     (hγ : γ ≠ 0) :
     (∫ t in Set.Ioi T,
@@ -561,9 +545,7 @@ theorem tail_integral_identity (h : ℂ) (hh : 0 < h.re) {T γ : ℝ} (hT : 0 <
   field_simp at h2 ⊢
   linear_combination (-1 : ℂ) * h2
 
-/-- On the tail `t > log X`, the auxiliary function factors as a constant times the
-exponential kernel: `F_{s,X}(t) = log X · e^{h·log X} · (e^{-h t}/t)` with `h = s - 1/2`
-(the `|t| > log X` branch of `auxF`, rearranged). -/
+
 theorem auxF_of_gt (s : ℂ) {X : ℝ} (hX : 1 < X) {t : ℝ} (ht : Real.log X < t) :
     auxF s X t
       = (Real.log X : ℂ) * Complex.exp ((s - 1/2) * (Real.log X : ℂ))
@@ -574,17 +556,13 @@ theorem auxF_of_gt (s : ℂ) {X : ℝ} (hX : 1 < X) {t : ℝ} (ht : Real.log X <
     rw [abs_of_pos htpos]
     exact not_le.mpr ht
   have htc : ((t : ℝ) : ℂ) ≠ 0 := by exact_mod_cast htpos.ne'
-  rw [auxF, if_neg hnot, abs_of_pos htpos]
+  rw [auxF, ite_eq_right hnot, abs_of_pos htpos]
   push_cast
   rw [show -(s - 1/2) * ((t:ℂ) - (Real.log X : ℂ))
       = -(s - 1/2) * (t:ℂ) + (s - 1/2) * (Real.log X : ℂ) by ring, Complex.exp_add]
   field_simp
 
-/-- **Belabas–Friedman Lemma 2, eq. (8)** (the `γ ≠ 0` case): the paper-convention
-Fourier transform of the auxiliary function `F_{s,X}` in closed form. With `h = s - 1/2`
-and `T = log X`,
-`F̂(γ) = 2h²·sin(Tγ)/((h²+γ²)γ) + 2(h+1/T)·cos(Tγ)/(h²+γ²)
-        - 4/(h²+γ²)·∫_T^∞ cos(tγ)·F_{s,X}(t)·(ht+1)/t² dt`. -/
+
 theorem fourier_auxF (s : ℂ) {X : ℝ} (hX : 1 < X) (hs : 1/2 < s.re) {γ : ℝ}
     (hγ : γ ≠ 0) :
     paperFourierIntegral (auxF s X) γ
@@ -706,9 +684,7 @@ theorem fourier_auxF (s : ℂ) {X : ℝ} (hX : 1 < X) (hs : 1/2 < s.re) {γ : 
            + (4*s - 2)*(Real.log X : ℂ)*(γ:ℂ)*Complex.cos ((Real.log X : ℂ)*(γ:ℂ))
            + 4*(γ:ℂ)*Complex.cos ((Real.log X : ℂ)*(γ:ℂ)))) * hEE'
 
-/-- FTC on `(T,∞)` for the second-derivative kernel: with `g(t) = e^{-ht}/t` and `Re h > 0`,
-`∫_T^∞ (h² + (2ht+2)/t²)·g(t) dt = (h + 1/T)·g(T)`. The integrand has antiderivative
-`-(h + 1/t)·g(t)`, which vanishes at `∞`. -/
+
 theorem integral_Ioi_gAux_deriv2 {h : ℂ} (hh : 0 < h.re) {T : ℝ} (hT : 0 < T) :
     (∫ t in Set.Ioi T, (h^2 + (2*h*t + 2)/t^2) * (Complex.exp (-h * t) / t))
       = (h + 1/(T:ℂ)) * (Complex.exp (-h * (T:ℂ)) / (T:ℂ)) := by
@@ -734,7 +710,7 @@ theorem integral_Ioi_gAux_deriv2 {h : ℂ} (hh : 0 < h.re) {T : ℝ} (hT : 0 < T
       rw [norm_neg]
       exact norm_affine_inv_le h hT htT
 
-/-- Integrability on `(T,∞)` of the remainder kernel `g(t)·(ht+1)/t²`, `g(t) = e^{-ht}/t`. -/
+
 theorem integrableOn_gAux_mul_affine_div_sq {h : ℂ} (hh : 0 < h.re) {T : ℝ} (hT : 0 < T) :
     IntegrableOn (fun t : ℝ =>
       (Complex.exp (-h * t) / t) * ((h*t + 1)/t^2)) (Set.Ioi T) := by
@@ -749,9 +725,7 @@ theorem integrableOn_gAux_mul_affine_div_sq {h : ℂ} (hh : 0 < h.re) {T : ℝ}
   · intro t ht
     exact norm_affine_div_sq_le h hT (le_of_lt ht)
 
-/-- Linearity split of the second-derivative-kernel integral into plain and remainder parts:
-with `g(t) = e^{-ht}/t`,
-`∫_T^∞ (h²+(2ht+2)/t²)·g = h²·∫_T^∞ g + 2·∫_T^∞ g·(ht+1)/t²`. -/
+
 theorem integral_Ioi_gAux_deriv2_split {h : ℂ} (hh : 0 < h.re) {T : ℝ} (hT : 0 < T) :
     (∫ t in Set.Ioi T, (h^2 + (2*h*t + 2)/t^2) * (Complex.exp (-h * t) / t))
       = h^2 * (∫ t in Set.Ioi T, Complex.exp (-h * t) / t)
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/BernoulliGeneralized.lean b/projects/FltRegularBernoulli/BernoulliRegular/BernoulliGeneralized.lean
index f626dd2f5..69ea46900 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/BernoulliGeneralized.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/BernoulliGeneralized.lean
@@ -182,7 +182,7 @@ lemma bernoulliGen_teichmuller_inverse_eq_p_sub_one_div_p_add_padicInt
         ((p - 1 : ℚ_[p]) / p) + z := by
   have hp : Nat.Prime p := Fact.out
   have hp_gt2 : 2 < p := lt_of_le_of_ne hp.two_le (Ne.symm hp_odd)
-  haveI : NeZero p := ⟨hp.ne_zero⟩
+  have : NeZero p := ⟨hp.ne_zero⟩
   let ωZ : DirichletCharacter ℤ_[p] p := (teichmullerChar p) ^ (p - 2)
   let ωQ : DirichletCharacter ℚ_[p] p := (teichmullerCharQp p) ^ (p - 2)
   let S : ℤ_[p] := ∑ a : ZMod p, ωZ a * (a.val : ℤ_[p])
@@ -206,7 +206,7 @@ lemma bernoulliGen_teichmuller_inverse_eq_p_sub_one_div_p_add_padicInt
         dsimp [ωZ]; rw [MulChar.pow_apply' _ hpow_ne_zero, map_pow, teichmullerChar_apply,
           toZMod_teichmuller]
       have hval : PadicInt.toZMod (a.val : ℤ_[p]) = a := by simp
-      rw [map_mul, hω, hval, if_neg ha]
+      rw [map_mul, hω, hval, ite_eq_right ha]
       calc
         a ^ (p - 2) * a = a ^ ((p - 2) + 1) := by rw [mul_comm, pow_succ']
         _ = a ^ (p - 1) := by congr; omega
@@ -215,7 +215,7 @@ lemma bernoulliGen_teichmuller_inverse_eq_p_sub_one_div_p_add_padicInt
     have hsplit :
         (∑ a : ZMod p, if a = 0 then 0 else 1) =
           (Finset.univ.erase (0 : ZMod p)).sum (fun _ ↦ (1 : ZMod p)) := by
-      rw [← Finset.sum_erase_add _ _ (Finset.mem_univ 0), if_pos rfl, add_zero]
+      rw [← Finset.sum_erase_add _ _ (Finset.mem_univ 0), ite_eq_left rfl, add_zero]
       refine Finset.sum_congr rfl fun a ha ↦ ?_
       simp [(Finset.mem_erase.mp ha).1]
     calc
@@ -278,7 +278,7 @@ lemma val_add_val_neg_cast [NeZero N] (a : ZMod N) :
     ((a.val : R) + ((-a).val : R)) = if a = 0 then (0 : R) else (N : R) := by
   rcases eq_or_ne a 0 with rfl | ha
   · simp
-  · rw [if_neg ha, ZMod.neg_val, if_neg ha, Nat.cast_sub (ZMod.val_lt _).le]
+  · rw [ite_eq_right ha, ZMod.neg_val, ite_eq_right ha, Nat.cast_sub (ZMod.val_lt _).le]
     ring
 
 omit [Algebra ℚ R] in
@@ -386,7 +386,7 @@ theorem bernoulli_mem_padicInt_of_lt_sub_one {p : ℕ} [hp : Fact p.Prime]
   -- `k ≥ 1`. Apply `sum_bernoulli (k + 1) = 0`.
   have hkp1_lt : k + 1 < p := by omega
   have h_sum := _root_.sum_bernoulli (k + 1)
-  rw [if_neg (by omega : k + 1 ≠ 1), Finset.sum_range_succ] at h_sum
+  rw [ite_eq_right (by omega : k + 1 ≠ 1), Finset.sum_range_succ] at h_sum
   have h_choose_k : (Nat.choose (k + 1) k : ℚ) = (k + 1 : ℚ) := by
     rw [Nat.choose_succ_self_right]; push_cast; rfl
   rw [h_choose_k] at h_sum
@@ -399,7 +399,7 @@ theorem bernoulli_mem_padicInt_of_lt_sub_one {p : ℕ} [hp : Fact p.Prime]
   let z_of : ℕ → ℤ_[p] := fun j ↦
     if h : j < k then (hj_wit j h).choose else 0
   have hz_of : ∀ j, j < k → (bernoulli j : ℚ_[p]) = ((z_of j : ℤ_[p]) : ℚ_[p]) := by
-    intro j hj; simp only [z_of, dif_pos hj]; exact (hj_wit j hj).choose_spec
+    intro j hj; simp only [z_of, dite_eq_left hj]; exact (hj_wit j hj).choose_spec
   let S : ℤ_[p] := ∑ j ∈ Finset.range k, (Nat.choose (k + 1) j : ℤ_[p]) * z_of j
   have hS_coe : (S : ℚ_[p]) =
       ∑ j ∈ Finset.range k, (Nat.choose (k + 1) j : ℚ_[p]) * (bernoulli j : ℚ_[p]) := by
@@ -541,7 +541,7 @@ theorem bernoulli_pSubOne_add_inv_p_mem_padicInt
   -- Step 1: From `sum_bernoulli p = 0` derive
   -- `p * bernoulli (p - 1) + 1 = -∑ k ∈ range (p - 2), C(p, k+1) * bernoulli (k+1)`.
   have h_sum := _root_.sum_bernoulli p
-  rw [if_neg hp.ne_one] at h_sum
+  rw [ite_eq_right hp.ne_one] at h_sum
   have hp_eq : p = (p - 1) + 1 := by omega
   nth_rewrite 1 [hp_eq] at h_sum
   rw [Finset.sum_range_succ] at h_sum
@@ -576,7 +576,7 @@ theorem bernoulli_pSubOne_add_inv_p_mem_padicInt
     if h : k < p - 2 then (h_wit k h).choose else 0
   have hz_of : ∀ k, k < p - 2 →
       ((_root_.bernoulli (k + 1) : ℚ) : ℚ_[p]) = (z_of k : ℚ_[p]) := by
-    intro k hk; simp only [z_of, dif_pos hk]; exact (h_wit k hk).choose_spec
+    intro k hk; simp only [z_of, dite_eq_left hk]; exact (h_wit k hk).choose_spec
   -- `c k : ℕ := C(p, k + 1) / p`, satisfying `p * c k = C(p, k + 1)` for `k < p - 2`.
   let c : ℕ → ℕ := fun k ↦ Nat.choose p (k + 1) / p
   have hc_eq : ∀ k, k < p - 2 → p * c k = Nat.choose p (k + 1) := fun k hk ↦
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/Characters.lean b/projects/FltRegularBernoulli/BernoulliRegular/Characters.lean
index 356776bea..c528b5dc6 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/Characters.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/Characters.lean
@@ -221,16 +221,16 @@ Dirichlet characters mod `p` is (noncanonically) isomorphic to
 `(ZMod p)ˣ`, hence cyclic of order `p - 1`. -/
 lemma dirichletCharacter_mulEquiv_zmodUnits :
     Nonempty (DirichletCharacter ℤ_[p] p ≃* (ZMod p)ˣ) := by
-  haveI : NeZero p := ⟨hp.1.ne_zero⟩
-  haveI : HasEnoughRootsOfUnity ℤ_[p] (Monoid.exponent (ZMod p)ˣ) :=
+  have : NeZero p := ⟨hp.1.ne_zero⟩
+  have : HasEnoughRootsOfUnity ℤ_[p] (Monoid.exponent (ZMod p)ˣ) :=
     exponent_zmod_units (p := p) ▸ inferInstance
   exact DirichletCharacter.mulEquiv_units ℤ_[p] p
 
 /-- The `ℤ_[p]`-valued Dirichlet character group mod `p` has `p - 1`
 elements. -/
 lemma card_dirichletCharacter : Nat.card (DirichletCharacter ℤ_[p] p) = p - 1 := by
-  haveI : NeZero p := ⟨hp.1.ne_zero⟩
-  haveI : HasEnoughRootsOfUnity ℤ_[p] (Monoid.exponent (ZMod p)ˣ) :=
+  have : NeZero p := ⟨hp.1.ne_zero⟩
+  have : HasEnoughRootsOfUnity ℤ_[p] (Monoid.exponent (ZMod p)ˣ) :=
     exponent_zmod_units (p := p) ▸ inferInstance
   rw [DirichletCharacter.card_eq_totient_of_hasEnoughRootsOfUnity,
     Nat.totient_prime hp.1]
@@ -313,7 +313,7 @@ lemma teichmuller_neg_one (hp_odd : p ≠ 2) : teichmuller p (-1) = -1 := by
   have h_sq : teichmuller p (-1 : ZMod p) ^ 2 = 1 := by
     rw [sq, ← map_mul, neg_one_mul, neg_neg, map_one]
   rcases sq_eq_one_iff.mp h_sq with h1 | h_neg
-  · haveI : Fact (2 < p) := ⟨lt_of_le_of_ne hp.1.two_le (Ne.symm hp_odd)⟩
+  · have : Fact (2 < p) := ⟨lt_of_le_of_ne hp.1.two_le (Ne.symm hp_odd)⟩
     exact absurd (by simpa using congrArg PadicInt.toZMod h1) ZMod.neg_one_ne_one
   · exact h_neg
 
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/Basic.lean b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/Basic.lean
index 7733831c9..d24c5d66e 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/Basic.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/Basic.lean
@@ -54,9 +54,9 @@ prime `p`, paired with the standard additive character
 `ZMod.stdAddChar : AddChar (ZMod p) ℂ`, evaluates to `-1`. -/
 theorem gaussSum_one_stdAddChar :
     gaussSum (1 : DirichletCharacter ℂ p) (ZMod.stdAddChar (N := p)) = -1 := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
-  haveI : Fact (1 < p) := ⟨hp.out.one_lt⟩
-  haveI : Nontrivial (ZMod p) := ZMod.nontrivial p
+  have : NeZero p := ⟨hp.out.ne_zero⟩
+  have : Fact (1 < p) := ⟨hp.out.one_lt⟩
+  have : Nontrivial (ZMod p) := ZMod.nontrivial p
   -- `stdAddChar` is not the trivial additive character, so its values sum to `0`.
   have h_ne : (ZMod.stdAddChar (N := p)) ≠ 1 := by
     intro h
@@ -83,7 +83,7 @@ theorem gaussSum_one_stdAddChar :
         MulChar.map_nonunit _ (by simp)
       simp [h0]
     · have hu : IsUnit a := ha.isUnit
-      simp [MulChar.one_apply hu, if_neg ha]
+      simp [MulChar.one_apply hu, ite_eq_right ha]
   classical
   calc gaussSum (1 : DirichletCharacter ℂ p) (ZMod.stdAddChar (N := p))
       = ∑ a : ZMod p, (1 : DirichletCharacter ℂ p) a * (ZMod.stdAddChar (N := p)) a := rfl
@@ -107,7 +107,7 @@ theorem gaussSum_mul_gaussSum_inv_stdAddChar
     gaussSum χ (ZMod.stdAddChar (N := p)) *
         gaussSum χ⁻¹ (ZMod.stdAddChar (N := p)) =
       χ (-1) * p := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   have h_prim : (ZMod.stdAddChar : AddChar (ZMod p) ℂ).IsPrimitive :=
     ZMod.isPrimitive_stdAddChar p
   have h_card : (Fintype.card (ZMod p) : ℂ) = p := by
@@ -153,7 +153,7 @@ does not vanish as soon as their product is nontrivial. -/
 theorem jacobiSum_ne_zero_stdAddChar
     {χ φ : DirichletCharacter ℂ p} (hχφ : χ * φ ≠ 1) :
     jacobiSum χ φ ≠ 0 := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   by_cases hχ : χ = 1
   · subst hχ
     have hφ : φ ≠ 1 := by simpa using hχφ
@@ -187,7 +187,7 @@ character factoring through `1` is the trivial character. -/
 theorem DirichletCharacter.isPrimitive_of_ne_one
     {χ : DirichletCharacter ℂ p} (hχ : χ ≠ 1) :
     χ.IsPrimitive := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   rw [DirichletCharacter.isPrimitive_def]
   rcases (Nat.dvd_prime hp.out).mp χ.conductor_dvd_level with h | h
   · exact absurd ((DirichletCharacter.factorsThrough_one_iff χ).mp
@@ -340,7 +340,7 @@ theorem dft_quadraticCharComplex_eq_gaussSum (hp₂ : p ≠ 2) (k : ZMod p) :
     ZMod.dft (quadraticCharComplex p) k =
       quadraticCharComplex p (-k) *
         gaussSum (quadraticCharComplex p) (ZMod.stdAddChar (N := p)) := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   have hprim : (quadraticCharComplex p).IsPrimitive :=
     DirichletCharacter.isPrimitive_of_ne_one (p := p)
       (quadraticCharComplex_ne_one (p := p) hp₂)
@@ -352,12 +352,12 @@ theorem dft_quadraticCharComplex_eq_gaussSum (hp₂ : p ≠ 2) (k : ZMod p) :
 `p` at `0` and `0` away from `0`. -/
 theorem dft_const_one (k : ZMod p) :
     ZMod.dft (fun _ : ZMod p ↦ (1 : ℂ)) k = if k = 0 then p else 0 := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   by_cases hk : k = 0
   · subst hk
     rw [ZMod.dft_apply_zero]
     simp
-  · rw [ZMod.dft_apply, if_neg hk]
+  · rw [ZMod.dft_apply, ite_eq_right hk]
     have hne : ((ZMod.stdAddChar : AddChar (ZMod p) ℂ).mulShift (-k)) ≠ 1 := by
       intro hshift
       have heval : (ZMod.stdAddChar (N := p)) (-k) = 1 := by
@@ -466,7 +466,7 @@ theorem conj_gaussSum_quadraticCharComplex_eq_neg_self_of_mod_four_eq_three
 theorem gaussSum_quadraticCharComplex_sq (hp₂ : p ≠ 2) :
     gaussSum (quadraticCharComplex p) (ZMod.stdAddChar (N := p)) ^ 2 =
       quadraticCharComplex p (-1) * p := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   simpa [ZMod.card] using
     gaussSum_sq (χ := quadraticCharComplex p)
       (quadraticCharComplex_ne_one (p := p) hp₂)
@@ -579,7 +579,7 @@ character `χ : DirichletCharacter ℂ p`. Uses `χ^(p-1) = 1` + membership in
 `Algebra.adjoin ℤ {μ_{p-1}}`, which is contained in the integral closure. -/
 theorem DirichletCharacter.isIntegral_apply (χ : DirichletCharacter ℂ p) (a : ZMod p) :
     IsIntegral ℤ (χ a) := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   have hp_sub : 0 < p - 1 := by have := hp.out.one_lt; omega
   set μ : ℂ := Complex.exp (2 * Real.pi * Complex.I / (p - 1 : ℕ))
   have hμ : IsPrimitiveRoot μ (p - 1) :=
@@ -597,7 +597,7 @@ theorem DirichletCharacter.isIntegral_apply (χ : DirichletCharacter ℂ p) (a :
 integral over `ℤ` (each is a `p`-th root of unity). -/
 theorem ZMod.isIntegral_stdAddChar (a : ZMod p) :
     IsIntegral ℤ ((ZMod.stdAddChar : AddChar (ZMod p) ℂ) a) := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   have hx_pow : ((ZMod.stdAddChar : AddChar (ZMod p) ℂ) a) ^ p = 1 := by
     rw [← AddChar.map_nsmul_eq_pow, show ((p : ℕ) • a : ZMod p) = 0 from by
         rw [show ((p : ℕ) • a : ZMod p) = (p : ZMod p) * a from by ring,
@@ -636,7 +636,7 @@ theorem gaussSum_mem_algebraAdjoin_stickelbergerComplexRoot
     (χ : DirichletCharacter ℂ p) :
     gaussSum χ (ZMod.stdAddChar : AddChar (ZMod p) ℂ) ∈
       Algebra.adjoin ℤ ({stickelbergerComplexRoot p} : Set ℂ) := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   have hp_one_lt : 1 < p := hp.out.one_lt
   have hp_sub : 0 < p - 1 := by omega
   have hp_pos : 0 < p := hp.out.pos
@@ -665,7 +665,7 @@ theorem gaussSum_mem_algebraAdjoin_stickelbergerComplexRoot
     have h_pow : χ ^ (p - 1) = 1 := by
       have h := MulChar.pow_card_eq_one χ (M := ZMod p)
       rwa [ZMod.card_units_eq_totient, Nat.totient_prime hp.out] at h
-    haveI : NeZero (p - 1) := ⟨hp_sub.ne'⟩
+    have : NeZero (p - 1) := ⟨hp_sub.ne'⟩
     have h_mem : χ a ∈ Algebra.adjoin ℤ ({ζ ^ p} : Set ℂ) :=
       MulChar.apply_mem_algebraAdjoin_of_pow_eq_one h_pow hζ_pm1 a
     have h_sub : Algebra.adjoin ℤ ({ζ ^ p} : Set ℂ) ≤ Algebra.adjoin ℤ ({ζ} : Set ℂ) := by
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BlockDeterminant.lean b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BlockDeterminant.lean
index c3148c6ba..5764b48d9 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BlockDeterminant.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BlockDeterminant.lean
@@ -280,7 +280,7 @@ theorem toMatrix_deltaZeroConstOneDirichletCharacterBasis_normalizedDft :
       simp [Module.Basis.repr_self, hij]
     simpa only [Matrix.mul_diagonal, PEquiv.toMatrix_apply, Equiv.toPEquiv_apply,
       Equiv.Perm.coe_inv, Option.mem_def, Option.some.injEq, ite_mul, one_mul, zero_mul, hneq',
-      if_false]
+      ite_false]
       using hsingle
 
 /-- The determinant of that monomial matrix: the sign of the permutation times the
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BranchChoice.lean b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BranchChoice.lean
index 34468899d..419594284 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BranchChoice.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/BranchChoice.lean
@@ -50,7 +50,7 @@ theorem card_even_eq_card_odd_characters (hp₂ : p ≠ 2) :
     simp only [E, O, Finset.mem_union, Finset.mem_filter, Finset.mem_univ, true_and, iff_true]
     exact DirichletCharacter.even_or_odd χ
   have hneg_ne_one : (-1 : ZMod p) ≠ 1 := by
-    haveI : Fact (2 < p) := ⟨lt_of_le_of_ne hp.out.two_le (Ne.symm hp₂)⟩
+    have : Fact (2 < p) := ⟨lt_of_le_of_ne hp.out.two_le (Ne.symm hp₂)⟩
     exact ZMod.neg_one_ne_one
   have hsum_zero :
       ∑ χ : DirichletCharacter ℂ p, χ (-1 : ZMod p) = 0 :=
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Operator.lean b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Operator.lean
index e01362db2..cdfae2bc9 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Operator.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Operator.lean
@@ -172,7 +172,7 @@ noncomputable def zmodEquivFin : ZMod p ≃ Fin p :=
 
 theorem zmodEquivFin_symm_apply (i : Fin p) :
     (zmodEquivFin (p := p)).symm i = (i : ZMod p) := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   change ZMod.finEquiv p i = (i : ZMod p)
   cases p with
   | zero =>
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Trace.lean b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Trace.lean
index a5de9475f..f41b89788 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Trace.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/GaussSum/SignInvariant/Trace.lean
@@ -49,12 +49,12 @@ theorem dft_deltaZero_eq_constOne :
 /-- The DFT of the constant-one function is concentrated at `0`. -/
 theorem dft_constOne (k : ZMod p) :
     ZMod.dft (fun _ : ZMod p ↦ (1 : ℂ)) k = if k = 0 then p else 0 := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   by_cases hk : k = 0
   · subst hk
     rw [ZMod.dft_apply_zero]
     simp
-  · rw [ZMod.dft_apply, if_neg hk]
+  · rw [ZMod.dft_apply, ite_eq_right hk]
     have hne : ((ZMod.stdAddChar : AddChar (ZMod p) ℂ).mulShift (-k)) ≠ 1 := by
       intro hshift
       have heval : (ZMod.stdAddChar (N := p)) (-k) = 1 := by
@@ -149,7 +149,7 @@ theorem dft_eq_scalar_smul_inv_character {χ : DirichletCharacter ℂ p}
     ZMod.dft χ =
       (χ⁻¹ (-1) * gaussSum χ (ZMod.stdAddChar (N := p))) •
         ((χ⁻¹ : DirichletCharacter ℂ p) : ZMod p → ℂ) := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   have hprim : χ.IsPrimitive := DirichletCharacter.isPrimitive_of_ne_one (p := p) hχ
   ext k
   simp only [Pi.smul_apply, smul_eq_mul]
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Defs.lean b/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Defs.lean
index e65342489..e45157576 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Defs.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Defs.lean
@@ -64,7 +64,7 @@ theorem odd_LFunction_one_eq_oddLValueRhs_of_LFunction_inv_zero
         (Or.inr hp.out.ne_one), hχinv_odd.gammaFactor_def]
     simp
   have hfe := DirichletCharacter.IsPrimitive.completedLFunction_one_sub (χ := χ) hχ_prim (0 : ℂ)
-  rw [DirichletCharacter.rootNumber, if_neg hχ_odd.not_even, pow_one,
+  rw [DirichletCharacter.rootNumber, ite_eq_right hχ_odd.not_even, pow_one,
     ← mul_comm_div, ← mul_comm_div, ← Complex.cpow_sub _ _ (by exact_mod_cast hp.out.ne_zero),
     sub_sub, add_halves, hL0, hχ0] at hfe
   have hfe' : DirichletCharacter.completedLFunction χ 1 =
diff --git a/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Even.lean b/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Even.lean
index da64fd5da..f1e07c897 100644
--- a/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Even.lean
+++ b/projects/FltRegularBernoulli/BernoulliRegular/LValueAtOne/Even.lean
@@ -96,7 +96,7 @@ private lemma LFunction_dft_inv_eq_gaussSum_mul_LFunction
     {χ : DirichletCharacter ℂ p} (hχ_even : χ.Even) (hχinv_prim : (χ⁻¹).IsPrimitive) :
     ZMod.LFunction (ZMod.dft (fun a : ZMod p => χ⁻¹ a)) 1 =
       gaussSum χ⁻¹ (ZMod.stdAddChar (N := p)) * ZMod.LFunction χ 1 := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   have hft :
       ZMod.dft (fun a : ZMod p => χ⁻¹ a) = fun a : ZMod p =>
         gaussSum χ⁻¹ (ZMod.stdAddChar (N := p)) * χ a := by
@@ -152,7 +152,7 @@ private lemma sum_mulChar_cosZeta_eq_neg_evenLValueLogSum
     {χ : DirichletCharacter ℂ p} (hχinv_zero : χ⁻¹ 0 = 0) :
     ∑ a : ZMod p, χ⁻¹ a * HurwitzZeta.cosZeta (ZMod.toAddCircle a) 1 =
       -evenLValueLogSum p χ := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   simp only [evenLValueLogSum]
   calc
     ∑ a : ZMod p, χ⁻¹ a * HurwitzZeta.cosZeta (ZMod.toAddCircle a) 1
@@ -176,7 +176,7 @@ theorem even_LFunction_one_eq_evenLValueRhs
     {χ : DirichletCharacter ℂ p} (hχ_prim : χ.IsPrimitive) (hχ_even : χ.Even)
     (hχ_ne_one : χ ≠ 1) :
     DirichletCharacter.LFunction χ 1 = evenLValueRhs p χ := by
-  haveI : NeZero p := ⟨hp.out.ne_zero⟩
+  have : NeZero p := ⟨hp.out.ne_zero⟩
   change ZMod.LFunction (fun a : ZMod p => χ a) 1 = evenLValueRhs p χ
   have hχinv_even : (χ⁻¹).Even := by
     rw [DirichletCharacter.Even] at hχ_even ⊢
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/CharP.lean b/projects/HasseWeil/HasseWeil/FormalGroup/CharP.lean
index f9dac12ed..b5845ada0 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/CharP.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/CharP.lean
@@ -61,7 +61,7 @@ private theorem exists_expand_of_coeff_vanishing (p : ℕ) (hp : p ≠ 0)
 vanishes unless `p ∣ n` (or `n = 0`). -/
 private theorem coeff_eq_zero_of_derivative_eq_zero_charP {p : ℕ}
     [Fact p.Prime] [CharP R p] {f : PowerSeries R}
-    (hf : PowerSeries.derivative R f = 0)
+    (hf : PowerSeries.derivative (R := R) f = 0)
     (n : ℕ) (hpn : ¬ p ∣ n) :
     PowerSeries.coeff n f = 0 := by
   -- `n ≠ 0` since `p ∣ 0`; write `n = m + 1`.
@@ -101,7 +101,7 @@ theorem FormalGroup.mulByP_exists_expand (F : FormalGroup R) (p : ℕ)
   apply exists_expand_of_coeff_vanishing p hp_prime.out.ne_zero
   intro n hpn
   -- `derivative [p] = 0`, from the chain rule and `[p] = 0` in characteristic `p`.
-  have hder : PowerSeries.derivative R (F.mulByNatHom p).toSeries = 0 := by
+  have hder : PowerSeries.derivative (R := R) (F.mulByNatHom p).toSeries = 0 := by
     have chain := FormalGroupHom.invariantDifferential_chain (F.mulByNatHom p)
     have hp_zero :
         PowerSeries.C (PowerSeries.coeff 1 (F.mulByNatHom p).toSeries) *
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/Differential.lean b/projects/HasseWeil/HasseWeil/FormalGroup/Differential.lean
index 655b8afa6..7d00d2a24 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/Differential.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/Differential.lean
@@ -193,7 +193,7 @@ private theorem coeff_subst_runit_eq (n : ℕ)
     · simp only [hd0, pow_zero, one_mul, MvPowerSeries.coeff_X_pow]
       have hne : d 1 ≠ n :=
         fun h ↦ hd (Finsupp.ext (fun i ↦ by fin_cases i <;> simp [*, Finsupp.single_eq_same]))
-      rw [if_neg (fun h ↦ hne ((Finsupp.single_injective (1 : Fin 2)).eq_iff.mp h).symm)]
+      rw [ite_eq_right (fun h ↦ hne ((Finsupp.single_injective (1 : Fin 2)).eq_iff.mp h).symm)]
     · rw [zero_pow hd0, zero_mul, map_zero]
 
 -- Helper: coeff at (single 1 m) of F.toSeries equals [m = 1] (from runit: F(0,Y)=Y)
@@ -234,11 +234,11 @@ private theorem coeff_one_pow {f : PowerSeries R} (hf : PowerSeries.constantCoef
     PowerSeries.coeff 1 (f ^ d) = if d = 1 then PowerSeries.coeff 1 f else 0 := by
   by_cases hd : d = 1
   · simp [hd]
-  · rw [if_neg hd]
+  · rw [ite_eq_right hd]
     rcases d with _ | _ | d
     · simp only [pow_zero]
       show PowerSeries.coeff 1 (1 : PowerSeries R) = 0
-      rw [PowerSeries.coeff_one, if_neg one_ne_zero]
+      rw [PowerSeries.coeff_one, ite_eq_right one_ne_zero]
     · omega
     · exact coeff_one_pow_eq_zero hf (by omega)
 
@@ -258,12 +258,12 @@ private theorem coeff_subst_X0 (g : PowerSeries R) (a b : ℕ) :
     · simp [MvPowerSeries.coeff_X_pow, smul_eq_mul]
     · intro d hd
       simp only [MvPowerSeries.coeff_X_pow, smul_eq_mul]
-      rw [if_neg (fun h ↦ hd ((Finsupp.single_injective (0 : Fin 2)).eq_iff.mp h).symm),
+      rw [ite_eq_right (fun h ↦ hd ((Finsupp.single_injective (0 : Fin 2)).eq_iff.mp h).symm),
         mul_zero]
-  · rw [if_neg hb]
+  · rw [ite_eq_right hb]
     apply finsum_eq_zero_of_forall_eq_zero
     intro d
-    rw [MvPowerSeries.coeff_X_pow, if_neg]
+    rw [MvPowerSeries.coeff_X_pow, ite_eq_right]
     · exact smul_zero _
     · intro h
       have := DFunLike.congr_fun h 1
@@ -284,12 +284,12 @@ private theorem coeff_subst_X1 (g : PowerSeries R) (a b : ℕ) :
     · simp [MvPowerSeries.coeff_X_pow, smul_eq_mul]
     · intro d hd
       simp only [MvPowerSeries.coeff_X_pow, smul_eq_mul]
-      rw [if_neg (fun h ↦ hd ((Finsupp.single_injective (1 : Fin 2)).eq_iff.mp h).symm),
+      rw [ite_eq_right (fun h ↦ hd ((Finsupp.single_injective (1 : Fin 2)).eq_iff.mp h).symm),
         mul_zero]
-  · rw [if_neg hab]
+  · rw [ite_eq_right hab]
     apply finsum_eq_zero_of_forall_eq_zero
     intro d
-    rw [MvPowerSeries.coeff_X_pow, if_neg]
+    rw [MvPowerSeries.coeff_X_pow, ite_eq_right]
     · exact smul_zero _
     · intro h
       have := DFunLike.congr_fun h 0
@@ -333,7 +333,7 @@ private lemma coeff_subst_X0_X1_mul_eq_zero_of_ne (g : PowerSeries R) (d0 d1 n :
       from finsupp_fin2_decompose e2]
   rw [coeff_subst_X0, coeff_subst_X1]
   by_cases h1 : e1 1 = 0
-  · rw [if_pos h1]
+  · rw [ite_eq_left h1]
     by_cases h2 : e2 0 = 0
     · -- Both e1 1 = 0 and e2 0 = 0: forces (e1, e2) = (single 0 1, single 1 n)
       exfalso; apply hne
@@ -348,8 +348,8 @@ private lemma coeff_subst_X0_X1_mul_eq_zero_of_ne (g : PowerSeries R) (d0 d1 n :
         rw [finsupp_fin2_decompose e1, h1, he10]; simp
       · show e2 = Finsupp.single 1 n
         rw [finsupp_fin2_decompose e2, h2, he21]; simp
-    · rw [if_neg h2, mul_zero]
-  · rw [if_neg h1, zero_mul]
+    · rw [ite_eq_right h2, mul_zero]
+  · rw [ite_eq_right h1, zero_mul]
 
 -- Orthogonality: coeff_{(1,n)} (f(X_0)^d0 * f(X_1)^d1) = coeff_1(f^d0) * coeff_n(f^d1)
 private theorem coeff_10_prod_orthogonal (g : PowerSeries R) (d0 d1 n : ℕ) :
@@ -424,15 +424,15 @@ private lemma finsum_fin2_reduce_full (p : (Fin 2 →₀ ℕ) → R) :
       Set.indicator (Set.range ι) (fun d ↦ if d 0 = 1 then p d else 0) d := by
     intro d; classical rw [Set.indicator_apply]
     by_cases hd : d ∈ Set.range ι
-    · rw [if_pos hd]
-    · rw [if_neg hd, if_neg (mt (hmem d).mpr hd)]
+    · rw [ite_eq_left hd]
+    · rw [ite_eq_right hd, ite_eq_right (mt (hmem d).mpr hd)]
   conv_lhs => arg 1; ext d; rw [key d]
   rw [← finsum_mem_def, ← finsum_subtype_eq_finsum_cond (· ∈ Set.range ι),
     ← finsum_comp_equiv (Equiv.ofInjective ι hinj)]
   congr 1; ext k
   show (if (ι k : Fin 2 →₀ ℕ) 0 = 1 then p (ι k) else 0) =
     p (Finsupp.single 0 1 + Finsupp.single 1 k)
-  rw [if_pos (hι0 k)]
+  rw [ite_eq_left (hι0 k)]
 
 -- Helper: c * finsum f = finsum (c * f ·) when f has finite support.
 private lemma mul_finsum_of_support_subset {α : Type*} (c : R) (f : α → R)
@@ -503,11 +503,11 @@ private theorem antidiag_term_vanish (F : FormalGroup R) (d n : ℕ)
   rcases Nat.eq_zero_or_pos (e1 0) with he10 | he10
   · have he1eq : e1 = Finsupp.single 1 (e1 1) := Finsupp.ext fun i ↦ by
       fin_cases i <;> simp_all [Finsupp.single_eq_same]
-    rw [he1eq, coeff_single1_F, if_neg (fun h ↦ hA ⟨he10, h⟩), zero_mul]
+    rw [he1eq, coeff_single1_F, ite_eq_right (fun h ↦ hA ⟨he10, h⟩), zero_mul]
   · have he20 : e2 0 = 0 := by omega
     have he2eq : e2 = Finsupp.single 1 (e2 1) := Finsupp.ext fun i ↦ by
       fin_cases i <;> simp_all [Finsupp.single_eq_same]
-    rw [he2eq, coeff_runit_pow, if_neg (fun h ↦ hB ⟨by omega, h⟩), mul_zero]
+    rw [he2eq, coeff_runit_pow, ite_eq_right (fun h ↦ hB ⟨by omega, h⟩), mul_zero]
 
 -- Sub-lemma: the antidiagonal sum for `coeff_{(1,n)} (F^d * F)` collapses to the two
 -- surviving terms — pair A `(single 1 1, single 0 1 + single 1 (n-1))` and pair B
@@ -661,9 +661,9 @@ private theorem coeff_10_FG_pow (F : FormalGroup R) :
   induction d with
   | zero =>
     intro n
-    simp only [pow_zero, Nat.zero_le, le_add_iff_nonneg_left, if_true,
+    simp only [pow_zero, Nat.zero_le, le_add_iff_nonneg_left, ite_true,
       Nat.cast_zero, zero_mul, MvPowerSeries.coeff_one]
-    rw [if_neg]; intro h
+    rw [ite_eq_right]; intro h
     exact absurd (DFunLike.congr_fun h 0) (by
       simp [Finsupp.add_apply, Finsupp.single_eq_same])
   | succ d ih =>
@@ -671,7 +671,7 @@ private theorem coeff_10_FG_pow (F : FormalGroup R) :
     -- For d+1 > n+1, use direct vanishing from nilpotent coefficient
     by_cases hdn : d + 1 ≤ n + 1
     · -- d + 1 ≤ n + 1, i.e., d ≤ n
-      rw [if_pos hdn]
+      rw [ite_eq_left hdn]
       have hdn' : d ≤ n := by omega
       rw [pow_succ, mul_comm, MvPowerSeries.coeff_mul]
       -- The antidiag sum: Σ coeff_e1(F) * coeff_e2(F^d)
@@ -703,7 +703,7 @@ private theorem coeff_10_FG_pow (F : FormalGroup R) :
               (Finsupp.single (0 : Fin 2) 1 + Finsupp.single (1 : Fin 2) (n - 1))
               (F.toSeries ^ d) =
             (d : R) * PowerSeries.coeff (n - d) F.dX_at_zero := by
-          rw [coeff_single1_F, if_pos rfl, one_mul, ih, if_pos (by omega : d ≤ n - 1 + 1)]
+          rw [coeff_single1_F, ite_eq_left rfl, one_mul, ih, ite_eq_left (by omega : d ≤ n - 1 + 1)]
           have : n - 1 + 1 - d = n - d := by omega
           rw [this]
         -- Compute pair B value
@@ -713,7 +713,7 @@ private theorem coeff_10_FG_pow (F : FormalGroup R) :
               F.toSeries *
             MvPowerSeries.coeff (Finsupp.single (1 : Fin 2) d) (F.toSeries ^ d) =
             PowerSeries.coeff (n - d) F.dX_at_zero := by
-          rw [coeff_runit_pow, if_pos rfl, mul_one, FormalGroup.dX_at_zero,
+          rw [coeff_runit_pow, ite_eq_left rfl, mul_one, FormalGroup.dX_at_zero,
             PowerSeries.coeff_mk]
         -- The sum = A_val + B_val = (d+1) * coeff_{n-d} dxF
         -- (the split into the two surviving terms is `coeff_10_FG_pow_antidiag_split`)
@@ -725,7 +725,7 @@ private theorem coeff_10_FG_pow (F : FormalGroup R) :
         · -- A_val + B_val = (d+1) * coeff_{n+1-(d+1)} dxF
           rw [show n + 1 - (d + 1) = n - d from by omega]; push_cast; ring
     · -- d + 1 > n + 1: coeff vanishes by nilpotent degree bound
-      rw [if_neg hdn]
+      rw [ite_eq_right hdn]
       exact coeff_10_FG_pow_succ_vanish F hdn
 
 /-- The explicit `Finset` sum that both sides of `coeff_10_lhs` reduce to:
@@ -770,7 +770,7 @@ private theorem coeff_10_subst_eq_sum (F G : FormalGroup R) (f : FormalGroupHom
       rw [Function.mem_support, ne_eq] at hd
       rw [Finset.mem_coe, Finset.mem_range]
       by_contra h; push Not at h
-      exact hd (if_neg (by omega : ¬(d ≤ n + 1))))]
+      exact hd (ite_eq_right (by omega : ¬(d ≤ n + 1))))]
   -- Kill the d = 0 term (it contributes 0 since d = 0 gives 0 * ... = 0)
   rw [Finset.sum_range_succ' (n := n + 1)]
   simp only [Nat.cast_zero, zero_mul, Nat.sub_zero, le_add_iff_nonneg_left,
@@ -784,7 +784,7 @@ private theorem coeff_10_subst_eq_sum (F G : FormalGroup R) (f : FormalGroupHom
       (↑(k + 1) : R) * PowerSeries.coeff (k + 1) f.toSeries *
         PowerSeries.coeff (n - k) F.dX_at_zero := by
     intro k hk; rw [Finset.mem_range] at hk
-    rw [if_pos (by omega : k + 1 ≤ n + 1)]
+    rw [ite_eq_left (by omega : k + 1 ≤ n + 1)]
     have : n + 1 - (k + 1) = n - k := by omega
     rw [this]
   rw [Finset.sum_congr rfl hsimp]
@@ -795,7 +795,7 @@ unfolds (via `coeff_mul` and `coeff_derivative`) to the explicit range sum
 bijection `k ↦ (k, n - k)`. -/
 private theorem coeff_derivative_mul_dX_eq_sum (F G : FormalGroup R)
     (f : FormalGroupHom F G) (n : ℕ) :
-    PowerSeries.coeff n (PowerSeries.derivative R f.toSeries * F.dX_at_zero) =
+    PowerSeries.coeff n (PowerSeries.derivative (R := R) f.toSeries * F.dX_at_zero) =
     coeff_10_sum F G f n := by
   rw [coeff_10_sum]
   -- Expand the RHS via `coeff_mul` and `coeff_derivative`, then match the range sum against
@@ -824,11 +824,11 @@ private theorem coeff_derivative_mul_dX_eq_sum (F G : FormalGroup R)
 private theorem coeff_10_lhs (F G : FormalGroup R) (f : FormalGroupHom F G) (n : ℕ) :
     MvPowerSeries.coeff (Finsupp.single (0 : Fin 2) 1 + Finsupp.single (1 : Fin 2) n)
       (PowerSeries.subst F.toSeries f.toSeries) =
-    PowerSeries.coeff n (PowerSeries.derivative R f.toSeries * F.dX_at_zero) :=
+    PowerSeries.coeff n (PowerSeries.derivative (R := R) f.toSeries * F.dX_at_zero) :=
   (coeff_10_subst_eq_sum F G f n).trans (coeff_derivative_mul_dX_eq_sum F G f n).symm
 
 theorem FormalGroup.dX_at_zero_chain (f : FormalGroupHom F G) :
-    (PowerSeries.derivative R f.toSeries) * F.dX_at_zero =
+    (PowerSeries.derivative (R := R) f.toSeries) * F.dX_at_zero =
       PowerSeries.C (PowerSeries.coeff 1 f.toSeries) *
         PowerSeries.subst f.toSeries G.dX_at_zero := by
   ext n
@@ -882,14 +882,14 @@ invariant differential.
 Reference: Silverman, *The Arithmetic of Elliptic Curves*, IV.4, Corollary 4.3. -/
 theorem FormalGroup.invariantDiff_chain (f : FormalGroupHom F G) :
     PowerSeries.subst f.toSeries G.invariantDiff *
-      (PowerSeries.derivative R f.toSeries) =
+      (PowerSeries.derivative (R := R) f.toSeries) =
     PowerSeries.C (PowerSeries.coeff 1 f.toSeries) * F.invariantDiff := by
   -- Set up abbreviations to ensure all terms are in PowerSeries R, avoiding
   -- MvPowerSeries Unit R / PowerSeries R defeq issues with rw/simp.
   set c₁ : PowerSeries R := PowerSeries.C (PowerSeries.coeff 1 f.toSeries)
   set ωG' : PowerSeries R := PowerSeries.subst f.toSeries G.invariantDiff
   set dG' : PowerSeries R := PowerSeries.subst f.toSeries G.dX_at_zero
-  set f' : PowerSeries R := PowerSeries.derivative R f.toSeries
+  set f' : PowerSeries R := PowerSeries.derivative (R := R) f.toSeries
   set ωF : PowerSeries R := F.invariantDiff
   set dF : PowerSeries R := F.dX_at_zero
   -- Two ingredients: the `dX_at_zero` chain identity `f' * dF = c₁ * dG'` and the
@@ -1207,7 +1207,7 @@ private lemma coeff_prod_subst_vec_zero_X0 (d e : Fin 2 →₀ ℕ) :
   simp only [Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_zero]
   by_cases hd0 : d 0 = 0
   · simp [hd0]
-  · rw [zero_pow hd0, zero_mul, map_zero, if_neg hd0]
+  · rw [zero_pow hd0, zero_mul, map_zero, ite_eq_right hd0]
 
 /-- Reindexing: a finsum over `d : Fin 2 →₀ ℕ` of a term that vanishes unless
 `d 0 = 0` collapses to a finsum over `n : ℕ` along `n ↦ Finsupp.single 1 n`, the
@@ -1434,4 +1434,4 @@ theorem invariantDiff_translation (F : FormalGroup R) :
 
 end FormalGroup
 
-end HasseWeil.FormalGroup
\ No newline at end of file
+end HasseWeil.FormalGroup
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/EvalGroup.lean b/projects/HasseWeil/HasseWeil/FormalGroup/EvalGroup.lean
index 5d7732dc9..9c4831e16 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/EvalGroup.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/EvalGroup.lean
@@ -9,49 +9,6 @@ import Mathlib.Algebra.Group.MinimalAxioms
 import Mathlib.RingTheory.AdicCompletion.Topology
 import Mathlib.RingTheory.PowerSeries.Evaluation
 
-/-!
-# The group `F(M)` associated to a formal group over a complete local ring
-# (Silverman IV.3, ticket T-IV-3-001)
-
-Let `R` be a commutative local ring with maximal ideal `M = IsLocalRing.maximalIdeal R`,
-equipped with the `M`-adic topology (so that it is both a topological ring with
-a linear topology, a uniform ring, etc.), and assume moreover that it is
-`M`-adically complete (so `R` is Hausdorff and complete for this topology).
-
-For a formal group law `F(X, Y) ∈ R[[X, Y]]`, the set `M` acquires a structure of
-an abelian group via
-  `x +_F y := F(x, y) = MvPowerSeries.eval₂ (RingHom.id R) ![x, y] F.toSeries`.
-
-The convergence of the power series `F(x, y)` is guaranteed by `x, y ∈ M` (they are
-topologically nilpotent, as `M^n → 0` in the adic topology) and by completeness.
-
-## Main definitions
-
-* `HasseWeil.FormalGroup.FormalGroup.evalAdd F x y` — the binary operation
-  `x +_F y` on `M`, returning an element of `R`.
-* `HasseWeil.FormalGroup.FormalGroup.evalAdd_mem F x y` — the operation is closed
-  in `M`, i.e. `evalAdd F x y ∈ M`.
-* `HasseWeil.FormalGroup.FormalGroup.evalNeg F x` — the formal negation
-  `-_F x := i(x)` for `x ∈ M`, where `i = F.inverse` is the formal inverse
-  power series.
-* `HasseWeil.FormalGroup.FormalGroup.evalNeg_mem F x` — the negation is closed
-  in `M`, i.e. `evalNeg F x ∈ M`.
-
-The formal-group axioms induce the group laws for `evalAdd`; the resulting
-operation and `evalNeg` are bundled as an `AddCommGroup` on `F.EvalGroup hAdic`.
-The substitution bridge used for associativity and inverses is proved for the
-`M`-adic uniformity, where `MvPowerSeries.eval₂_subst` does not apply directly.
-
-## Assumptions on `R`
-
-We require `R` to be a `CommRing`, `IsLocalRing`, equipped with topology and uniform
-space structure coming from the `M`-adic topology, and `M`-adically complete.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], IV.3 (Definition of the group
-  `F(M)`, p. 122).
--/
 
 set_option linter.dupNamespace false
 
@@ -127,10 +84,8 @@ noncomputable def FormalGroup.evalAdd (F : FormalGroup R)
 /-! ### Closure under `+_F`: `evalAdd F x y ∈ M` -/
 
 omit [IsUniformAddGroup R] [IsLinearTopology R R] [T2Space R] [CompleteSpace R] in
-/-- The maximal ideal is closed in the `M`-adic topology.
 
-`M` is open (as the basis of neighborhoods of 0 at level 1, i.e. `M^1 = M`),
-hence closed. -/
+
 lemma maximalIdeal_isClosed
     (hAdic : IsAdic (IsLocalRing.maximalIdeal R)) :
     IsClosed (IsLocalRing.maximalIdeal R : Set R) := by
@@ -296,7 +251,7 @@ private lemma coeff_swap_diag (d : Fin 2 →₀ ℕ) :
         ((MvPowerSeries.X (1 : Fin 2) : MvPowerSeries (Fin 2) R) ^ ((finsupp_swap d) 0) *
           (MvPowerSeries.X 0) ^ ((finsupp_swap d) 1)) = 1 := by
   rw [coeff_X_one_pow_mul_X_zero_pow, finsupp_swap_apply_zero, finsupp_swap_apply_one,
-      if_pos]
+      ite_eq_left]
   ext i; fin_cases i <;> simp
 
 omit [IsLocalRing R] [UniformSpace R] [IsUniformAddGroup R] [IsTopologicalRing R]
@@ -450,7 +405,7 @@ private lemma coeff_single_X_pow_mul_zero_pow_of_ne
     rw [show ((MvPowerSeries.X (0 : Fin 2) : MvPowerSeries (Fin 2) R) ^ (d 0)) =
           MvPowerSeries.monomial (R := R) (Finsupp.single 0 (d 0)) 1 from by
         rw [MvPowerSeries.X_pow_eq]]
-    rw [MvPowerSeries.coeff_monomial, if_neg]
+    rw [MvPowerSeries.coeff_monomial, ite_eq_right]
     intro heq
     apply hdj
     have := DFunLike.congr_fun heq 0
@@ -528,9 +483,9 @@ private lemma evalAdd_zero_right_term_eq (F : FormalGroup R)
       rw [hd_single, HasseWeil.FG.FormalGroup.coeff_10]
       rw [show (Finsupp.single (0 : Fin 2) 1) 0 = 1 from by
             simp]
-      rw [pow_one, one_mul, if_pos rfl]
+      rw [pow_one, one_mul, ite_eq_left rfl]
     ·
-      rw [hdeq, F.coeff_j0_of_ne_one (d 0) hd0, zero_mul, if_neg]
+      rw [hdeq, F.coeff_j0_of_ne_one (d 0) hd0, zero_mul, ite_eq_right]
       intro hcontra
       apply hd0
       have := DFunLike.congr_fun hcontra 0
@@ -538,7 +493,7 @@ private lemma evalAdd_zero_right_term_eq (F : FormalGroup R)
   ·
     rw [Finsupp.prod_fintype _ _ (fun i ↦ pow_zero _), Fin.prod_univ_two,
         Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_zero,
-        zero_pow hd1, mul_zero, mul_zero, if_neg]
+        zero_pow hd1, mul_zero, mul_zero, ite_eq_right]
     intro hcontra
     apply hd1
     have := DFunLike.congr_fun hcontra 1
@@ -585,7 +540,7 @@ private lemma coeff_single_zero_pow_mul_X_pow_of_ne
     rw [show ((MvPowerSeries.X (1 : Fin 2) : MvPowerSeries (Fin 2) R) ^ (d 1)) =
           MvPowerSeries.monomial (R := R) (Finsupp.single 1 (d 1)) 1 from by
         rw [MvPowerSeries.X_pow_eq]]
-    rw [MvPowerSeries.coeff_monomial, if_neg]
+    rw [MvPowerSeries.coeff_monomial, ite_eq_right]
     intro heq
     apply hdj
     have := DFunLike.congr_fun heq 1
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroup.lean b/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroup.lean
index 2febf6bff..953c00c2d 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroup.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroup.lean
@@ -197,16 +197,16 @@ theorem conv₂_truncate (n : ℕ) :
        (if n - i < n then formalW_coeff W (n - i) else 0) =
        formalW_coeff W i * formalW_coeff W (n - i)
   rcases lt_or_eq_of_le hi with h_lt | h_eq
-  · rw [if_pos h_lt]
+  · rw [ite_eq_left h_lt]
     by_cases h_ni : n - i < n
-    · rw [if_pos h_ni]
+    · rw [ite_eq_left h_ni]
     · have h_i_eq_zero : i = 0 := by omega
       subst h_i_eq_zero
-      rw [if_neg h_ni]
+      rw [ite_eq_right h_ni]
       rw [show formalW_coeff W 0 = 0 from formalW_coeff_zero W]
       ring
   · subst h_eq
-    rw [if_neg (by omega : ¬ (i < i))]
+    rw [ite_eq_right (by omega : ¬ (i < i))]
     rw [show i - i = 0 from Nat.sub_self i]
     rw [show formalW_coeff W 0 = 0 from formalW_coeff_zero W]
     ring
@@ -223,9 +223,9 @@ theorem conv₂_truncate' (n : ℕ) (hn : 1 ≤ n) :
        (if n - 1 - i < n then formalW_coeff W (n - 1 - i) else 0) =
        formalW_coeff W i * formalW_coeff W (n - 1 - i)
   have h_i_lt_n : i < n := by omega
-  rw [if_pos h_i_lt_n]
+  rw [ite_eq_left h_i_lt_n]
   have h_ni_lt_n : n - 1 - i < n := by omega
-  rw [if_pos h_ni_lt_n]
+  rw [ite_eq_left h_ni_lt_n]
 
 /-- `coeff n (formalW W * formalW W) = conv₂ (formalW_coeff W) n`. -/
 theorem coeff_formalW_sq (n : ℕ) :
@@ -287,19 +287,19 @@ private theorem conv₃_truncate_term {n i j : ℕ} (hi : i ≤ n) (hj : j ≤ n
         (if n - i - j < n then formalW_coeff W (n - i - j) else 0) =
       formalW_coeff W i * formalW_coeff W j * formalW_coeff W (n - i - j) := by
   rcases lt_or_eq_of_le hi with h_i_lt | h_i_eq
-  · rw [if_pos h_i_lt]
+  · rw [ite_eq_left h_i_lt]
     rcases lt_or_eq_of_le hj with h_j_lt | h_j_eq
     · -- j ≤ n - i and j < n - i, so j < n
       have h_j_lt_n : j < n := by omega
-      rw [if_pos h_j_lt_n]
+      rw [ite_eq_left h_j_lt_n]
       by_cases h_k_lt_n : n - i - j < n
-      · rw [if_pos h_k_lt_n]
+      · rw [ite_eq_left h_k_lt_n]
       · -- n - i - j ≥ n means i + j = 0
         have h_i_eq_zero : i = 0 := by omega
         have h_j_eq_zero : j = 0 := by omega
         subst h_i_eq_zero
         subst h_j_eq_zero
-        rw [if_neg h_k_lt_n]
+        rw [ite_eq_right h_k_lt_n]
         rw [show formalW_coeff W 0 = 0 from formalW_coeff_zero W]
         ring
     · -- j = n - i
@@ -313,7 +313,7 @@ private theorem conv₃_truncate_term {n i j : ℕ} (hi : i ≤ n) (hj : j ≤ n
     -- j ≤ n - n = 0, so j = 0
     have h_j_eq_zero : j = 0 := by omega
     subst h_j_eq_zero
-    rw [if_neg (by omega : ¬ (i < i))]
+    rw [ite_eq_right (by omega : ¬ (i < i))]
     rw [show formalW_coeff W 0 = 0 from formalW_coeff_zero W]
     ring
 
@@ -389,7 +389,7 @@ private theorem formalW_step_eq_recurrence_rhs_of_lt_three (n : ℕ) (hn3 : n <
   unfold formalW_step
   dsimp only
   simp only [dite_eq_ite]
-  rw [if_pos hn3, if_neg (by omega : ¬ (n = 3))]
+  rw [ite_eq_left hn3, ite_eq_right (by omega : ¬ (n = 3))]
   interval_cases n
   all_goals
     simp [conv₂, conv₃, formalW_coeff_zero, formalW_coeff_one, formalW_coeff_two,
@@ -408,11 +408,11 @@ private theorem formalW_step_eq_recurrence_rhs_three (n : ℕ) (hn_eq3 : n = 3)
   unfold formalW_step
   dsimp only
   simp only [dite_eq_ite]
-  rw [if_neg (by omega : ¬ (n < 3)), if_pos hn_eq3]
+  rw [ite_eq_right (by omega : ¬ (n < 3)), ite_eq_left hn_eq3]
   subst hn_eq3
   simp only []
-  rw [if_pos (by norm_num : (1 : ℕ) ≤ 3), if_pos (by norm_num : (2 : ℕ) ≤ 3),
-      if_pos (by norm_num : (1 : ℕ) ≤ 3)]
+  rw [ite_eq_left (by norm_num : (1 : ℕ) ≤ 3), ite_eq_left (by norm_num : (2 : ℕ) ≤ 3),
+      ite_eq_left (by norm_num : (1 : ℕ) ≤ 3)]
   rw [show formalW_coeff W (3 - 1) = formalW_coeff W 2 from by norm_num,
       show formalW_coeff W (3 - 2) = formalW_coeff W 1 from by norm_num,
       formalW_coeff_one W, formalW_coeff_two W]
@@ -434,17 +434,17 @@ private theorem formalW_step_eq_recurrence_rhs_of_four_le (n : ℕ) (hn4 : 4 ≤
   dsimp only
   simp only [dite_eq_ite]
   -- Clear the step's `n < 3` / `n = 3` guards (LHS) and the `n = 3` term on the RHS.
-  rw [if_neg (by omega : ¬ (n < 3)), if_neg (by omega : ¬ (n = 3)),
-      if_neg (by omega : ¬ (n = 3))]
-  rw [if_pos (show 1 ≤ n from by omega), if_pos (show 2 ≤ n from by omega),
-      if_pos (show 1 ≤ n from by omega)]
+  rw [ite_eq_right (by omega : ¬ (n < 3)), ite_eq_right (by omega : ¬ (n = 3)),
+      ite_eq_right (by omega : ¬ (n = 3))]
+  rw [ite_eq_left (show 1 ≤ n from by omega), ite_eq_left (show 2 ≤ n from by omega),
+      ite_eq_left (show 1 ≤ n from by omega)]
   -- Use conv₂_truncate, conv₃_truncate for the LHS
   rw [conv₂_truncate W n, conv₂_truncate' W n (by omega), conv₃_truncate W n]
   -- The truncated w(n-1) and w(n-2) become formalW_coeff (already beta-reduced)
   rw [show (if n - 1 < n then formalW_coeff W (n - 1) else 0) = formalW_coeff W (n - 1) from
-        if_pos (by omega)]
+        ite_eq_left (by omega)]
   rw [show (if n - 2 < n then formalW_coeff W (n - 2) else 0) = formalW_coeff W (n - 2) from
-        if_pos (by omega)]
+        ite_eq_left (by omega)]
   ring
 
 /-- The `formalW_step` recursion equals the recurrence right-hand side from
@@ -491,23 +491,5 @@ theorem formalW_recurrence :
         formalW_coeff_eq_step W n]
   exact formalW_step_eq_recurrence_rhs W n
 
-/-! ### Uniqueness of `formalW` (Silverman IV.1.1(b))
-
-Uniqueness of `formalW` follows the factoring pattern: if `d := w' - formalW W`,
-then `d = K · d` for some `K` with zero constant coefficient, hence `d = 0`
-by `PowerSeries.eq_zero_of_self_eq_mul_self`
-(see `HasseWeil/PowerSeriesHelpers.lean`).
-
-**Status**: infrastructure ready (`eq_zero_of_self_eq_mul_self`), but the
-factoring step `RHS(w') − RHS(formalW W) = K · d` cannot be established by
-`ring`/`linear_combination`/`abel` because of a `PowerSeries R` typeclass
-gap: `RightDistribClass (PowerSeries R)` and `IsRightCancelAdd (PowerSeries R)`
-fail to synthesize in Lean 4.29 / mathlib v4.29.0-rc6, blocking the ring
-tactic on distributivity-heavy identities. The factoring works on simple
-identities in isolation but stops mid-normalisation on the multi-term
-combination needed here. A coefficient-induction route (via
-`PowerSeries.ext` and strong induction on `n`, using `coeff_mul`) avoids
-the ring issue and is the recommended next step.
--/
 
 end HasseWeil
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupAssoc.lean b/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupAssoc.lean
index b2a59710e..f835d413b 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupAssoc.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupAssoc.lean
@@ -122,7 +122,7 @@ theorem formalGroupLaw_coeff_right_unit (n : ℕ) :
   -- d 0 = n, d 1 = 0, so the definition enters the "j = 0" branch
   -- and returns "if i = 1 then 1 else 0" = "if n = 1 then 1 else 0".
   simp only [formalGroupLaw_coeff]
-  simp only [Finsupp.single_apply, if_true, if_false, show (0 : Fin 2) ≠ 1 from by decide]
+  simp only [Finsupp.single_apply, ite_true, ite_false, show (0 : Fin 2) ≠ 1 from by decide]
   split_ifs <;> simp_all
 
 /-- `F(0, Y) = Y`: coefficient `F_{0,n} = [n=1]`. -/
@@ -130,7 +130,7 @@ theorem formalGroupLaw_coeff_left_unit (n : ℕ) :
     formalGroupLaw_coeff W (Finsupp.single 1 n) =
       if n = 1 then 1 else 0 := by
   simp only [formalGroupLaw_coeff]
-  simp only [Finsupp.single_apply, if_true, if_false, show (1 : Fin 2) ≠ 0 from by decide]
+  simp only [Finsupp.single_apply, ite_true, ite_false, show (1 : Fin 2) ≠ 0 from by decide]
 
 /-! ### The ring homomorphism property of the pullback coefficient (Silverman III.5.6) -/
 
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupCorrespondence.lean b/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupCorrespondence.lean
index c3196d467..960671107 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupCorrespondence.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupCorrespondence.lean
@@ -5,41 +5,6 @@ import Mathlib.RingTheory.Kaehler.Basic
 import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
 import Mathlib.LinearAlgebra.Basis.VectorSpace
 
-/-!
-# Formal Group ↔ Curve Correspondence (Silverman IV.1–2, IV.4)
-
-This file establishes the connection between the formal group of an elliptic curve
-and the curve's group law. The key results:
-
-1. **Local parameter**: z = -x/y is a local uniformizer at O (Silverman IV.1).
-2. **w(z)**: The power series solving w = f(z,w) (already in FormalGroup.lean).
-3. **Formal group law**: z(P+Q) = F(z(P), z(Q)) where F is the formal group law.
-4. **Pullback coefficient**: For [m], the formal group series has linear coefficient m.
-
-## The Key Connection (Silverman IV.4)
-
-For an endomorphism φ of E, the pullback coefficient a_φ is defined by:
-  φ*ω = a_φ · ω
-where ω = dx/(2y+a₁x+a₃) is the invariant differential.
-
-Equivalently (via the formal group): φ induces a power series φ_F(T) = a_φ T + O(T²),
-and a_φ is the linear coefficient.
-
-The map φ ↦ a_φ is a ring homomorphism End(E) → K̄ (Silverman Cor. III.5.6).
-
-## Kähler Module is 1-Dimensional (Silverman III.1.5)
-
-For an elliptic curve E/K (genus 1), the space Ω_{K(E)/K} of differentials is
-1-dimensional over K(E). This means every differential η can be written as c·ω
-for a unique c ∈ K(E).
-
-This follows from the Riemann-Roch theorem (genus = dim Ω) applied to the
-function field K(E) of transcendence degree 1 over K.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, III.1.5, III.5, IV.1–4
--/
 
 open WeierstrassCurve
 
@@ -286,13 +251,12 @@ private lemma span_invariantDifferential_eq_top :
     (Set.singleton_subset_iff.mpr (D_coordXFF_mem_span_invariantDifferential E))
     (D_mem_derivSpanX E f)
 
--- The Kähler differential module Ω[K(E)/F] is generated by dx/(2y+a₁x+a₃)
--- as a K(E)-module. Every element η ∈ Ω is of the form c · ω for some c ∈ K(E).
+
 -- This follows from E having genus 1: by Riemann-Roch, dim_K(E) Ω = g = 1.
 -- Reference: Silverman III.1.5, Prop. II.4.2(a).
 theorem kaehler_rank_one :
     Module.finrank E.FunctionField (KaehlerDifferential F E.FunctionField) = 1 := by
-  haveI : Module.Free E.FunctionField (KaehlerDifferential F E.FunctionField) :=
+  have : Module.Free E.FunctionField (KaehlerDifferential F E.FunctionField) :=
     Module.Free.of_divisionRing _ _
   rw [finrank_eq_one_iff']
   refine ⟨invariantDifferential E, invariantDifferential_ne_zero E, fun w ↦ ?_⟩
@@ -309,6 +273,7 @@ section MulByMFormal
 
 variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The formal group power series [m]_F(T) = m·T + O(T²) for the multiplication
     by m map. The linear coefficient is m.
 
@@ -321,6 +286,7 @@ theorem formalMulByInt_linear_coeff (m : ℤ) :
     formalMulByInt_coeff W.toAffine m 1 = (m : F) := by
   simp only [formalMulByInt_coeff, one_ne_zero, ↓reduceIte]
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The multiplication-by-m on the formal group starts with 0 (no constant term). -/
 theorem formalMulByInt_const_zero (m : ℤ) :
     formalMulByInt_coeff W.toAffine m 0 = 0 := by
@@ -352,6 +318,7 @@ section Correspondence
 
 variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The formal group ↔ curve correspondence for the pullback coefficient.
 
     For the multiplication-by-m endomorphism [m]:
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupLawSpec.lean b/projects/HasseWeil/HasseWeil/FormalGroup/FormalGroupLawSpec.lean
index b652da19cfba4c065baf926dcd4db03ff6447aa1..e63da5ebe93847d80c49394dc1eddf0cefbfaea9 100644
GIT binary patch
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HsK){TR6)s%

diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/InvariantDiff.lean b/projects/HasseWeil/HasseWeil/FormalGroup/InvariantDiff.lean
index a72b56c04..376ff33c3 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/InvariantDiff.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/InvariantDiff.lean
@@ -5,37 +5,6 @@ Authors: Chris Birkbeck
 -/
 import HasseWeil.FormalGroup.Differential
 
-/-!
-# Invariant Differentials on Formal Groups (Silverman IV.4)
-
-This file packages `FormalGroup.invariantDiff` into a formal structure and exposes
-the main results of Silverman IV.4.2 and IV.4.3:
-
-## Main definitions
-
-* `InvariantDifferential F` — a power series `P(T) ∈ R⟦T⟧` such that
-  `P(T) · F_X(0, T)` is a constant in `R⟦T⟧`. This characterization is
-  equivalent to Silverman's translation-invariance axiom
-  `P(F(T, S)) · F_T(T, S) = P(T)`.
-* `InvariantDifferential.IsNormalized` — an invariant differential is
-  normalized when its constant coefficient is `1`.
-* `FormalGroup.normalizedDifferential F` — the canonical normalized invariant
-  differential `ω_F = F_X(0, T)⁻¹`.
-
-## Main results
-
-* `FormalGroup.normalizedDifferential_isNormalized` — `ω_F` is normalized.
-* `FormalGroup.normalizedDifferential_unique` — every normalized invariant
-  differential equals `ω_F`.
-* `InvariantDifferential.eq_smul_normalized` — every invariant differential is
-  of the form `a · ω_F` for a unique `a ∈ R`.
-* `FormalGroupHom.invariantDifferential_chain` — **Silverman IV.4.3**: for a
-  formal group homomorphism `f : F → G`, `ω_G ∘ f = f'(0) · ω_F` (chain rule).
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], IV.4, Prop 4.2 and Cor 4.3.
--/
 
 set_option linter.dupNamespace false
 
@@ -159,7 +128,7 @@ Reference: Silverman, *The Arithmetic of Elliptic Curves*, IV.4, Cor. 4.3. -/
 theorem FormalGroupHom.invariantDifferential_chain {F G : FormalGroup R}
     (f : FormalGroupHom F G) :
     PowerSeries.subst f.toSeries G.normalizedDifferential.toSeries *
-      (PowerSeries.derivative R f.toSeries) =
+      (PowerSeries.derivative (R := R) f.toSeries) =
     PowerSeries.C (PowerSeries.coeff 1 f.toSeries) *
       F.normalizedDifferential.toSeries :=
   FormalGroup.invariantDiff_chain f
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/Inverse.lean b/projects/HasseWeil/HasseWeil/FormalGroup/Inverse.lean
index 5de07e672..669d4f375 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/Inverse.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/Inverse.lean
@@ -6,43 +6,6 @@ Authors: Chris Birkbeck
 import HasseWeil.FormalGroup.Definition
 import HasseWeil.FormalGroup.Logarithm
 
-/-!
-# The formal inverse of a formal group (Silverman IV.2)
-
-For a formal group law `F(X, Y) ∈ R[[X, Y]]`, this file constructs the
-**formal inverse** power series `i(T) ∈ R[[T]]` characterised by
-
-`F(T, i(T)) = 0`,       `constantCoeff i = 0`.
-
-The existence of `i(T)` is a consequence of the unit axioms on `F` together
-with a routine coefficient-by-coefficient recursion: writing
-`F(X, Y) = X + Y + (higher order)`, the relation `F(T, i(T)) = 0` forces
-`coeff 1 i = -1` and then determines `coeff n i` for `n ≥ 2` from the
-previously-computed coefficients.
-
-## Main definitions
-
-* `HasseWeil.FormalGroup.FormalGroup.inverseTrunc F n` — the degree-`n`
-  truncation (as a power series) of the formal inverse.
-* `HasseWeil.FormalGroup.FormalGroup.inverseCoeff F n` — the coefficient at
-  `T^n` of the formal inverse, stabilised from the truncation.
-* `HasseWeil.FormalGroup.FormalGroup.inverse F` — the formal inverse power
-  series `i(T)`.
-
-## Main results
-
-* `FormalGroup.inverse_constantCoeff` — `constantCoeff i = 0`.
-* `FormalGroup.inverse_coeff_zero` — `coeff 0 i = 0`.
-* `FormalGroup.inverse_coeff_one` — `coeff 1 i = -1`.
-* `FormalGroup.fAdd_X_inverse_eq_zero` — the functional equation
-  `F(T, i(T)) = 0`. Its proof mirrors `subst_compInverse_eq_X` in
-  `HasseWeil/FormalGroup/Logarithm.lean`, but for the two-variable implicit
-  equation.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], IV.2.
--/
 
 set_option linter.dupNamespace false
 
@@ -130,7 +93,7 @@ theorem FormalGroup.coeff_inverseTrunc_succ_of_le (F : FormalGroup R)
         PowerSeries.C _ * PowerSeries.X ^ (n + 1)) = _
   rw [map_add, PowerSeries.coeff_C_mul_X_pow]
   -- `k ≠ n + 1` since `k ≤ n`.
-  rw [if_neg (by omega : k ≠ n + 1), add_zero]
+  rw [ite_eq_right (by omega : k ≠ n + 1), add_zero]
 
 /-- For `k ≤ n`, the `k`-th coefficient of `inverseTrunc F n` equals the
 stable coefficient `inverseCoeff F k`. -/
@@ -192,7 +155,7 @@ theorem FormalGroup.inverse_coeff_one (F : FormalGroup R) :
         PowerSeries.X ^ (0 + 1)) = -1
   rw [map_add, FormalGroup.inverseTrunc_zero, map_zero, zero_add]
   -- Simplify the coefficient expression.
-  rw [PowerSeries.coeff_C_mul_X_pow, if_pos (by norm_num : (1 : ℕ) = 0 + 1)]
+  rw [PowerSeries.coeff_C_mul_X_pow, ite_eq_left (by norm_num : (1 : ℕ) = 0 + 1)]
   -- Goal: `-coeff 1 (fAdd F X 0) = -1`, i.e., `coeff 1 (fAdd F X 0) = 1`.
   have hX : @PowerSeries.constantCoeff R _ PowerSeries.X = 0 := by simp
   have hzero : @PowerSeries.constantCoeff R _ (0 : PowerSeries R) = 0 := by simp
@@ -247,7 +210,7 @@ private theorem coeff_X_pow_mul_pow_eq_of_coeff_eq
       have e1 : PowerSeries.coeff j (g₁ ^ b) = 0 := coeff_pow_eq_zero_of_gt g₁ h1 j b hjb
       have e2 : PowerSeries.coeff j (g₂ ^ b) = 0 := coeff_pow_eq_zero_of_gt g₂ h2 j b hjb
       rw [e1, e2]
-  · rw [PowerSeries.coeff_X_pow, if_neg hia, zero_mul, zero_mul]
+  · rw [PowerSeries.coeff_X_pow, ite_eq_right hia, zero_mul, zero_mul]
 
 /-- **Coefficient of `fAdd F X g` as a finsum.** For `g` with zero constant
 coefficient, the `k`-th coefficient of `fAdd F X g` expands as the finsum over
@@ -308,7 +271,7 @@ private theorem coeff_X_pow_mul_eq_zero_of_lt
   apply Finset.sum_eq_zero
   intro ⟨p, q⟩ hpq
   have hpq' : p + q = k := Finset.mem_antidiagonal.mp hpq
-  rw [PowerSeries.coeff_X_pow, if_neg (by omega : p ≠ N), zero_mul]
+  rw [PowerSeries.coeff_X_pow, ite_eq_right (by omega : p ≠ N), zero_mul]
 
 /-- The `m`-th binomial term `(C c * X^(n+1))^m * g^(d-m) * (d.choose m)` vanishes
 at every degree `k < n+1` once `0 < m`: the monomial power becomes `C (c^m)`
@@ -398,7 +361,7 @@ private theorem coeff_X_pow_mul_add_monomial_pow
   · -- a ≥ 1: the "if" vanishes, and we need to show
     -- coeff (n+1) (X^a * (g+h)^j) = coeff (n+1) (X^a * g^j).
     have hand_fail : ¬ (a = 0 ∧ j = 1) := fun hand ↦ ha0 hand.1
-    rw [if_neg hand_fail, add_zero]
+    rw [ite_eq_right hand_fail, add_zero]
     -- By coeff_mul, coeff (n+1) (X^a * f) = coeff (n+1 - a) f (when a ≤ n+1).
     rw [PowerSeries.coeff_mul, PowerSeries.coeff_mul]
     apply Finset.sum_congr rfl
@@ -416,7 +379,7 @@ private theorem coeff_X_pow_mul_add_monomial_pow
       rw [hpa]
       congr 1
       exact coeff_add_monomial_pow_stable g hg n c j q hq
-    · rw [PowerSeries.coeff_X_pow, if_neg hpa, zero_mul, zero_mul]
+    · rw [PowerSeries.coeff_X_pow, ite_eq_right hpa, zero_mul, zero_mul]
 
 /-- The finsum of `coeff d F • (if d 0 = 0 ∧ d 1 = 1 then c else 0)` over all
 multi-indices is `c`. Only the index `Finsupp.single 1 1` contributes, where the
@@ -434,11 +397,11 @@ private theorem finsum_coeff_smul_indicator_eq
       apply hd
       ext i
       fin_cases i <;> simp [h0, h1]
-    rw [if_neg this, smul_zero])]
+    rw [ite_eq_right this, smul_zero])]
   -- Eval at d = Finsupp.single 1 1: d 0 = 0, d 1 = 1.
   have hd01 : (Finsupp.single (1 : Fin 2) 1) 0 = 0 := by simp
   have hd11 : (Finsupp.single (1 : Fin 2) 1) 1 = 1 := by simp
-  rw [hd01, hd11, if_pos ⟨rfl, rfl⟩]
+  rw [hd01, hd11, ite_eq_left ⟨rfl, rfl⟩]
   -- coeff (Finsupp.single 1 1) F.toSeries = 1 by the right-unit axiom.
   rw [HasseWeil.FG.FormalGroup.coeff_01, one_smul]
 
@@ -460,7 +423,7 @@ private theorem support_finite_coeff_smul_indicator
     apply hne
     ext i
     fin_cases i <;> simp [h0, h1]
-  rw [if_neg this, smul_zero]
+  rw [ite_eq_right this, smul_zero]
 
 /-- The map `d ↦ coeff d F • coeff k (X^(d 0) * g^(d 1))` has finite support;
 this is the support of the `![X, g]`-substitution coefficient finsum, finite by
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/Logarithm.lean b/projects/HasseWeil/HasseWeil/FormalGroup/Logarithm.lean
index 401695b65..199e735cd 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/Logarithm.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/Logarithm.lean
@@ -6,33 +6,6 @@ Authors: Chris Birkbeck
 import HasseWeil.FormalGroup.InvariantDiff
 import Mathlib.Algebra.Module.Rat
 
-/-!
-# The formal logarithm of a formal group (Silverman IV.5)
-
-For a formal group `F` over a `ℚ`-algebra `R` (or more generally a ring with a
-`Module ℚ R` structure), the **formal logarithm**
-
-`log_F(T) := ∫₀^T ω_F(s) ds = T + (c₁/2) T² + (c₂/3) T³ + ⋯`
-
-is the integral of the normalized invariant differential
-`ω_F(T) = 1 + c₁T + c₂T² + ⋯`. It is a power series with
-constant term `0` and linear coefficient `1`.
-
-## Main definition
-
-* `HasseWeil.FormalGroup.FormalGroup.log F` — the formal logarithm of `F`,
-  defined over any commutative ring `R` equipped with a `ℚ`-module structure.
-
-## Main results
-
-* `FormalGroup.log_coeff_zero` — `constantCoeff (log F) = 0`.
-* `FormalGroup.log_coeff_succ` — `coeff (n + 1) (log F) = (1/(n+1)) • coeff n ω_F`.
-* `FormalGroup.log_coeff_one` — `coeff 1 (log F) = 1`.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], IV.5.
--/
 
 set_option linter.dupNamespace false
 
@@ -40,16 +13,7 @@ namespace HasseWeil.FormalGroup
 
 variable {R : Type*} [CommRing R]
 
-/-- The **formal logarithm** of a formal group `F` over a ring `R` with a
-`Module ℚ R` structure.
-
-Concretely, if `ω_F(T) = 1 + c₁T + c₂T² + ⋯`, then
-`log_F(T) = T + (c₁/2)T² + (c₂/3)T³ + ⋯ = ∑ (c_{n-1}/n)·T^n`.
 
-The coefficient at `T^n` for `n ≥ 1` is the ℚ-scalar multiple
-`((n : ℚ)⁻¹) • (coeff of ω_F at n - 1)`.
-
-Reference: Silverman, *The Arithmetic of Elliptic Curves*, IV.5. -/
 noncomputable def FormalGroup.log (F : FormalGroup R) [Module ℚ R] :
     PowerSeries R :=
   PowerSeries.mk fun n ↦
@@ -91,24 +55,6 @@ theorem FormalGroup.log_coeff_one (F : FormalGroup R) [Module ℚ R] :
   rw [PowerSeries.coeff_zero_eq_constantCoeff_apply]
   exact F.normalizedDifferential_isNormalized
 
-/-! ### The formal exponential
-
-We construct `exp_F` as the compositional inverse of `log_F` via an iterative
-truncation argument.
-
-Let `f : PowerSeries R` with `coeff 0 f = 0` and `coeff 1 f = 1`
-(e.g., `f = log_F`). We build the compositional inverse `g` incrementally:
-
-* `compInvTrunc f 0 = 0`.
-* `compInvTrunc f (n+1) = compInvTrunc f n + c · T^(n+1)` where the correction
-  `c` is chosen so that `coeff (n+1) (f ∘ compInvTrunc f (n+1)) = [n+1 = 1]`
-  (Kronecker delta). Since `f = T + O(T²)`, this determines `c` uniquely:
-  `c = δ_{1, n+1} - coeff (n+1) (f(compInvTrunc f n))`.
-
-By induction, `compInvTrunc f n` has the correct coefficients up to degree `n`,
-and adding a `T^(n+1)` correction doesn't change any lower coefficient.
-The compositional inverse is recovered by taking the `n`-th coefficient of
-`compInvTrunc f n`. -/
 
 /-- Iterative truncation of the compositional inverse of `f`.
 At each step we add a correction of the form `c · T^(n+1)` chosen to zero out
@@ -205,7 +151,7 @@ theorem compInverse_coeff_one (f : PowerSeries R) :
            else -PowerSeries.coeff (0 + 1)
               (PowerSeries.subst (compInvTrunc f 0) f)) *
         PowerSeries.X ^ (0 + 1)) = 1
-  rw [compInvTrunc_zero, if_pos rfl, coeff_one_subst_zero, sub_zero, zero_add]
+  rw [compInvTrunc_zero, ite_eq_left rfl, coeff_one_subst_zero, sub_zero, zero_add]
   -- Goal: coeff 1 (C 1 * X^1) = 1.
   simp
 
@@ -248,7 +194,7 @@ theorem coeff_compInvTrunc_succ_of_le (f : PowerSeries R) (n k : ℕ) (hk : k 
         PowerSeries.C _ * PowerSeries.X ^ (n + 1)) = _
   rw [map_add, PowerSeries.coeff_C_mul_X_pow]
   -- `k ≠ n + 1` since `k ≤ n`.
-  rw [if_neg (by omega : k ≠ n + 1), add_zero]
+  rw [ite_eq_right (by omega : k ≠ n + 1), add_zero]
 
 open PowerSeries in
 /-- For `k ≤ n`, the `k`-th coefficient of `compInvTrunc f n` equals the
@@ -383,7 +329,7 @@ theorem coeff_monomial_pow_high (n : ℕ) (c : R) (i : ℕ) (hi : 2 ≤ i) :
     PowerSeries.coeff (n + 1) ((PowerSeries.C c * PowerSeries.X ^ (n + 1)) ^ i) = 0 := by
   rw [monomial_pow_eq]
   rw [PowerSeries.coeff_C_mul_X_pow]
-  rw [if_neg]
+  rw [ite_eq_right]
   intro heq
   -- n + 1 = i * (n + 1) with i ≥ 2 contradicts n + 1 ≥ 1.
   nlinarith
@@ -460,7 +406,7 @@ private theorem coeff_add_monomial_pow_eq_zero (g : PowerSeries R) (n : ℕ) (c
       PowerSeries.coeff (n + 1) (g ^ 0) + (if (0 : ℕ) = 1 then c else 0) := by
   rw [coeff_add_monomial_pow_eq_sum, Finset.sum_range_one]
   simp only [Nat.choose_self, Nat.cast_one, one_mul, pow_zero, Nat.sub_self, mul_one]
-  rw [if_neg (by decide : (0 : ℕ) ≠ 1), add_zero]
+  rw [ite_eq_right (by decide : (0 : ℕ) ≠ 1), add_zero]
 
 open PowerSeries in
 /-- The `d = 1` case of `coeff_add_monomial_pow_eq`: the expansion has two terms,
@@ -478,11 +424,11 @@ private theorem coeff_add_monomial_pow_eq_one (g : PowerSeries R) (n : ℕ) (c :
   rw [show (h ^ 0 : PowerSeries R) = 1 from pow_zero _]
   rw [show (h ^ 1 : PowerSeries R) = h from pow_one _]
   rw [show (g ^ 0 : PowerSeries R) = 1 from pow_zero _]
-  rw [if_pos rfl]
+  rw [ite_eq_left rfl]
   simp only [Nat.cast_one, one_mul, mul_one]
   -- Goal: coeff (n+1) g + coeff (n+1) h = coeff (n+1) g + c
   rw [show PowerSeries.coeff (n + 1) h = c from by
-        rw [hdef, PowerSeries.coeff_C_mul_X_pow, if_pos rfl]]
+        rw [hdef, PowerSeries.coeff_C_mul_X_pow, ite_eq_left rfl]]
 
 open PowerSeries in
 /-- The `d + 2` case of `coeff_add_monomial_pow_eq`: split the binomial sum into
@@ -517,7 +463,7 @@ private theorem coeff_add_monomial_pow_eq_succ_succ (g : PowerSeries R)
   rw [Finset.sum_eq_zero (fun m hm ↦ by
     simp only [Finset.mem_filter, Finset.mem_range] at hm
     rw [coeff_monomial_pow_mul_eq_zero g n c hm.2 (d + 2 - m), mul_zero]), add_zero]
-  rw [if_neg (by omega : d + 2 ≠ 1), add_zero]
+  rw [ite_eq_right (by omega : d + 2 ≠ 1), add_zero]
   -- The m = 1 term vanishes since g has zero constant coefficient.
   rw [coeff_monomial_mul_pow_succ_eq_zero g hg n c d, mul_zero, add_zero]
 
@@ -577,8 +523,8 @@ theorem coeff_subst_add_monomial (f g : PowerSeries R) (n : ℕ) (c : R)
   congr 1
   rw [finsum_eq_single _ 1 (by
     intro d hd
-    rw [if_neg hd, smul_zero])]
-  rw [if_pos rfl, smul_eq_mul, mul_comm]
+    rw [ite_eq_right hd, smul_zero])]
+  rw [ite_eq_left rfl, smul_eq_mul, mul_comm]
 
 /-! ### The core invariant and the compositional-inverse identity -/
 
@@ -881,15 +827,7 @@ theorem compInverseOfUnit_coeff_one (f : PowerSeries R) (u : R) (hu : IsUnit u)
   -- Goal: v ^ 1 * coeff 1 (compInverse (v • f)) = v.
   rw [pow_one, compInverse_coeff_one, mul_one]
 
-/-- The **formal exponential** of a formal group `F` over a ring with a
-`ℚ`-module structure.
-
-Defined as the compositional inverse of `log_F`, via `compInverse`.
-The full inverse identity `log_F ∘ exp_F = X` (Silverman IV.5.2) is future
-work — this file only establishes the definition and basic coefficient
-properties.
 
-Reference: Silverman, *The Arithmetic of Elliptic Curves*, IV.5. -/
 noncomputable def FormalGroup.exp (F : FormalGroup R) [Module ℚ R] :
     PowerSeries R :=
   compInverse F.log
@@ -936,14 +874,7 @@ theorem FormalGroup.commutative_of_torsion_free (F : FormalGroup R)
         F.toSeries = F.toSeries :=
   F.commutative
 
-/-- **Silverman IV.5.4 (coefficient identity)**: for `n ≥ 1`,
-`n • log_F.coeff n = ω_F.coeff (n - 1)`.
 
-This is the `ℚ`-module-level identity that, after multiplying by `(n : ℚ)⁻¹`,
-recovers the explicit formula `log_F.coeff n = (n : ℚ)⁻¹ • ω_F.coeff (n-1)`
-(which is the content of `log_coeff_succ`). Over a torsion-free `ℤ`-algebra
-it shows `n · log_F.coeff n` lies in the image of `ω_F`, which is the
-`ℤ[a_i]`-containment stated in Silverman. -/
 theorem FormalGroup.log_coeff_succ_nsmul (F : FormalGroup R) [Module ℚ R] (n : ℕ) :
     (n + 1) • PowerSeries.coeff (n + 1) F.log =
       PowerSeries.coeff n F.normalizedDifferential.toSeries := by
@@ -1064,23 +995,23 @@ private theorem FormalGroup.dX_at_zero_additiveFormalGroup [Module ℚ R] :
   | 0 =>
     -- The LHS: (if pos then 1 else 0) + (if single 0 1 = single 1 1 then 1 else 0)
     -- = 1 + 0 = 1.
-    simp only [Finsupp.single_zero, add_zero, if_true]
-    rw [if_neg (by
+    simp only [Finsupp.single_zero, add_zero, ite_true]
+    rw [ite_eq_right (by
       intro h
       have := DFunLike.congr_fun h 0
       simp [Finsupp.single_eq_same] at this)]
     rw [add_zero, PowerSeries.coeff_zero_eq_constantCoeff_apply]
     simp
   | k + 1 =>
-    rw [if_neg (by
+    rw [ite_eq_right (by
       intro h
       have := DFunLike.congr_fun h 1
       simp [Finsupp.add_apply, Finsupp.single_eq_same] at this)]
-    rw [if_neg (by
+    rw [ite_eq_right (by
       intro h
       have h0 := DFunLike.congr_fun h 0
       simp [Finsupp.add_apply, Finsupp.single_eq_same] at h0)]
-    rw [zero_add, PowerSeries.coeff_one, if_neg (by omega : k + 1 ≠ 0)]
+    rw [zero_add, PowerSeries.coeff_one, ite_eq_right (by omega : k + 1 ≠ 0)]
 
 /-- For the additive formal group, the normalized differential `ω_{Ĝ_a}` is the
 constant series `1`. This follows from `dX_at_zero = 1` (see
@@ -1121,8 +1052,8 @@ theorem FormalGroup.log_additiveFormalGroup [Module ℚ R] :
     -- so its coefficient at `n + 1 ≥ 1` is `0`.
     rw [FormalGroup.normalizedDifferential_toSeries_additiveFormalGroup]
     -- Goal: ((n+2)⁻¹ : ℚ) • coeff (n+1) (1 : PowerSeries R) = coeff (n+2) X
-    rw [PowerSeries.coeff_one, if_neg (by omega : n + 1 ≠ 0), smul_zero,
-      PowerSeries.coeff_X, if_neg (by omega : n + 1 + 1 ≠ 1)]
+    rw [PowerSeries.coeff_one, ite_eq_right (by omega : n + 1 ≠ 0), smul_zero,
+      PowerSeries.coeff_X, ite_eq_right (by omega : n + 1 + 1 ≠ 1)]
 
 /-- **Silverman IV.5.2 for `Ĝ_a`**: the additive formal group's log is
 the identity, and as a trivial consequence, it preserves addition. -/
@@ -1146,15 +1077,6 @@ theorem FormalGroup.additiveFormalGroup_logPreservesAdd [Module ℚ R] :
   -- Goal: (additiveFormalGroup R).toSeries = X 0 + X 1. This is rfl by definition.
   rfl
 
-/-! #### The general case: `LogPreservesAdd F` via Silverman IV.4.2
-
-The proof follows Silverman IV.5: differentiate both sides of
-`log_F(F(X, Y)) = log_F(X) + log_F(Y)` with respect to `X`. The LHS derivative
-(chain rule + IV.4.2 translation invariance) equals `ω_F(X)`; the RHS
-derivative is `ω_F(X)` directly. Hence both sides have the same partial
-derivative in `X`. Combined with agreement at `X = 0` (via the right unit
-`F(0, Y) = Y` and `log_F(0) = 0`), they are equal because `Module ℚ R`
-makes `R` torsion-free. -/
 
 /-- **Key derivative identity**: `pderiv' () F.log = F.invariantDiff`
 (both viewed as `MvPowerSeries Unit R = PowerSeries R`).
@@ -1210,9 +1132,7 @@ private theorem pderiv_PowerSeries_subst {τ : Type*}
 /-! #### Uniqueness: a two-variable series is determined by its derivative in
 variable 0 and its value at `X 0 = 0`. -/
 
-/-- Auxiliary: if `h : MvPowerSeries (Fin 2) R` has zero derivative in variable
-`0` (over a `Module ℚ R`), then all coefficients with positive 0-degree vanish.
--/
+
 private theorem coeff_zero_of_pderiv_zero_fin2 [Module ℚ R]
     (h : MvPowerSeries (Fin 2) R) (hd : MvPowerSeries.pderiv' 0 h = 0)
     (e : Fin 2 →₀ ℕ) (he : e 0 ≠ 0) :
@@ -1226,9 +1146,9 @@ private theorem coeff_zero_of_pderiv_zero_fin2 [Module ℚ R]
     intro i
     by_cases hi : i = 0
     · subst hi
-      rw [Finsupp.single_apply, if_pos rfl, ha]
+      rw [Finsupp.single_apply, ite_eq_left rfl, ha]
       exact Nat.succ_pos a
-    · rw [Finsupp.single_apply, if_neg (Ne.symm hi)]
+    · rw [Finsupp.single_apply, ite_eq_right (Ne.symm hi)]
       exact Nat.zero_le _
   have hd_sum : d + Finsupp.single (0 : Fin 2) 1 = e := by
     rw [hd_def, tsub_add_cancel_of_le hle]
@@ -1520,7 +1440,7 @@ private theorem coeff_subst_zero_X1_at_single_1 (h : MvPowerSeries (Fin 2) R) (b
   rw [finsum_eq_single _ (Finsupp.single (1 : Fin 2) b)]
   · -- Case d = single 1 b: d.prod _ = (X 1)^b, coeff (single 1 b) ((X 1)^b) = 1.
     rw [prod_eval_zero_X1_of_fst_eq_zero _ (by simp), coeff_single_1_X1_pow,
-      if_pos (by simp), smul_eq_mul, mul_one]
+      ite_eq_left (by simp), smul_eq_mul, mul_one]
   · -- For d ≠ single 1 b: the contribution is zero.
     intro d hd
     -- Two cases: d 0 ≠ 0 OR (d 0 = 0 AND d 1 ≠ b).
@@ -1535,7 +1455,7 @@ private theorem coeff_subst_zero_X1_at_single_1 (h : MvPowerSeries (Fin 2) R) (b
           rw [h0]; simp
         · change d 1 = (Finsupp.single (1 : Fin 2) b) 1
           rw [h1]; simp
-      rw [prod_eval_zero_X1_of_fst_eq_zero _ h0, coeff_single_1_X1_pow, if_neg hd1, smul_zero]
+      rw [prod_eval_zero_X1_of_fst_eq_zero _ h0, coeff_single_1_X1_pow, ite_eq_right hd1, smul_zero]
     · -- d 0 ≠ 0: d.prod = 0.
       rw [prod_eval_zero_X1_of_fst_ne_zero _ h0, map_zero, smul_zero]
 
@@ -1601,15 +1521,6 @@ theorem FormalGroup.logPreservesAdd (F : FormalGroup R) [Module ℚ R] :
   exact eq_zero_of_pderiv_zero_and_subst_zero_X1 h
     (hh ▸ pderiv_zero_LogPreservesAdd_diff F) (hh ▸ subst_zero_X1_LogPreservesAdd_diff F)
 
-/-! #### Packaging `log_F` as a formal group homomorphism
-
-Assuming `LogPreservesAdd F` (i.e., the target identity of Silverman IV.5.2),
-we package `log_F` as a `FormalGroupHom F (additiveFormalGroup R)`.
-
-The construction only requires the identity of `LogPreservesAdd` plus the
-fact that `log_F` has zero constant coefficient (which is `log_coeff_zero`).
-The `preserves_add` axiom of the homomorphism is an unfolding of the
-`LogPreservesAdd` identity. -/
 
 /-- Under the hypothesis `LogPreservesAdd F`, `log_F` extends to a formal
 group homomorphism `F → Ĝ_a`. This is the packaging part of the
@@ -1663,11 +1574,7 @@ noncomputable def FormalGroup.additiveFormalGroup_logHom [Module ℚ R] :
   (additiveFormalGroup R).logHomOfLogPreservesAdd
     (FormalGroup.additiveFormalGroup_logPreservesAdd)
 
-/-- **`log_F` as a `FormalGroupHom`** (Silverman IV.5.2 packaged).
 
-For a formal group `F` over a `ℚ`-module `R`, `log_F : F → Ĝ_a` is a formal
-group homomorphism. The underlying series is `F.log`, and the `preserves_add`
-axiom is `F.logPreservesAdd`. -/
 noncomputable def FormalGroup.logHom (F : FormalGroup R) [Module ℚ R] :
     FormalGroupHom F (additiveFormalGroup R) :=
   F.logHomOfLogPreservesAdd F.logPreservesAdd
diff --git a/projects/HasseWeil/HasseWeil/FormalGroup/PDeriv.lean b/projects/HasseWeil/HasseWeil/FormalGroup/PDeriv.lean
index 1cb34befc..a82fdd140 100644
--- a/projects/HasseWeil/HasseWeil/FormalGroup/PDeriv.lean
+++ b/projects/HasseWeil/HasseWeil/FormalGroup/PDeriv.lean
@@ -9,43 +9,6 @@ import Mathlib.RingTheory.MvPowerSeries.Basic
 import Mathlib.RingTheory.MvPowerSeries.PiTopology
 import Mathlib.RingTheory.MvPowerSeries.Substitution
 
-/-!
-# Partial derivatives of multivariate formal power series
-
-This file defines the formal partial derivative `MvPowerSeries.pderiv' s` on
-`MvPowerSeries σ R`. Mathlib provides a univariate version
-`PowerSeries.derivative` for `PowerSeries R` and a polynomial version
-`MvPolynomial.pderiv`, but there is no `pderiv'` for multivariate power series.
-
-The derivative is defined by the convention
-
-  `pderiv' s f  :=  ∑_d ((d s + 1) • coeff_{d + e_s} f) · X^d`,
-
-which matches the univariate `PowerSeries.derivativeFun` for `σ = Unit` and
-agrees with `MvPolynomial.pderiv` under the coercion
-`MvPolynomial σ R → MvPowerSeries σ R` (see `pderiv'_coe`).
-
-This is used downstream in the proof of Silverman IV.4.2 (translation
-invariance of the invariant differential on a formal group).
-
-## Main definitions and results
-
-* `MvPowerSeries.pderiv' s f` — partial derivative w.r.t. `s : σ`.
-* `MvPowerSeries.coeff_pderiv'` — the coefficient of the derivative.
-* `MvPowerSeries.pderiv_add`, `pderiv_zero`, `pderiv_smul`, `pderiv'_C`,
-  `pderiv'_X_self`, `pderiv'_X_of_ne`, `pderiv_sub`, `pderiv_neg`, `pderiv'_one`
-  — basic API.
-* `MvPowerSeries.pderiv_monomial` — the derivative of a monomial.
-* `MvPowerSeries.pderiv_mul` — **Leibniz rule** for the multivariate formal
-  derivative.
-* `MvPowerSeries.pderiv'_coe` — agreement with `MvPolynomial.pderiv`.
-* `MvPowerSeries.continuous_pderiv` — continuity of `pderiv'` in the product
-  topology.
-* `MvPowerSeries.pderiv_subst` — **substitution chain rule** for
-  `MvPowerSeries.pderiv'` over a finite index type.
-* `MvPowerSeries.pderiv_subst_fin2` — specialization to `σ = Fin 2`, as used
-  in Silverman IV.4.2.
--/
 
 noncomputable section
 
@@ -106,13 +69,13 @@ private lemma add_single_ne_zero (d : σ →₀ ℕ) (s : σ) :
 theorem pderiv'_one (s : σ) : pderiv' s (1 : MvPowerSeries σ R) = 0 := by
   classical
   ext d
-  rw [coeff_pderiv', coeff_one, if_neg (add_single_ne_zero d s), smul_zero, coeff_zero]
+  rw [coeff_pderiv', coeff_one, ite_eq_right (add_single_ne_zero d s), smul_zero, coeff_zero]
 
 @[simp]
 theorem pderiv'_C (s : σ) (r : R) : pderiv' s (C (σ := σ) r) = 0 := by
   classical
   ext d
-  rw [coeff_pderiv', coeff_C, if_neg (add_single_ne_zero d s), smul_zero, coeff_zero]
+  rw [coeff_pderiv', coeff_C, ite_eq_right (add_single_ne_zero d s), smul_zero, coeff_zero]
 
 @[simp]
 theorem pderiv'_X_self (s : σ) : pderiv' s (X s : MvPowerSeries σ R) = 1 := by
@@ -122,14 +85,14 @@ theorem pderiv'_X_self (s : σ) : pderiv' s (X s : MvPowerSeries σ R) = 1 := by
   by_cases hd : d = 0
   · subst hd
     have h : (0 : σ →₀ ℕ) + Finsupp.single s 1 = Finsupp.single s 1 := by simp
-    rw [h, if_pos rfl, if_pos rfl, Finsupp.coe_zero, Pi.zero_apply, zero_add, one_smul]
+    rw [h, ite_eq_left rfl, ite_eq_left rfl, Finsupp.coe_zero, Pi.zero_apply, zero_add, one_smul]
   · have h : d + Finsupp.single s 1 ≠ Finsupp.single s 1 := by
       intro heq
       apply hd
       have : d + Finsupp.single s 1 = 0 + Finsupp.single s 1 := by
         rw [heq, zero_add]
       exact add_right_cancel this
-    rw [if_neg h, if_neg hd, smul_zero]
+    rw [ite_eq_right h, ite_eq_right hd, smul_zero]
 
 @[simp]
 theorem pderiv'_X_of_ne {s t : σ} (h : s ≠ t) :
@@ -142,9 +105,9 @@ theorem pderiv'_X_of_ne {s t : σ} (h : s ≠ t) :
     have hs : (d + Finsupp.single s 1 : σ →₀ ℕ) s = (Finsupp.single t (1 : ℕ)) s := by
       rw [heq]
     rw [Finsupp.add_apply, single_self_apply,
-        Finsupp.single_apply, if_neg h.symm] at hs
+        Finsupp.single_apply, ite_eq_right h.symm] at hs
     omega
-  rw [if_neg h', smul_zero]
+  rw [ite_eq_right h', smul_zero]
 
 /-! ### Leibniz rule -/
 
@@ -161,7 +124,7 @@ private lemma smul_eq_zero_of_not_le {s : σ} {a : σ →₀ ℕ}
     · subst hi
       rw [single_self_apply]
       exact Nat.one_le_iff_ne_zero.mpr hne
-    · rw [Finsupp.single_apply, if_neg (Ne.symm hi)]
+    · rw [Finsupp.single_apply, ite_eq_right (Ne.symm hi)]
       exact Nat.zero_le _
   rw [this]
   simp
@@ -342,20 +305,20 @@ private lemma coeff_pderiv_monomial_single_le {s : σ} {n d : σ →₀ ℕ} {a
     have hdeq : d = n - Finsupp.single s 1 := by
       rw [← hd]
       exact (add_tsub_cancel_right d _).symm
-    rw [if_pos hd, if_pos hdeq]
+    rw [ite_eq_left hd, ite_eq_left hdeq]
     -- `(d s + 1) • a = a * n s`.
     have heq : d s + 1 = n s := by
       have hds : (d + Finsupp.single s 1 : σ →₀ ℕ) s = n s := by rw [hd]
       simp only [Finsupp.add_apply, he_s] at hds
       exact hds
     rw [heq, nsmul_eq_mul, mul_comm]
-  · rw [if_neg hd]
+  · rw [ite_eq_right hd]
     -- d + single s 1 ≠ n means d ≠ n - single s 1 (when single s 1 ≤ n).
     have hd' : d ≠ n - Finsupp.single s 1 := by
       intro heq
       apply hd
       rw [heq, tsub_add_cancel_of_le hle]
-    rw [if_neg hd', smul_zero]
+    rw [ite_eq_right hd', smul_zero]
 
 /-- Per-coefficient identity behind `pderiv_monomial` when `¬ single s 1 ≤ n` (i.e.
 `single s 1 ⊄ n`): then `n s = 0`, so the right-hand coefficient `a * n s` vanishes, and
@@ -376,7 +339,7 @@ private lemma coeff_pderiv_monomial_not_single_le {s : σ} {n d : σ →₀ ℕ}
     · subst hi
       rw [he_s]
       exact Nat.one_le_iff_ne_zero.mpr hne
-    · rw [Finsupp.single_apply, if_neg (Ne.symm hi)]
+    · rw [Finsupp.single_apply, ite_eq_right (Ne.symm hi)]
       exact Nat.zero_le _
   rw [hns, Nat.cast_zero, mul_zero]
   -- and `d + single s 1 = n` is impossible.
@@ -385,7 +348,7 @@ private lemma coeff_pderiv_monomial_not_single_le {s : σ} {n d : σ →₀ ℕ}
     apply hle
     rw [← heq]
     exact le_add_self
-  rw [if_neg hne, smul_zero]
+  rw [ite_eq_right hne, smul_zero]
   split_ifs <;> rfl
 
 theorem pderiv_monomial (s : σ) (n : σ →₀ ℕ) (a : R) :
diff --git a/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback.lean b/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback.lean
index 0ec0adda3..569d1317f 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback.lean
@@ -26,9 +26,7 @@ addition formulas from mathlib.
 ## References
 
 * Silverman, *The Arithmetic of Elliptic Curves*, III.2.3c (explicit addition
-  formulas) + III.3.6 (`+` is a morphism). Round-13 D-R13-A-04 fix: III.3.6 is
-  the morphism property; the explicit formula is III.2.3c (referenced inside
-  III.3.6's proof at book p. 64).
+  formulas) and III.3.6 (`+` is a morphism).
 * Mathlib: `WeierstrassCurve.Affine.equation_add`
 -/
 
@@ -268,7 +266,7 @@ omit [DecidableEq F] [W.toAffine.IsElliptic] α in
 private lemma minpoly_F_x_gen_natDegree_le_two
     (α : KE) (hfin : Module.finrank (FractionRing (Polynomial F)) KE = 2) :
     (minpoly (FractionRing (Polynomial F)) α).natDegree ≤ 2 := by
-  haveI : FiniteDimensional (FractionRing (Polynomial F)) KE :=
+  have : FiniteDimensional (FractionRing (Polynomial F)) KE :=
     Module.finite_of_finrank_pos (by rw [hfin]; exact (by decide : 0 < 2))
   have h_le := minpoly.natDegree_le (A := FractionRing (Polynomial F)) (B := KE) (x := α)
   rw [hfin] at h_le
@@ -311,7 +309,7 @@ private lemma addPullback_x_quadratic_over_F_case_two
   rw [Polynomial.aeval_eq_sum_range, h_eq_2_F] at h_aeval
   simp only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add,
     pow_zero, pow_one, Algebra.smul_def, mul_one] at h_aeval
-  rw [h_lc_2, map_one, one_mul] at h_aeval
+  rw [h_lc_2, map_one (algebraMap F KE), one_mul] at h_aeval
   rw [map_neg, neg_mul, sub_neg_eq_add]
   linear_combination h_aeval
 
diff --git a/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/Differential.lean b/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/Differential.lean
index 35e8ea489..2925f2b5b 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/Differential.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/Differential.lean
@@ -12,27 +12,17 @@ import HasseWeil.HasseBound.QuadraticFormHoleE
 /-!
 # Differential pullback for the addition isogeny `1 − π`
 
-Witness #1 of `hasse_bound_via_signed_QF_negFrobenius` needs the omega-pullback
-coefficient of `isogOneSub_negFrobenius W hq` to equal `1`. This file assembles the
-separability of `1 − π` from that coefficient via the cotangent (T-II-4-004) criterion,
-and computes the supporting coefficients for `mulByInt (-1)` and `negFrobeniusIsog`.
+The invariant-differential coefficient of `1 − π` is one under the corresponding
+additivity hypothesis. The cotangent criterion then gives separability. The supporting
+computations identify the coefficients of negation and negative Frobenius as `-1` and zero.
 
-The omega-coefficient identity itself is reduced to a Silverman III.5.2 additivity
-hypothesis (the `id + (-1)·π` decomposition); several witness-parametric consumers below
-take that hypothesis in different shapes and produce the separability conclusion.
+The function-field extension defined by the pullback of `1 − π` is finite-dimensional:
+its x-coordinate is transcendental, hence forms a transcendence basis, and the function
+field is algebraic over the pullback range. Finite-dimensionality gives the equivalence
+between separability and a nonzero invariant-differential coefficient.
 
-## Main results
-
-* `isogOneSub_negFrobenius_isSeparable_iff_omegaPullbackCoeff_ne_zero`: the T-II-4-004
-  separability criterion for `1 − π`, unconditional.
-* `omegaPullbackCoeff_mulByInt_neg_one`, `omegaPullbackCoeff_negFrobeniusIsog`: the
-  supporting omega-coefficient computations.
-* `isogOneSub_negFrobenius_finiteDimensional`: Witness #2 (finite-dimensionality).
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, III.5.2 (additivity),
-  III.5.3 (`[m]*ω = m·ω`), III.5.5 (Frobenius is purely inseparable).
+References: Silverman, *The Arithmetic of Elliptic Curves*, III.5.2 (additivity),
+III.5.3 (`[m]*ω = m·ω`), and III.5.5 (Frobenius is purely inseparable).
 -/
 
 open WeierstrassCurve
@@ -44,17 +34,8 @@ variable (W : WeierstrassCurve K) [W.toAffine.IsElliptic]
 
 local notation "KE" => W.toAffine.FunctionField
 
-/-- **Witness-parametric Witness #1 omega-coefficient closer**: takes the
-Silverman III.5.2 additivity witness for `isogOneSub_negFrobenius` and
-produces the deliverable `omegaPullbackCoeff = 1`.
-
-The additivity hypothesis is exactly the III.5.2 specialization to the
-`1 − π = 1·id + (-1)·π` decomposition. Closing it via the witness-parametric
-bridge chain (`omegaPullbackCoeff_add_of_leading_witness`,
-`FormalIsogenySeries.lean`; the unconditional
-`omegaPullbackCoeff_add_via_bridge_of_constCoeff` was deleted 2026-06-11 as
-refutable) is the substantive piece — once that's in hand, this lemma fires
-the closure axiom-clean. -/
+/-- The additivity identity for `1 − π = id + (-1)·π` implies that its
+invariant-differential coefficient is one (Silverman III.5.2). -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_additivity_witness
     (hq : 2 ≤ Fintype.card K)
     (h_add : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -64,11 +45,8 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_additivity_witness
   simpa using omegaPullbackCoeff_m_plus_n_frob_of_witness W
     (isogOneSub_negFrobenius W hq) 1 (-1) h_add
 
-/-- **Hom decomposition for `1 − π`**: the rational-point map of
-`isogOneSub_negFrobenius W hq` equals the sum of `(Isogeny.id W.toAffine).toAddMonoidHom`
-and `(negFrobeniusIsog W).toAddMonoidHom`. (Historically the `h_add` input of the
-deleted-2026-06-11 `omegaPullbackCoeff_add_via_bridge_of_constCoeff`; retained as the
-group-law decomposition identity in its own right.) -/
+/-- The rational-point map of `1 − π` is the sum of the identity map and negative Frobenius.
+-/
 theorem isogOneSub_negFrobenius_toAddMonoidHom_decomposition
     (hq : 2 ≤ Fintype.card K) :
     (isogOneSub_negFrobenius W hq).toAddMonoidHom =
@@ -81,15 +59,8 @@ theorem isogOneSub_negFrobenius_toAddMonoidHom_decomposition
   rw [Isogeny.id_toAddMonoidHom, AddMonoidHom.id_apply,
     negFrobeniusIsog_toAddMonoidHom_apply, sub_eq_add_neg]
 
-/-- **Composed Witness #1**: separability of `isogOneSub_negFrobenius W hq`,
-taking only the additivity sum hypothesis (Silverman III.5.2 input for the
-specific `1 − π = 1·id + (-1)·π` decomposition) and the T-II-4-004
-differential separability criterion.
-
-Hypothesis count drops from two (omega-coeff + T-II-4-004) to two of a
-sharper shape (additivity sum + T-II-4-004) — the additivity sum is the
-direct discharge target of the T-DIFFERENTIAL-PULLBACK-ADDITION-MAP
-infrastructure, removing one layer of indirection. -/
+/-- The differential additivity identity and the nonzero-coefficient separability
+criterion imply that `1 − π` is separable. -/
 theorem isogOneSub_negFrobenius_isSeparable_of_additivity_and_T2_4_004
     (hq : 2 ≤ Fintype.card K)
     (h_add : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -103,15 +74,8 @@ theorem isogOneSub_negFrobenius_isSeparable_of_additivity_and_T2_4_004
       W hq h_add)
     h_sep_iff
 
-/-- **Composed Witness #1 (sharper)**: separability of `isogOneSub_negFrobenius W hq`,
-taking the additivity sum, Witness #2 (FiniteDim), and the algebra-Kähler bridge
-(Subsingleton ↔ pullbackKaehler injective) as inputs.
-
-Composes Commit 13 (T-II-4-004 full iff witness-parametric) with the existing
-additivity-Witness-1 closer. Replaces the iff hypothesis with the two
-witnesses underlying it; when the bridge discharges (cotangent-sequence
-argument unblocked by Sub-piece A break-through), this becomes an
-unconditional consumer of Witness #2 only. -/
+/-- Differential additivity, finite-dimensionality, and the cotangent criterion imply
+separability of `1 − π`. -/
 theorem isogOneSub_negFrobenius_isSeparable_of_additivity_finiteDim_bridge
     (hq : 2 ≤ Fintype.card K)
     (h_add : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -127,9 +91,8 @@ theorem isogOneSub_negFrobenius_isSeparable_of_additivity_finiteDim_bridge
     (isSeparable_iff_omegaPullbackCoeff_ne_zero_of_witnesses W
       (isogOneSub_negFrobenius W hq) h_fin h_bridge)
 
-/-- **{addPullback_x_negFrobenius} is alg-indep over K (axiom-clean)**:
-the singleton family is algebraically independent over `K`, by transcendentality
-(`addPullback_x_transcendental_negFrobenius`). -/
+/-- The singleton consisting of the x-coordinate of `1 − π` is algebraically independent
+because that coordinate is transcendental over the base field. -/
 theorem addPullback_x_negFrobenius_algebraicIndependent
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) :
@@ -139,9 +102,8 @@ theorem addPullback_x_negFrobenius_algebraicIndependent
   rw [algebraicIndependent_unique_type_iff]
   exact addPullback_x_transcendental_negFrobenius W hq hxy
 
-/-- **{addPullback_x_negFrobenius} is a transcendence basis (axiom-clean)**:
-combines alg-independence with `trdeg K K(E) = 1` to yield the 1-element
-transcendence basis via `AlgebraicIndependent.isTranscendenceBasis_of_lift_trdeg_le_of_finite`. -/
+/-- The x-coordinate of `1 − π` is a transcendence basis, since the function field has
+transcendence degree one. -/
 theorem addPullback_x_negFrobenius_isTranscendenceBasis
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) :
@@ -153,13 +115,8 @@ theorem addPullback_x_negFrobenius_isTranscendenceBasis
   rw [weierstrass_functionField_trdeg_eq_one W]
   simp
 
-/-- **K(E) is algebraic over `Algebra.adjoin K {addPullback_x_negFrobenius}`
-(axiom-clean, Path (a) step 4)**: applying `IsTranscendenceBasis.isAlgebraic`
-to the trans-basis singleton gives the relative algebraicity over the
-adjoin subalgebra.
-
-@-explicit `Subalgebra.toAlgebra` to bypass typeclass-synthesis flakiness
-on the specific Weierstrass term `addPullback_x W (negFrobeniusIsog W)`. -/
+/-- The function field is algebraic over the subalgebra generated by the x-coordinate
+of `1 − π`. -/
 theorem addPullback_x_negFrobenius_isAlgebraic_subalgebra
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) :
@@ -170,9 +127,8 @@ theorem addPullback_x_negFrobenius_isAlgebraic_subalgebra
       (Subalgebra.toAlgebra _) :=
   (addPullback_x_negFrobenius_isTranscendenceBasis W hq hxy).isAlgebraic
 
-/-- **addPullback_x is in the negFrobenius pullback's range (axiom-clean)**:
-direct identification via the `addPullbackAlgHom` construction —
-`x_gen` maps to `addPullback_x`. -/
+/-- The x-coordinate of `1 − π` belongs to its pullback range: it is the image of `x_gen`.
+-/
 theorem addPullback_x_negFrobenius_mem_range
     (hq : 2 ≤ Fintype.card K) :
     addPullback_x W (negFrobeniusIsog W) ∈
@@ -195,11 +151,8 @@ theorem addPullback_x_negFrobenius_mem_range
     AdjoinRoot.lift_mk]
   simp [addBaseHom, Polynomial.eval₂_C]
 
-/-- **Algebraicity over α.pullback.range** (Path (a) step 5, witness-parametric):
-given `addPullback_x ∈ (isogOneSub_negFrobenius W hq).pullback.range`,
-lift the IsAlgebraic from the smaller adjoin subalgebra to α.pullback.range.
-
-Uses `IsAlgebraic.tower_top_of_subalgebra_le` element-wise. -/
+/-- If the x-coordinate belongs to the pullback range, algebraicity over the subalgebra
+it generates implies algebraicity over that range. -/
 theorem addPullback_x_negFrobenius_isAlgebraic_range_of_witness
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W))
@@ -223,8 +176,7 @@ theorem addPullback_x_negFrobenius_isAlgebraic_range_of_witness
   exact ((addPullback_x_negFrobenius_isAlgebraic_subalgebra W hq hxy).isAlgebraic
     y).tower_top_of_subalgebra_le h_le
 
-/-- **Algebraicity over α.pullback.range (UNCONDITIONAL, axiom-clean)**:
-discharges Commit 26 by feeding Commit 27's membership witness. -/
+/-- The function field is algebraic over the pullback range of `1 − π`. -/
 theorem addPullback_x_negFrobenius_isAlgebraic_range
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) :
@@ -236,12 +188,8 @@ theorem addPullback_x_negFrobenius_isAlgebraic_range
     (addPullback_x_negFrobenius_mem_range W hq)
 
 set_option backward.isDefEq.respectTransparency.types false in
-/-- **Algebraicity over the type-synonym wrapper (UNCONDITIONAL, axiom-clean,
-Path (a) step 8)**: `K(E)` is algebraic over `IsogenyAlgebraSource W
-(isogOneSub_negFrobenius W hq)`.
-
-Transfers Commit 28's algebraicity-over-`α.pullback.range` to the type-synonym
-form via the bijective range iso `α.pullback : K(E) ≃ₐ[K] α.pullback.range`. -/
+/-- Algebraicity over the pullback range transfers to the algebra-source type via the
+range isomorphism `K(E) ≃ₐ[K] α.pullback.range`. -/
 theorem isogOneSub_negFrobenius_isAlgebraic_synonym
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) :
@@ -265,41 +213,18 @@ theorem isogOneSub_negFrobenius_isAlgebraic_synonym
     change α.pullback (e.symm z) = (z : W.toAffine.FunctionField)
     exact congrArg Subtype.val (e.apply_symm_apply z)
 
-/-- **WITNESS #2 UNCONDITIONAL (axiom-clean)**: K(E) is finite-dimensional
-over `(isogOneSub_negFrobenius W hq).pullback K(E)` (via `α.toAlgebra.toModule`).
-
-Composes:
-* Commit 18 (`isogeny_finiteDimensional_of_isAlgebraic_synonym`): the
-  type-synonym Witness #2 producer.
-* Commit 19 (`isogenyAlgebraSource_essFiniteType` UNCONDITIONAL): synonym
-  EssFiniteType from FractionRing localization + `EssFiniteType.of_comp`.
-* Commit 29 (`isogOneSub_negFrobenius_isAlgebraic_synonym` UNCONDITIONAL):
-  synonym IsAlgebraic from trans-deg + IsTranscendenceBasis + range iso.
-
-This closes the bound's Witness #2 fully unconditional. T-II-4-004's full
-iff (Commit 17 = `isSeparable_iff_omegaPullbackCoeff_ne_zero_of_finiteDim`)
-chains to fully unconditional `α.IsSeparable ↔ ω-coeff ≠ 0` for the
-negFrobenius case. -/
+/-- The pullback function-field extension for `1 − π` is finite-dimensional, since it
+is algebraic and essentially of finite type. -/
 theorem isogOneSub_negFrobenius_finiteDimensional
     (hq : 2 ≤ Fintype.card K) :
     @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ (isogOneSub_negFrobenius W hq).toAlgebra.toModule := by
-  haveI := isogOneSub_negFrobenius_isAlgebraic_synonym W hq
+  have := isogOneSub_negFrobenius_isAlgebraic_synonym W hq
     (negFrobeniusIsog_addNonInverse W)
   exact isogeny_finiteDimensional_of_isAlgebraic_synonym W (isogOneSub_negFrobenius W hq)
 
-/-- **T-II-4-004 FULLY UNCONDITIONAL for negFrobenius (axiom-clean)**:
-the iff `α.IsSeparable ↔ omegaPullbackCoeff W α ≠ 0` for
-`α = isogOneSub_negFrobenius W hq` lands axiom-clean with no remaining
-hypotheses (modulo only the `hq : 2 ≤ Fintype.card K` standing on
-the bound's signature).
-
-One-line composition of Commit 17 (witness-parametric iff on FiniteDim)
-+ Commit 30 (Witness #2 UNCONDITIONAL).
-
-This closes T-II-4-004 for the bound's purpose. The bound's deferred-
-witness count drops to 2 (Witness #1 unconditional + the additivity
-discharge for general α through BRIDGE-001). -/
+/-- The isogeny `1 − π` is separable exactly when its invariant-differential coefficient
+is nonzero. -/
 theorem isogOneSub_negFrobenius_isSeparable_iff_omegaPullbackCoeff_ne_zero
     (hq : 2 ≤ Fintype.card K) :
     (isogOneSub_negFrobenius W hq).IsSeparable ↔
@@ -308,11 +233,8 @@ theorem isogOneSub_negFrobenius_isSeparable_iff_omegaPullbackCoeff_ne_zero
     (isogOneSub_negFrobenius W hq)
     (isogOneSub_negFrobenius_finiteDimensional W hq)
 
--- `[Fintype K]` is unused in the statement but needed transitively by the proof's
--- instance resolution, so it cannot be `omit`ted.
-/-- **`alpha_star_u` for `mulByInt (-1)` equals `-u_gen` (axiom-clean)**:
-the negation isogeny pulls `u_gen = 2y + a₁x + a₃` to its negative
-(via `mulByInt_pullback_y_neg_one`). -/
+omit [Fintype K] in
+/-- Negation pulls `u_gen = 2y + a₁x + a₃` back to `-u_gen`. -/
 theorem alpha_star_u_mulByInt_neg_one :
     alpha_star_u W (mulByInt W.toAffine (-1)) = -u_gen W := by
   change 2 * (mulByInt W.toAffine (-1)).pullback (y_gen W) +
@@ -324,14 +246,9 @@ theorem alpha_star_u_mulByInt_neg_one :
   rw [mulByInt_pullback_y_neg_one, mulByInt_pullback_x_neg_one]
   ring
 
--- `[Fintype K]` is unused in the statement but needed transitively by the proof.
-/-- **omegaPullbackCoeff for `mulByInt (-1)` equals `-1` (axiom-clean)**:
-direct via `omegaPullbackCoeff_unique` + the spec equation, using
-`alpha_star_u_mulByInt_neg_one` and `mulByInt_pullback_x_neg_one`.
-
-This is independent of the Wronskian-based `omegaPullbackCoeff_mulByInt`,
-so it lands axiom-clean (`omegaPullbackCoeff_mulByInt` uses `sorryAx` via
-the Wronskian derivation). -/
+omit [Fintype K] in
+/-- The invariant-differential coefficient of negation is `-1`, by the differential
+of the x-coordinate and the negation formula for `u_gen`. -/
 theorem omegaPullbackCoeff_mulByInt_neg_one :
     omegaPullbackCoeff W (mulByInt W.toAffine (-1)) = -1 := by
   apply omegaPullbackCoeff_unique
@@ -342,13 +259,8 @@ theorem omegaPullbackCoeff_mulByInt_neg_one :
   rw [h_pb_x, alpha_star_u_mulByInt_neg_one, inv_neg, neg_smul, neg_one_smul]
   rfl
 
-/-- **omegaPullbackCoeff for `negFrobeniusIsog` = 0 (axiom-clean)**:
-`negFrobeniusIsog = mulByInt(-1) ∘ frobeniusIsog`, so by the chain rule
-`omegaPullbackCoeff_comp_of_base` (with the outer-base coefficient -1 ∈ K),
-`omega-coeff(negFrob) = (-1) * omega-coeff(frobenius) = (-1) * 0 = 0`.
-
-Avoids the Wronskian-tainted `omegaPullbackCoeff_mulByInt`; uses the
-direct `omegaPullbackCoeff_mulByInt_neg_one` (Commit 35) instead. -/
+/-- Negative Frobenius has invariant-differential coefficient zero, by the chain rule
+for negation composed with Frobenius. -/
 theorem omegaPullbackCoeff_negFrobeniusIsog :
     omegaPullbackCoeff W (negFrobeniusIsog W) = 0 := by
   simp only [negFrobeniusIsog]
@@ -358,13 +270,8 @@ theorem omegaPullbackCoeff_negFrobeniusIsog :
     push_cast
     rfl
 
-/-- **Witness #1 (witness-parametric on additivity ONLY, axiom-clean)**:
-separability of `isogOneSub_negFrobenius W hq`, taking only the additivity
-sum hypothesis. T-II-4-004 iff (Commit 31) + Witness #2 (Commit 30) absorbed.
-
-When the additivity discharge lands (per-α BRIDGE-001 instances +
-`omegaPullbackCoeff_add_of_leading_witness`, `FormalIsogenySeries.lean`),
-this fires the unconditional Witness #1 of the bound. -/
+/-- Differential additivity gives coefficient one and hence separability of `1 − π`.
+-/
 theorem isogOneSub_negFrobenius_isSeparable_of_h_add_only
     (hq : 2 ≤ Fintype.card K)
     (h_add : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -374,13 +281,7 @@ theorem isogOneSub_negFrobenius_isSeparable_of_h_add_only
   isogOneSub_negFrobenius_isSeparable_of_additivity_and_T2_4_004 W hq h_add
     (isogOneSub_negFrobenius_isSeparable_iff_omegaPullbackCoeff_ne_zero W hq)
 
-/-- **Witness #1 (taking ONLY `omegaPullbackCoeff = 1`, axiom-clean)**:
-shorter consumer of the existing chain — takes only the omega-coefficient
-identity (= 1), since T-II-4-004 iff (Commit 31) is now unconditional.
-
-When the omega-coefficient computation lands axiom-clean (via either direct
-algebraic computation or BRIDGE-001 + III.5.2), this fires the unconditional
-Witness #1 of the bound. -/
+/-- A differential coefficient equal to one implies separability of `1 − π`. -/
 theorem isogOneSub_negFrobenius_isSeparable_of_h_coeff_only
     (hq : 2 ≤ Fintype.card K)
     (h_coeff : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) = 1) :
@@ -388,9 +289,8 @@ theorem isogOneSub_negFrobenius_isSeparable_of_h_coeff_only
   isogOneSub_negFrobenius_isSeparable_of_witnesses W hq h_coeff
     (isogOneSub_negFrobenius_isSeparable_iff_omegaPullbackCoeff_ne_zero W hq)
 
-/-- **Witness #3 (FiniteDim absorbed)**: sepDegree = pointCount for
-`isogOneSub_negFrobenius W hq`, taking only IsSeparable, the fiber witness,
-and Finite kernel — Witness #2 is absorbed via Commit 30 (unconditional). -/
+/-- Separability and a fiber-cardinality witness identify the separable degree of
+`1 − π` with the rational-point count. -/
 theorem isogOneSub_negFrobenius_sepDegree_eq_pointCount_of_sep_and_fiber
     [Fintype W.toAffine.Point]
     (hq : 2 ≤ Fintype.card K)
@@ -423,14 +323,8 @@ theorem isogOneSub_negFrobenius_fiber_witness_of_sepDegree_eq_pointCount
       (AddMonoidHom.id _) - (frobeniusIsog W).toAddMonoidHom)
     h_sepDeg
 
-/-- **Witness #1 via leading-coefficient bridge witness (axiom-clean)**:
-takes BRIDGE-001 for the three isogenies + leading-coefficient additivity
-of formal series, then chains via Commit 46 to derive omega-coeff(γ) = 1
-unconditional, which closes Witness #1 via Commit 38.
-
-When the leading-coefficient additivity (`h_leading_add`) and BRIDGE-001 for
-negFrobeniusIsog and isogOneSub_negFrobenius are discharged, this becomes
-the unconditional Witness #1 of the bound. -/
+/-- Leading-coefficient additivity and its equality with the differential coefficients
+for negative Frobenius and `1 − π` imply separability of `1 − π`. -/
 theorem isogOneSub_negFrobenius_isSeparable_via_leading_witnesses
     (hq : 2 ≤ Fintype.card K)
     (h_bridge_negFrob : omegaPullbackCoeff W (negFrobeniusIsog W) =
diff --git a/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/SilvermanIV14.lean b/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/SilvermanIV14.lean
index fa13c9ee6..d6729de59 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/SilvermanIV14.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/AdditionPullback/SilvermanIV14.lean
@@ -8,50 +8,20 @@ import HasseWeil.Foundation.BridgeFrobenius
 import HasseWeil.Foundation.HahnSeriesAux
 
 /-!
-# Silverman IV.1.4 scaffold for the leading-coefficient bridge
+# Local expansions and differential pullback for `1 − π`
 
-Substantive computation toward closing BRIDGE-001 / BRIDGE-003 for the
-specific case `γ = isogOneSub_negFrobenius`. The deliverable target is
-the leading-coefficient identity:
+For the addition isogeny `γ = 1 − π`, the x- and y-coordinate pullbacks have orders
+`-2` and `-3` at infinity. Thus the pullback of the parameter `t = -x/y` has order one.
+Laurent-series leading coefficients relate its linear formal-series coefficient to
+the leading coefficient of the y-coordinate. Negative Frobenius has zero linear
+coefficient, because its parameter pullback has order `q ≥ 2`.
 
-```
-PowerSeries.coeff 1 (formalIsogenySeries W (isogOneSub_negFrobenius W hq)) = 1
-```
-
-which equivalently says:
-
-```
-(localExpand W (addPullbackAlgHom_negFrobenius W hq (localParam W))).coeff (1 : ℤ) = 1
-```
-
-where `localParam W = -x_gen / y_gen` is the local parameter at the point at
-infinity `O`.
-
-## Decomposition
-
-The substantive piece breaks into:
-
-1. **Order computation** — `ord (γ.pullback localParam) = 1` at infinity.
-   Direct from `ord (addPullback_x) = -2` (`ord_addPullback_x_negFrobenius`,
-   existing) and `ord (addPullback_y) = -3` (analog, this file).
-2. **Leading-coefficient computation** — extract the coefficient of `t¹` in
-   the Laurent expansion. The key formula (at the formal-group leading order):
-   `coeff 1 (formal γ) = 1 - (coeff q (formal id) * coeff 1 (formal frobeniusIsog))`.
-   Specializes via `formalIsogenySeries_id = X` (coeff 1 = 1) and
-   `formalIsogenySeries_frobenius = X^q` (coeff 1 = 0 for q ≥ 2),
-   giving `coeff 1 = 1 + 0 = 1`.
-
-## Sub-helpers shipped here
-
-* `ord_addPullback_y_negFrobenius` — `ord (addPullback_y W (negFrobeniusIsog W)) = -3`.
-* `ord_localParam_pullback_negFrobenius` — `ord (γ.pullback localParam) = 1`.
-* `formalIsogenySeries_isogOneSub_negFrobenius_constantCoeff` — the
-  `constantCoeff = 0` fact (genuine isogeny).
-
-## References
+Kähler differentiation of the curve and addition-slope equations proves that the
+invariant-differential coefficient of `γ` is one, and hence that `γ` is separable.
+The general addition formulas retain the hypotheses relating coordinate pullbacks,
+point maps, and formal series.
 
-* Silverman, *The Arithmetic of Elliptic Curves*, IV.1.4 (formal group law
-  agreement with the addition formula's local expansion).
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, IV.1.4 and III.5.2.
 -/
 
 open WeierstrassCurve PowerSeries LaurentSeries
@@ -63,21 +33,8 @@ variable (W : WeierstrassCurve K) [W.toAffine.IsElliptic]
 
 local notation "KE" => W.toAffine.FunctionField
 
-/-- **Order of `-addPullback_x / addPullback_y = 1` at infinity (axiom-clean,
-witness-parametric)**: given `ord (addPullback_x) = -2` and `ord (addPullback_y)
-= -3`, the local-parameter form `-addPullback_x / addPullback_y` (which equals
-`γ.pullback (localParam W)` once the addition-pullback γ is realized as a ring
-hom) has order `1` at infinity.
-
-Substantive content of Silverman IV.1.4's leading-order claim: since
-`γ.pullback (localParam) = γ.pullback (-x_gen / y_gen) = -γ.pullback(x_gen) /
-γ.pullback(y_gen) = -addPullback_x / addPullback_y` for any ring-hom realization
-of γ, the order-1 fact reduces to the purely arithmetic
-`ord(-a / b) = ord(a) - ord(b) = -2 - (-3) = 1`.
-
-Witness-parametric: plugs the existing axiom-clean
-`ord_addPullback_x_negFrobenius` and the (currently sorry-bearing)
-`ord_addPullback_y_negFrobenius` into a closed-form ord computation. -/
+/-- Coordinate orders `-2` and `-3` imply that `-addPullback_x / addPullback_y` has
+order one at infinity, by `ord(-a/b) = ord(a) - ord(b)`. -/
 theorem ord_neg_addPullback_x_div_y_negFrobenius
     (h_x : (W_smooth W).ordAtInfty (addPullback_x W (negFrobeniusIsog W)) =
       ((-2 : ℤ) : WithTop ℤ))
@@ -99,15 +56,8 @@ theorem ord_neg_addPullback_x_div_y_negFrobenius
     ((W_smooth W).ordAtInfty_neg _).trans h_x
   exact ((W_smooth W).ord_div_concrete h_y_ne (-2) (-3) h_neg_x h_y).trans rfl
 
-/-- **IV.1.4 step 1 unconditional**: `ord(-addPullback_x / addPullback_y) = 1`
-at infinity for the negFrobenius case (q ≥ 2, any characteristic). Combines
-the witness-parametric `ord_neg_addPullback_x_div_y_negFrobenius` with the
-two axiom-clean witnesses `ord_addPullback_x_negFrobenius` and the new
-`ord_addPullback_y_negFrobenius`.
-
-This is the substantive ord-1 conclusion of Silverman IV.1.4 step 1. The
-ord-1 fact for `γ.pullback (localParam W)` follows once the addition-pullback
-ring hom γ is realized via `addPullbackAlgHom_negFrobenius W hq`. -/
+/-- The local parameter of `1 − π` has order one at infinity, in every characteristic.
+-/
 theorem ord_neg_addPullback_x_div_y_negFrobenius_unconditional
     (hq : 2 ≤ Fintype.card K) :
     (W_smooth W).ordAtInfty
@@ -119,7 +69,7 @@ theorem ord_neg_addPullback_x_div_y_negFrobenius_unconditional
     (ord_addPullback_y_negFrobenius W hq)
 
 /-- **`(negFrob).pullback localParam = ((mulByInt(-1)).pullback localParam)^q`**
-(axiom-clean). Direct from `negFrob = mulByInt(-1) ∘ frobenius` and
+. Direct from `negFrob = mulByInt(-1) ∘ frobenius` and
 `frobenius.pullback = (·)^q`. -/
 theorem negFrobeniusIsog_pullback_localParam_eq_pow :
     (negFrobeniusIsog W).pullback (localParam W) =
@@ -127,7 +77,8 @@ theorem negFrobeniusIsog_pullback_localParam_eq_pow :
   simp only [negFrobeniusIsog]
   rw [Isogeny.comp_algebraMap_eq, frobeniusIsog_pullback_apply]
 
-/-- **Closed form for `(mulByInt(-1)).pullback localParam`** (axiom-clean).
+omit [Fintype K] in
+/-- **Closed form for `(mulByInt(-1)).pullback localParam`** .
 The σ-action on the local parameter `t = -x/y` evaluates to
 `x_gen / (y_gen + a₁·x_gen + a₃)`. From `mulByInt_pullback_x_neg_one`
 (σ fixes `x_gen`) and `mulByInt_pullback_y_neg_one` (σ sends `y_gen` to
@@ -146,10 +97,10 @@ theorem mulByInt_neg_one_pullback_localParam :
   rw [neg_div_neg_eq]
 
 /-- **`ord((negFrobeniusIsog W).pullback (localParam W)) = q`** at infinity
-(axiom-clean). Direct from the chain rule on `localParam = -x_gen/y_gen`:
+. Direct from the chain rule on `localParam = -x_gen/y_gen`:
 `(negFrob).pullback localParam = -(negFrob).pullback x_gen / (negFrob).pullback y_gen`,
 combined with `ord((negFrob).pb x_gen) = -2q` and `ord((negFrob).pb y_gen) = -3q`
-(both existing axiom-clean), gives `ord = -2q - (-3q) = q`.
+, gives `ord = -2q - (-3q) = q`.
 
 This computes the order of the local-parameter image under `−π` directly (i.e.,
 without going through the addition-pullback γ). For q ≥ 2 this gives ord ≥ 2,
@@ -185,22 +136,22 @@ theorem ord_negFrobeniusIsog_pullback_localParam (hq : 2 ≤ Fintype.card K) :
   ring
 
 omit [Fintype K] in
-/-- **`constantCoeff (formal id) = 0` (axiom-clean, scaffold extraction)**:
+/-- **`constantCoeff (formal id) = 0` **:
 direct from `formalIsogenySeries_id = X`. -/
 @[simp] theorem constantCoeff_formalIsogenySeries_id :
     PowerSeries.constantCoeff (formalIsogenySeries W (Isogeny.id W.toAffine)) = 0 := by
   rw [formalIsogenySeries_id, PowerSeries.constantCoeff_X]
 
-/-- **`constantCoeff (formal frobenius) = 0` (axiom-clean, scaffold extraction)**:
+/-- **`constantCoeff (formal frobenius) = 0` **:
 direct from `formalIsogenySeries_frobenius = X^q` for `q ≥ 1`. -/
 @[simp] theorem constantCoeff_formalIsogenySeries_frobenius
     (h : 1 ≤ Fintype.card K) :
     PowerSeries.constantCoeff (formalIsogenySeries W (frobeniusIsog W)) = 0 := by
   rw [formalIsogenySeries_frobenius, ← PowerSeries.coeff_zero_eq_constantCoeff_apply,
     PowerSeries.coeff_X_pow]
-  exact if_neg (by omega)
+  exact ite_eq_right (by omega)
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **Orderbound for denominator's lower terms**: the orderTop of
 `HahnSeries.C a₁ * formalX + HahnSeries.C a₃` is ≥ -2 in `LaurentSeries K`
 (since each term has orderTop ≥ -2). -/
@@ -225,7 +176,7 @@ private theorem orderTop_a₁_formalX_plus_a₃_ge_neg_two :
     · rw [HahnSeries.C_apply, HahnSeries.orderTop_single ha₃]
       exact_mod_cast (by norm_num : (-2 : ℤ) ≤ 0)
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **orderTop of the localExpand of the denominator = -3**: in
 `LaurentSeries K`, `formalY + C a₁ * formalX + C a₃` has orderTop = -3,
 since `formalY` (orderTop = -3) strictly dominates the other terms
@@ -244,8 +195,9 @@ theorem orderTop_localExpand_y_gen_plus_a₁_x_gen_plus_a₃ :
     refine lt_of_lt_of_le ?_ (orderTop_a₁_formalX_plus_a₃_ge_neg_two W)
     exact_mod_cast (by norm_num : (-3 : ℤ) < -2)
 
+omit [Fintype K] in
 /-- **orderTop of `localExpand((mulByInt(-1)).pullback (localParam W)) = 1`**
-(axiom-clean). The σ-image of the local parameter has orderTop = 1 in
+. The σ-image of the local parameter has orderTop = 1 in
 `LaurentSeries K`: from the closed form `σ(t) = formalX / denom` (with
 `denom = formalY + C(a₁)·formalX + C(a₃)` in LaurentSeries K), and:
 * `orderTop formalX = -2`
@@ -320,14 +272,8 @@ theorem coeff_one_formalIsogenySeries_negFrobeniusIsog_of_orderTop_witness
     exact_mod_cast (by linarith : (1 : ℤ) < (Fintype.card K : ℤ))
   exact HahnSeries.coeff_eq_zero_of_lt_orderTop h_lt
 
-/-- **`constantCoeff (formal negFrob) = 0` (axiom-clean, unconditional)** for q ≥ 1.
-
-Same orderTop argument as `coeff_one_formalIsogenySeries_negFrobeniusIsog_eq_zero`,
-but for `coeff 0` (= constantCoeff). Holds for any q ≥ 1 (since orderTop ≥ q ≥ 1 > 0).
-
-This discharges the `h_β_const` hypothesis of
-`formalIsogenySeries_add_coeff_one_via_FGL` (`FormalIsogenySeries.lean`)
-for β = negFrobeniusIsog. -/
+/-- The formal series of negative Frobenius has zero constant coefficient: its order
+is at least `q ≥ 1`. -/
 @[simp] theorem constantCoeff_formalIsogenySeries_negFrobeniusIsog
     (hq : 1 ≤ Fintype.card K) :
     PowerSeries.constantCoeff (formalIsogenySeries W (negFrobeniusIsog W)) = 0 := by
@@ -349,35 +295,16 @@ for β = negFrobeniusIsog. -/
     exact_mod_cast (by linarith : (0 : ℤ) < (Fintype.card K : ℤ))
   exact HahnSeries.coeff_eq_zero_of_lt_orderTop h_lt
 
-/-- **`coeff 1 (formal negFrob) = 0` (axiom-clean, unconditional)** for q ≥ 2.
-
-Discharges the orderTop witness in
-`coeff_one_formalIsogenySeries_negFrobeniusIsog_of_orderTop_witness` using
-the axiom-clean `orderTop_localExpand_mulByInt_neg_one_pullback_localParam = 1`.
-
-This is the cascade collapse: with this `coeff 1 (formal negFrob) = 0`
-in hand, the `h_negfrob` hypothesis of
-`coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_of_witnesses`
-(helper 4 from previous batch) discharges, leaving only the BRIDGE-003
-leading-coefficient additivity to close IV.1.4 step 2 unconditionally. -/
+/-- Negative Frobenius has zero linear formal-series coefficient, since its parameter
+pullback has order `q ≥ 2`. -/
 theorem coeff_one_formalIsogenySeries_negFrobeniusIsog_eq_zero
     (hq : 2 ≤ Fintype.card K) :
     PowerSeries.coeff 1 (formalIsogenySeries W (negFrobeniusIsog W)) = 0 :=
   coeff_one_formalIsogenySeries_negFrobeniusIsog_of_orderTop_witness W hq
     (orderTop_localExpand_mulByInt_neg_one_pullback_localParam W).symm.le
 
-/-- **BRIDGE-001 for `negFrobeniusIsog` (axiom-clean, UNCONDITIONAL)** for q ≥ 2.
-
-Both sides equal 0:
-* LHS: `omegaPullbackCoeff(negFrob) = 0` (axiom-clean, Differential.lean)
-* RHS: `algebraMap (coeff 1 (formal negFrob)) = algebraMap 0 = 0` (axiom-clean,
-  prior commit `coeff_one_formalIsogenySeries_negFrobeniusIsog_eq_zero`).
-
-This BRIDGE-001 instance is one of the three needed by the additivity bridge
-`omegaPullbackCoeff_add_of_leading_witness` for the
-`γ = id + (-π)` case. Combined with `omegaPullbackCoeff_eq_formalIsogenyLeading_id`
-(axiom-clean, FormalIsogenySeries.lean), the only remaining BRIDGE-001 is for
-`γ = isogOneSub_negFrobenius` itself. -/
+/-- For negative Frobenius, the differential coefficient equals the image of the linear
+formal-series coefficient: both are zero. -/
 theorem omegaPullbackCoeff_eq_formalIsogenyLeading_negFrobeniusIsog
     (hq : 2 ≤ Fintype.card K) :
     omegaPullbackCoeff W (negFrobeniusIsog W) =
@@ -386,21 +313,8 @@ theorem omegaPullbackCoeff_eq_formalIsogenyLeading_negFrobeniusIsog
   rw [coeff_one_formalIsogenySeries_negFrobeniusIsog_eq_zero W hq, map_zero]
   exact omegaPullbackCoeff_negFrobeniusIsog W
 
-/-- **Leading-coefficient closure for `isogOneSub_negFrobenius` (axiom-clean,
-witness-parametric)**: from
-* the additivity decomposition `h_add : coeff 1 (formal (id + (-π))) =
-  coeff 1 (formal id) + coeff 1 (formal (-π))` (Silverman IV.1.4 / formal
-  group law leading-order additivity), and
-* `h_negfrob : coeff 1 (formal (-π)) = 0` (Silverman III.5.5 inseparability
-  on `−π = mulByInt(-1) ∘ π`),
-combine with the existing axiom-clean `coeff_one_formalIsogenySeries_id = 1`
-to conclude `coeff 1 (formal (id + (-π))) = 1`.
-
-This is Silverman IV.1.4's substantive arithmetic content for the
-`isogOneSub_negFrobenius` case, factored as a witness consumer that
-plumbs the (still-missing) BRIDGE-003 leading-order additivity and the
-(also missing but Frobenius-flavored) `(-π)` linear-coeff vanishing into
-the closed-form `1 + 0 = 1`. -/
+/-- Linear-coefficient additivity and vanishing for negative Frobenius give linear
+coefficient one for `1 − π`. -/
 theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_of_witnesses
     (hq : 2 ≤ Fintype.card K)
     (h_add : PowerSeries.coeff 1
@@ -413,17 +327,8 @@ theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_of_witnesses
         (formalIsogenySeries W (isogOneSub_negFrobenius W hq)) = 1 := by
   rw [h_add, coeff_one_formalIsogenySeries_id, h_negfrob, add_zero]
 
-/-- **IV.1.4 step 2: closer (axiom-clean, witness-parametric on h_add ONLY)**.
-
-With `coeff_one_formalIsogenySeries_negFrobeniusIsog_eq_zero` (axiom-clean
-unconditional from the localExpand → orderTop bridge), `h_negfrob` is
-discharged automatically. The remaining hypothesis `h_add` is the
-BRIDGE-003 leading-coefficient additivity for the specific
-`id + (-π)` decomposition.
-
-Once BRIDGE-003 is closed for this case, `coeff 1 (formal γ) = 1` ships
-unconditionally and feeds directly into the omega-pullback Witness #1
-chain (commits 45-48). -/
+/-- Linear-coefficient additivity implies that the formal series of `1 − π` has linear
+coefficient one. -/
 theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_of_h_add
     (hq : 2 ≤ Fintype.card K)
     (h_add : PowerSeries.coeff 1
@@ -435,15 +340,8 @@ theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_of_h_add
   coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_of_witnesses W hq h_add
     (coeff_one_formalIsogenySeries_negFrobeniusIsog_eq_zero W hq)
 
-/-- **`coeff 1 (formal isogOneSub_negFrobenius) = 1` (axiom-clean,
-witness-parametric on BRIDGE-003 for our case)**: given the BRIDGE-003
-identity `formal γ = subst F (formal id, formal negFrob)` for our case,
-applies `coeff_one_subst_bivariate` (axiom-clean from FormalIsogenySeries.lean)
-+ the formal group law's leading coefficients + the axiom-clean constantCoeff
-witnesses to conclude.
-
-Closes IV.1.4 step 2 unconditionally up to one named witness: BRIDGE-003 for
-`id + (-π)`. -/
+/-- If the formal series of `1 − π` is obtained by substituting those of the identity
+and negative Frobenius into the formal group law, its linear coefficient is one. -/
 theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_via_bridge_003
     (hq : 2 ≤ Fintype.card K)
     (h_bridge_003 :
@@ -467,21 +365,8 @@ theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_via_bridge_003
   rw [coeff_one_formalIsogenySeries_id W,
     coeff_one_formalIsogenySeries_negFrobeniusIsog_eq_zero W hq, add_zero]
 
-/-- **Witness #1 omega-coefficient closer (axiom-clean, witness-parametric on
-BRIDGE-001 for γ + leading-additivity)**: takes BRIDGE-001 for the addition
-isogeny γ = `isogOneSub_negFrobenius` and the leading-coefficient additivity
-for `id + (-π)`, and produces `omegaPullbackCoeff(γ) = 1`.
-
-Composes:
-* `omegaPullbackCoeff_add_of_leading_witness` (axiom-clean, FormalIsogenySeries.lean)
-* `omegaPullbackCoeff_eq_formalIsogenyLeading_id` (axiom-clean)
-* `omegaPullbackCoeff_eq_formalIsogenyLeading_negFrobeniusIsog` (axiom-clean,
-  this file)
-* `omegaPullbackCoeff_id`, `omegaPullbackCoeff_negFrobeniusIsog`
-  (axiom-clean, prior commits)
-
-Result: `omegaPullbackCoeff(γ) = omegaPullbackCoeff(id) + omegaPullbackCoeff(negFrob)
-= 1 + 0 = 1`. -/
+/-- Equality of the differential and linear formal-series coefficients for `1 − π`,
+together with linear-coefficient additivity, gives differential coefficient one. -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_bridge_and_leading
     (hq : 2 ≤ Fintype.card K)
     (h_bridge_γ : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -500,12 +385,8 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_bridge_and_leading
   rw [h_omega_add, omegaPullbackCoeff_id W, omegaPullbackCoeff_negFrobeniusIsog W,
     add_zero]
 
-/-- **Witness #1 (witness-parametric on BRIDGE-001 for γ + leading-additivity,
-axiom-clean)**: separability of `isogOneSub_negFrobenius W hq`, taking only
-the BRIDGE-001 for γ + leading-coefficient additivity. T-II-4-004 absorbed.
-
-Once BRIDGE-001 for γ + the leading-additivity (= BRIDGE-003 specialization)
-land, this fires the unconditional Witness #1 of the Hasse-Weil bound. -/
+/-- The differential/formal-series coefficient identity and linear-coefficient
+additivity imply separability of `1 − π`. -/
 theorem isogOneSub_negFrobenius_isSeparable_of_bridge_and_leading
     (hq : 2 ≤ Fintype.card K)
     (h_bridge_γ : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -520,17 +401,8 @@ theorem isogOneSub_negFrobenius_isSeparable_of_bridge_and_leading
     (omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_bridge_and_leading W hq
       h_bridge_γ h_leading_add)
 
-/-- **Witness #1 closer (axiom-clean) taking BRIDGE-001 for γ + BRIDGE-003
-for our case**: with both substantive witnesses in hand, produces
-`(isogOneSub_negFrobenius W hq).IsSeparable`.
-
-Chain:
-1. BRIDGE-003 for our case → `coeff 1 (formal γ) = 1`
-   (via `coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_via_bridge_003`).
-2. Leading-additivity follows trivially: `1 = 1 + 0`.
-3. BRIDGE-001 for γ + leading-additivity → `omegaPullbackCoeff(γ) = 1`
-   (via the existing `isogOneSub_negFrobenius_isSeparable_of_bridge_and_leading`).
-4. T-II-4-004 (axiom-clean) → IsSeparable. -/
+/-- A formal group law substitution identity and equality of the differential and
+linear formal-series coefficients imply separability of `1 − π`. -/
 theorem isogOneSub_negFrobenius_isSeparable_of_bridge_001_γ_and_bridge_003
     (hq : 2 ≤ Fintype.card K)
     (h_bridge_001_γ : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -557,12 +429,8 @@ theorem isogOneSub_negFrobenius_isSeparable_of_bridge_001_γ_and_bridge_003
   exact isogOneSub_negFrobenius_isSeparable_of_bridge_and_leading W hq
     h_bridge_001_γ h_leading_add
 
-/-- **omegaPullbackCoeff(γ) = 1 axiom-clean given BRIDGE-001 for γ + BRIDGE-003 for our case**.
-Direct consequence: derives `coeff 1 (formal γ) = 1` from BRIDGE-003 via the
-just-shipped via_bridge_003 closer, then leading-additivity follows trivially
-(both sides = 1), then the existing
-`omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_bridge_and_leading`
-fires. -/
+/-- Formal group law substitution and the differential/formal-series coefficient
+identity give differential coefficient one for `1 − π`. -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_bridge_001_γ_and_bridge_003
     (hq : 2 ≤ Fintype.card K)
     (h_bridge_001_γ : omegaPullbackCoeff W (isogOneSub_negFrobenius W hq) =
@@ -589,14 +457,15 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_bridge_001_γ_and_
   exact omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_bridge_and_leading W hq
     h_bridge_001_γ h_leading_add
 
-/-- **Sub-helper 1**: `localExpand(x_gen - π·x_gen) = formalX - formalX^q`
-(axiom-clean). Direct from `localExpand` ring hom + `frobeniusIsog_pullback_apply`. -/
+/-- `localExpand(x_gen - π·x_gen) = formalX - formalX^q`
+. Direct from `localExpand` ring hom + `frobeniusIsog_pullback_apply`. -/
 theorem localExpand_x_gen_sub_frobenius_pullback_x_gen :
     localExpand W (x_gen W - (frobeniusIsog W).pullback (x_gen W)) =
       formalX W - (formalX W) ^ Fintype.card K := by
   rw [frobeniusIsog_pullback_apply, map_sub, map_pow, localExpand_x_gen]
 
-/-- **Sub-helper 2**: `(formalX - formalX^q).orderTop = -2q` for q ≥ 2.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX - formalX^q).orderTop = -2q` for q ≥ 2.
 For q ≥ 2, `formalX^q` strictly dominates `formalX` (orderTop -2q < -2),
 so by `orderTop_add_eq_right`, the sum's orderTop = orderTop(-formalX^q) = -2q. -/
 theorem orderTop_formalX_sub_formalX_pow (hq : 2 ≤ Fintype.card K) :
@@ -620,14 +489,15 @@ theorem orderTop_formalX_sub_formalX_pow (hq : 2 ≤ Fintype.card K) :
     nlinarith
   rw [HahnSeries.orderTop_add_eq_right h_lt, hX_neg_pow_orderTop]
 
-/-- **Sub-helper 3**: `localExpand((x_gen - π·x_gen)²) = (formalX - formalX^q)²`.
+/-- `localExpand((x_gen - π·x_gen)²) = (formalX - formalX^q)²`.
 Direct from `localExpand` ring hom + `localExpand_x_gen_sub_frobenius_pullback_x_gen`. -/
 theorem localExpand_x_gen_sub_frob_pullback_x_gen_sq :
     localExpand W ((x_gen W - (frobeniusIsog W).pullback (x_gen W)) ^ 2) =
       (formalX W - (formalX W) ^ Fintype.card K) ^ 2 := by
   rw [map_pow, localExpand_x_gen_sub_frobenius_pullback_x_gen]
 
-/-- **Sub-helper 4**: `((formalX - formalX^q)²).orderTop = -4q` for q ≥ 2.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `((formalX - formalX^q)²).orderTop = -4q` for q ≥ 2.
 By `orderTop_mul` + `orderTop_formalX_sub_formalX_pow`. -/
 theorem orderTop_formalX_sub_formalX_pow_sq (hq : 2 ≤ Fintype.card K) :
     ((formalX W - (formalX W) ^ Fintype.card K) ^ 2 : LaurentSeries K).orderTop =
@@ -641,7 +511,7 @@ theorem orderTop_formalX_sub_formalX_pow_sq (hq : 2 ≤ Fintype.card K) :
   congr 1
   ring
 
-/-- **Sub-helper 5**: `localExpand(x_gen · (π·x_gen)²) = formalX^(2q+1)`.
+/-- `localExpand(x_gen · (π·x_gen)²) = formalX^(2q+1)`.
 The dominant term of `addPullbackNumerator_reduced_negFrobenius` at infinity. -/
 theorem localExpand_x_gen_mul_frob_pullback_x_gen_sq :
     localExpand W (x_gen W * (frobeniusIsog W).pullback (x_gen W) ^ 2) =
@@ -649,7 +519,8 @@ theorem localExpand_x_gen_mul_frob_pullback_x_gen_sq :
   rw [frobeniusIsog_pullback_apply, ← pow_mul, map_mul, map_pow, localExpand_x_gen]
   ring
 
-/-- **Sub-helper 6**: `(formalX^(2q+1)).orderTop = -4q - 2` for q ≥ 1.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX^(2q+1)).orderTop = -4q - 2` for q ≥ 1.
 Direct from `formalX_pow_orderTop`. -/
 theorem orderTop_formalX_pow_two_q_plus_one :
     ((formalX W) ^ (2 * Fintype.card K + 1) : LaurentSeries K).orderTop =
@@ -659,13 +530,14 @@ theorem orderTop_formalX_pow_two_q_plus_one :
   push_cast
   ring
 
-/-- **Sub-helper 7**: `(formalX^(2q+1)).leadingCoeff = 1`.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX^(2q+1)).leadingCoeff = 1`.
 Direct from `formalX_pow_leadingCoeff`. -/
 theorem leadingCoeff_formalX_pow_two_q_plus_one :
     ((formalX W) ^ (2 * Fintype.card K + 1) : LaurentSeries K).leadingCoeff = 1 :=
   formalX_pow_leadingCoeff W (2 * Fintype.card K + 1)
 
-/-- **Sub-helper 8**: `(localExpand(x_gen · (π·x_gen)²)).orderTop = -2 - 4q`. Composes
+/-- `(localExpand(x_gen · (π·x_gen)²)).orderTop = -2 - 4q`. Composes
 the dominant-term identification with the orderTop computation. -/
 theorem orderTop_localExpand_x_gen_mul_frob_pullback_x_gen_sq :
     (localExpand W (x_gen W * (frobeniusIsog W).pullback (x_gen W) ^ 2)).orderTop =
@@ -673,7 +545,7 @@ theorem orderTop_localExpand_x_gen_mul_frob_pullback_x_gen_sq :
   rw [localExpand_x_gen_mul_frob_pullback_x_gen_sq W]
   exact orderTop_formalX_pow_two_q_plus_one W
 
-/-- **Sub-helper 9**: `(localExpand(x_gen · (π·x_gen)²)).leadingCoeff = 1`. The
+/-- `(localExpand(x_gen · (π·x_gen)²)).leadingCoeff = 1`. The
 leading coefficient at infinity of the dominant term is 1 (rather than 0 or
 some other value), which feeds into the leading-coefficient analysis for
 `addPullback_x` and ultimately `coeff 1 (formal γ) = 1`. -/
@@ -682,8 +554,8 @@ theorem leadingCoeff_localExpand_x_gen_mul_frob_pullback_x_gen_sq :
   rw [localExpand_x_gen_mul_frob_pullback_x_gen_sq W]
   exact leadingCoeff_formalX_pow_two_q_plus_one W
 
-omit [Fintype K] in
-/-- **Sub-helper 10**: `(formalY^n).orderTop = -3 * n`.
+omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalY^n).orderTop = -3 * n`.
 Mirror of `formalX_pow_orderTop`. -/
 theorem formalY_pow_orderTop (n : ℕ) :
     ((formalY W) ^ n : LaurentSeries K).orderTop =
@@ -704,14 +576,15 @@ theorem formalY_pow_orderTop (n : ℕ) :
     push_cast
     ring
 
-/-- **Sub-helper 11**: `localExpand(y_gen · π·y_gen) = formalY · formalY^q`.
+/-- `localExpand(y_gen · π·y_gen) = formalY · formalY^q`.
 Direct from `localExpand` ring hom + `frobeniusIsog_pullback_apply`. -/
 theorem localExpand_y_gen_mul_frob_pullback_y_gen :
     localExpand W (y_gen W * (frobeniusIsog W).pullback (y_gen W)) =
       formalY W * (formalY W) ^ Fintype.card K := by
   rw [frobeniusIsog_pullback_apply, map_mul, map_pow, localExpand_y_gen]
 
-/-- **Sub-helper 12**: `(formalY · formalY^q).orderTop = -3 - 3q` for q ≥ 1.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalY · formalY^q).orderTop = -3 - 3q` for q ≥ 1.
 Direct from `orderTop_mul` + `formalY_orderTop` + `formalY_pow_orderTop`. -/
 theorem orderTop_formalY_mul_formalY_pow :
     (formalY W * (formalY W) ^ Fintype.card K : LaurentSeries K).orderTop =
@@ -719,7 +592,7 @@ theorem orderTop_formalY_mul_formalY_pow :
   rw [HahnSeries.orderTop_mul, formalY_orderTop, formalY_pow_orderTop, ← WithTop.coe_add,
     show ((-3 : ℤ) + -3 * (Fintype.card K : ℤ)) = -3 - 3 * (Fintype.card K : ℤ) from by ring]
 
-/-- **Sub-helper 13**: `localExpand(x_gen² · π·x_gen) = formalX^(q+2)`.
+/-- `localExpand(x_gen² · π·x_gen) = formalX^(q+2)`.
 The "term 6" of `addPullbackNumerator_reduced_negFrobenius`: `x² · π·x` with
 orderTop -4 - 2q (sub-dominant). -/
 theorem localExpand_x_gen_sq_mul_frob_pullback_x_gen :
@@ -729,7 +602,8 @@ theorem localExpand_x_gen_sq_mul_frob_pullback_x_gen :
   congr 1
   ring
 
-/-- **Sub-helper 14**: `(formalX^(q+2)).orderTop = -4 - 2q`. -/
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX^(q+2)).orderTop = -4 - 2q`. -/
 theorem orderTop_formalX_pow_q_plus_two :
     ((formalX W) ^ (Fintype.card K + 2) : LaurentSeries K).orderTop =
       (((-4 - 2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) := by
@@ -738,7 +612,7 @@ theorem orderTop_formalX_pow_q_plus_two :
   push_cast
   ring
 
-/-- **Sub-helper 15**: `localExpand(x_gen · π·x_gen) = formalX^(q+1)`.
+/-- `localExpand(x_gen · π·x_gen) = formalX^(q+1)`.
 The "x · π·x" component (used in term 8 of `addPullbackNumerator_reduced`). -/
 theorem localExpand_x_gen_mul_frob_pullback_x_gen :
     localExpand W (x_gen W * (frobeniusIsog W).pullback (x_gen W)) =
@@ -746,7 +620,8 @@ theorem localExpand_x_gen_mul_frob_pullback_x_gen :
   rw [frobeniusIsog_pullback_apply, map_mul, map_pow, localExpand_x_gen,
     show Fintype.card K + 1 = 1 + Fintype.card K from by ring, pow_add, pow_one]
 
-/-- **Sub-helper 16**: `(formalX^(q+1)).orderTop = -2 - 2q`. -/
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX^(q+1)).orderTop = -2 - 2q`. -/
 theorem orderTop_formalX_pow_q_plus_one :
     ((formalX W) ^ (Fintype.card K + 1) : LaurentSeries K).orderTop =
       (((-2 - 2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) := by
@@ -755,14 +630,15 @@ theorem orderTop_formalX_pow_q_plus_one :
   push_cast
   ring
 
-/-- **Sub-helper 17**: `localExpand(x_gen + π·x_gen) = formalX + formalX^q`.
+/-- `localExpand(x_gen + π·x_gen) = formalX + formalX^q`.
 Term 1's "x + π·x" component (used in `a₄ · (x + π·x)`). -/
 theorem localExpand_x_gen_add_frob_pullback_x_gen :
     localExpand W (x_gen W + (frobeniusIsog W).pullback (x_gen W)) =
       formalX W + (formalX W) ^ Fintype.card K := by
   rw [frobeniusIsog_pullback_apply, map_add, map_pow, localExpand_x_gen]
 
-/-- **Sub-helper 18**: `(formalX + formalX^q).orderTop ≥ -2q` for q ≥ 1.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX + formalX^q).orderTop ≥ -2q` for q ≥ 1.
 Via `min_orderTop_le_orderTop_add` + `formalX_orderTop` + `formalX_pow_orderTop`. -/
 theorem orderTop_formalX_add_formalX_pow_ge_neg_two_q (hq : 2 ≤ Fintype.card K) :
     (((-2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -777,14 +653,15 @@ theorem orderTop_formalX_add_formalX_pow_ge_neg_two_q (hq : 2 ≤ Fintype.card K
   · -- -2q ≤ orderTop formalX^q = -2q
     rw [formalX_pow_orderTop]
 
-/-- **Sub-helper 19**: `localExpand(y_gen + π·y_gen) = formalY + formalY^q`.
+/-- `localExpand(y_gen + π·y_gen) = formalY + formalY^q`.
 Term 3's "y + π·y" component (used in `a₃ · (y + π·y)`). -/
 theorem localExpand_y_gen_add_frob_pullback_y_gen :
     localExpand W (y_gen W + (frobeniusIsog W).pullback (y_gen W)) =
       formalY W + (formalY W) ^ Fintype.card K := by
   rw [frobeniusIsog_pullback_apply, map_add, map_pow, localExpand_y_gen]
 
-/-- **Sub-helper 20**: `(formalY + formalY^q).orderTop ≥ -3q` for q ≥ 1.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalY + formalY^q).orderTop ≥ -3q` for q ≥ 1.
 Via `min_orderTop_le_orderTop_add` + `formalY_orderTop` + `formalY_pow_orderTop`. -/
 theorem orderTop_formalY_add_formalY_pow_ge_neg_three_q (hq : 2 ≤ Fintype.card K) :
     (((-3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -797,13 +674,14 @@ theorem orderTop_formalY_add_formalY_pow_ge_neg_three_q (hq : 2 ≤ Fintype.card
     nlinarith
   · rw [formalY_pow_orderTop]
 
-/-- **Sub-helper 21**: `localExpand(x · π·y) = formalX · formalY^q`. -/
+/-- `localExpand(x · π·y) = formalX · formalY^q`. -/
 theorem localExpand_x_gen_mul_frob_pullback_y_gen :
     localExpand W (x_gen W * (frobeniusIsog W).pullback (y_gen W)) =
       formalX W * (formalY W) ^ Fintype.card K := by
   rw [frobeniusIsog_pullback_apply, map_mul, map_pow, localExpand_x_gen, localExpand_y_gen]
 
-/-- **Sub-helper 22**: `(formalX · formalY^q).orderTop = -2 - 3q`. -/
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX · formalY^q).orderTop = -2 - 3q`. -/
 theorem orderTop_formalX_mul_formalY_pow :
     (formalX W * (formalY W) ^ Fintype.card K : LaurentSeries K).orderTop =
       (((-2 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) := by
@@ -811,13 +689,14 @@ theorem orderTop_formalX_mul_formalY_pow :
     show ((-2 : ℤ) + (-3 : ℤ) * (Fintype.card K : ℤ)) =
         -2 - 3 * (Fintype.card K : ℤ) from by ring]
 
-/-- **Sub-helper 23**: `localExpand(π·x · y) = formalX^q · formalY`. -/
+/-- `localExpand(π·x · y) = formalX^q · formalY`. -/
 theorem localExpand_frob_pullback_x_gen_mul_y_gen :
     localExpand W ((frobeniusIsog W).pullback (x_gen W) * y_gen W) =
       (formalX W) ^ Fintype.card K * formalY W := by
   rw [frobeniusIsog_pullback_apply, map_mul, map_pow, localExpand_x_gen, localExpand_y_gen]
 
-/-- **Sub-helper 24**: `(formalX^q · formalY).orderTop = -3 - 2q`. -/
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX^q · formalY).orderTop = -3 - 2q`. -/
 theorem orderTop_formalX_pow_mul_formalY :
     ((formalX W) ^ Fintype.card K * formalY W : LaurentSeries K).orderTop =
       (((-3 - 2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) := by
@@ -825,7 +704,8 @@ theorem orderTop_formalX_pow_mul_formalY :
     show ((-2 : ℤ) * (Fintype.card K : ℤ) + (-3 : ℤ)) =
         -3 - 2 * (Fintype.card K : ℤ) from by ring]
 
-/-- **Sub-helper 25**: `(formalX·formalY^q + formalX^q·formalY).orderTop ≥ -2 - 3q`
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX·formalY^q + formalX^q·formalY).orderTop ≥ -2 - 3q`
 for q ≥ 2. Via `min_orderTop_le_orderTop_add`, since `-2-3q ≤ min(-2-3q, -3-2q)` for q ≥ 2. -/
 theorem orderTop_formalX_mul_formalY_pow_plus_formalX_pow_mul_formalY_ge
     (hq : 2 ≤ Fintype.card K) :
@@ -841,7 +721,7 @@ theorem orderTop_formalX_mul_formalY_pow_plus_formalX_pow_mul_formalY_ge
     nlinarith
 
 omit [Fintype K] in
-/-- **Sub-helper 26**: `(HahnSeries.C c).orderTop ≥ 0` for any `c : K`.
+/-- `(HahnSeries.C c).orderTop ≥ 0` for any `c : K`.
 Helper for term 2 (`2·a₆`) and similar constant-coefficient bounds. -/
 theorem orderTop_HahnSeries_C_ge_zero (c : K) :
     (0 : WithTop ℤ) ≤ (HahnSeries.C c : LaurentSeries K).orderTop := by
@@ -851,8 +731,8 @@ theorem orderTop_HahnSeries_C_ge_zero (c : K) :
   · rw [HahnSeries.C_apply, HahnSeries.orderTop_single hc]
     rfl
 
-omit [Fintype K] in
-/-- **Sub-helper 27**: `localExpand W (algebraMap K KE c) = HahnSeries.C c`.
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `localExpand W (algebraMap K KE c) = HahnSeries.C c`.
 Direct from `localExpand_algebraMap` + `HahnSeries.ofPowerSeries_C`. The
 constant-coefficient bridge: `algebraMap K KE` factors through `localExpand`
 to give a constant Hahn series. -/
@@ -861,8 +741,8 @@ theorem localExpand_algebraMap_eq_C (c : K) :
   rw [localExpand_algebraMap]
   exact HahnSeries.ofPowerSeries_C c
 
-omit [Fintype K] in
-/-- **Sub-helper 28**: `(localExpand W (algebraMap K KE c * f)).orderTop ≥
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(localExpand W (algebraMap K KE c * f)).orderTop ≥
 (localExpand W f).orderTop`. The constant factor `algebraMap K KE c` has
 orderTop ≥ 0 in the local expansion (it's a constant Hahn series), so
 multiplication by it does not decrease the orderTop. Bridge for terms 1, 3, 5
@@ -877,8 +757,8 @@ theorem orderTop_localExpand_algebraMap_mul_ge (c : K) (f : KE) :
     _ ≤ (HahnSeries.C c : LaurentSeries K).orderTop + (localExpand W f).orderTop := by
         gcongr
 
-omit [Fintype K] in
-/-- **Sub-helper 29**: `(localExpand W (2 * f)).orderTop ≥ (localExpand W f).orderTop`.
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(localExpand W (2 * f)).orderTop ≥ (localExpand W f).orderTop`.
 Bridge for terms 4, 7 of `addPullbackNumerator_reduced_negFrobenius` (the `2·...`
 terms). Uses `2 * f = f + f` to dodge characteristic-2 considerations. -/
 theorem orderTop_localExpand_two_mul_ge (f : KE) :
@@ -887,15 +767,14 @@ theorem orderTop_localExpand_two_mul_ge (f : KE) :
   refine le_trans ?_ HahnSeries.min_orderTop_le_orderTop_add
   exact le_min (le_refl _) (le_refl _)
 
-omit [Fintype K] in
-/-- **Sub-helper 30**: `(localExpand W (-f)).orderTop = (localExpand W f).orderTop`.
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(localExpand W (-f)).orderTop = (localExpand W f).orderTop`.
 Negation preserves orderTop. -/
 theorem orderTop_localExpand_neg (f : KE) :
     (localExpand W (-f)).orderTop = (localExpand W f).orderTop := by
   rw [map_neg, HahnSeries.orderTop_neg]
 
-/-- **Sub-helper 31**:
-`localExpand((negFrob).pullback y_gen) = -formalY^q - C(a₁)·formalX^q - C(a₃)`.
+/-- `localExpand((negFrob).pullback y_gen) = -formalY^q - C(a₁)·formalX^q - C(a₃)`.
 Direct from `negFrobeniusIsog_pullback_y_gen` + ring-hom facts. -/
 theorem localExpand_negFrobeniusIsog_pullback_y_gen :
     localExpand W ((negFrobeniusIsog W).pullback (y_gen W)) =
@@ -907,7 +786,8 @@ theorem localExpand_negFrobeniusIsog_pullback_y_gen :
     localExpand_y_gen, localExpand_x_gen, localExpand_algebraMap_eq_C,
     localExpand_algebraMap_eq_C]
 
-/-- **Sub-helper 32**: `(C(a₁)·formalX^q + C(a₃)).orderTop ≥ -2q` for q ≥ 1. -/
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(C(a₁)·formalX^q + C(a₃)).orderTop ≥ -2q` for q ≥ 1. -/
 theorem orderTop_C_a₁_mul_formalX_pow_plus_C_a₃_ge :
     (((-2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
       (HahnSeries.C W.a₁ * (formalX W) ^ Fintype.card K +
@@ -929,7 +809,7 @@ theorem orderTop_C_a₁_mul_formalX_pow_plus_C_a₃_ge :
     have : (Fintype.card K : ℤ) ≥ 0 := by positivity
     nlinarith
 
-/-- **Sub-helper 33**: `(localExpand((negFrob).pullback y_gen)).orderTop = -3q`
+/-- `(localExpand((negFrob).pullback y_gen)).orderTop = -3q`
 for q ≥ 2. The `formalY^q` term dominates strictly. -/
 theorem orderTop_localExpand_negFrobeniusIsog_pullback_y_gen
     (hq : 2 ≤ Fintype.card K) :
@@ -962,8 +842,8 @@ theorem orderTop_localExpand_negFrobeniusIsog_pullback_y_gen
     nlinarith
   rw [HahnSeries.orderTop_add_eq_left h_lt, h_neg_y_pow]
 
-omit [Fintype K] in
-/-- **Sub-helper 34**: `(C(a₃) · formalY).orderTop ≥ -3` for q ≥ 1. -/
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(C(a₃) · formalY).orderTop ≥ -3` for q ≥ 1. -/
 theorem orderTop_C_a₃_mul_formalY_ge :
     (((-3 : ℤ)) : WithTop ℤ) ≤
       (HahnSeries.C W.a₃ * formalY W : LaurentSeries K).orderTop := by
@@ -974,7 +854,8 @@ theorem orderTop_C_a₃_mul_formalY_ge :
   · rw [HahnSeries.C_apply, HahnSeries.orderTop_single ha₃,
         show ((0 : ℤ) : WithTop ℤ) = (0 : WithTop ℤ) from rfl, zero_add]
 
-/-- **Sub-helper 35**: `(C(a₁) · (formalY · formalX^q)).orderTop ≥ -2-3q`
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(C(a₁) · (formalY · formalX^q)).orderTop ≥ -2-3q`
 for q ≥ 1. -/
 theorem orderTop_C_a₁_mul_formalY_mul_formalX_pow_ge (hq : 1 ≤ Fintype.card K) :
     (((-2 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -995,7 +876,7 @@ theorem orderTop_C_a₁_mul_formalY_mul_formalX_pow_ge (hq : 1 ≤ Fintype.card
     have h_q : (1 : ℤ) ≤ (Fintype.card K : ℤ) := by exact_mod_cast hq
     omega
 
-/-- **Sub-helper 36**: `localExpand(y · negFrob.π·y) = -formalY · formalY^q -
+/-- `localExpand(y · negFrob.π·y) = -formalY · formalY^q -
 C(a₁) · (formalY · formalX^q) - C(a₃) · formalY`. -/
 theorem localExpand_y_gen_mul_negFrobeniusIsog_pullback_y_gen :
     localExpand W (y_gen W * (negFrobeniusIsog W).pullback (y_gen W)) =
@@ -1005,11 +886,11 @@ theorem localExpand_y_gen_mul_negFrobeniusIsog_pullback_y_gen :
   rw [map_mul, localExpand_negFrobeniusIsog_pullback_y_gen, localExpand_y_gen]
   ring
 
-/-- **Sub-helper 37**: `(localExpand(y · negFrob.π·y)).orderTop ≥ -3-3q` for q ≥ 2.
+/-- `(localExpand(y · negFrob.π·y)).orderTop ≥ -3-3q` for q ≥ 2.
 Each of the three terms has orderTop ≥ -3-3q:
-* `-formalY · formalY^q`: = -3-3q (sub-helper 12).
-* `-C(a₁) · (formalY · formalX^q)`: ≥ -2-3q ≥ -3-3q (sub-helper 35).
-* `-C(a₃) · formalY`: ≥ -3 ≥ -3-3q for q ≥ 0 (sub-helper 34). -/
+* `-formalY · formalY^q`: = -3-3q.
+* `-C(a₁) · (formalY · formalX^q)`: ≥ -2-3q ≥ -3-3q.
+* `-C(a₃) · formalY`: ≥ -3 ≥ -3-3q for q ≥ 0. -/
 theorem orderTop_localExpand_y_gen_mul_negFrobeniusIsog_pullback_y_gen_ge
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1041,7 +922,7 @@ theorem orderTop_localExpand_y_gen_mul_negFrobeniusIsog_pullback_y_gen_ge
       have h_q_nn : (0 : ℤ) ≤ (Fintype.card K : ℤ) := by positivity
       linarith
 
-/-- **Sub-helper 38**: `(localExpand(y_gen + (negFrob).pullback y_gen)).orderTop = -3q`
+/-- `(localExpand(y_gen + (negFrob).pullback y_gen)).orderTop = -3q`
 for q ≥ 2. The `(negFrob).pullback y_gen` part dominates strictly (orderTop -3q
 < -3 for q ≥ 2). -/
 theorem orderTop_localExpand_y_gen_add_negFrobeniusIsog_pullback_y_gen
@@ -1063,7 +944,7 @@ theorem orderTop_localExpand_y_gen_add_negFrobeniusIsog_pullback_y_gen
     nlinarith
   rw [HahnSeries.orderTop_add_eq_right h_lt, h_negFrob_y]
 
-/-- **Sub-helper 39**: `(localExpand(a₃ · (y_gen + (negFrob).pullback y_gen))).orderTop ≥ -3q`. -/
+/-- `(localExpand(a₃ · (y_gen + (negFrob).pullback y_gen))).orderTop ≥ -3q`. -/
 theorem orderTop_localExpand_a₃_mul_y_gen_add_negFrobeniusIsog_pullback_y_gen_ge
     (hq : 2 ≤ Fintype.card K) :
     (((-3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1072,7 +953,7 @@ theorem orderTop_localExpand_a₃_mul_y_gen_add_negFrobeniusIsog_pullback_y_gen_
   refine le_trans ?_ (orderTop_localExpand_algebraMap_mul_ge W W.a₃ _)
   rw [orderTop_localExpand_y_gen_add_negFrobeniusIsog_pullback_y_gen W hq]
 
-/-- **Sub-helper 40**: `(localExpand(x_gen + (negFrob).pullback x_gen)).orderTop ≥ -2q`
+/-- `(localExpand(x_gen + (negFrob).pullback x_gen)).orderTop ≥ -2q`
 for q ≥ 2. Uses the fact that `(negFrob).pullback x_gen = (frob).pullback x_gen`. -/
 theorem orderTop_localExpand_x_gen_add_negFrobeniusIsog_pullback_x_gen_ge
     (hq : 2 ≤ Fintype.card K) :
@@ -1082,7 +963,7 @@ theorem orderTop_localExpand_x_gen_add_negFrobeniusIsog_pullback_x_gen_ge
   rw [negFrobeniusIsog_pullback_x_gen, localExpand_x_gen_add_frob_pullback_x_gen]
   exact orderTop_formalX_add_formalX_pow_ge_neg_two_q W hq
 
-/-- **Sub-helper 41**: `(localExpand(a₄ · (x_gen + (negFrob).pullback x_gen))).orderTop ≥ -2q`. -/
+/-- `(localExpand(a₄ · (x_gen + (negFrob).pullback x_gen))).orderTop ≥ -2q`. -/
 theorem orderTop_localExpand_a₄_mul_x_gen_add_negFrobeniusIsog_pullback_x_gen_ge
     (hq : 2 ≤ Fintype.card K) :
     (((-2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1091,8 +972,8 @@ theorem orderTop_localExpand_a₄_mul_x_gen_add_negFrobeniusIsog_pullback_x_gen_
   refine le_trans ?_ (orderTop_localExpand_algebraMap_mul_ge W W.a₄ _)
   exact orderTop_localExpand_x_gen_add_negFrobeniusIsog_pullback_x_gen_ge W hq
 
-omit [Fintype K] in
-/-- **Sub-helper 42**: `(localExpand(2·a₆)).orderTop ≥ 0`. -/
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(localExpand(2·a₆)).orderTop ≥ 0`. -/
 theorem orderTop_localExpand_two_mul_a₆_ge :
     (0 : WithTop ℤ) ≤
       (localExpand W ((2 : W.toAffine.FunctionField) *
@@ -1101,7 +982,7 @@ theorem orderTop_localExpand_two_mul_a₆_ge :
   rw [localExpand_algebraMap_eq_C]
   exact orderTop_HahnSeries_C_ge_zero W.a₆
 
-/-- **Sub-helper 43**: `(localExpand(2 · y_gen · (negFrob).π·y)).orderTop ≥ -3-3q`. -/
+/-- `(localExpand(2 · y_gen · (negFrob).π·y)).orderTop ≥ -3-3q`. -/
 theorem orderTop_localExpand_two_mul_y_gen_mul_negFrob_π_y_ge
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1114,7 +995,7 @@ theorem orderTop_localExpand_two_mul_y_gen_mul_negFrob_π_y_ge
   refine le_trans ?_ (orderTop_localExpand_two_mul_ge W _)
   exact orderTop_localExpand_y_gen_mul_negFrobeniusIsog_pullback_y_gen_ge W hq
 
-/-- **Sub-helper 44**: `(localExpand(x_gen² · (negFrob).π·x)).orderTop = -4 - 2q`. -/
+/-- `(localExpand(x_gen² · (negFrob).π·x)).orderTop = -4 - 2q`. -/
 theorem orderTop_localExpand_x_gen_sq_mul_negFrobeniusIsog_pullback_x_gen :
     (localExpand W (x_gen W ^ 2 *
         (negFrobeniusIsog W).pullback (x_gen W))).orderTop =
@@ -1122,7 +1003,7 @@ theorem orderTop_localExpand_x_gen_sq_mul_negFrobeniusIsog_pullback_x_gen :
   rw [negFrobeniusIsog_pullback_x_gen, localExpand_x_gen_sq_mul_frob_pullback_x_gen]
   exact orderTop_formalX_pow_q_plus_two W
 
-/-- **Sub-helper 45**: `(localExpand(x_gen · ((negFrob).π·x)²)).orderTop = -2 - 4q`.
+/-- `(localExpand(x_gen · ((negFrob).π·x)²)).orderTop = -2 - 4q`.
 The DOMINANT term of `addPullbackNumerator_reduced_negFrobenius`. -/
 theorem orderTop_localExpand_x_gen_mul_negFrobeniusIsog_pullback_x_gen_sq :
     (localExpand W (x_gen W *
@@ -1131,7 +1012,7 @@ theorem orderTop_localExpand_x_gen_mul_negFrobeniusIsog_pullback_x_gen_sq :
   rw [negFrobeniusIsog_pullback_x_gen]
   exact orderTop_localExpand_x_gen_mul_frob_pullback_x_gen_sq W
 
-/-- **Sub-helper 46**: `(localExpand(2 · a₂ · x_gen · (negFrob).π·x)).orderTop ≥ -2 - 2q`. -/
+/-- `(localExpand(2 · a₂ · x_gen · (negFrob).π·x)).orderTop ≥ -2 - 2q`. -/
 theorem orderTop_localExpand_two_mul_a₂_mul_x_gen_mul_negFrob_π_x_ge :
     (((-2 - 2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
       (localExpand W ((2 : W.toAffine.FunctionField) *
@@ -1149,7 +1030,7 @@ theorem orderTop_localExpand_two_mul_a₂_mul_x_gen_mul_negFrob_π_x_ge :
     localExpand_x_gen_mul_frob_pullback_x_gen]
   exact (orderTop_formalX_pow_q_plus_one W).symm.le
 
-/-- **Sub-helper 47**: `(localExpand(x · (negFrob).π·y)).orderTop = -2 - 3q` for q ≥ 2. -/
+/-- `(localExpand(x · (negFrob).π·y)).orderTop = -2 - 3q` for q ≥ 2. -/
 theorem orderTop_localExpand_x_gen_mul_negFrob_pullback_y_gen
     (hq : 2 ≤ Fintype.card K) :
     (localExpand W (x_gen W * (negFrobeniusIsog W).pullback (y_gen W))).orderTop =
@@ -1159,16 +1040,16 @@ theorem orderTop_localExpand_x_gen_mul_negFrob_pullback_y_gen
     show ((-2 : ℤ) + (-3 : ℤ) * (Fintype.card K : ℤ)) =
         -2 - 3 * (Fintype.card K : ℤ) from by ring]
 
-/-- **Sub-helper 48**: `(localExpand((negFrob).π·x · y)).orderTop = -3 - 2q`. -/
+/-- `(localExpand((negFrob).π·x · y)).orderTop = -3 - 2q`. -/
 theorem orderTop_localExpand_negFrob_pullback_x_gen_mul_y_gen :
     (localExpand W ((negFrobeniusIsog W).pullback (x_gen W) * y_gen W)).orderTop =
       (((-3 - 2 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) := by
   rw [negFrobeniusIsog_pullback_x_gen, localExpand_frob_pullback_x_gen_mul_y_gen]
   exact orderTop_formalX_pow_mul_formalY W
 
-/-- **Sub-helper 49** (Term 5 inner): `(localExpand(x · negFrob.π·y +
+/-- Term 5 inner: `(localExpand(x · negFrob.π·y +
 negFrob.π·x · y)).orderTop ≥ -3-3q` for q ≥ 2. The `min_orderTop_le_orderTop_add`
-extracts both summands. Sub-helper 47 = -2-3q ≥ -3-3q, sub-helper 48 = -3-2q ≥ -3-3q. -/
+The summands have orders `-2-3q` and `-3-2q`, both at least `-3-3q`. -/
 theorem orderTop_localExpand_x_gen_mul_negFrob_π_y_plus_negFrob_π_x_mul_y_gen_ge
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1185,8 +1066,8 @@ theorem orderTop_localExpand_x_gen_mul_negFrob_π_y_plus_negFrob_π_x_mul_y_gen_
     refine WithTop.coe_le_coe.mpr ?_
     linarith
 
-/-- **Sub-helper 50** (Term 5 outer): `(localExpand(a₁ · (x · negFrob.π·y +
-negFrob.π·x · y))).orderTop ≥ -3-3q` for q ≥ 2. Composition of sub-helpers 28 + 49. -/
+/-- Term 5 outer: `(localExpand(a₁ · (x · negFrob.π·y +
+negFrob.π·x · y))).orderTop ≥ -3-3q` for q ≥ 2. Multiplication by a base-field constant preserves the lower order bound. -/
 theorem orderTop_localExpand_a₁_mul_x_negFrob_π_y_plus_negFrob_π_x_y_ge
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1196,8 +1077,8 @@ theorem orderTop_localExpand_a₁_mul_x_negFrob_π_y_plus_negFrob_π_x_y_ge
   refine le_trans ?_ (orderTop_localExpand_algebraMap_mul_ge W W.a₁ _)
   exact orderTop_localExpand_x_gen_mul_negFrob_π_y_plus_negFrob_π_x_mul_y_gen_ge W hq
 
-/-- **Sub-helper 51** (Term 1 promoted): `(localExpand(a₄ · (x_gen +
-negFrob.π·x))).orderTop ≥ -3-3q` for q ≥ 1. From sub-helper 41 (≥ -2q). -/
+/-- Term 1 promoted: `(localExpand(a₄ · (x_gen +
+negFrob.π·x))).orderTop ≥ -3-3q` for q ≥ 1. The sharper lower bound is `-2q`. -/
 theorem orderTop_localExpand_a₄_mul_x_gen_add_negFrob_π_x_promoted
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1208,7 +1089,8 @@ theorem orderTop_localExpand_a₄_mul_x_gen_add_negFrob_π_x_promoted
   refine WithTop.coe_le_coe.mpr ?_
   linarith
 
-/-- **Sub-helper 52** (Term 2 promoted): `(localExpand(2·a₆)).orderTop ≥ -3-3q`. -/
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- Term 2 promoted: `(localExpand(2·a₆)).orderTop ≥ -3-3q`. -/
 theorem orderTop_localExpand_two_mul_a₆_promoted
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1219,8 +1101,8 @@ theorem orderTop_localExpand_two_mul_a₆_promoted
   refine WithTop.coe_le_coe.mpr ?_
   linarith
 
-/-- **Sub-helper 53** (Term 3 promoted): `(localExpand(a₃ · (y_gen +
-negFrob.π·y))).orderTop ≥ -3-3q`. From sub-helper 39 (≥ -3q). -/
+/-- Term 3 promoted: `(localExpand(a₃ · (y_gen +
+negFrob.π·y))).orderTop ≥ -3-3q`. The sharper lower bound is `-3q`. -/
 theorem orderTop_localExpand_a₃_mul_y_gen_add_negFrob_π_y_promoted
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1231,8 +1113,8 @@ theorem orderTop_localExpand_a₃_mul_y_gen_add_negFrob_π_y_promoted
   refine WithTop.coe_le_coe.mpr ?_
   linarith
 
-/-- **Sub-helper 54** (Term 6 promoted): `(localExpand(x² · negFrob.π·x)).orderTop ≥ -3-3q`.
-From sub-helper 44 (= -4-2q). -/
+/-- Term 6 promoted: `(localExpand(x² · negFrob.π·x)).orderTop ≥ -3-3q`.
+The exact order is `-4-2q`. -/
 theorem orderTop_localExpand_x_gen_sq_mul_negFrob_π_x_promoted
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1243,8 +1125,8 @@ theorem orderTop_localExpand_x_gen_sq_mul_negFrob_π_x_promoted
   refine WithTop.coe_le_coe.mpr ?_
   linarith
 
-/-- **Sub-helper 55** (Term 8 promoted): `(localExpand(2·a₂·x·negFrob.π·x)).orderTop ≥ -3-3q`.
-From sub-helper 46 (≥ -2-2q). -/
+/-- Term 8 promoted: `(localExpand(2·a₂·x·negFrob.π·x)).orderTop ≥ -3-3q`.
+The sharper lower bound is `-2-2q`. -/
 theorem orderTop_localExpand_two_a₂_x_negFrob_π_x_promoted
     (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1257,14 +1139,14 @@ theorem orderTop_localExpand_two_a₂_x_negFrob_π_x_promoted
   linarith
 
 omit [Fintype K] [DecidableEq K] in
-/-- **Sub-helper 56**: `add_ge_of_both_ge` — addition preserves orderTop bounds. -/
+/-- `add_ge_of_both_ge` — addition preserves orderTop bounds. -/
 theorem orderTop_add_ge_of_both_ge {a b : LaurentSeries K} {n : WithTop ℤ}
     (ha : n ≤ a.orderTop) (hb : n ≤ b.orderTop) :
     n ≤ (a + b).orderTop :=
   le_trans (le_min ha hb) HahnSeries.min_orderTop_le_orderTop_add
 
 omit [Fintype K] [DecidableEq K] in
-/-- **Sub-helper 57**: `sub_ge_of_both_ge` — subtraction preserves orderTop bounds.
+/-- `sub_ge_of_both_ge` — subtraction preserves orderTop bounds.
 Uses `HahnSeries.orderTop_neg` to convert `b` ↔ `-b`. -/
 theorem orderTop_sub_ge_of_both_ge {a b : LaurentSeries K} {n : WithTop ℤ}
     (ha : n ≤ a.orderTop) (hb : n ≤ b.orderTop) :
@@ -1274,7 +1156,7 @@ theorem orderTop_sub_ge_of_both_ge {a b : LaurentSeries K} {n : WithTop ℤ}
   rw [HahnSeries.orderTop_neg]
   exact hb
 
-/-- **Sub-helper 58** (Cumulative sum): `(localExpand(rest)).orderTop ≥ -3-3q`,
+/-- Cumulative sum: `(localExpand(rest)).orderTop ≥ -3-3q`,
 where `rest` is `addPullbackNumerator_reduced_negFrobenius` minus the dominant
 term `x · (negFrob.π·x)²`. Sequential `add/sub_ge_of_both_ge` chain over
 the 7 sub-dominant term bounds. -/
@@ -1339,7 +1221,7 @@ private theorem addPullbackNumerator_reduced_negFrobenius_split_eq :
 
 /-- The dominant term `localExpand(x_gen · (negFrob.π·x)²)` has strictly smaller
 `orderTop` than the `rest` (for `q ≥ 2`): `-2-4q < -3-3q`. Combines the dominant
-order (sub-helper 45) with the `rest ≥ -3-3q` bound (sub-helper 58). -/
+order with the `rest ≥ -3-3q` bound. -/
 private theorem orderTop_localExpand_dominant_lt_rest_negFrobenius
     (hq : 2 ≤ Fintype.card K) :
     (localExpand W (x_gen W *
@@ -1374,8 +1256,8 @@ private theorem orderTop_localExpand_dominant_lt_rest_negFrobenius
 for q ≥ 2.
 
 Algebraic decomposition: the full expression is `rest + dominant`, where
-* `dominant = x_gen · (negFrob.π·x)²` has orderTop = -2-4q (sub-helper 45)
-* `rest` (the other 7 terms) has orderTop ≥ -3-3q (sub-helper 58)
+* `dominant = x_gen · (negFrob.π·x)²` has orderTop = -2-4q
+* `rest` (the other 7 terms) has orderTop ≥ -3-3q
 
 For q ≥ 2: -2-4q < -3-3q, so `orderTop_add_eq_right` extracts the dominant
 orderTop. -/
@@ -1442,7 +1324,7 @@ private lemma orderTop_localExpand_dominant_lt_rest (hq : 2 ≤ Fintype.card K)
 /-- **Leading coefficient companion**:
 `(localExpand(addPullbackNumerator_reduced_negFrobenius)).leadingCoeff = 1`
 for q ≥ 2. The leading coefficient of the dominant term `x_gen · (negFrob.π·x)²`
-is 1 (sub-helper 9 / `leadingCoeff_localExpand_x_gen_mul_frob_pullback_x_gen_sq`),
+is 1 by `leadingCoeff_localExpand_x_gen_mul_frob_pullback_x_gen_sq`,
 which propagates through the strict-non-arch via `leadingCoeff_add_eq_right`. -/
 theorem leadingCoeff_localExpand_addPullbackNumerator_reduced_negFrobenius_eq
     (hq : 2 ≤ Fintype.card K) :
@@ -1454,7 +1336,8 @@ theorem leadingCoeff_localExpand_addPullbackNumerator_reduced_negFrobenius_eq
     negFrobeniusIsog_pullback_x_gen]
   exact leadingCoeff_localExpand_x_gen_mul_frob_pullback_x_gen_sq W
 
-/-- **Sub-helper 61**: `(formalX - formalX^q).leadingCoeff = -1` for q ≥ 2.
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- `(formalX - formalX^q).leadingCoeff = -1` for q ≥ 2.
 The dominant term `-formalX^q` (orderTop -2q < -2) determines the leading
 coefficient via `leadingCoeff_add_eq_right`. -/
 theorem leadingCoeff_formalX_sub_formalX_pow (hq : 2 ≤ Fintype.card K) :
@@ -1469,16 +1352,16 @@ theorem leadingCoeff_formalX_sub_formalX_pow (hq : 2 ≤ Fintype.card K) :
   rw [HahnSeries.leadingCoeff_add_eq_right h_lt, HahnSeries.leadingCoeff_neg,
     formalX_pow_leadingCoeff]
 
-/-- **Sub-helper 62**: `((formalX - formalX^q)²).leadingCoeff = 1` for q ≥ 2.
-Direct from `HahnSeries.leadingCoeff_mul` (squared) and sub-helper 61
-(`(-1)² = 1`). -/
+omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- Squaring `formalX - formalX^q`, whose leading coefficient is `-1`, gives leading
+coefficient one. -/
 theorem leadingCoeff_formalX_sub_formalX_pow_sq (hq : 2 ≤ Fintype.card K) :
     ((formalX W - (formalX W) ^ Fintype.card K) ^ 2 : LaurentSeries K).leadingCoeff = 1 := by
   rw [sq, HahnSeries.leadingCoeff_mul, leadingCoeff_formalX_sub_formalX_pow W hq]
   ring
 
-/-- **Sub-helper 63**: `(localExpand((x_gen - negFrob.π·x)²)).orderTop = -4q` for q ≥ 2.
-Bridges via `negFrobeniusIsog_pullback_x_gen` to the frobenius-side sub-helper 4. -/
+/-- `(localExpand((x_gen - negFrob.π·x)²)).orderTop = -4q` for q ≥ 2.
+Negative Frobenius has the same x-coordinate pullback as Frobenius. -/
 theorem orderTop_localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq
     (hq : 2 ≤ Fintype.card K) :
     (localExpand W ((x_gen W -
@@ -1487,7 +1370,7 @@ theorem orderTop_localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq
   rw [negFrobeniusIsog_pullback_x_gen, localExpand_x_gen_sub_frob_pullback_x_gen_sq]
   exact orderTop_formalX_sub_formalX_pow_sq W hq
 
-/-- **Sub-helper 64**: `(localExpand((x_gen - negFrob.π·x)²)).leadingCoeff = 1` for q ≥ 2. -/
+/-- `(localExpand((x_gen - negFrob.π·x)²)).leadingCoeff = 1` for q ≥ 2. -/
 theorem leadingCoeff_localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq
     (hq : 2 ≤ Fintype.card K) :
     (localExpand W ((x_gen W -
@@ -1495,7 +1378,7 @@ theorem leadingCoeff_localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq
   rw [negFrobeniusIsog_pullback_x_gen, localExpand_x_gen_sub_frob_pullback_x_gen_sq]
   exact leadingCoeff_formalX_sub_formalX_pow_sq W hq
 
-/-- **Sub-helper 65**: `(localExpand((x_gen - negFrob.π·x)²)) ≠ 0` for q ≥ 2.
+/-- `(localExpand((x_gen - negFrob.π·x)²)) ≠ 0` for q ≥ 2.
 Direct from orderTop being a finite value. -/
 theorem localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq_ne_zero
     (hq : 2 ≤ Fintype.card K) :
@@ -1508,7 +1391,7 @@ theorem localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq_ne_zero
   rw [orderTop_localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq W hq] at h_top
   exact WithTop.coe_ne_top h_top
 
-/-- **Sub-helper 65b**: `(x_gen - negFrob.π·x) ≠ 0` in K(E). Derived from the
+/-- `(x_gen - negFrob.π·x) ≠ 0` in K(E). Derived from the
 LaurentSeries-side fact `(formalX - formalX^q).orderTop = -2q ≠ ⊤`. -/
 private theorem x_gen_sub_negFrob_pullback_x_gen_ne_zero_local
     (hq : 2 ≤ Fintype.card K) :
@@ -1569,7 +1452,7 @@ theorem leadingCoeff_localExpand_addPullback_x_negFrobenius_eq
     leadingCoeff_localExpand_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq W hq]
   ring
 
-/-- **Sub-helper 66**: `localExpand(addPullback_x_negFrobenius) ≠ 0` for q ≥ 2.
+/-- `localExpand(addPullback_x_negFrobenius) ≠ 0` for q ≥ 2.
 Direct from orderTop being a finite value. -/
 theorem localExpand_addPullback_x_negFrobenius_ne_zero
     (hq : 2 ≤ Fintype.card K) :
@@ -1580,11 +1463,11 @@ theorem localExpand_addPullback_x_negFrobenius_ne_zero
   rw [orderTop_localExpand_addPullback_x_negFrobenius_eq W hq] at h_top
   exact WithTop.coe_ne_top h_top
 
-/-- **Sub-helper 67**: `(localExpand(addPullback_x_negFrobenius)).coeff (-2) = 1` for q ≥ 2.
+/-- `(localExpand(addPullback_x_negFrobenius)).coeff (-2) = 1` for q ≥ 2.
 Direct extraction of the leading coefficient at the leading exponent
 via `HahnSeries.coeff_untop_eq_leadingCoeff` (when orderTop = -2 finite).
 
-This is the ACTUAL coefficient value at the most-negative exponent, the
+This is the coefficient value at the most-negative exponent, the
 LaurentSeries-side input for the IV.1.4 step 2 leading-coefficient
 computation. -/
 theorem coeff_neg_two_localExpand_addPullback_x_negFrobenius_eq
@@ -1602,7 +1485,7 @@ theorem coeff_neg_two_localExpand_addPullback_x_negFrobenius_eq
   rw [← h_untop, HahnSeries.coeff_untop_eq_leadingCoeff]
   exact h_leadingCoeff
 
-/-- **Sub-helper 68**: `((localExpand addPullback_x)^2).orderTop = -4`. -/
+/-- `((localExpand addPullback_x)^2).orderTop = -4`. -/
 theorem orderTop_localExpand_addPullback_x_negFrobenius_sq
     (hq : 2 ≤ Fintype.card K) :
     ((localExpand W (addPullback_x W (negFrobeniusIsog W))) ^ 2 :
@@ -1612,7 +1495,7 @@ theorem orderTop_localExpand_addPullback_x_negFrobenius_sq
     orderTop_localExpand_addPullback_x_negFrobenius_eq W hq, ← WithTop.coe_add]
   congr 1
 
-/-- **Sub-helper 69**: `((localExpand addPullback_x)^3).orderTop = -6`. -/
+/-- `((localExpand addPullback_x)^3).orderTop = -6`. -/
 theorem orderTop_localExpand_addPullback_x_negFrobenius_cube
     (hq : 2 ≤ Fintype.card K) :
     ((localExpand W (addPullback_x W (negFrobeniusIsog W))) ^ 3 :
@@ -1623,7 +1506,7 @@ theorem orderTop_localExpand_addPullback_x_negFrobenius_cube
     orderTop_localExpand_addPullback_x_negFrobenius_eq W hq, ← WithTop.coe_add]
   congr 1
 
-/-- **Sub-helper 70**: `localExpand(addPullback_y)` is nonzero for q ≥ 2.
+/-- `localExpand(addPullback_y)` is nonzero for q ≥ 2.
 Direct from `localExpand` being a ring hom to a field + addPullback_y ≠ 0
 (via `ord_addPullback_y_negFrobenius`). -/
 theorem localExpand_addPullback_y_negFrobenius_ne_zero
@@ -1638,7 +1521,7 @@ theorem localExpand_addPullback_y_negFrobenius_ne_zero
   rw [h_y_eq_zero] at h_y_ord
   exact WithTop.top_ne_coe ((W_smooth W).ordAtInfty_zero.symm.trans h_y_ord)
 
-/-- **Sub-helper 71**: localExpand of the curve equation
+/-- localExpand of the curve equation
 `Y² + a₁·X·Y + a₃·Y = X³ + a₂·X² + a₄·X + a₆` applied to the addPullback
 coordinates. Direct from `addPullback_equation` + `localExpand` ring hom.
 
@@ -1675,7 +1558,7 @@ theorem localExpand_addPullback_curve_equation :
   rw [ha₁, ha₂, ha₃, ha₄, ha₆] at h_le
   exact h_le
 
-/-- **Sub-helper 72**: `(localExpand(a₂·X² + a₄·X + a₆)).orderTop ≥ -4` for q ≥ 2.
+/-- `(localExpand(a₂·X² + a₄·X + a₆)).orderTop ≥ -4` for q ≥ 2.
 The non-cubic RHS terms have orderTop bounded by -4 (since `a₂·X²` has
 orderTop ≥ -4 = orderTop X² when a₂ ≠ 0, and `a₄·X` has orderTop ≥ -2 ≥ -4,
 and `a₆` has orderTop ≥ 0 ≥ -4). -/
@@ -1713,7 +1596,7 @@ theorem orderTop_localExpand_RHS_lower_terms_ge
     refine le_trans ?_ (orderTop_HahnSeries_C_ge_zero W.a₆)
     exact_mod_cast (by norm_num : (-4 : ℤ) ≤ 0)
 
-/-- **Sub-helper 73**: `(localExpand(X³ + a₂·X² + a₄·X + a₆)).orderTop = -6`
+/-- `(localExpand(X³ + a₂·X² + a₄·X + a₆)).orderTop = -6`
 for q ≥ 2. Strict non-arch with `X³` dominating (orderTop -6) and the other
 terms bounded ≥ -4. -/
 theorem orderTop_localExpand_addPullback_RHS_eq
@@ -1751,8 +1634,7 @@ theorem orderTop_localExpand_addPullback_RHS_eq
   exact (HahnSeries.orderTop_add_eq_left h_lt).trans
     (orderTop_localExpand_addPullback_x_negFrobenius_cube W hq)
 
-/-- **Sub-helper 74** (Y-side LHS orderTop = -6): direct from sub-helper 73 +
-the curve equation (sub-helper 71). -/
+/-- Y-side LHS orderTop = -6: the curve equation identifies its order with the cubic right-hand side. -/
 theorem orderTop_localExpand_addPullback_LHS_eq
     (hq : 2 ≤ Fintype.card K) :
     ((localExpand W (addPullback_y W (negFrobeniusIsog W)))^2 +
@@ -1766,8 +1648,8 @@ theorem orderTop_localExpand_addPullback_LHS_eq
   rw [localExpand_addPullback_curve_equation W]
   exact orderTop_localExpand_addPullback_RHS_eq W hq
 
-/-- **Sub-helper 75** (extract m): `localExpand(addPullback_y).orderTop = (m : ℤ)`
-for some integer `m`. Uses sub-helper 70 (≠ 0). -/
+/-- extract m: `localExpand(addPullback_y).orderTop = (m : ℤ)`
+for some integer `m`. The series is nonzero, so its order is finite. -/
 theorem exists_m_orderTop_localExpand_addPullback_y_negFrobenius
     (hq : 2 ≤ Fintype.card K) :
     ∃ m : ℤ, (localExpand W (addPullback_y W (negFrobeniusIsog W))).orderTop =
@@ -1779,7 +1661,7 @@ theorem exists_m_orderTop_localExpand_addPullback_y_negFrobenius
   obtain ⟨m, hm⟩ := WithTop.ne_top_iff_exists.mp h_ne_top
   exact ⟨m, hm.symm⟩
 
-/-- **Sub-helper 76** (Y² orderTop = 2m): with `m = orderTop(Y)`, `Y²` has orderTop `2m`. -/
+/-- Y² orderTop = 2m: with `m = orderTop(Y)`, `Y²` has orderTop `2m`. -/
 theorem orderTop_localExpand_addPullback_y_negFrobenius_sq_eq_two_m
     (_hq : 2 ≤ Fintype.card K) (m : ℤ)
     (hm : (localExpand W (addPullback_y W (negFrobeniusIsog W))).orderTop =
@@ -1790,7 +1672,7 @@ theorem orderTop_localExpand_addPullback_y_negFrobenius_sq_eq_two_m
   rw [sq, HahnSeries.orderTop_mul, hm, ← WithTop.coe_add]
   congr 1; ring
 
-/-- **Sub-helper 77** (X·Y orderTop = -2 + m): -/
+/-- X·Y orderTop = -2 + m: -/
 theorem orderTop_localExpand_addPullback_x_mul_y_negFrobenius_eq
     (hq : 2 ≤ Fintype.card K) (m : ℤ)
     (hm : (localExpand W (addPullback_y W (negFrobeniusIsog W))).orderTop =
@@ -1858,7 +1740,7 @@ private theorem orderTop_a3_mul_y_negFrobenius_ge (m : ℤ) (h_m_ge : -2 ≤ m)
     rw [zero_add]
     refine WithTop.coe_le_coe.mpr ?_; linarith
 
-/-- **Sub-helper 78** (rule out m ≥ -2): if `m ≥ -2`, then LHS orderTop ≥ -4,
+/-- rule out m ≥ -2: if `m ≥ -2`, then LHS orderTop ≥ -4,
 contradicting LHS orderTop = -6. -/
 theorem m_le_neg_three_orderTop_localExpand_addPullback_y_negFrobenius
     (hq : 2 ≤ Fintype.card K) (m : ℤ)
@@ -1876,7 +1758,7 @@ theorem m_le_neg_three_orderTop_localExpand_addPullback_y_negFrobenius
   have h46 : (-4 : ℤ) ≤ -6 := by exact_mod_cast h_lhs_ge
   omega
 
-/-- **Sub-helper 79** (Y² strictly dominates a₁·X·Y, given m ≤ -3): for `m ≤ -3`,
+/-- Y² strictly dominates a₁·X·Y, given m ≤ -3: for `m ≤ -3`,
 `Y²` has orderTop = `2m ≤ -6 < -2 + m = orderTop(a₁·X·Y)`. -/
 theorem orderTop_y_sq_lt_a1xy_negFrobenius
     (hq : 2 ≤ Fintype.card K) (m : ℤ) (h_m_le : m ≤ -3)
@@ -1907,7 +1789,7 @@ theorem orderTop_y_sq_lt_a1xy_negFrobenius
     refine WithTop.coe_lt_coe.mpr ?_
     linarith
 
-/-- **Sub-helper 80** (Y² strictly dominates a₃·Y, given m ≤ -3): for `m ≤ -3`,
+/-- Y² strictly dominates a₃·Y, given m ≤ -3: for `m ≤ -3`,
 `Y²` has orderTop = `2m ≤ -6 < m = orderTop(a₃·Y)` (since `2m < m` for `m ≤ -3`). -/
 theorem orderTop_y_sq_lt_a3y_negFrobenius
     (hq : 2 ≤ Fintype.card K) (m : ℤ) (h_m_le : m ≤ -3)
@@ -1976,8 +1858,8 @@ theorem orderTop_localExpand_addPullback_y_negFrobenius_eq
   congr 1
   omega
 
-/-- **Sub-helper 84**: leadingCoeff of `(localExpand X)^3 = 1`, derived from
-sub-helper main companion (`leadingCoeff(localExpand X) = 1`) + leadingCoeff_mul. -/
+/-- leadingCoeff of `(localExpand X)^3 = 1`, derived from
+`leadingCoeff(localExpand X) = 1` and multiplicativity of leading coefficients. -/
 theorem leadingCoeff_localExpand_addPullback_x_negFrobenius_cube_eq
     (hq : 2 ≤ Fintype.card K) :
     ((localExpand W (addPullback_x W (negFrobeniusIsog W))) ^ 3 :
@@ -1987,7 +1869,7 @@ theorem leadingCoeff_localExpand_addPullback_x_negFrobenius_cube_eq
     leadingCoeff_localExpand_addPullback_x_negFrobenius_eq W hq]
   ring
 
-/-- **Sub-helper 85**: leadingCoeff of RHS = 1.
+/-- leadingCoeff of RHS = 1.
 
 Strict-dominance of `X³` on RHS (orderTop -6 < -4 of other terms) gives
 `leadingCoeff(RHS) = leadingCoeff(X³) = 1` via `leadingCoeff_add_eq_left`. -/
@@ -2025,7 +1907,7 @@ theorem leadingCoeff_localExpand_addPullback_RHS_eq
   rw [HahnSeries.leadingCoeff_add_eq_left h_lt]
   exact leadingCoeff_localExpand_addPullback_x_negFrobenius_cube_eq W hq
 
-/-- **Sub-helper 86**: leadingCoeff of LHS = 1.
+/-- leadingCoeff of LHS = 1.
 
 LHS = RHS (curve equation), so leadingCoeff equal. -/
 theorem leadingCoeff_localExpand_addPullback_LHS_eq
@@ -2040,7 +1922,7 @@ theorem leadingCoeff_localExpand_addPullback_LHS_eq
   rw [localExpand_addPullback_curve_equation W]
   exact leadingCoeff_localExpand_addPullback_RHS_eq W hq
 
-/-- **Sub-helper 87**: leadingCoeff of `Y²` extracted from LHS via strict-dominance.
+/-- leadingCoeff of `Y²` extracted from LHS via strict-dominance.
 
 For the actual y-orderTop = -3, `Y²` (orderTop = -6) strictly dominates `a₁·X·Y`
 (orderTop = -5) and `a₃·Y` (orderTop = -3). So `leadingCoeff(LHS) = leadingCoeff(Y²)`. -/
@@ -2080,7 +1962,7 @@ theorem leadingCoeff_localExpand_addPullback_y_negFrobenius_sq_eq_one
   exact h_lhs_lc
 
 /-- **MAIN Y-SIDE LEADING-COEFFICIENT**: `(leadingCoeff(localExpand addPullback_y))² = 1`
-for q ≥ 2. From `leadingCoeff(Y²) = 1` (sub-helper 87) + `leadingCoeff_mul`. -/
+for q ≥ 2. From `leadingCoeff(Y²) = 1` + `leadingCoeff_mul`. -/
 theorem leadingCoeff_localExpand_addPullback_y_negFrobenius_sq
     (hq : 2 ≤ Fintype.card K) :
     ((localExpand W (addPullback_y W (negFrobeniusIsog W))).leadingCoeff)^2 = 1 := by
@@ -2089,7 +1971,7 @@ theorem leadingCoeff_localExpand_addPullback_y_negFrobenius_sq
   rw [sq]
   exact h
 
-/-- **Sub-helper 89**: `leadingCoeff(localExpand addPullback_y) ≠ 0` for q ≥ 2.
+/-- `leadingCoeff(localExpand addPullback_y) ≠ 0` for q ≥ 2.
 Direct from the squared identity = 1 (so leadingCoeff = ±1). -/
 theorem leadingCoeff_localExpand_addPullback_y_negFrobenius_ne_zero
     (hq : 2 ≤ Fintype.card K) :
@@ -2099,7 +1981,7 @@ theorem leadingCoeff_localExpand_addPullback_y_negFrobenius_ne_zero
   rw [h, sq, mul_zero] at h_sq
   exact zero_ne_one h_sq
 
-/-- **Sub-helper 89b**: `addPullbackAlgHom_negFrobenius (x_gen) = addPullback_x`. -/
+/-- `addPullbackAlgHom_negFrobenius (x_gen) = addPullback_x`. -/
 theorem addPullbackAlgHom_negFrobenius_x_gen_eq
     (hq : 2 ≤ Fintype.card K) :
     addPullbackAlgHom_negFrobenius W hq (x_gen W) =
@@ -2119,7 +2001,7 @@ theorem addPullbackAlgHom_negFrobenius_x_gen_eq
   rw [AdjoinRoot.lift_mk]
   simp [addBaseHom, Polynomial.eval₂_C]
 
-/-- **Sub-helper 89c**: `addPullbackAlgHom_negFrobenius (y_gen) = addPullback_y`. -/
+/-- `addPullbackAlgHom_negFrobenius (y_gen) = addPullback_y`. -/
 theorem addPullbackAlgHom_negFrobenius_y_gen_eq
     (hq : 2 ≤ Fintype.card K) :
     addPullbackAlgHom_negFrobenius W hq (y_gen W) =
@@ -2141,8 +2023,7 @@ theorem addPullbackAlgHom_negFrobenius_y_gen_eq
   rw [AdjoinRoot.lift_mk]
   simp [addBaseHom, Polynomial.eval₂_X]
 
-/-- **Sub-helper 90**:
-`localExpand(γ.pullback localParam) = -localExpand(addPullback_x) / localExpand(addPullback_y)`
+/-- `localExpand(γ.pullback localParam) = -localExpand(addPullback_x) / localExpand(addPullback_y)`
 for γ = isogOneSub_negFrobenius. Direct ring-hom + AlgHom.commutes_div + map_div₀. -/
 theorem localExpand_isogOneSub_negFrobenius_pullback_localParam
     (hq : 2 ≤ Fintype.card K) :
@@ -2155,7 +2036,7 @@ theorem localExpand_isogOneSub_negFrobenius_pullback_localParam
     addPullbackAlgHom_negFrobenius_x_gen_eq W hq,
     addPullbackAlgHom_negFrobenius_y_gen_eq W hq]
 
-/-- **Sub-helper 91**: `(localExpand(γ.pullback localParam)).orderTop = 1` for
+/-- `(localExpand(γ.pullback localParam)).orderTop = 1` for
 γ = isogOneSub_negFrobenius. Via `orderTop_div`: `-2 - (-3) = 1`. -/
 theorem orderTop_localExpand_isogOneSub_negFrobenius_pullback_localParam
     (hq : 2 ≤ Fintype.card K) :
@@ -2169,7 +2050,7 @@ theorem orderTop_localExpand_isogOneSub_negFrobenius_pullback_localParam
     orderTop_localExpand_addPullback_y_negFrobenius_eq W hq]
   rfl
 
-/-- **Sub-helper 92**: `(localExpand(γ.pullback localParam)).leadingCoeff = -1 / leadingCoeff(Y)`
+/-- `(localExpand(γ.pullback localParam)).leadingCoeff = -1 / leadingCoeff(Y)`
 for γ = isogOneSub_negFrobenius. Via `leadingCoeff_div` + `leadingCoeff_neg`. -/
 theorem leadingCoeff_localExpand_isogOneSub_negFrobenius_pullback_localParam
     (hq : 2 ≤ Fintype.card K) :
@@ -2181,7 +2062,7 @@ theorem leadingCoeff_localExpand_isogOneSub_negFrobenius_pullback_localParam
   rw [HahnSeries.leadingCoeff_neg,
     leadingCoeff_localExpand_addPullback_x_negFrobenius_eq W hq]
 
-/-- **Sub-helper 93**: `coeff 1 (formal γ) = -1 / leadingCoeff(Y)` for γ = isogOneSub_negFrobenius.
+/-- `coeff 1 (formal γ) = -1 / leadingCoeff(Y)` for γ = isogOneSub_negFrobenius.
 
 Via `formalIsogenySeries_coeff` + `coeff_untop_eq_leadingCoeff` at orderTop = 1.
 This is the LaurentSeries-side computation of `coeff 1 (formal γ)` modulo
@@ -2209,8 +2090,8 @@ theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_eq_neg_one_div_lea
     HahnSeries.coeff_untop_eq_leadingCoeff]
   exact h_Z_lc
 
-/-- **Sub-helper 94** (witness-parametric coeff 1 = 1): with `leadingCoeff(Y) = -1`
-witness, `coeff 1 (formalIsogenySeries γ) = 1` axiom-clean. -/
+/-- witness-parametric coeff 1 = 1: with `leadingCoeff(Y) = -1`
+witness, `coeff 1 (formalIsogenySeries γ) = 1`. -/
 theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_eq_one_of_y_lc
     (hq : 2 ≤ Fintype.card K)
     (h_y_lc : (localExpand W (addPullback_y W (negFrobeniusIsog W))).leadingCoeff = -1) :
@@ -2220,9 +2101,8 @@ theorem coeff_one_formalIsogenySeries_isogOneSub_negFrobenius_eq_one_of_y_lc
   -- Goal: -1 / -1 = 1
   norm_num
 
-/-- **Sub-helper 95** (Witness #1 closer via leadingCoeff(Y) witness + BRIDGE-001 γ):
-takes `leadingCoeff(Y) = -1` as the LaurentSeries-side witness and BRIDGE-001
-for γ as the K(E)-side witness, fires Witness #1 unconditional. -/
+/-- A y-coordinate Laurent leading coefficient of `-1`, together with equality of the
+differential and linear formal-series coefficients, implies separability of `1 − π`. -/
 theorem isogOneSub_negFrobenius_isSeparable_via_y_lc_and_bridge_001
     (hq : 2 ≤ Fintype.card K)
     (h_y_lc : (localExpand W (addPullback_y W (negFrobeniusIsog W))).leadingCoeff = -1)
@@ -2241,8 +2121,8 @@ theorem isogOneSub_negFrobenius_isSeparable_via_y_lc_and_bridge_001
   exact isogOneSub_negFrobenius_isSeparable_of_bridge_and_leading W hq
     h_bridge_001_γ h_leading_add
 
-/-- **Sub-helper 96** (omegaPullbackCoeff = 1 via leadingCoeff(Y) + BRIDGE-001 γ):
-companion to 95 producing `omegaPullbackCoeff(γ) = 1`. -/
+/-- A y-coordinate Laurent leading coefficient of `-1` and the differential/formal-series
+coefficient identity give differential coefficient one for `1 − π`. -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_y_lc_and_bridge_001
     (hq : 2 ≤ Fintype.card K)
     (h_y_lc : (localExpand W (addPullback_y W (negFrobeniusIsog W))).leadingCoeff = -1)
@@ -2261,12 +2141,8 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_y_lc_and_bridge_00
   exact omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_of_bridge_and_leading W hq
     h_bridge_001_γ h_leading_add
 
-/-- **Sub-helper 96b** (sign determination — char 2 case axiom-clean):
-in characteristic 2, `(localExpand addPullback_y).leadingCoeff = -1`
-because `c_y² = 1` (sub-helper 88) and `1 = -1` in char 2 (via `CharTwo.neg_eq`).
-
-This pins the sign for characteristic 2 axiom-clean, closing the
-sign-determination work for q = 2^k case. -/
+/-- In characteristic two the y-coordinate Laurent leading coefficient is `-1`,
+since its square is one and `1 = -1`. -/
 theorem leadingCoeff_localExpand_addPullback_y_negFrobenius_eq_neg_one_char_two
     [CharP K 2]
     (hq : 2 ≤ Fintype.card K) :
@@ -2294,8 +2170,8 @@ theorem leadingCoeff_localExpand_addPullback_y_negFrobenius_eq_neg_one_char_two
     pow_eq_zero_iff (n := 2) (by norm_num : 2 ≠ 0) |>.mp h_c_plus_one_sq
   exact eq_neg_of_add_eq_zero_left h_c_plus_one
 
-/-- **Sub-helper 98** (Witness #1 char 2): in characteristic 2, the y-side
-sign pin is automatic via 96b. Witness #1 fires with only BRIDGE-001 for γ. -/
+/-- In characteristic two, equality of the differential and linear formal-series
+coefficients implies separability of `1 − π`. -/
 theorem isogOneSub_negFrobenius_isSeparable_char_two
     [CharP K 2]
     (hq : 2 ≤ Fintype.card K)
@@ -2307,7 +2183,7 @@ theorem isogOneSub_negFrobenius_isSeparable_char_two
     (leadingCoeff_localExpand_addPullback_y_negFrobenius_eq_neg_one_char_two W hq)
     h_bridge_001_γ
 
-/-- **Sub-helper 99** (omegaPullbackCoeff(γ) = 1 char 2). -/
+/-- omegaPullbackCoeff(γ) = 1 char 2. -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_char_two
     [CharP K 2]
     (hq : 2 ≤ Fintype.card K)
@@ -2319,10 +2195,8 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_char_two
     (leadingCoeff_localExpand_addPullback_y_negFrobenius_eq_neg_one_char_two W hq)
     h_bridge_001_γ
 
-/-- **Sub-helper 101** (BRIDGE-001 for γ char-2 from omega witness): takes
-`omegaPullbackCoeff(γ) = 1` directly and discharges BRIDGE-001 for γ
-axiom-clean. The LaurentSeries side gives `coeff 1 (formal γ) = 1` via
-y-lc (sub-helper 96b) + ratio chain. -/
+/-- In characteristic two, differential coefficient one implies its equality with the
+linear formal-series coefficient, which is also one. -/
 theorem bridge_001_γ_isogOneSub_negFrobenius_char_two
     [CharP K 2]
     (hq : 2 ≤ Fintype.card K)
@@ -2336,7 +2210,7 @@ theorem bridge_001_γ_isogOneSub_negFrobenius_char_two
   exact (map_one _).symm
 
 omit [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 104** (Frobenius D vanishing): for finite K with characteristic
+/-- Frobenius D vanishing: for finite K with characteristic
 p (and `q = #K` a power of p), the Kähler differential of `x_gen^q` vanishes.
 
 Proof: `D(x^q) = q · x^(q-1) · D(x)` (Leibniz pow), and `(q : K) = 0` since
@@ -2355,9 +2229,8 @@ theorem kaehler_D_x_gen_pow_card_eq_zero
     rw [Nat.cast_smul_eq_nsmul, smul_assoc]]
   rw [h_card_zero, zero_smul, zero_smul]
 
-/-- **Sub-helper 105** (Frobenius D pullback vanishing): the Kähler differential
-of `(frobeniusIsog).pullback x_gen` vanishes. Direct from sub-helper 104 +
-`frobeniusIsog_pullback_apply` (= `x_gen^q`). -/
+/-- The Kähler differential of the Frobenius pullback of `x_gen` vanishes, because that
+pullback is a q-th power. -/
 theorem kaehler_D_frobeniusIsog_pullback_x_gen
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     KaehlerDifferential.D K W.toAffine.FunctionField
@@ -2365,7 +2238,7 @@ theorem kaehler_D_frobeniusIsog_pullback_x_gen
   rw [frobeniusIsog_pullback_apply]
   exact kaehler_D_x_gen_pow_card_eq_zero W p
 
-/-- **Sub-helper 106** (Frobenius D pullback y vanishing): the Kähler differential
+/-- Frobenius D pullback y vanishing: the Kähler differential
 of `(frobeniusIsog).pullback y_gen` vanishes. Same proof shape as 105. -/
 theorem kaehler_D_frobeniusIsog_pullback_y_gen
     (p : ℕ) [Fact p.Prime] [CharP K p] :
@@ -2382,13 +2255,13 @@ theorem kaehler_D_frobeniusIsog_pullback_y_gen
     rw [Nat.cast_smul_eq_nsmul, smul_assoc]]
   rw [h_card_zero, zero_smul, zero_smul]
 
-/-- **Sub-helper 107** (negFrobenius D pullback x vanishing): the Kähler differential
+/-- negFrobenius D pullback x vanishing: the Kähler differential
 of `(negFrobeniusIsog).pullback x_gen` vanishes.
 
 `negFrobeniusIsog = mulByInt(-1) ∘ frobeniusIsog`, so
 `(negFrob).pullback x_gen = frobenius.pullback ((mulByInt(-1)).pullback x_gen) =
 frobenius.pullback x_gen = x_gen^q` (since `[-1]` fixes `x_gen`).
-Then by sub-helper 104, `D(x_gen^q) = 0`. -/
+The differential of a q-th power vanishes. -/
 theorem kaehler_D_negFrobeniusIsog_pullback_x_gen
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     KaehlerDifferential.D K W.toAffine.FunctionField
@@ -2396,13 +2269,13 @@ theorem kaehler_D_negFrobeniusIsog_pullback_x_gen
   rw [negFrobeniusIsog_pullback_x_gen]
   exact kaehler_D_frobeniusIsog_pullback_x_gen W p
 
-/-- **Sub-helper 108** (negFrobenius D pullback y vanishing): the Kähler differential
+/-- negFrobenius D pullback y vanishing: the Kähler differential
 of `(negFrobeniusIsog).pullback y_gen` vanishes.
 
 `(negFrob).pullback y_gen = -π·y_gen - a₁·π·x_gen - a₃` (where π·· =
 frobeniusIsog.pullback). The differential operator `D` is a K-derivation,
 so `D(constant) = 0` for `a₃ ∈ K`, and `D(a₁ · f) = a₁ · D(f)` for `a₁ ∈ K`,
-and `D(-f) = -D(f)`. Combined with sub-helpers 105-106, all three terms
+and `D(-f) = -D(f)`. Vanishing of both Frobenius coordinate differentials shows that all three terms
 vanish. -/
 theorem kaehler_D_negFrobeniusIsog_pullback_y_gen
     (p : ℕ) [Fact p.Prime] [CharP K p] :
@@ -2419,7 +2292,7 @@ theorem kaehler_D_negFrobeniusIsog_pullback_y_gen
   simp
 
 omit [Fintype K] in
-/-- **Sub-helper 109** (D(addPullback_x) formula via slope).
+/-- D(addPullback_x) formula via slope.
 
 For any α : Isogeny W W with `D((α).pullback x_gen) = 0` (i.e., α has its
 x-pullback Kähler-flat, true for Frobenius / negFrobenius in characteristic
@@ -2452,9 +2325,9 @@ theorem kaehler_D_addPullback_x_via_slope_witness
         (Nat.cast_smul_eq_nsmul (R := KE) 2 _).symm]
   rw [smul_smul]
 
-/-- **Sub-helper 110** (D(addPullback_x) for negFrobenius — specialized).
+/-- D(addPullback_x) for negFrobenius — specialized.
 
-Direct from sub-helper 109 + sub-helper 107: for the addition-pullback
+Differentiating the addition formula and using the vanishing Frobenius differential gives, for the addition-pullback
 under negFrobenius, `D(addPullback_x) = (2·ℓ + a₁) • D(ℓ) - D(x_gen)`
 where ℓ = addSlope W (negFrobeniusIsog W).
 
@@ -2473,7 +2346,7 @@ theorem kaehler_D_addPullback_x_negFrobenius
     (kaehler_D_negFrobeniusIsog_pullback_x_gen W p)
 
 omit [Fintype K] in
-/-- **Route B core (III.5.2), general slope-differential formula** (no `D(α*x)=0`
+/-- **General slope-differential formula** (no `D(α*x)=0`
 hypothesis): for any `α`, the Kähler differential of the `id + α` addition-pullback
 x-coordinate is the slope term minus `D(x_gen)` minus `D(α*x)`. Generalizes
 `kaehler_D_addPullback_x_via_slope_witness` (which drops the last term under the
@@ -2502,19 +2375,8 @@ theorem kaehler_D_addPullback_x_general
   rw [show (2 : ℕ) • (ℓ • D ℓ) = ((2 : KE)) • (ℓ • D ℓ) from
         (Nat.cast_smul_eq_nsmul (R := KE) 2 _).symm, smul_smul]
 
-/-- **Sub-helper 111** (ω(γ) = 1 via Kähler witness).
-
-Closing-arc witness consumer: given that
-  `(α*(u))⁻¹ • D(addPullback_x) = invariantDifferential`
-(equivalently, `α*(u) • ω = D(addPullback_x)` for ω = invariantDifferential),
-conclude `omegaPullbackCoeff(γ) = 1`. Direct from
-`omegaPullbackCoeff_spec` + `omegaPullbackCoeff_unique`.
-
-The hypothesis is the SUBSTANTIVE Kähler identity in K(E) that III.5.2 / IV.1.4
-carries out: relating the slope's differential to the addition-pullback's
-differential. Once this identity is closed (via, e.g., the curve equation
-matching at orderTop -6 and the slope-of-formal-identity lemma), `ω(γ) = 1`
-follows axiom-clean and the Hasse-Weil bound discharges via sub-helper 103. -/
+/-- The Kähler identity `(α*u)⁻¹ • D(α*x) = ω` gives differential coefficient one.
+This follows from the defining identity and uniqueness of the coefficient. -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_kaehler_witness
     (hq : 2 ≤ Fintype.card K)
     (h_kaehler :
@@ -2528,12 +2390,8 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_kaehler_witness
   simp only [x_gen] at h_kaehler
   exact h_kaehler
 
-/-- **Sub-helper 113** (Kähler ω(γ) = 1 via pullbackKaehler witness — closing-arc).
-
-The cleaner reformulation of sub-helper 111 using `pullbackKaehler` directly:
-given `γ.pullbackKaehler ω = ω`, conclude `omegaPullbackCoeff(γ) = 1`.
-
-Direct from `pullbackKaehler_invariantDifferential` + `omegaPullbackCoeff_unique`. -/
+/-- An isogeny fixing the invariant differential has differential coefficient one.
+-/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_pullbackKaehler_witness
     (hq : 2 ≤ Fintype.card K)
     (h_pK : (isogOneSub_negFrobenius W hq).pullbackKaehler
@@ -2544,7 +2402,7 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_pullbackKaehler_wi
   exact h_pK
 
 omit [Fintype K] in
-/-- **Sub-helper 114** (id.pullbackKaehler ω = ω, axiom-clean).
+/-- id.pullbackKaehler ω = ω.
 
 The identity isogeny acts trivially on the invariant differential. Direct
 from `pullbackKaehler_invariantDifferential (Isogeny.id W.toAffine)` +
@@ -2554,28 +2412,15 @@ theorem pullbackKaehler_invariantDifferential_id :
       invariantDifferential W.toAffine := by
   rw [Isogeny.pullbackKaehler_invariantDifferential, omegaPullbackCoeff_id, one_smul]
 
-/-- **Sub-helper 115** ((-π).pullbackKaehler ω = 0, axiom-clean).
-
-The negation-of-Frobenius isogeny annihilates the invariant differential
-(at the cotangent level). Direct from `pullbackKaehler_invariantDifferential
-(negFrobeniusIsog W)` + `omegaPullbackCoeff_negFrobeniusIsog = 0`
-(Differential.lean line 430, axiom-clean). -/
+/-- Negative Frobenius annihilates the invariant differential, because its
+invariant-differential coefficient is zero. -/
 theorem pullbackKaehler_invariantDifferential_negFrobeniusIsog :
     (negFrobeniusIsog W).pullbackKaehler (invariantDifferential W.toAffine) = 0 := by
   rw [Isogeny.pullbackKaehler_invariantDifferential,
       omegaPullbackCoeff_negFrobeniusIsog, zero_smul]
 
-/-- **Sub-helper 116** (γ.pullbackKaehler ω = ω via III.5.2 additivity witness — axiom-clean).
-
-Given the III.5.2 additivity at the differential level for our specific
-decomposition `γ = id + (-π)`:
-  `γ.pullbackKaehler ω = id.pullbackKaehler ω + (-π).pullbackKaehler ω`,
-combine with sub-helpers 114-115 (axiom-clean: `id.pullbackKaehler ω = ω` and
-`(-π).pullbackKaehler ω = 0`) to conclude `γ.pullbackKaehler ω = ω + 0 = ω`.
-
-This is the closing-arc commit: the only remaining substantive input is the
-III.5.2 differential additivity for our specific case, which is the III.5.2
-content for elliptic curve isogenies. -/
+/-- Differential additivity for `id + (-π)` gives pullback of the invariant differential
+equal to itself, since the identity fixes it and negative Frobenius annihilates it. -/
 theorem pullbackKaehler_invariantDifferential_isogOneSub_negFrobenius_via_additivity_witness
     (hq : 2 ≤ Fintype.card K)
     (h_add : (isogOneSub_negFrobenius W hq).pullbackKaehler
@@ -2587,11 +2432,8 @@ theorem pullbackKaehler_invariantDifferential_isogOneSub_negFrobenius_via_additi
   rw [h_add, pullbackKaehler_invariantDifferential_id W,
       pullbackKaehler_invariantDifferential_negFrobeniusIsog W, add_zero]
 
-/-- **Sub-helper 117** (ω(γ) = 1 via III.5.2 additivity witness — closing-arc).
-
-Composes sub-helper 113 (Kähler witness → ω(γ) = 1) with sub-helper 116
-(III.5.2 additivity → γ.pullbackKaehler ω = ω). The single substantive
-input is the III.5.2 differential additivity for our specific decomposition. -/
+/-- Additivity of the pullback of the invariant differential for `id + (-π)` gives
+differential coefficient one. -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_additivity_witness
     (hq : 2 ≤ Fintype.card K)
     (h_add : (isogOneSub_negFrobenius W hq).pullbackKaehler
@@ -2603,20 +2445,9 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one_via_additivity_witness
     (pullbackKaehler_invariantDifferential_isogOneSub_negFrobenius_via_additivity_witness
       W hq h_add)
 
-/-- **Sub-helper 119** (Kähler witness via slope-derivative witness — closing-arc).
-
-Given `D(addSlope W (negFrobeniusIsog W)) = c • D(x_gen)` for some `c : K(E)`
-(derived from differentiating the slope formula `(y_gen - π·y) / (x_gen - π·x)`
-with the Frobenius differential vanishing — sub-helpers 107, 108) and the
-algebraic identity in K(E):
-  `(2ℓ + a₁) · c - 1 = (alpha_star_u γ) · u_gen⁻¹`
-(equivalently, after clearing denominators: the K(E) identity that the III.5.2
-content carries out), conclude the Kähler witness needed by sub-helper 111.
-
-This factors the III.5.2 differential additivity through an explicit K(E)
-identity. The slope-derivative witness is the Kähler-differential form of
-the curve-tangent calculation; the K(E) identity is the substantive arithmetic
-content of III.5.2 for our `id + (-π)` decomposition. -/
+/-- A slope derivative `D(ℓ) = c • D(x)` and the coefficient identity
+`((2ℓ + a₁)c - 1)u = α*u` imply `(α*u)⁻¹ • D(α*x) = ω` for `α = 1 − π`.
+This expresses differential additivity through an identity in the function field. -/
 theorem kaehler_witness_via_slope_deriv_witness
     (p : ℕ) [Fact p.Prime] [CharP K p]
     (hq : 2 ≤ Fintype.card K)
@@ -2665,14 +2496,8 @@ theorem kaehler_witness_via_slope_deriv_witness
   linear_combination h_KE_identity
 
 omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 120** (Weierstrass equation in K(E), axiom-clean):
-`y_gen² + a₁·x_gen·y_gen + a₃·y_gen = x_gen³ + a₂·x_gen² + a₄·x_gen + a₆`
-in K(E). Direct from the affine equation `Affine.Equation` for the generic
-point on `W_KE`, with W_KE coefficients lifted to algebraMap form.
-
-This is the foundational K(E) identity that the curve-equation Kähler
-differential identity (sub-helper 121) builds upon. No derivation involved
-yet — purely the equation in K(E). -/
+/-- The generic coordinates satisfy the Weierstrass equation in the function field.
+-/
 theorem weierstrass_equation_in_KE :
     y_gen W ^ 2 + algebraMap K KE W.a₁ * x_gen W * y_gen W +
         algebraMap K KE W.a₃ * y_gen W =
@@ -2684,14 +2509,8 @@ theorem weierstrass_equation_in_KE :
   exact h_gen
 
 omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 121** (Kähler form of curve equation, axiom-clean foundation):
-literal D-application of the Weierstrass equation in K(E):
-`D(y² + a₁xy + a₃y) = D(x³ + a₂x² + a₄x + a₆)` in `Ω[K(E)/K]`.
-
-Direct from `congrArg D` applied to `weierstrass_equation_in_KE` (sub-helper
-120). The substantive Kähler identity `(a₃ + 2y + a₁x)·Dy = (3x² + 2a₂x +
-a₄ - a₁y)·Dx` follows by Leibniz expansion; the explicit ℕ-smul ↔ KE-smul
-conversion in that expansion is technical and deferred. -/
+/-- Applying the universal derivation to the generic Weierstrass equation gives
+`D(y² + a₁xy + a₃y) = D(x³ + a₂x² + a₄x + a₆)` in `Ω[K(E)/K]`. -/
 theorem kaehler_D_weierstrass_equation_K_E :
     KaehlerDifferential.D K W.toAffine.FunctionField
         (y_gen W ^ 2 +
@@ -2706,9 +2525,8 @@ theorem kaehler_D_weierstrass_equation_K_E :
     (weierstrass_equation_in_KE W)
 
 omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 122** (D(y²) via pow_two — axiom-clean KE-smul-only):
-`D(y_gen²) = y_gen • D(y_gen) + y_gen • D(y_gen)`. Wall-break: avoids
-`Derivation.leibniz_pow`'s ℕ-smul. -/
+/-- D(y²) via pow_two — KE-smul-only:
+`D(y_gen²) = y_gen • D(y_gen) + y_gen • D(y_gen)`. The product rule expresses both summands using function-field scalar multiplication. -/
 theorem kaehler_D_y_gen_sq :
     KaehlerDifferential.D K W.toAffine.FunctionField (y_gen W ^ 2) =
       y_gen W • KaehlerDifferential.D K W.toAffine.FunctionField (y_gen W) +
@@ -2717,7 +2535,7 @@ theorem kaehler_D_y_gen_sq :
     (KaehlerDifferential.D K W.toAffine.FunctionField).leibniz (y_gen W) (y_gen W)]
 
 omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 123** (D(x²) via pow_two — axiom-clean):
+/-- D(x²) via pow_two —:
 `D(x_gen²) = x_gen • D(x_gen) + x_gen • D(x_gen)`. -/
 theorem kaehler_D_x_gen_sq :
     KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W ^ 2) =
@@ -2727,7 +2545,7 @@ theorem kaehler_D_x_gen_sq :
     (KaehlerDifferential.D K W.toAffine.FunctionField).leibniz (x_gen W) (x_gen W)]
 
 omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 124** (D(x³) via pow expansion + leibniz — axiom-clean):
+/-- D(x³) via pow expansion + leibniz —:
 `D(x_gen³) = x_gen² • D(x_gen) + x_gen • D(x_gen²)`. -/
 theorem kaehler_D_x_gen_cube :
     KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W ^ 3) =
@@ -2739,9 +2557,8 @@ theorem kaehler_D_x_gen_cube :
     (KaehlerDifferential.D K W.toAffine.FunctionField).leibniz (x_gen W ^ 2) (x_gen W)]
 
 omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 125** (D-distribution of Weierstrass equation LHS, axiom-clean):
-fully Leibniz-expanded LHS in KE-smul-only form. Uses sub-helper 122 (D(y²)
-bypass) + Derivation.leibniz + D.map_algebraMap. -/
+/-- Applying the product rule to the left-hand side of the Weierstrass equation gives
+its expansion using function-field scalar multiplication. -/
 theorem kaehler_D_weierstrass_LHS_expanded :
     KaehlerDifferential.D K W.toAffine.FunctionField
         (y_gen W ^ 2 +
@@ -2770,8 +2587,8 @@ theorem kaehler_D_weierstrass_LHS_expanded :
   simp only [smul_zero, add_zero]
 
 omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
-/-- **Sub-helper 126** (D-distribution of Weierstrass equation RHS, axiom-clean):
-fully Leibniz-expanded RHS. Uses sub-helpers 123, 124. -/
+/-- D-distribution of Weierstrass equation RHS:
+fully Leibniz-expanded RHS. The product rule expands the square and cube. -/
 theorem kaehler_D_weierstrass_RHS_expanded :
     KaehlerDifferential.D K W.toAffine.FunctionField
         (x_gen W ^ 3 +
@@ -2803,17 +2620,10 @@ theorem kaehler_D_weierstrass_RHS_expanded :
       (KaehlerDifferential.D K W.toAffine.FunctionField).map_algebraMap W.a₆]
   simp only [smul_zero, add_zero]
 
-/-- **Sub-helper 127** (curve-equation Kähler identity, axiom-clean):
-
+omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
+/-- Differentiating the Weierstrass equation gives
 `(a₃ + 2y + a₁x) • D(y) = (3x² + 2a₂x + a₄ - a₁y) • D(x)` in `Ω[K(E)/K]`.
-
-Combines sub-helper 121 (D(LHS) = D(RHS) via congrArg) + 125 (LHS expansion)
-+ 126 (RHS expansion) + `linear_combination`. All KE-smul throughout —
-ℕ-smul wall bypassed via sub-helpers 122-124.
-
-This is the SUBSTANTIVE K(E) Kähler identity. NO witnesses. Foundational
-for sub-helper 128 (D(slope)) and the III.5.2 invariant-differential
-additivity. -/
+-/
 theorem kaehler_curve_equation_K_E :
     (algebraMap K KE W.a₃ + (2 : KE) * y_gen W +
         algebraMap K KE W.a₁ * x_gen W) •
@@ -2823,10 +2633,7 @@ theorem kaehler_curve_equation_K_E :
         algebraMap K KE W.a₄ -
         algebraMap K KE W.a₁ * y_gen W) •
         KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W) := by
-  -- Sixth wall-break attempt: pre-rewrite goal `(2 : KE) * y_gen` to
-  -- `y_gen + y_gen` using ring identities BEFORE expansion, eliminating the
-  -- (2 : KE) literal at the source. Then both sides use only +-additive
-  -- composition + KE-smul, no `(2 : ℕ)` casts anywhere.
+  -- Expand the integer coefficients as sums in the function field before differentiating.
   set Dx := KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W)
   set Dy := KaehlerDifferential.D K W.toAffine.FunctionField (y_gen W)
   -- Substitute literals with explicit sums in K(E).
@@ -2851,24 +2658,9 @@ theorem kaehler_curve_equation_K_E :
   -- Now both sides are sums of (KE-coefficient) • (Dx or Dy). Match via abel.
   linear_combination (norm := abel) h_eq
 
-/-- **Sub-helper 128** (D(addSlope) for negFrobenius — substantive K(E) identity, axiom-clean):
-
-The Kähler derivative of `addSlope W (negFrobeniusIsog W) =
-(y_gen - π·y) / (x_gen - π·x)` satisfies the closed-form identity:
-
-```
-(x_gen - π·x)² • D(addSlope) = (x_gen - π·x) • D(y_gen) - (y_gen - π·y) • D(x_gen)
-```
-
-where π·x = (negFrob).pullback x_gen, π·y = (negFrob).pullback y_gen.
-
-Proof: apply `Derivation.leibniz_div` to `addSlope_negFrobeniusIsog_eq`,
-simplify via `D(π·x) = 0` (sub-helper 107), `D(π·y) = 0` (sub-helper 108),
-cancel Den² · Den⁻² = 1.
-
-This is the SUBSTANTIVE K(E) Kähler identity for the addition slope —
-NO witnesses, NO hypotheses beyond Mathlib's quotient rule and the
-existing Frobenius-differential vanishing (axiom-clean). -/
+/-- The quotient rule and vanishing of Frobenius differentials give
+`Den² • D(ℓ) = Den • D(y) - N • D(x)` for the addition slope, where
+`Den = x - (-π)*x` and `N = y - (-π)*y`. -/
 theorem kaehler_D_addSlope_negFrobenius
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     (x_gen W - (negFrobeniusIsog W).pullback (x_gen W)) ^ 2 •
@@ -2902,7 +2694,7 @@ theorem kaehler_D_addSlope_negFrobenius
   rw [one_smul]
 
 omit [Fintype K] in
-/-- **Route B core (III.5.2), general slope differential** (no Frobenius flatness): for any `α`
+/-- **General slope differential** (no Frobenius flatness): for any `α`
 with `x_gen ≠ α*x_gen` (non-doubling), the Kähler derivative of `addSlope = (y−α*y)/(x−α*x)`
 satisfies `Den²·D(addSlope) = Den·(D(y)−D(α*y)) − N·(D(x)−D(α*x))` where `N = y−α*y`,
 `Den = x−α*x`. Generalizes `kaehler_D_addSlope_negFrobenius` (which used `D(π*x)=D(π*y)=0`). -/
@@ -2931,10 +2723,9 @@ theorem kaehler_D_addSlope_general
     show Den ^ 2 * Den⁻¹ ^ 2 = 1 from by rw [← mul_pow, mul_inv_cancel₀ hDen_ne, one_pow],
     one_smul]
 
-/-- **Sub-helper 129** (D(γ.pullback x_gen) cleared form, axiom-clean):
+/-- D(γ.pullback x_gen) cleared form:
 
-Combines sub-helper 110 (`D(addPullback_x) = (2ℓ + a₁) • D(ℓ) - D(x_gen)`)
-with sub-helper 128 (`Den² • D(addSlope) = Den • D(y_gen) - N • D(x_gen)`)
+Combines the differentiated addition formula with the denominator-cleared slope derivative
 to give a single closed-form identity for D(γ.pullback x_gen) involving
 D(y_gen) and D(x_gen) (with Den² as the multiplier).
 
@@ -2949,7 +2740,7 @@ where ℓ = addSlope, Den = x_gen - π·x, N = y_gen - π·y.
 
 This is the substantive D(γ.pullback x_gen) reduction in K(E), expressed
 purely in terms of the curve generators' Kähler differentials and Den, N
-without any inverses. Foundational for the III.5.2 cascade closure. -/
+without any inverses. This is a differential form of Silverman III.5.2. -/
 theorem kaehler_D_addPullback_x_negFrobenius_cleared
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     (x_gen W - (negFrobeniusIsog W).pullback (x_gen W)) ^ 2 •
@@ -2963,14 +2754,14 @@ theorem kaehler_D_addPullback_x_negFrobenius_cleared
           KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W)) -
       (x_gen W - (negFrobeniusIsog W).pullback (x_gen W)) ^ 2 •
         KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W) := by
-  -- Multiply sub-helper 110 by Den² and use sub-helper 128.
+  -- Multiply the x-coordinate differential by Den² and substitute the slope derivative.
   rw [kaehler_D_addPullback_x_negFrobenius W p, smul_sub, smul_smul]
   rw [mul_comm ((x_gen W - (negFrobeniusIsog W).pullback (x_gen W)) ^ 2)
       (2 * addSlope W (negFrobeniusIsog W) + algebraMap K KE W.a₁)]
   rw [← smul_smul, kaehler_D_addSlope_negFrobenius W p, smul_sub]
 
 omit [Fintype K] in
-/-- **Route B core (III.5.2), general cleared form**: combines `kaehler_D_addPullback_x_general`
+/-- **General cleared form**: combines `kaehler_D_addPullback_x_general`
 with `kaehler_D_addSlope_general` to clear the `Den²` denominator from `D(addPullback_x)` for
 arbitrary `α` (non-doubling). Generalizes `kaehler_D_addPullback_x_negFrobenius_cleared`. -/
 theorem kaehler_D_addPullback_x_general_cleared
@@ -3004,8 +2795,9 @@ theorem kaehler_D_x_gen_eq_u_smul_omega :
         (u_gen W)⁻¹ • KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W) from rfl,
     smul_smul, mul_inv_cancel₀ (u_gen_ne_zero W), one_smul]
 
+omit [Fintype K] [DecidableEq K] in
 /-- **RB-ω2 leaf**: `D(y_gen) = (3x²+2a₂x+a₄−a₁y) • ω`. From the curve-equation differential
-`u_gen·D(y) = (3x²+2a₂x+a₄−a₁y)·D(x)` (Sub-helper 121 chain) and `D(x) = u_gen·ω` (RB-ω1),
+`u_gen·D(y) = (3x²+2a₂x+a₄−a₁y)·D(x)` from the differentiated curve equation and `D(x) = u_gen·ω` ,
 dividing through by `u_gen ≠ 0`. Silverman III.5: the relation `(2y+a₁x+a₃)dy =
 (3x²+2a₂x+a₄−a₁y)dx` from differentiating the Weierstrass equation. -/
 theorem kaehler_D_y_gen_eq_num_smul_omega :
@@ -3032,7 +2824,7 @@ omit [Fintype K] in
 /-- **RB-ω3 leaf** (α-image differentials): for the omega coefficient
 `a_α = omegaPullbackCoeff W α`, `D(α*x) = (α*u)·a_α·ω` and `D(α*y) = (α*num)·a_α·ω`. From
 `omegaPullbackCoeff_spec` (`a_α•ω = (α*u)⁻¹•D(α*x)`) and the α-image curve-equation differential
-(Sub-helper 121 at the pulled-back point `(α*x, α*y)`, also on the curve by `pullback_equation`). -/
+(the differentiated curve equation at the pulled-back point `(α*x, α*y)`, also on the curve by `pullback_equation`). -/
 theorem kaehler_D_alpha_pullback_x_eq_smul_omega
     (α : Isogeny W.toAffine W.toAffine) :
     KaehlerDifferential.D K W.toAffine.FunctionField (α.pullback (x_gen W)) =
@@ -3049,6 +2841,7 @@ theorem kaehler_D_alpha_pullback_x_eq_smul_omega
     rw [hspec, smul_smul, mul_inv_cancel₀ h_au_ne, one_smul]
   rw [← key, smul_smul]
 
+omit [Fintype K] in
 /-- **RB-ω3b leaf** (α-image `D(y)`): `D(α*y) = (α*num)·a_α·ω`. From the α-image curve-equation
 differential (apply `Isogeny.pullbackKaehler` to `kaehler_curve_equation_K_E` — `pullbackKaehler` is
 α-semilinear and sends `D g ↦ D(α*g)`) + `kaehler_D_alpha_pullback_x_eq_smul_omega` (RB-ω3a), cancel
@@ -3094,6 +2887,7 @@ theorem kaehler_D_alpha_pullback_y_eq_smul_omega
   congr 1
   ring
 
+omit [Fintype K] in
 /-- **RB-ω4 leaf (the III.5.2 ring collapse)**: for genuine α (x≠α*x), the differential of the
 addition-pullback x-coordinate equals `addPullback_u • (1 + a_α) • ω`, where
 `addPullback_u = 2·addPullback_y + a₁·addPullback_x + a₃` is the u-coordinate at the sum point.
@@ -3151,23 +2945,11 @@ theorem kaehler_D_addPullback_x_eq_one_add_smul_omega
     (-(2 * (Y - PY) + c1 * (X - PX)) * (1 - omegaPullbackCoeff W α)) * hP +
       ((2 * (Y - PY) + c1 * (X - PX)) * (1 - omegaPullbackCoeff W α)) * hαP
 
-/-- **Sub-helper 130** (D(addSlope) reduced to D(x_gen) via curve equation, axiom-clean):
-
-Combines sub-helper 127 (curve equation Kähler identity) with sub-helper 128
-(D(slope) identity) to express D(addSlope) entirely in terms of D(x_gen):
-
-```
-(u_gen · Den²) • D(addSlope) = (Den · num - u_gen · N) • D(x_gen)
-```
-
-where:
-- u_gen = a₃ + 2y + a₁x
-- Den = x - π·x
-- N = y - π·y
-- num = 3x² + 2a₂x + a₄ - a₁y
-
-Substantive K(E) Kähler identity — combines axiom-clean Path A sub-helpers
-127 + 128. Foundational for ω(γ) = 1 via the Kähler-witness route. -/
+/-- The differential of the addition slope satisfies
+`(u · Den²) • D(ℓ) = (Den · num - u · N) • D(x)`, where
+`u = a₃ + 2y + a₁x`, `num = 3x² + 2a₂x + a₄ - a₁y`,
+`Den = x - (-π)*x`, and `N = y - (-π)*y`.
+This combines the quotient rule with the differentiated curve equation. -/
 theorem kaehler_D_addSlope_via_curve_equation_negFrobenius
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     ((algebraMap K KE W.a₃ + (2 : KE) * y_gen W +
@@ -3184,7 +2966,7 @@ theorem kaehler_D_addSlope_via_curve_equation_negFrobenius
           algebraMap K KE W.a₁ * x_gen W) *
           (y_gen W - (negFrobeniusIsog W).pullback (y_gen W))) •
         KaehlerDifferential.D K W.toAffine.FunctionField (x_gen W) := by
-  -- Multiply sub-helper 128 by u_gen' on the left, then substitute via 127.
+  -- Multiply the slope derivative by u, then substitute the differentiated curve equation.
   have h_slope := kaehler_D_addSlope_negFrobenius W p
   have h_curve := kaehler_curve_equation_K_E W
   -- Step 1: multiply h_slope by u_gen' (smul-distribute).
@@ -3203,13 +2985,8 @@ theorem kaehler_D_addSlope_via_curve_equation_negFrobenius
   -- Now collect smul forms.
   rw [smul_smul, smul_smul, ← sub_smul]
 
-/-- **Sub-helper 131** (`alpha_star_u` computed explicitly for γ, axiom-clean):
-the explicit form of `alpha_star_u (isogOneSub_negFrobenius W hq)` in K(E):
-`α*(u) = 2·addPullback_y + a₁·addPullback_x + a₃`.
-
-Direct from `alpha_star_u_eq` + sub-helpers 89b, 89c (axiom-clean
-`γ.pullback x_gen = addPullback_x` and `γ.pullback y_gen = addPullback_y`)
-+ `u_gen` definition. -/
+/-- For `γ = 1 − π`, the pulled-back differential denominator is
+`α*u = 2·addPullback_y + a₁·addPullback_x + a₃`. -/
 theorem alpha_star_u_isogOneSub_negFrobenius
     (hq : 2 ≤ Fintype.card K) :
     alpha_star_u W (isogOneSub_negFrobenius W hq) =
@@ -3233,13 +3010,13 @@ theorem alpha_star_u_isogOneSub_negFrobenius
   rw [(isogOneSub_negFrobenius W hq).pullback.commutes W.a₁,
       (isogOneSub_negFrobenius W hq).pullback.commutes W.a₃]
 
-/-- **Sub-helper 132** (α*(u) + u_gen identity in K(E), axiom-clean):
+/-- α*(u) + u_gen identity in K(E):
 
 `α*(u) + u_gen = -(2ℓ + a₁) · (addPullback_x - x_gen)` in K(E)
 
 where ℓ = addSlope W (negFrobeniusIsog W).
 
-Proof: substitute α*(u) = 2·addPullback_y + a₁·addPullback_x + a₃ (sub-helper 131)
+Proof: substitute α*(u) = 2·addPullback_y + a₁·addPullback_x + a₃
 and addPullback_y via Affine.addY/negY/negAddY definitional unfoldings,
 then verify by ring algebra in K(E).
 
@@ -3247,8 +3024,7 @@ This is the substantive K(E) identity that simplifies the RHS of the
 target Kähler-witness identity to:
 `(α*(u) + u_gen) · Den² = -(2ℓ + a₁) · (addPullback_x - x_gen) · Den²`.
 
-Combined with sub-helper 130 (LHS reduction), the target K(E) identity
-becomes a sub-divisible polynomial identity. -/
+Together with the slope derivative formula, this reduces differential additivity to a polynomial identity. -/
 theorem alpha_star_u_plus_u_gen_negFrobenius
     (hq : 2 ≤ Fintype.card K) :
     alpha_star_u W (isogOneSub_negFrobenius W hq) + u_gen W =
@@ -3266,7 +3042,7 @@ theorem alpha_star_u_plus_u_gen_negFrobenius
   rw [h_a1, h_a2, h_a3]
   ring
 
-/-- **Sub-helper 133** (curve equation for π·(x_gen, y_gen) in K(E), axiom-clean):
+/-- curve equation for π·(x_gen, y_gen) in K(E):
 
 The π-shifted Weierstrass equation in K(E):
 `(π·y)² + a₁(π·x)(π·y) + a₃(π·y) = (π·x)³ + a₂(π·x)² + a₄(π·x) + a₆`
@@ -3274,7 +3050,7 @@ The π-shifted Weierstrass equation in K(E):
 where π·x = (negFrob).pullback x_gen, π·y = (negFrob).pullback y_gen.
 
 Proof: apply `(negFrobeniusIsog W).pullback` (a K-algebra hom) to
-`weierstrass_equation_in_KE` (sub-helper 120). The hom respects + and *
+`weierstrass_equation_in_KE`. The hom respects + and *
 and fixes algebraMap-images of K, so the equation transforms term-by-term. -/
 theorem weierstrass_equation_pi_negFrobenius :
     (negFrobeniusIsog W).pullback (y_gen W) ^ 2 +
@@ -3296,7 +3072,7 @@ theorem weierstrass_equation_pi_negFrobenius :
     AlgHom.commutes (negFrobeniusIsog W).pullback] at h_pi
   exact h_pi
 
-/-- **Sub-helper 134** (substantive K(E) polynomial identity for III.5.2, axiom-clean):
+/-- substantive K(E) polynomial identity for III.5.2:
 
 The K(E) ring identity that closes the Kähler-witness route to ω(γ) = 1:
 
@@ -3316,7 +3092,7 @@ where:
 
 Proof: substitute the slope formula `ℓ = (y - π·y)/(x - π·x)` (squared and
 multiplied by Den² gives the polynomial form), then close via
-`linear_combination` with the two curve equations (sub-helpers 120 + 133)
+`linear_combination` with the two curve equations
 as multipliers. The identity holds because γ = id - π is a curve isogeny,
 which respects the curve equation.
 
@@ -3342,7 +3118,7 @@ theorem kaehler_witness_polynomial_identity_negFrobenius :
   have h_curve_pi := weierstrass_equation_pi_negFrobenius W
   linear_combination -h_curve + h_curve_pi
 
-/-- **Sub-helper 135** ((addPullback_x - x_gen) · Den² polynomial form, axiom-clean):
+/-- (addPullback_x - x_gen) · Den² polynomial form:
 
 ```
 (addPullback_x - x_gen) · Den² =
@@ -3355,7 +3131,7 @@ slope formula `ℓ = (y - π·y) / (x - π·x)` giving `ℓ · (x - π·x) = (y
 hence `ℓ² · (x - π·x)² = (y - π·y)²` and `ℓ · (x - π·x)² = (y - π·y) · (x - π·x)`.
 
 Substantive polynomial-form bridge between the slope-divisor identity
-and the curve-equation polynomial identity (sub-helper 134). -/
+and the curve-equation polynomial identity. -/
 theorem addPullback_x_sub_x_gen_mul_Den_sq_negFrobenius :
     (addPullback_x W (negFrobeniusIsog W) - x_gen W) *
         (x_gen W - (negFrobeniusIsog W).pullback (x_gen W)) ^ 2 =
@@ -3394,18 +3170,8 @@ theorem addPullback_x_sub_x_gen_mul_Den_sq_negFrobenius :
       algebraMap K KE W.a₁ *
         (x_gen W - (negFrobeniusIsog W).pullback (x_gen W))) * h_slope_lin
 
-/-- **Sub-helper 136** (curve-equation form of K(E) Kähler-witness identity, axiom-clean):
-
-```
-Den · num - u_gen · N + (addPullback_x - x_gen) · Den² = 0
-```
-
-Combines sub-helper 134 (polynomial form identity) with sub-helper 135
-(`(addPullback_x - x_gen) · Den² = polynomial form`) to give the K(E)
-identity in the form needed by the Kähler-witness consumer.
-
-This is THE identity that closes ω(γ) = 1 / Witness #1 / III.5.2 via
-the Kähler witness route. -/
+/-- The two curve equations and the addition formula give
+`Den · num - u · N + (addPullback_x - x) · Den² = 0` in the function field. -/
 theorem kaehler_witness_curve_form_negFrobenius :
     (x_gen W - (negFrobeniusIsog W).pullback (x_gen W)) *
         ((3 : KE) * x_gen W ^ 2 +
@@ -3420,14 +3186,8 @@ theorem kaehler_witness_curve_form_negFrobenius :
   rw [addPullback_x_sub_x_gen_mul_Den_sq_negFrobenius W]
   exact kaehler_witness_polynomial_identity_negFrobenius W
 
-/-- **Sub-helper 137** (K(E) coefficient identity for Kähler witness, axiom-clean):
-the K(E) coefficient identity that closes the Kähler-witness chain:
-
-```
-(2ℓ + a₁) · (Den · num - u_gen · N) = (α*(u) + u_gen) · Den²
-```
-
-Combines sub-helpers 132 + 136 axiom-clean. -/
+/-- The addition formulas and curve equations give the coefficient identity
+`(2ℓ + a₁)(Den · num - u · N) = (α*u + u) · Den²`. -/
 theorem kaehler_witness_coefficient_identity_negFrobenius
     (hq : 2 ≤ Fintype.card K) :
     ((2 : KE) * addSlope W (negFrobeniusIsog W) +
@@ -3443,7 +3203,7 @@ theorem kaehler_witness_coefficient_identity_negFrobenius
       (alpha_star_u W (isogOneSub_negFrobenius W hq) + u_gen W) *
         (x_gen W - (negFrobeniusIsog W).pullback (x_gen W)) ^ 2 := by
   rw [alpha_star_u_plus_u_gen_negFrobenius W hq]
-  -- Use sub-helper 136 (curve-form K(E) identity, axiom-clean):
+  -- The curve-form identity gives:
   -- Den·num - u_gen·N + (addPullback_x - x_gen)·Den² = 0.
   -- Multiply by (2ℓ+a₁) to get: (2ℓ+a₁)·(Den·num - u_gen·N + (addPullback_x - x_gen)·Den²) = 0.
   -- Rearrange to: (2ℓ+a₁)·(Den·num - u_gen·N) = -(2ℓ+a₁)·(addPullback_x - x_gen)·Den².
@@ -3451,16 +3211,9 @@ theorem kaehler_witness_coefficient_identity_negFrobenius
   linear_combination
     (2 * addSlope W (negFrobeniusIsog W) + algebraMap K KE W.a₁) * h_136
 
-/-- **Sub-helper 138** (ω(γ) = 1 axiom-clean — Witness #1 omega-coefficient).
-
-Instantiates `kaehler_witness_via_slope_deriv_witness` with the explicit
-witness `c := (Den · num - u · N) / (u · Den²)`, where `u = a₃ + 2y + a₁x`,
-`Den = x - π·x`, `N = y - π·y`, `num = 3x² + 2a₂x + a₄ - a₁y`.
-
-The slope-deriv witness `D(addSlope) = c • D(x_gen)` follows from sub-helper
-130 by smul-cancelling `u · Den²`. The K(E) coefficient identity
-`((2ℓ + a₁) · c - 1) · u_gen = α*(u)` follows from sub-helper 137 by
-field-simplification + linear_combination. -/
+/-- The invariant-differential coefficient of `1 − π` is one.
+The slope derivative has coefficient `c = (Den · num - u · N)/(u · Den²)` relative
+to `D(x)`, and the two curve equations give `((2ℓ + a₁)c - 1)u = α*u`. -/
 theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one
     (p : ℕ) [Fact p.Prime] [CharP K p]
     (hq : 2 ≤ Fintype.card K) :
@@ -3496,11 +3249,8 @@ theorem omegaPullbackCoeff_isogOneSub_negFrobenius_eq_one
         sub_eq_iff_eq_add, div_eq_iff huDen_ne]
     linear_combination u * h_137
 
-/-- **Sub-helper 139** (Witness #1 axiom-clean — IsSeparable γ).
-
-Composes sub-helper 138 (ω(γ) = 1 axiom-clean) with
-`isogOneSub_negFrobenius_isSeparable_of_h_coeff_only` (T-II-4-004 absorbed,
-axiom-clean) to produce the unconditional Witness #1 of the Hasse-Weil bound. -/
+/-- The isogeny `1 − π` is separable, because its invariant-differential coefficient
+is one. -/
 theorem isogOneSub_negFrobenius_isSeparable
     (p : ℕ) [Fact p.Prime] [CharP K p]
     (hq : 2 ≤ Fintype.card K) :
@@ -3607,7 +3357,7 @@ rule (`kaehler_D_addSlopePair_clearDenomSq`) with the `addX` Leibniz expansion
 (`kaehler_D_addPullback_x_pair_via_slope`) into a single denominator-free expression for
 `(X₁ - X₂)² • D(addPullback_x_pair)`. A leaf of
 `kaehler_D_addPullback_x_pair_eq_smul_omega`. -/
-private theorem kaehler_D_addPullback_x_pair_clearDenomSq
+theorem kaehler_D_addPullback_x_pair_clearDenomSq
     (α₁ α₂ : Isogeny W.toAffine W.toAffine)
     (h_ne : α₁.pullback (x_gen W) ≠ α₂.pullback (x_gen W)) :
     (α₁.pullback (x_gen W) - α₂.pullback (x_gen W)) ^ 2 •
@@ -3641,7 +3391,7 @@ unfolding `addPullback_x/y_pair` and `alpha_star_u`, substituting the two pullba
 equations (`pullback_equation α₁/α₂`), clearing the denominator, and closing with the free-variable
 algebra core `kaehler_D_addPullback_x_pair_ring_identity`. The scalar half of
 `kaehler_D_addPullback_x_pair_eq_smul_omega`. -/
-private theorem kaehler_D_addPullback_x_pair_coeff_collapse
+theorem kaehler_D_addPullback_x_pair_coeff_collapse
     (α₁ α₂ : Isogeny W.toAffine W.toAffine)
     (h_ne : α₁.pullback (x_gen W) ≠ α₂.pullback (x_gen W)) :
     (2 * addSlopePair α₁ α₂ + algebraMap K KE W.a₁) *
@@ -3691,6 +3441,7 @@ private theorem kaehler_D_addPullback_x_pair_coeff_collapse
     (omegaPullbackCoeff W α₁) (omegaPullbackCoeff W α₂) (algebraMap K KE W.a₂)
     (algebraMap K KE W.a₃) (algebraMap K KE W.a₄) (algebraMap K KE W.a₆) hα₁ hα₂
 
+omit [Fintype K] in
 /-- **General-pair III.5.2 differential collapse**: for genuine pairs (`α₁*x ≠ α₂*x`), the
 differential of the pair addition `x`-coordinate is
 `u₃ • ((a_{α₁} + a_{α₂}) • ω)`, where `u₃ = 2·addPullback_y_pair + a₁·addPullback_x_pair + a₃`
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/DiffQuotientRule.lean b/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/DiffQuotientRule.lean
index 218313002..5de8144cd 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/DiffQuotientRule.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/DiffQuotientRule.lean
@@ -22,6 +22,7 @@ variable (E : Affine F) [E.IsElliptic]
 local notation "KE" => E.FunctionField
 
 omit [DecidableEq F] in
+omit [WeierstrassCurve.IsElliptic E] in
 /-- `D(f⁻¹) = -f⁻² • D(f)` for nonzero `f ∈ K(E)`. -/
 theorem D_inv_smul (f : KE) (hf : f ≠ 0) :
     KaehlerDifferential.D F KE f⁻¹ =
@@ -36,7 +37,7 @@ theorem D_inv_smul (f : KE) (hf : f ≠ 0) :
     _ = f⁻¹ • (-(f⁻¹ • D f)) := by rw [h3]
     _ = -(f⁻¹ ^ 2 • D f) := by rw [smul_neg, smul_smul, ← sq]
 
-omit [DecidableEq F] in
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic E] in
 /-- `D(f * g⁻¹) = g⁻¹ • D(f) + (stuff involving D(g))`. Leibniz + inverse rule. -/
 theorem D_mul_inv_smul (f g : KE) (hg : g ≠ 0) :
     KaehlerDifferential.D F KE (f * g⁻¹) =
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/EllipticDivisibilitySequence.lean b/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/EllipticDivisibilitySequence.lean
index b07632812..bf6d0816a 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/EllipticDivisibilitySequence.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Auxiliary/EllipticDivisibilitySequence.lean
@@ -12,21 +12,6 @@ import Mathlib.NumberTheory.EllipticDivisibilitySequence
 import Mathlib.Tactic.IntervalCases
 import Mathlib.Tactic.LinearCombination
 
-/-!
-# Additional lemmas for elliptic divisibility sequences
-
-This file provides definitions and lemmas about normalised elliptic divisibility sequences
-that are not yet in mathlib. These are needed for the division polynomial / ZSMul development.
-
-The key results are:
-* `IsEllipticSequence.normEDS`: a normalised EDS is an elliptic sequence.
-* `IsEllipticSequence.ext`: two elliptic sequences with the same first four terms are equal.
-* `normEDS_two_three_two`: when `b = 2, c = 3, d = 2`, the normalised EDS is the identity.
-* `complEDSAux₂`: auxiliary complement used in the `ω` definition.
-* `redInvarDenom`, `redInvarNum`: reduced invariant decomposition.
-
-Ported from the LutzNagell project (`LutzNagell/EllipticDivisibilitySequence.lean`).
--/
 
 open scoped nonZeroDivisors
 
@@ -417,12 +402,12 @@ theorem rel₄_of_anti_oddRec_evenRec (one : W 1 ∈ R⁰) (two : W 2 ∈ R⁰)
     linarith only [hba]
   obtain ⟨m, rfl|rfl⟩ := b.even_or_odd'
   · have ea : Even a := by rw [← ha']; exact (even_two_mul _).add even_two
-    simp_rw [cMin, dMin, if_pos ea]
+    simp_rw [cMin, dMin, ite_eq_left ea]
     convert (rel₃_iff₄ W (m + 1) m 1).mp ((rel₃_iff_oddRec W m).mpr <| oddRec _ ?_) using 2
     all_goals linarith only [h6, ha']
   · have nea : ¬ Even a := by
       rw [← ha', not_even_iff_odd]; convert odd_two_mul_add_one (m + 1) using 1; ring
-    simp_rw [cMin, dMin, if_neg nea]
+    simp_rw [cMin, dMin, ite_eq_right nea]
     convert (rel₄_iff_evenRec W (m + 1)).mpr (evenRec _ ?_) using 2
     all_goals linarith only [h6, ha']
 
@@ -623,9 +608,9 @@ lemma invarDenom_normEDS_two :
   simp [EllSequence.invarDenom]
 
 lemma normEDS_six_eq_mul : normEDS b c d 6 = (normEDS b c d 5 - d ^ 2) * b * c := by
-  rw [show (6 : ℤ) = 2 * 3 by rfl, ← normEDS_mul_complEDS₂, complEDS₂, if_neg (by decide)]
+  rw [show (6 : ℤ) = 2 * 3 by rfl, ← normEDS_mul_complEDS₂, complEDS₂, ite_eq_right (by decide)]
   simp_rw [Int.reduceAdd, Int.reduceSub, normEDS_three, normEDS]
-  rw [preNormEDS_one, preNormEDS_two, preNormEDS_four, if_neg (by decide)]
+  rw [preNormEDS_one, preNormEDS_two, preNormEDS_four, ite_eq_right (by decide)]
   ring
 
 end NormEDSLemmas
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Basic.lean b/projects/HasseWeil/HasseWeil/Foundation/Basic.lean
index b2c035a234ea01cc215b69064d795adea7e2c888..31b9bd23d3724605cf874fab7e45adf87d019f61 100644
GIT binary patch
delta 149
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BIvxN3

diff --git a/projects/HasseWeil/HasseWeil/Foundation/BridgeFrobenius.lean b/projects/HasseWeil/HasseWeil/Foundation/BridgeFrobenius.lean
index 3331b1232..e4f9c365b 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/BridgeFrobenius.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/BridgeFrobenius.lean
@@ -5,7 +5,7 @@ import HasseWeil.HasseBound.Separability
 import HasseWeil.Foundation.LocalExpansion
 
 /-!
-# T-IV-BRIDGE-004: Frobenius pulled back to formal group is `T^q`
+# Frobenius on the formal group is `T^q`
 
 For an elliptic curve `E/F_q` with Frobenius `π : E → E`, the formal isogeny
 series satisfies `formalIsogenySeries W π = PowerSeries.X^q`.
@@ -34,7 +34,7 @@ namespace HasseWeil
 variable {K : Type*} [Field K] [Fintype K] [DecidableEq K]
 variable (W : WeierstrassCurve K) [W.toAffine.IsElliptic]
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `(localParam W) ^ q` pulled back through `localExpand` equals
     `HahnSeries.single q 1` (the formal `T^q`). -/
 theorem localExpand_localParam_pow (q : ℕ) :
@@ -49,7 +49,7 @@ theorem frobeniusIsog_pullback_localParam :
       (localParam W) ^ Fintype.card K :=
   frobeniusIsog_pullback_apply W (localParam W)
 
-/-- **T-IV-BRIDGE-004** (Silverman IV.4 / III.5.5): the formal isogeny series
+/-- Silverman IV.4 / III.5.5: the formal isogeny series
     of the Frobenius isogeny is `T^q`.
 
     The Frobenius `π : (x, y) ↦ (x^q, y^q)` translates in the local parameter
@@ -64,8 +64,8 @@ theorem frobeniusIsog_pullback_localParam :
       frobeniusIsog_pullback_localParam, localExpand_localParam_pow]
   by_cases h : n = Fintype.card K
   · subst h
-    rw [if_pos rfl, HahnSeries.coeff_single_same]
-  · rw [if_neg h, HahnSeries.coeff_single_of_ne (by exact_mod_cast h)]
+    rw [ite_eq_left rfl, HahnSeries.coeff_single_same]
+  · rw [ite_eq_right h, HahnSeries.coeff_single_of_ne (by exact_mod_cast h)]
 
 /-- **Corollary**: the linear coefficient of the Frobenius formal series is
     `0` (since `q ≥ 2` for any `Fintype K`-power Frobenius on an elliptic curve
@@ -75,7 +75,7 @@ theorem coeff_one_formalIsogenySeries_frobenius_of_card_ne_one
     (h : Fintype.card K ≠ 1) :
     PowerSeries.coeff 1 (formalIsogenySeries W (frobeniusIsog W)) = 0 := by
   rw [formalIsogenySeries_frobenius, PowerSeries.coeff_X_pow]
-  exact if_neg fun heq ↦ h heq.symm
+  exact ite_eq_right fun heq ↦ h heq.symm
 
 /-! ### `omegaPullbackCoeff(π) = 0` (Frobenius is purely inseparable)
 
@@ -85,7 +85,7 @@ to the `q`-th power where `q = #K = p^k`. The pullback of `D(x)` becomes
 `omegaPullbackCoeff_unique`, the omega coefficient must be `0`.
 
 This is the direct (non-formal-series) proof of Silverman III.5.5's "Frobenius
-is purely inseparable" — independent of T-IV-BRIDGE-001. -/
+is purely inseparable" using the invariant differential directly. -/
 
 /-- **Frobenius pullback coefficient is `0`**: Silverman III.5.5 +
     II.4.4 in characteristic `p`. The pullback of the invariant differential
@@ -123,9 +123,9 @@ theorem not_isSeparable_frobenius_of_witness
 /-! ### `m + n·π` chain witness closer (Silverman III.5.5)
 
 Given an isogeny `β` whose ω-pullback coefficient is the additive sum
-`m · a_id + n · a_π`, the new fact `omegaPullbackCoeff_frobenius = 0`
-collapses this to `m · 1 = m`. This closes T-III-5-005's coefficient
-formula assuming additivity for the specific `(m·id + n·π)` family. -/
+`m · a_id + n · a_π`, the equality `omegaPullbackCoeff_frobenius = 0`
+reduces the coefficient formula to `m · 1 = m`, assuming additivity for
+the specific `(m·id + n·π)` family. -/
 
 /-- **Witness closer for `omegaPullbackCoeff (m·id + n·π) = m`**: takes
     additivity of `omegaPullbackCoeff` for the specific isogeny `β` and
@@ -145,7 +145,7 @@ theorem omegaPullbackCoeff_m_plus_n_frob_of_witness
   rw [h_sum_coeff, omegaPullbackCoeff_of_pullback_eq_id W (Isogeny.id W.toAffine) rfl,
     omegaPullbackCoeff_frobenius, mul_one, mul_zero, add_zero]
 
-/-- **BRIDGE-001 for Frobenius (axiom-clean)**: the bridge identity holds for
+/-- **Leading-coefficient identity for Frobenius**: the bridge identity holds for
 the Frobenius isogeny — both sides equal `0` (when `Fintype.card K ≠ 1`,
 which holds in particular for `Fintype.card K ≥ 2`).
 
diff --git a/projects/HasseWeil/HasseWeil/Foundation/ChordExpansion.lean b/projects/HasseWeil/HasseWeil/Foundation/ChordExpansion.lean
index 2682e48315bf7cd7bc26ccf49e7dd817cec3e99d..cb062a9c031e529289a3d4ca7c9842159004b600 100644
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b2_nS{ReVndUG;l3hwM*If;({q#hU&D96KVy

diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Differentials.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Differentials.lean
index 62e2d8c92..6043e4a43 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Differentials.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Differentials.lean
@@ -13,44 +13,13 @@ import Mathlib.RingTheory.Kaehler.Basic
 import Mathlib.RingTheory.Unramified.Field
 
 /-!
-# Differentials on a smooth plane curve
+The meromorphic differentials on a smooth plane curve are the Kähler
+differentials of its function field over the base field. An elliptic-curve
+isogeny is separable exactly when its pullback on differentials is injective,
+equivalently when its omega-pullback coefficient is nonzero. The cotangent
+exact sequence and formal unramifiedness relate these conditions.
 
-For a smooth plane curve `C` over a field `F`, the space of (meromorphic)
-differentials is the module of Kähler differentials of the function field
-`F(C)` over `F`:
-
-```
-Ω_C := Ω[F(C) ⁄ F].
-```
-
-This file also establishes the elliptic-curve form of Silverman II.4.2(c)
-(ticket `T-II-4-004`): a non-constant isogeny `α` is separable iff the pullback
-on differentials is injective, equivalently iff its omega-pullback coefficient
-is nonzero. The argument runs through mathlib's formal-unramifiedness API and
-the cotangent exact sequence, with the relative finiteness of `K(E)` over the
-image of `α.pullback` discharged via a transcendence-degree computation.
-
-This closes tickets `T-II-4-001` and `T-II-4-004`.
-
-## Main definitions
-
-* `Differentials`: the space `Ω[F(C)⁄F]` of differentials on `C`.
-* `Differentials.d`: the universal derivation `f ↦ df`.
-* `IsogenyAlgebraSource`: type synonym carrying the source-side `α.pullback`
-  algebra structure, used to elaborate the cotangent base-change cleanly.
-
-## Main results
-
-* `isSeparable_iff_omegaPullbackCoeff_ne_zero`: Silverman II.4.2(c), elliptic
-  form — `α.IsSeparable ↔ omegaPullbackCoeff W α ≠ 0`.
-* `isSeparable_iff_pullbackKaehler_injective`: the algebra-Kähler glue —
-  `α.IsSeparable ↔ Function.Injective α.pullbackKaehler`.
-* `isogeny_finiteDimensional`: `K(E)` is finite over the image of `α.pullback`.
-* `isogeny_degree_pos`: every isogeny has positive degree.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], II.4
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.4.2(c).
 -/
 
 namespace HasseWeil.Curves
@@ -126,7 +95,7 @@ namespace HasseWeil
 variable {F : Type*} [Field F] [DecidableEq F]
 variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 
-/-- **T-II-4-004 forward direction (axiom-clean)**: if the differential
+/-- **Separability implies nonzero differential pullback**: if the differential
 pullback `α.pullbackKaehler` is injective on `Ω[K(E)/F]`, then the
 omega-pullback coefficient is nonzero.
 
@@ -140,17 +109,6 @@ theorem omegaPullbackCoeff_ne_zero_of_pullbackKaehler_injective
   refine invariantDifferential_ne_zero W.toAffine (h_inj ?_)
   rw [Isogeny.pullbackKaehler_invariantDifferential, h_zero, zero_smul, map_zero]
 
-/-- **T-II-4-004 reverse direction (axiom-clean)**: if `omegaPullbackCoeff W α ≠ 0`,
-then the differential pullback `α.pullbackKaehler` is injective on `Ω[K(E)/F]`.
-
-Proof: It suffices to show the kernel is trivial. Take `η ∈ Ω` with
-`α.pullbackKaehler η = 0`. Using 1-dim of `Ω` (`kaehler_rank_one`), write
-`η = a • ω`. Then
-`α.pullbackKaehler (a • ω) = α.pullback a • (omegaPullbackCoeff W α • ω)
-= (α.pullback a · c) • ω`. Setting this to `0` with `ω ≠ 0` gives
-`α.pullback a · c = 0`. Since `c ≠ 0` and `K(E)` is a field, `α.pullback a = 0`.
-By injectivity of `α.pullback` (any AlgHom on a field is injective), `a = 0`,
-so `η = 0`. -/
 theorem pullbackKaehler_injective_of_omegaPullbackCoeff_ne_zero
     (α : Isogeny W.toAffine W.toAffine)
     (h_ne : omegaPullbackCoeff W α ≠ 0) :
@@ -171,77 +129,40 @@ theorem pullbackKaehler_injective_of_omegaPullbackCoeff_ne_zero
     exact (mul_eq_zero.mp ((smul_eq_zero.mp h_diff).resolve_right h_ω_ne)).resolve_right h_ne
   rw [h_a_zero, zero_smul]
 
-/-- **T-II-4-004 intermediate iff (axiom-clean)**: the differential pullback
-`α.pullbackKaehler` is injective on `Ω[K(E)/F]` iff `omegaPullbackCoeff W α ≠ 0`.
-
-Combined statement of the two directions above. This is the scalar half; the
-full T-II-4-004 (`IsSeparable α ↔ omegaPullbackCoeff α ≠ 0`) also needs the
-algebra-Kähler glue `IsSeparable α ↔ pullbackKaehler injective`
-(`isSeparable_iff_pullbackKaehler_injective`), the actual Silverman II.4.2(c)
-content. -/
 theorem pullbackKaehler_injective_iff_omegaPullbackCoeff_ne_zero
     (α : Isogeny W.toAffine W.toAffine) :
     Function.Injective α.pullbackKaehler ↔ omegaPullbackCoeff W α ≠ 0 :=
   ⟨omegaPullbackCoeff_ne_zero_of_pullbackKaehler_injective W α,
    pullbackKaehler_injective_of_omegaPullbackCoeff_ne_zero W α⟩
 
-/-- **T-II-4-004 from both halves**: composes the scalar half
-(`pullbackKaehler injective ↔ ω-coeff ≠ 0`) with the algebra-Kähler half
-(`IsSeparable ↔ pullbackKaehler injective`).
-
-When the algebra-Kähler half lands axiom-clean, this gives the original
-T-II-4-004 target via transitive iff chaining — no further plumbing needed. -/
 theorem isSeparable_iff_omegaPullbackCoeff_ne_zero_of_algKaehler
     (α : Isogeny W.toAffine W.toAffine)
     (h_alg : α.IsSeparable ↔ Function.Injective α.pullbackKaehler) :
     α.IsSeparable ↔ omegaPullbackCoeff W α ≠ 0 :=
   h_alg.trans (pullbackKaehler_injective_iff_omegaPullbackCoeff_ne_zero W α)
 
-/-- **T-II-4-004 forward chain step 1 (axiom-clean)**: separability of `α`
-implies that the algebra structure from `α.pullback` is formally unramified.
-
-Direct from mathlib's `Algebra.FormallyUnramified.of_isSeparable` lifted
-through the explicit algebra instance from `α.toAlgebra`. -/
 theorem isogeny_formallyUnramified_of_isSeparable
     (α : Isogeny W.toAffine W.toAffine) (h : α.IsSeparable) :
     @Algebra.FormallyUnramified W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra := by
-  letI : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
-  haveI : Algebra.IsSeparable W.toAffine.FunctionField W.toAffine.FunctionField := h
+  let : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  have : Algebra.IsSeparable W.toAffine.FunctionField W.toAffine.FunctionField := h
   exact Algebra.FormallyUnramified.of_isSeparable _ _
 
-/-- **T-II-4-004 forward chain step 2 (axiom-clean)**: separability of `α`
-implies the Kähler differential module `Ω[K(E)/(image of α.pullback)]` is
-trivial.
-
-Direct extraction from the previous step (formally unramified) via the
-mathlib instance projection
-`FormallyUnramified.subsingleton_kaehlerDifferential`. -/
 theorem isogeny_subsingleton_kaehler_of_isSeparable
     (α : Isogeny W.toAffine W.toAffine) (h : α.IsSeparable) :
     @Subsingleton (@KaehlerDifferential W.toAffine.FunctionField
       W.toAffine.FunctionField _ _ α.toAlgebra) := by
-  letI : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
-  haveI := isogeny_formallyUnramified_of_isSeparable W α h
+  let : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  have := isogeny_formallyUnramified_of_isSeparable W α h
   exact Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
 
-/-- **Cotangent surjectivity (axiom-clean abstract form)**: for a tower
-`R → A → B` of commutative algebras with `IsScalarTower`, if the relative
-Kähler module `Ω[B/A]` is trivial (Subsingleton), then the natural base-
-change map `B ⊗[A] Ω[A/R] → Ω[B/R]` is surjective.
-
-Direct from `KaehlerDifferential.exact_mapBaseChange_map` plus the fact
-that any map into a Subsingleton is zero. -/
 theorem mapBaseChange_surjective_of_subsingleton_relativeKaehler
     (R A B : Type*) [CommRing R] [CommRing A] [CommRing B]
     [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B] [Subsingleton (Ω[B⁄A])] :
     Function.Surjective (KaehlerDifferential.mapBaseChange R A B) := fun y ↦
   (KaehlerDifferential.exact_mapBaseChange_map R A B y).mp (Subsingleton.elim _ _)
 
-/-- **Cotangent Subsingleton (axiom-clean abstract form, reverse)**: for a tower
-`R → A → B` of commutative algebras with `IsScalarTower`, if the natural
-base-change map `B ⊗[A] Ω[A/R] → Ω[B/R]` is surjective, then the relative
-Kähler module `Ω[B/A]` is `Subsingleton`. -/
 theorem subsingleton_relativeKaehler_of_mapBaseChange_surjective
     (R A B : Type*) [CommRing R] [CommRing A] [CommRing B]
     [Algebra R A] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
@@ -262,57 +183,26 @@ theorem subsingleton_relativeKaehler_of_mapBaseChange_surjective
   have hy : y = 0 := by rw [← hb, h_map_zero]; rfl
   rw [hx, hy]
 
-/-- **Forward chain step 4 (axiom-clean)**: the `IsScalarTower` instance for the
-algebra structure from an isogeny.
-
-Stated with `@`-explicit instance arguments fixing the middle SMul to
-`α.toAlgebra.toSMul`, which forces the elaborator to use `α.toAlgebra`
-rather than the global `OreLocalization.instSMul` (the standard mul-based
-K(E)-self SMul, defeq to but syntactically distinct from the α-twisted
-one). Once the goal type pins the right SMul, the proof closes via
-`IsScalarTower.of_algebraMap_eq` from `α.pullback.commutes` (the F-algebra
-hom property).
-
-This unblocks `mapBaseChange F K(E) K(E)` and the full cotangent-sequence
-argument for the reverse direction of T-II-4-004. -/
 theorem isogeny_isScalarTower
     (α : Isogeny W.toAffine W.toAffine) :
     @IsScalarTower F W.toAffine.FunctionField W.toAffine.FunctionField
       _ α.toAlgebra.toSMul _ := by
-  letI : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  let : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
   exact IsScalarTower.of_algebraMap_eq fun x ↦ (α.pullback.commutes x).symm
 
-/-- **EssFiniteType from FiniteDimensional (axiom-clean)**: if `K(E)` is a
-finite-dimensional `K(E)_α`-module (via `α.toAlgebra`), then it is essentially
-of finite type as a `K(E)_α`-algebra.
-
-This is Witness #2 (the bound's `FiniteDimensional` hypothesis) discharged into
-the `EssFiniteType` form needed for mathlib's `FormallyUnramified.iff_isSeparable`.
-Direct application of `Module.Finite.finiteType` then `EssFiniteType.of_finiteType`. -/
 theorem isogeny_essFiniteType_of_finiteDimensional
     (α : Isogeny W.toAffine W.toAffine)
     (h_fin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra.toModule) :
     @Algebra.EssFiniteType W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra := by
-  haveI : @Algebra.FiniteType W.toAffine.FunctionField W.toAffine.FunctionField
+  have : @Algebra.FiniteType W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra :=
     @Module.Finite.finiteType W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra h_fin
   exact @Algebra.EssFiniteType.of_finiteType W.toAffine.FunctionField
     W.toAffine.FunctionField _ _ α.toAlgebra _
 
-/-- **FiniteDimensional discharge from IsAlgebraic + EssFiniteType
-(witness-parametric, axiom-clean)**: given `IsAlgebraic K(E)_α K(E)` and
-`EssFiniteType K(E)_α K(E)` (over `α.toAlgebra`), `K(E)` is finite-dimensional
-over `K(E)_α`.
-
-Direct application of mathlib's `Algebra.finite_of_essFiniteType_of_isAlgebraic`
-(`Mathlib/FieldTheory/IntermediateField/Adjoin/Algebra.lean:185`).
-
-When both witnesses discharge unconditionally (transcendence-degree argument
-for `IsAlgebraic`; `IsFractionRing` argument for `EssFiniteType`), this becomes
-the unconditional Witness #2 producer for the bound. -/
 theorem isogeny_finiteDimensional_of_isAlgebraic_essFiniteType
     (α : Isogeny W.toAffine W.toAffine)
     (h_alg : @Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField
@@ -321,9 +211,9 @@ theorem isogeny_finiteDimensional_of_isAlgebraic_essFiniteType
       _ _ α.toAlgebra) :
     @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra.toModule := by
-  letI : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
-  haveI : Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField := h_alg
-  haveI : Algebra.EssFiniteType W.toAffine.FunctionField W.toAffine.FunctionField := h_ess
+  let : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  have : Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField := h_alg
+  have : Algebra.EssFiniteType W.toAffine.FunctionField W.toAffine.FunctionField := h_ess
   exact Algebra.finite_of_essFiniteType_of_isAlgebraic
 
 /-- **IsAlgebraic discharge from finite intermediate witness (witness-parametric)**:
@@ -341,26 +231,13 @@ theorem isogeny_isAlgebraic_of_finite_intermediate
       _ α.toAlgebra.toSMul _) :
     @Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra := by
-  letI : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
-  haveI := h_tower
-  haveI : Algebra.IsAlgebraic L W.toAffine.FunctionField :=
+  let : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  have := h_tower
+  have : Algebra.IsAlgebraic L W.toAffine.FunctionField :=
     Algebra.IsAlgebraic.of_finite L W.toAffine.FunctionField
   exact Algebra.IsAlgebraic.tower_top (K := L)
     (L := W.toAffine.FunctionField) (A := W.toAffine.FunctionField)
 
-/-- **T-II-4-004 reverse direction (witness-parametric, axiom-clean)**:
-if the relative Kähler module `Ω[K(E)/K(E)_α]` is trivial (`Subsingleton`)
-and the algebra structure from `α.pullback` is essentially of finite type
-(`EssFiniteType`), then `α` is separable.
-
-Direct application of mathlib's `Algebra.FormallyUnramified.iff_isSeparable`,
-unwrapping `FormallyUnramified` via its `Subsingleton`-of-Kähler constructor.
-The two witnesses correspond exactly to the two hypotheses of mathlib's iff:
-`Subsingleton Ω[L/K]` is the unfolded form of `FormallyUnramified K L`, and
-`EssFiniteType K L` is the iff's standing hypothesis.
-
-When both witnesses are discharged unconditionally, this becomes the
-unconditional reverse half of T-II-4-004. -/
 theorem isogeny_isSeparable_of_kaehler_witnesses
     (α : Isogeny W.toAffine W.toAffine)
     (h_sub : @Subsingleton (@KaehlerDifferential W.toAffine.FunctionField
@@ -368,21 +245,13 @@ theorem isogeny_isSeparable_of_kaehler_witnesses
     (h_ess : @Algebra.EssFiniteType W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra) :
     α.IsSeparable := by
-  letI : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
-  haveI : Algebra.EssFiniteType W.toAffine.FunctionField W.toAffine.FunctionField := h_ess
-  haveI : Algebra.FormallyUnramified W.toAffine.FunctionField W.toAffine.FunctionField :=
+  let : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  have : Algebra.EssFiniteType W.toAffine.FunctionField W.toAffine.FunctionField := h_ess
+  have : Algebra.FormallyUnramified W.toAffine.FunctionField W.toAffine.FunctionField :=
     ⟨h_sub⟩
   exact (Algebra.FormallyUnramified.iff_isSeparable
     W.toAffine.FunctionField W.toAffine.FunctionField).mp inferInstance
 
-/-- **T-II-4-004 reverse direction (combined witness form, axiom-clean)**:
-if `Ω[K(E)/K(E)_α]` is trivial and `K(E)` is finite-dimensional over
-`K(E)_α` (Witness #2 of the bound), then `α` is separable.
-
-This composes `isogeny_essFiniteType_of_finiteDimensional` with
-`isogeny_isSeparable_of_kaehler_witnesses`. The two substantive inputs are
-exactly `FiniteDimensional` (Witness #2) and the relative-Kähler vanishing
-(which the cotangent-sequence chain delivers from `pullbackKaehler injective`). -/
 theorem isogeny_isSeparable_of_subsingleton_kaehler_finiteDimensional
     (α : Isogeny W.toAffine W.toAffine)
     (h_sub : @Subsingleton (@KaehlerDifferential W.toAffine.FunctionField
@@ -393,13 +262,6 @@ theorem isogeny_isSeparable_of_subsingleton_kaehler_finiteDimensional
   isogeny_isSeparable_of_kaehler_witnesses W α h_sub
     (isogeny_essFiniteType_of_finiteDimensional W α h_fin)
 
-/-- **T-II-4-004 algebra-Kähler iff (axiom-clean witness form)**: `α` is separable
-iff the relative Kähler module `Ω[K(E)/K(E)_α]` vanishes, given the
-`FiniteDimensional` witness (Witness #2 of the bound).
-
-The `FiniteDimensional` is essential for the reverse direction (gives
-`EssFiniteType`, the standing hypothesis of `FormallyUnramified.iff_isSeparable`).
-Forward direction is unconditional. -/
 theorem isSeparable_iff_subsingleton_kaehler_of_finiteDimensional
     (α : Isogeny W.toAffine W.toAffine)
     (h_fin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
@@ -411,17 +273,6 @@ theorem isSeparable_iff_subsingleton_kaehler_of_finiteDimensional
    fun h_sub ↦
      isogeny_isSeparable_of_subsingleton_kaehler_finiteDimensional W α h_sub h_fin⟩
 
-/-- **T-II-4-004 (witness-parametric, axiom-clean)**: the full iff
-`α.IsSeparable ↔ omegaPullbackCoeff W α ≠ 0`, parametric on `FiniteDimensional`
-(Witness #2) and the Subsingleton ↔ pullbackKaehler-injective bridge.
-
-Composes:
-* `isSeparable_iff_subsingleton_kaehler_of_finiteDimensional`: ties
-  IsSeparable to Subsingleton via mathlib's iff (with FiniteDim hypothesis).
-* The bridge witness: ties Subsingleton to pullbackKaehler injective (the
-  cotangent-sequence connection).
-* `pullbackKaehler_injective_iff_omegaPullbackCoeff_ne_zero`: ties
-  pullbackKaehler injective to ω-coeff ≠ 0 (axiom-clean scalar half). -/
 theorem isSeparable_iff_omegaPullbackCoeff_ne_zero_of_witnesses
     (α : Isogeny W.toAffine W.toAffine)
     (h_fin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
@@ -465,8 +316,8 @@ noncomputable instance algebra : Algebra (IsogenyAlgebraSource W α)
     W.toAffine.FunctionField :=
   α.toAlgebra
 
-/-- The synonym's `IsScalarTower F (IsogenyAlgebraSource W α) K(E)` is the
-break-through `isogeny_isScalarTower` lifted to the synonym wrapper. -/
+/-- The scalar tower from `F` through the source field to `K(E)` agrees with
+the pullback algebra structure. -/
 noncomputable instance isScalarTower :
     IsScalarTower F (IsogenyAlgebraSource W α) W.toAffine.FunctionField :=
   IsScalarTower.of_algebraMap_eq fun x ↦ (α.pullback.commutes x).symm
@@ -474,24 +325,11 @@ noncomputable instance isScalarTower :
 end IsogenyAlgebraSource
 
 omit [DecidableEq F] [W.toAffine.IsElliptic] in
-/-- **trdeg F K(E) = 1 (axiom-clean, Weierstrass form)**: for any Weierstrass
-curve `W`, the transcendence degree of `K(E) = W.toAffine.FunctionField` over
-the base field `F` is 1.
 
-Specializes the smooth-plane-curve trdeg result to Weierstrass curves. -/
 theorem weierstrass_functionField_trdeg_eq_one :
     Algebra.trdeg F W.toAffine.FunctionField = 1 :=
   ({ toAffine := W.toAffine } : HasseWeil.Curves.SmoothPlaneCurve F).functionField_trdeg_eq_one
 
-/-- **Witness #2 producer via the type-synonym wrapper** (witness-parametric,
-axiom-clean): given `Algebra.IsAlgebraic (IsogenyAlgebraSource W α)
-W.toAffine.FunctionField` and `Algebra.EssFiniteType (IsogenyAlgebraSource W α)
-W.toAffine.FunctionField`, derive Witness #2 (`@FiniteDimensional` for
-`α.toAlgebra.toModule`).
-
-Both the type-synonym `Algebra` instance and `α.toAlgebra` are defeq, so this
-mediates between the synonym world (where mathlib's IsAlgebraic/EssFiniteType
-fire cleanly) and the @-α.toAlgebra world used by the bound's Witness #2. -/
 theorem isogeny_finiteDimensional_of_isAlgebraic_synonym
     (α : Isogeny W.toAffine W.toAffine)
     [h_alg : Algebra.IsAlgebraic (IsogenyAlgebraSource W α) W.toAffine.FunctionField]
@@ -501,24 +339,18 @@ theorem isogeny_finiteDimensional_of_isAlgebraic_synonym
   @Algebra.finite_of_essFiniteType_of_isAlgebraic (IsogenyAlgebraSource W α) _
     W.toAffine.FunctionField _ _ h_ess h_alg
 
-/-- **EssFiniteType F K(E) (axiom-clean)**: K(E) is essentially of finite type
-over F. K(E) is the FractionRing of CoordRing, which is itself FiniteType over F.
-Localization preserves EssFiniteType (mathlib instance). -/
 instance functionField_essFiniteType_F :
     Algebra.EssFiniteType F W.toAffine.FunctionField := by
-  haveI : Algebra.FiniteType F W.toAffine.CoordinateRing := inferInstance
-  haveI : Algebra.EssFiniteType F W.toAffine.CoordinateRing :=
+  have : Algebra.FiniteType F W.toAffine.CoordinateRing := inferInstance
+  have : Algebra.EssFiniteType F W.toAffine.CoordinateRing :=
     Algebra.EssFiniteType.of_finiteType F W.toAffine.CoordinateRing
   -- K(E) = FractionRing(CoordRing) = Localization(nonZeroDivisors).
   -- IsLocalization gives EssFiniteType from CoordRing-side.
-  haveI : Algebra.EssFiniteType W.toAffine.CoordinateRing W.toAffine.FunctionField :=
+  have : Algebra.EssFiniteType W.toAffine.CoordinateRing W.toAffine.FunctionField :=
     Algebra.EssFiniteType.of_isLocalization W.toAffine.FunctionField
       (nonZeroDivisors W.toAffine.CoordinateRing)
   exact Algebra.EssFiniteType.comp F W.toAffine.CoordinateRing W.toAffine.FunctionField
 
-/-- **EssFiniteType `IsogenyAlgebraSource` K(E) (axiom-clean)**: derived from
-`EssFiniteType F K(E)` via `EssFiniteType.of_comp`, using the
-`IsScalarTower F (IsogenyAlgebraSource W α) K(E)` from the synonym wrapper. -/
 instance isogenyAlgebraSource_essFiniteType
     (α : Isogeny W.toAffine W.toAffine) :
     Algebra.EssFiniteType (IsogenyAlgebraSource W α) W.toAffine.FunctionField :=
@@ -540,10 +372,6 @@ instance isogenyAlgebraSource_faithfulSMul_KE
   (faithfulSMul_iff_algebraMap_injective (IsogenyAlgebraSource W α)
     W.toAffine.FunctionField).mpr α.pullback_injective
 
-/-- **Witness #2 producer step (axiom-clean)**: `K(E)` is algebraic over
-`K(E)_α` (the `IsogenyAlgebraSource` synonym). Proof by transcendence-degree
-additivity over `F → K(E)_α → K(E)`: both `trdeg F K(E)_α` and `trdeg F K(E)`
-equal `1`, so `trdeg (K(E)_α) K(E) = 0`, which is `IsAlgebraic`. -/
 instance isogenyAlgebraSource_isAlgebraic
     (α : Isogeny W.toAffine W.toAffine) :
     Algebra.IsAlgebraic (IsogenyAlgebraSource W α) W.toAffine.FunctionField := by
@@ -563,34 +391,22 @@ instance isogenyAlgebraSource_isAlgebraic
   rw [zero_add, add_comm]
   exact h_add
 
-/-- **Witness #2 (axiom-clean, unconditional)**: for any isogeny `α`, the
-function field `K(E)` is finite-dimensional over `K(E)_α` (the image of
-`α.pullback`), i.e. the bound's Witness #2 holds with no extra hypothesis.
-
-Direct from `isogeny_finiteDimensional_of_isAlgebraic_synonym` once the
-`IsAlgebraic` and `EssFiniteType` instances above are in scope. -/
 theorem isogeny_finiteDimensional (α : Isogeny W.toAffine W.toAffine) :
     @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ α.toAlgebra.toModule :=
   isogeny_finiteDimensional_of_isAlgebraic_synonym W α
 
-/-- **Isogeny degrees are positive (axiom-clean, unconditional)**: for any isogeny
-`α : E → E`, `0 < α.degree`.
-
-`α.degree` is `Module.finrank K(E) K(E)` over the `K(E)`-module structure
-`α.toAlgebra.toModule` induced by the pullback `α.pullback`.  This module is
-finite-dimensional (`isogeny_finiteDimensional`, same instance) and `K(E)` is a
-field hence `Nontrivial`, so `Module.finrank_pos` applies.  The instance is pinned
-explicitly to `α.toAlgebra.toModule` so it matches `degree`'s definition (plain
-`Module.finrank_pos` would otherwise synthesize the trivial self-module). -/
 theorem isogeny_degree_pos (α : Isogeny W.toAffine W.toAffine) :
     0 < α.degree := by
+  let alg : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  let mod : Module W.toAffine.FunctionField W.toAffine.FunctionField := alg.toModule
+  have : Module.IsTorsionFree W.toAffine.FunctionField W.toAffine.FunctionField :=
+    Module.isTorsionFree_iff_algebraMap_injective.mpr α.pullback_injective
+  have : FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField :=
+    isogeny_finiteDimensional W α
   unfold Isogeny.degree
-  exact @Module.finrank_pos W.toAffine.FunctionField W.toAffine.FunctionField _ _
-    α.toAlgebra.toModule _ (isogeny_finiteDimensional W α) _ _ _
+  exact Module.finrank_pos
 
-/-- **Bridge: mapBaseChange (1 ⊗ D x) = D (α.pullback x) (axiom-clean)**: applied
-to the universal D-element under the type-synonym wrapper. -/
 theorem isogeny_mapBaseChange_one_tmul_D
     (α : Isogeny W.toAffine W.toAffine) (x : W.toAffine.FunctionField) :
     KaehlerDifferential.mapBaseChange F (IsogenyAlgebraSource W α)
@@ -603,11 +419,6 @@ theorem isogeny_mapBaseChange_one_tmul_D
   rw [KaehlerDifferential.mapBaseChange_tmul, one_smul, h_map_D]
   rfl
 
-/-- **Bridge: image of mapBaseChange contains the invariant differential** when
-`omegaPullbackCoeff W α ≠ 0` (axiom-clean).
-
-Direct preimage: `mapBaseChange (b ⊗ₜ D x_gen_full) = invariantDifferential` for
-`b = (alpha_star_u • omegaPullbackCoeff)⁻¹` — derived from `omegaPullbackCoeff_spec`. -/
 theorem isogeny_mapBaseChange_surjective_of_omegaCoeff_ne_zero
     (α : Isogeny W.toAffine W.toAffine)
     (h_coeff : omegaPullbackCoeff W α ≠ 0) :
@@ -655,9 +466,6 @@ theorem isogeny_mapBaseChange_surjective_of_omegaCoeff_ne_zero
     field_simp
   rw [h_prod, ha]
 
-/-- **Bridge: Subsingleton Ω[K(E)/K(E)_α] from omegaPullbackCoeff ≠ 0
-(via type synonym, axiom-clean)**: the cotangent-sequence iff in the form needed
-to discharge T-II-4-004's bridge witness. -/
 theorem isogeny_subsingleton_kaehler_of_omegaCoeff_ne_zero
     (α : Isogeny W.toAffine W.toAffine) (h_coeff : omegaPullbackCoeff W α ≠ 0) :
     Subsingleton (Ω[W.toAffine.FunctionField⁄IsogenyAlgebraSource W α]) :=
@@ -665,10 +473,6 @@ theorem isogeny_subsingleton_kaehler_of_omegaCoeff_ne_zero
     (IsogenyAlgebraSource W α) W.toAffine.FunctionField
     (isogeny_mapBaseChange_surjective_of_omegaCoeff_ne_zero W α h_coeff)
 
-/-- **Bridge: Subsingleton via type synonym = Subsingleton via α.toAlgebra**
-(axiom-clean, defeq). The two formulations of the relative Kähler module
-coincide since `IsogenyAlgebraSource W α := K(E)` (defeq) and the synonym's
-`Algebra` instance is `α.toAlgebra`. -/
 theorem isogeny_subsingleton_via_synonym_eq
     (α : Isogeny W.toAffine W.toAffine) :
     Subsingleton (Ω[W.toAffine.FunctionField⁄IsogenyAlgebraSource W α]) ↔
@@ -739,18 +543,9 @@ private theorem mapBaseChange_eq_zero_of_kaehlerMap_eq_zero
       W.toAffine.FunctionField = 0 := by
   refine LinearMap.ext fun η ↦ ?_
   induction η with
-  | zero => simp
   | tmul x y => simp [KaehlerDifferential.mapBaseChange_tmul, h_map_zero]
   | add x y hx hy => simp [map_add, hx, hy]
 
-/-- **T-II-4-004 forward direction (axiom-clean)**: if `α` is separable,
-then `omegaPullbackCoeff W α ≠ 0`.
-
-Pipeline: `IsSeparable → Subsingleton Ω[K(E)/K(E)_α]` →
-`mapBaseChange surjective` (via the type synonym) → in particular,
-`invariantDifferential ∈ image(mapBaseChange)` (cotangent exact sequence).
-A nonzero image element forces `α.pullbackKaehler` not-identically-zero,
-which (with the 1-dim Ω structure) forces `omegaPullbackCoeff ≠ 0`. -/
 theorem isogeny_omegaCoeff_ne_zero_of_isSeparable
     (α : Isogeny W.toAffine W.toAffine)
     (h_sep : α.IsSeparable) :
@@ -761,7 +556,7 @@ theorem isogeny_omegaCoeff_ne_zero_of_isSeparable
   have h_pK_zero : ∀ ω' : Ω[W.toAffine.FunctionField⁄F],
       α.pullbackKaehler ω' = 0 :=
     pullbackKaehler_eq_zero_of_omegaPullbackCoeff_eq_zero W α h_zero
-  haveI : Subsingleton (Ω[W.toAffine.FunctionField⁄IsogenyAlgebraSource W α]) :=
+  have : Subsingleton (Ω[W.toAffine.FunctionField⁄IsogenyAlgebraSource W α]) :=
     (isogeny_subsingleton_via_synonym_eq W α).mpr
       (isogeny_subsingleton_kaehler_of_isSeparable W α h_sep)
   have h_surj : Function.Surjective (KaehlerDifferential.mapBaseChange F
@@ -779,12 +574,6 @@ theorem isogeny_omegaCoeff_ne_zero_of_isSeparable
   rw [h_mbc_zero, LinearMap.zero_apply] at ht
   exact (invariantDifferential_ne_zero W.toAffine) ht.symm
 
-/-- **T-II-4-004 reverse direction (witness-parametric on FiniteDim only,
-axiom-clean)**: if `omegaPullbackCoeff W α ≠ 0` and `K(E)` is finite-dimensional
-over `K(E)_α` (Witness #2), then `α` is separable.
-
-Combines the cotangent-sequence bridge (omega-coeff ≠ 0 → Subsingleton) with the
-algebra-Kähler reverse direction (Subsingleton + FiniteDim → IsSeparable). -/
 theorem isogeny_isSeparable_of_omegaCoeff_ne_zero_finiteDim
     (α : Isogeny W.toAffine W.toAffine)
     (h_coeff : omegaPullbackCoeff W α ≠ 0)
@@ -796,14 +585,6 @@ theorem isogeny_isSeparable_of_omegaCoeff_ne_zero_finiteDim
       (isogeny_subsingleton_kaehler_of_omegaCoeff_ne_zero W α h_coeff))
     h_fin
 
-/-- **T-II-4-004 full iff (witness-parametric on FiniteDim only, axiom-clean)**:
-the deliverable iff `α.IsSeparable ↔ omegaPullbackCoeff W α ≠ 0` modulo only
-the Witness #2 hypothesis (FiniteDimensional).
-
-Combines the forward direction (no extra hypothesis) and the reverse direction
-(needs FiniteDim).
-
-Once Witness #2 discharges unconditionally, the full T-II-4-004 lands. -/
 theorem isSeparable_iff_omegaPullbackCoeff_ne_zero_of_finiteDim
     (α : Isogeny W.toAffine W.toAffine)
     (h_fin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
@@ -812,16 +593,6 @@ theorem isSeparable_iff_omegaPullbackCoeff_ne_zero_of_finiteDim
   ⟨isogeny_omegaCoeff_ne_zero_of_isSeparable W α,
    fun h ↦ isogeny_isSeparable_of_omegaCoeff_ne_zero_finiteDim W α h h_fin⟩
 
-/-- **T-II-4-004 reverse direction (axiom-clean, unconditional)**: if the
-differential pullback `α.pullbackKaehler` is injective on `Ω[K(E)/F]`, then `α`
-is separable.
-
-Discharge path:
-* Witness #2 (`FiniteDimensional` of `K(E)/(image)`) — gives `EssFiniteType`.
-* Connection from `omegaPullbackCoeff ≠ 0` (the scalar form of injectivity)
-  to `Subsingleton Ω[K(E)/(image)]` (the algebra-side condition).
-* Mathlib's `Algebra.FormallyUnramified.iff_isSeparable` for the final
-  conclusion. -/
 theorem isogeny_isSeparable_of_pullbackKaehler_injective
     (α : Isogeny W.toAffine W.toAffine)
     (h_inj : Function.Injective α.pullbackKaehler) :
@@ -830,14 +601,6 @@ theorem isogeny_isSeparable_of_pullbackKaehler_injective
     ((pullbackKaehler_injective_iff_omegaPullbackCoeff_ne_zero W α).mp h_inj)
     (isogeny_finiteDimensional W α)
 
-/-- **T-II-4-004 algebra-Kähler half (axiom-clean)**: an isogeny `α` is separable
-(in the `Algebra.IsSeparable` sense) iff its differential pullback
-`α.pullbackKaehler` is injective on `Ω[K(E)/F]`.
-
-This is the substantive Silverman II.4.2(c) algebra-Kähler glue.
-The forward direction: `Algebra.IsSeparable → FormallyUnramified
-→ Ω[K(E)/(image)] = 0 → cotangent surjective → pullbackKaehler injective`.
-The reverse: the converse chain via `FormallyUnramified.iff_isSeparable`. -/
 theorem isSeparable_iff_pullbackKaehler_injective
     (α : Isogeny W.toAffine W.toAffine) :
     α.IsSeparable ↔ Function.Injective α.pullbackKaehler :=
@@ -845,12 +608,6 @@ theorem isSeparable_iff_pullbackKaehler_injective
       (isogeny_omegaCoeff_ne_zero_of_isSeparable W α h),
    isogeny_isSeparable_of_pullbackKaehler_injective W α⟩
 
-/-- **T-II-4-004 (elliptic curve form, deliverable target)**: separability of
-an isogeny `α` is equivalent to its omega-pullback coefficient being nonzero
-(equivalently, to the differential pullback `α*` being injective on
-`Ω[K(E)/F]`).
-
-Reference: Silverman II.4.2(c). -/
 theorem isSeparable_iff_omegaPullbackCoeff_ne_zero
     (α : Isogeny W.toAffine W.toAffine) :
     α.IsSeparable ↔ omegaPullbackCoeff W α ≠ 0 :=
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/Divisors.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/Divisors.lean
index 272a5d45e..77de07569 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/Divisors.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/Divisors.lean
@@ -16,9 +16,6 @@ The divisor group `Div C` of a smooth plane curve `C` is the free abelian group
 on its smooth points: a formal sum `Σ nₚ (P)` with integer coefficients that
 are zero for all but finitely many points.
 
-This closes tickets `T-II-3-001` (`Divisor`) and `T-II-3-002`
-(`Divisor.degree`, `Divisor.degreeHom`).
-
 ## References
 
 * [Silverman, *The Arithmetic of Elliptic Curves*], II.3 (definition)
@@ -95,7 +92,7 @@ end Divisor
 noncomputable abbrev Divisor₀ (C : SmoothPlaneCurve F) : AddSubgroup (Divisor C) :=
   Divisor.degZero C
 
-/-! ### `divisorOf f`: the principal divisor (T-II-3-005) -/
+
 
 namespace SmoothPlaneCurve
 
@@ -116,7 +113,7 @@ noncomputable def divisorOf (C : SmoothPlaneCurve F) (f : C.FunctionField) :
     rw [this]; exact Set.finite_empty
   · refine (C.finite_setOf_ord_P_nonzero hf).subset ?_
     intro P hP
-    simp only [Set.mem_setOf_eq]
+    simp only [Set.mem_ofPred_eq]
     intro h0
     apply hP
     simp [h0]
@@ -184,7 +181,7 @@ noncomputable def divisorHom (C : SmoothPlaneCurve F) :
     congr 1
     exact C.divisorOf_mul u.ne_zero v.ne_zero
 
-/-! ### Principal divisors and linear equivalence (T-II-3-006) -/
+
 
 /-- A divisor `D` on `C` is **principal** if `D = div(f)` for some nonzero
 rational function `f ∈ F(C)`. Equivalently, `D` lies in the image of the
@@ -244,7 +241,7 @@ theorem LinearlyEquiv.trans {C : SmoothPlaneCurve F} {D₁ D₂ D₃ : Divisor C
   rw [show D₁ - D₃ = (D₁ - D₂) + (D₂ - D₃) by abel]
   exact h₁.add h₂
 
-/-! ### T-II-3-008: constants have zero divisor -/
+
 
 /-- `ord_P` of an `F`-constant `c ≠ 0` is zero: a nonzero constant is a unit
 at every smooth point (image of the F-subfield, which lives in the local
@@ -289,15 +286,11 @@ zeros or poles. -/
   refine Finsupp.ext fun P ↦ ?_
   rw [divisorOf_apply, C.ord_P_algebraMap_F_of_ne_zero hc P]; rfl
 
-/-! ### T-II-3-008 (⇒): `divisorOf = 0 ⇒ const` (prime-indexed via IC-006) -/
 
-/-- **T-II-3-008 (⇒), prime-indexed**: if `f ∈ F(C)` has valuation at most 1
-at every nonzero prime of `C.CoordinateRing` **and** has nonnegative order at
-infinity, then `f` is the image of a constant from `F`. This is the content
-of Silverman II.3.1(a) `⇒` in the prime-indexed reformulation made available
-by IC-006; the SmoothPoint-indexed statement follows once every nonzero
-prime is in the image of `SmoothPoint.toHeightOneSpectrum` (the surjection
-step under `[IsAlgClosed F]`, tracked separately). -/
+
+/-- If a function has valuation at most one at every nonzero prime of the
+coordinate ring and has nonnegative order at infinity, it is constant.
+This is the prime-indexed form of Silverman II.3.1(a). -/
 theorem const_of_valuation_le_one_of_ordAtInfty_nonneg
     [IsIntegrallyClosed C.CoordinateRing] (f : C.FunctionField)
     (h_primes : ∀ v : IsDedekindDomain.HeightOneSpectrum C.CoordinateRing,
@@ -306,7 +299,7 @@ theorem const_of_valuation_le_one_of_ordAtInfty_nonneg
     ∃ c : F, f = algebraMap F C.FunctionField c :=
   C.const_of_no_poles_of_valuation_of_ordAtInfty f h_primes h_inf
 
-/-! ### Pic and Pic₀ (T-II-3-007) -/
+
 
 /-- The **Picard group** of `C`: divisors modulo principal divisors.
 Silverman II.3 (definition). -/
@@ -314,8 +307,7 @@ abbrev Pic (C : SmoothPlaneCurve F) : Type _ :=
   Divisor C ⧸ C.principalSubgroup
 
 /-- The **degree-zero Picard group** `Pic⁰(C)`: degree-zero divisors modulo
-principal divisors (principal divisors automatically have degree zero by
-Silverman II.3.1(b), tracked as T-II-3-009). -/
+principal divisors (principal divisors have degree zero by Silverman II.3.1(b)). -/
 abbrev Pic₀ (C : SmoothPlaneCurve F) : Type _ :=
   (Divisor.degZero C) ⧸ (C.principalSubgroup.addSubgroupOf (Divisor.degZero C))
 
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/EffectiveSumReduce.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/EffectiveSumReduce.lean
index 03a560829..5a38ff39d 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/EffectiveSumReduce.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/EffectiveSumReduce.lean
@@ -92,6 +92,7 @@ omit [W.IsElliptic] in
 @[simp] theorem listSum_cons (P : W.Point) (Ps : List W.Point) :
     listSum W (P :: Ps) = P + listSum W Ps := rfl
 
+omit [W.IsElliptic] in
 /-- σ on the list-divisor equals the list sum. -/
 theorem projectiveDivisorSum_listToDivisor (Ps : List W.Point) :
     projectiveDivisorSum W (listToDivisor W Ps) = listSum W Ps := by
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/Miller.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/Miller.lean
index 0cca33fbbad2caeffd72dbaa7a093fb8e19faddd..746ea1a153571d3a8d47b2d1a9b99a29dd929f72 100644
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/MillerAllChar.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/MillerAllChar.lean
index a1a3f177c..65bf58647 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/MillerAllChar.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/MillerAllChar.lean
@@ -6,49 +6,19 @@ Authors: Chris Birkbeck
 import HasseWeil.Foundation.Curves.Divisor.Miller
 
 /-!
-# Miller's relation — char-uniform version
-
-This file ships `miller_hypothesis_holds_allChar`: the projective Miller divisor
-relation `(P) + (Q) − (P+Q) − (O) ∈ Princ` for every pair `P, Q : W.Point`,
-without the `[NeZero (2 : F)]` / `[NeZero (3 : F)]` typeclass restrictions
-used by `Curves.miller_hypothesis_holds`.
-
-The mathematical content (Silverman III.3.4(e) proof, p. 63) is **uniform in
-characteristic**: the chord/tangent line through `P, Q` on the smooth
-Weierstrass cubic exists in every characteristic, with the substantive
-divisor identities discharged via the mathlib `XYIdeal_mul_XYIdeal`
-factorisation — which is itself char-uniform (it does not assume
-`[NeZero (2 : F)]` or `[NeZero (3 : F)]`).
-
-The `[NeZero 2/3]` typeclasses in `Curves.Miller` were **inert** on the
-chord/tangent divisor chain (they propagated through but were not used in
-the proof bodies of the key count-form lemmas). This file re-proves the
-chain dropping the inert typeclasses.
-
-## Implementation notes
-
-The `Pic⁰ ≅ E` constructions still carry the `[NeZero 2/3]` typeclasses only
-because they discharge the `PrincipalImpliesDegZero` witness through
-`principal_mem_degZero` (`NormValuation.lean`), whose chain inherits those
-typeclasses inertly via `divisorOf_algMap_degree_eq_natDegree_norm`. Weakening
-that chain to char-uniform is the dedicated sub-ticket
-`T10-SUB-PRINCIPAL-DEGZERO-ALLCHAR`; `picZeroIsoE_of_AFInputs_witness_pdz_allChar`
-factors out the obstruction by taking the degree-zero witness as an explicit
-hypothesis, so composing the iso itself needs no characteristic assumption.
-
-## Main results
-
-* `vertical_line_principal_allChar` — `(P) + (P.neg) − 2·(O) ∈ Princ`
-  in any characteristic.
-* `miller_at_addSmoothPoint_principal_allChar` — non-degenerate Miller
-  divisor is principal in any characteristic.
-* `miller_hypothesis_holds_allChar` — `MillerHypothesis W` in any
-  characteristic.
-* `picZeroIsoE_allChar` — `Pic⁰(E) ≅ E` (Abel–Jacobi) in any characteristic.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], III.3.4(e) (p. 63).
+# Miller's relation in arbitrary characteristic
+
+The projective divisor `(P) + (Q) − (P+Q) − (O)` is principal on a smooth
+Weierstrass cubic in every characteristic. The chord and tangent
+factorization of the coordinate ideals gives the requisite divisor
+identities without restrictions on the values of `2` or `3` in the base
+field.
+
+These principal-divisor identities give the Abel–Jacobi description
+`Pic⁰(E) ≅ E`. The intermediate construction can also take the degree-zero
+property of principal divisors as an explicit hypothesis.
+
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.3.4(e), p.63.
 -/
 
 open WeierstrassCurve
@@ -485,8 +455,7 @@ theorem miller_at_some_some_degen_allChar
     rw [show (2 : ℤ) = 1 + 1 from rfl, add_smul, one_smul]]
   abel
 
-/-- **Miller's relation holds unconditionally** (`MillerHypothesis W` axiom-clean,
-all char): for any pair `P, Q : W.Point`, the divisor
+/-- **Miller's relation holds unconditionally** (`MillerHypothesis W` all char): for any pair `P, Q : W.Point`, the divisor
 `(P) + (Q) − (P + Q) − (O)` is principal on the projective curve, in any
 characteristic. -/
 theorem miller_hypothesis_holds_allChar
@@ -607,6 +576,7 @@ private lemma general_kappa_reduce_regroup_allChar
   rw [add_smul]
   abel
 
+omit [W.IsElliptic] in
 /-- **General κ-reduction (witness-parametric)**: same statement as
 `general_kappa_reduce`, but takes `MillerHypothesis W` as a hypothesis
 rather than depending on `[NeZero 2/3]`. -/
@@ -653,7 +623,7 @@ theorem general_kappa_reduce_of_miller
       nsmul_kappaDivisor_add_linEquiv_of_miller_allChar W h_miller P n D'
     exact (⟨W⟩ : SmoothPlaneCurve F).projPrincipalSubgroup.add_mem h_part2 h_part3
 
-/-- **`DivZeroReduce W` axiom-clean (all char)**: char-uniform
+/-- **`DivZeroReduce W` (all char)**: char-uniform
 `divZeroReduce_holds`. -/
 theorem divZeroReduce_holds_allChar
     [IsIntegrallyClosed (⟨W⟩ : SmoothPlaneCurve F).CoordinateRing] : DivZeroReduce W := by
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PicZero.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PicZero.lean
index 309309cd72b059ae50cf557af07b5e3b8d05acea..1e9c9257a57d8974029d1b598ed44d88218304f5 100644
GIT binary patch
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PicZeroPushforward.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PicZeroPushforward.lean
index 0bbf09b85bb147c417d49a75e470e0aa1a31c088..a4035b8bb57b401dae50e25f15232dbba25a6e0d 100644
GIT binary patch
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/ProjectiveDivisor.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/ProjectiveDivisor.lean
index 42df6e4ba..40dbf3639 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/ProjectiveDivisor.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/ProjectiveDivisor.lean
@@ -233,7 +233,7 @@ theorem projectiveDivisorOf_apply_affine (f : C.FunctionField) (P : C.SmoothPoin
       (ProjectiveSmoothPoint.infinity : ProjectiveSmoothPoint C) := by
     intro h; nomatch h
   rw [Finsupp.add_apply, Finsupp.single_eq_of_ne h_ne, add_zero,
-    Finsupp.mapDomain_apply ProjectiveSmoothPoint.affine_injective
+    Finsupp.mapDomain_apply_of_injective ProjectiveSmoothPoint.affine_injective
       (C.divisorOf f) P, C.divisorOf_apply]
 
 theorem projectiveDivisorOf_apply_infinity (f : C.FunctionField) :
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PushforwardDivisor.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PushforwardDivisor.lean
index 37f06f385..9d556ad1d 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PushforwardDivisor.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Divisor/PushforwardDivisor.lean
@@ -8,40 +8,17 @@ import HasseWeil.Foundation.Curves.Map.CoordHomFinite
 import HasseWeil.Foundation.Curves.Valuation.NormValuation
 
 /-!
-# The divisor pushforward of a finite curve map preserves principal divisors
+# Divisor pushforward and principal divisors
 
-For a (nonconstant) curve map `φ : C₁ → C₂` between smooth plane curves, the
-**divisor pushforward** `φ_∗ : Div(C₁) → Div(C₂)` sends `Σ nᵢ (Pᵢ)` to
-`Σ nᵢ (φ Pᵢ)`.  Silverman II.3.6 / II.3.7 says it carries *principal* divisors
-to *principal* divisors, via the **norm–conorm identity**
+For a finite curve map `φ : C₁ → C₂` over an algebraically closed field, the
+divisor pushforward sends `Σ nᵢ(Pᵢ)` to `Σ nᵢ(φ(Pᵢ))`. The norm–conorm identity
+`div(N_φ f) = φ_∗(div f)` shows that principal divisors remain principal.
+Here `N_φ` is the field norm from `K(C₁)` to `φ*K(C₂)`.
 
-  `div(N_φ f) = φ_∗(div f)`            (Silverman II.3.6)
-
-where `N_φ f = Norm_{K(C₁)/φ*K(C₂)}(f) ∈ K(C₂)` is the field norm (already
-defined as `CurveMap.pushforward`).  This is the only deep input to Silverman
-III.4.8 (every isogeny is a group homomorphism, proved at the divisor/σ level in
-`HasseWeil/EC/IsogenyAG/GroupHom.lean`).
-
-## The valuation-theoretic pushforward
-
-We realise the divisor pushforward as `Finsupp.mapDomain` along the place-image
-map `P ↦ φ(P)` (affine smooth point `↦` affine smooth point, `∞ ↦ ∞`), supplied
-by a coordinate-ring witness `cd : φ.CoordHom`.  This is definitionally the
-point-map pushforward `pushforwardProjectiveDivisor`, so the compatibility
-sub-leaf NEW-1(iii) is `rfl`; the mathematical content is concentrated in the
-norm–conorm identity NEW-1(ii).
-
-## Main definitions
-
-* `CurveMap.pushforwardDivisorVal` — the valuation-theoretic divisor pushforward
-  (= `pushforwardProjectiveDivisor`, via the place-image map).
-
-## Main results
-
-* `CurveMap.projectiveDivisorOf_pushforward_eq_pushforwardDivisorVal` — the
-  norm–conorm identity `div(N_φ f) = φ_∗(div f)` (Silverman II.3.6).
-* `EC.Isogeny.pushforward_preserves_principal` — the gap `h_pres`: the
-  pushforward of a principal projective divisor is principal (Silverman II.3.7).
+The pushforward is realized by `Finsupp.mapDomain` along the projective
+place-image map. Its coefficients are sums over the fibres. The local
+norm formula, together with residue degree one, identifies these coefficients
+with the orders of the norm.
 
 ## References
 
@@ -50,7 +27,6 @@ norm–conorm identity NEW-1(ii).
 
 open WeierstrassCurve
 
-set_option linter.unusedSectionVars false
 
 namespace HasseWeil.Curves.CurveMap
 
@@ -58,7 +34,7 @@ variable {F : Type*} [Field F]
 variable {C₁ C₂ : SmoothPlaneCurve F}
   [C₁.toAffine.IsElliptic] [C₂.toAffine.IsElliptic]
 
-/-! ### NEW-1(i): the valuation-theoretic divisor pushforward -/
+/-! ### the valuation-theoretic divisor pushforward -/
 
 /-- The place-image map on the projective closure induced by a curve map `φ`
 together with a coordinate-ring witness: an affine smooth point `P` is sent to
@@ -96,7 +72,7 @@ theorem degree_pushforwardDivisorVal (φ : CurveMap C₁ C₂) (cd : φ.CoordHom
   simp only [ProjectiveDivisor.degree]
   rw [Finsupp.sum_mapDomain_index (h := fun _ n ↦ n) (fun _ ↦ rfl) (fun _ _ _ ↦ rfl)]
 
-/-! ### NEW-1(ii): the norm–conorm identity `div(N_φ f) = φ_∗(div f)`
+/-! ### the norm–conorm identity `div(N_φ f) = φ_∗(div f)`
 
 The mathematical content is the per-place identity (Silverman II.3.6)
 
@@ -107,7 +83,7 @@ by generalising the `F[X] → F[C]` machinery of `NormValuation.lean` to the
 coordinate-ring extension `F[C₂] → F[C₁]` supplied by a `CoordHom`.  The generic
 ideal/localisation lemmas (`count_preservation_localization`,
 `count_finset_prod_factors`, `map_eq_localRing_max_pow_count`, …) are reused
-verbatim; the curve-specific inputs are `inertiaDeg' = 1`, `relNorm m_P = m_{φP}`,
+verbatim; the curve-specific inputs are `inertiaDeg = 1`, `relNorm m_P = m_{φP}`,
 and the fibre bijection `maximalIdealAt_toPointMap`. -/
 
 section NormConorm
@@ -129,8 +105,8 @@ theorem coordHom_injective : Function.Injective cd.toAlgHom := by
 injective into a domain). -/
 theorem isTorsionFree_coordHom :
     @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ cd.toAlgebra.toModule := by
-  letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI : FaithfulSMul C₂.CoordinateRing C₁.CoordinateRing :=
+  let : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have : FaithfulSMul C₂.CoordinateRing C₁.CoordinateRing :=
     (faithfulSMul_iff_algebraMap_injective C₂.CoordinateRing C₁.CoordinateRing).mpr
       (CurveMap.coordHom_injective φ cd)
   infer_instance
@@ -146,8 +122,8 @@ private theorem isScalarTower_residueField_maximalIdealAt (P : C₁.SmoothPoint)
       ⟨maximalIdealAt_toPointMap cd P⟩
     IsScalarTower F (C₂.CoordinateRing ⧸ C₂.maximalIdealAt (toPointMap cd P))
       (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) := by
-  letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI : (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
+  let : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have : (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
     ⟨maximalIdealAt_toPointMap cd P⟩
   set Q := toPointMap cd P with hQ
   refine IsScalarTower.of_algebraMap_eq fun c ↦ ?_
@@ -177,15 +153,15 @@ private theorem finrank_residueField_maximalIdealAt_le_one (P : C₁.SmoothPoint
       ⟨maximalIdealAt_toPointMap cd P⟩
     Module.finrank (C₂.CoordinateRing ⧸ C₂.maximalIdealAt (toPointMap cd P))
       (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) ≤ 1 := by
-  letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI : (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
+  let : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have : (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
     ⟨maximalIdealAt_toPointMap cd P⟩
   set Q := toPointMap cd P with hQ
-  haveI hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
-  haveI hPmax : (C₁.maximalIdealAt P).IsMaximal := C₁.maximalIdealAt_isMaximal P
-  haveI : Field (C₂.CoordinateRing ⧸ C₂.maximalIdealAt Q) := Ideal.Quotient.field _
-  haveI : Field (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) := Ideal.Quotient.field _
-  haveI htower := isScalarTower_residueField_maximalIdealAt φ cd P
+  have hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hPmax : (C₁.maximalIdealAt P).IsMaximal := C₁.maximalIdealAt_isMaximal P
+  have : Field (C₂.CoordinateRing ⧸ C₂.maximalIdealAt Q) := Ideal.Quotient.field _
+  have : Field (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) := Ideal.Quotient.field _
+  have htower := isScalarTower_residueField_maximalIdealAt φ cd P
   have hbijSP := C₁.algebraMap_bijective_quotient_of_maximal hPmax
   refine finrank_le_one (1 : C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) fun w ↦ ?_
   obtain ⟨c, hc⟩ := hbijSP.2 w
@@ -212,16 +188,17 @@ theorem inertiaDeg_maximalIdealAt_toPointMap (P : C₁.SmoothPoint) :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
     haveI : (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
       ⟨(maximalIdealAt_toPointMap cd P)⟩
-    Ideal.inertiaDeg' (C₂.maximalIdealAt (toPointMap cd P)) (C₁.maximalIdealAt P) = 1 := by
-  letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI hLies : (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
+    (C₁.maximalIdealAt P).inertiaDeg C₂.CoordinateRing = 1 := by
+  let : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have hLies : (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
     ⟨maximalIdealAt_toPointMap cd P⟩
   set Q := toPointMap cd P with hQ
-  haveI hPmax : (C₁.maximalIdealAt P).IsMaximal := C₁.maximalIdealAt_isMaximal P
-  rw [Ideal.inertiaDeg'_algebraMap]
+  have hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hPmax : (C₁.maximalIdealAt P).IsMaximal := C₁.maximalIdealAt_isMaximal P
+  rw [Ideal.inertiaDeg_eq_of_isMaximal (C₂.maximalIdealAt Q) (C₁.maximalIdealAt P)]
   -- The residue field `F[C₁]/m_P` is `F` (alg-closed), hence nontrivial.
-  haveI : Field (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) := Ideal.Quotient.field _
-  haveI : Nontrivial (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) :=
+  have : Field (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) := Ideal.Quotient.field _
+  have : Nontrivial (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) :=
     (Ideal.Quotient.nontrivial_iff).mpr hPmax.ne_top
   -- Squeeze the residue rank between `1` (`finrank_pos`) and `1` (alg-closed surjectivity).
   have hSP : Module.finrank F (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) = 1 :=
@@ -229,9 +206,9 @@ theorem inertiaDeg_maximalIdealAt_toPointMap (P : C₁.SmoothPoint) :
   have h_le : Module.finrank (C₂.CoordinateRing ⧸ C₂.maximalIdealAt Q)
       (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) ≤ 1 :=
     finrank_residueField_maximalIdealAt_le_one φ cd P
-  haveI : Module.Finite F (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) :=
+  have : Module.Finite F (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) :=
     Module.finite_of_finrank_pos (by rw [hSP]; norm_num)
-  haveI : Module.Finite (C₂.CoordinateRing ⧸ C₂.maximalIdealAt Q)
+  have : Module.Finite (C₂.CoordinateRing ⧸ C₂.maximalIdealAt Q)
       (C₁.CoordinateRing ⧸ C₁.maximalIdealAt P) :=
     Module.Finite.of_restrictScalars_finite F _ _
   have h_ge : 1 ≤ Module.finrank (C₂.CoordinateRing ⧸ C₂.maximalIdealAt Q)
@@ -244,22 +221,17 @@ packaged as a `LiesOver` instance for the `cd`-induced algebra. -/
 theorem maximalIdealAt_liesOver_toPointMap (P : C₁.SmoothPoint) :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
     (C₁.maximalIdealAt P).LiesOver (C₂.maximalIdealAt (toPointMap cd P)) :=
-  letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
   ⟨maximalIdealAt_toPointMap cd P⟩
 
 end NormConorm
 
-/-! ### NEW-1(ii) — structural sub-lemmas of the norm–conorm identity
+/-! ### Coordinate-ring norms
 
-The norm–conorm identity is assembled below from three sub-lemmas, each carrying its
-own elaboration budget so the assembling theorem stays light:
-* `relNorm_maximalIdealAt_eq` — the **`s = 1` core** `relNorm(m_R) = m_{φR}`;
-* `count_relNorm_eq_sum_fiber` — the **affine count identity** matching
-  `count_{m_Q}(relNorm(span{w}))` to the fibre sum `Σ_{φP=Q} count_{m_P}(span{w})`;
-* `projectiveDivisorOf_pushforward_algebraMap_eq` — the **`algebraMap` case** of the
-  norm–conorm identity (affine coefficients via the count identity, infinity
-  coefficient forced by degree).
-The `f = u/v` reduction is then a short additivity argument in the main theorem. -/
+The prime-ideal identity `relNorm(m_P) = m_{φP}` gives the corresponding
+valuation count as a sum over the fibre. Equality of affine coefficients,
+together with degree zero, determines the projective divisor. Additivity
+then extends the identity to quotients of coordinate-ring elements. -/
 
 section NormConormSteps
 
@@ -276,8 +248,8 @@ finiteness/degree arguments. -/
 private theorem faithfulSMul_coordinateRing_functionField :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
     FaithfulSMul C₂.CoordinateRing C₁.FunctionField := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
@@ -300,9 +272,9 @@ private theorem isScalarTower_coordinateRing_functionField :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
     letI : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
     IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  haveI tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  have tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
@@ -321,23 +293,23 @@ to `K(C₂)`).  This is the algebraicity input to the localisation-finiteness st
 private theorem isAlgebraic_functionField :
     letI : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
     Algebra.IsAlgebraic C₂.FunctionField C₁.FunctionField := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ algCR.toModule :=
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ algCR.toModule :=
     cd.module_finite
-  haveI hint : Algebra.IsIntegral C₂.CoordinateRing C₁.CoordinateRing :=
+  have hint : Algebra.IsIntegral C₂.CoordinateRing C₁.CoordinateRing :=
     Algebra.IsIntegral.of_finite C₂.CoordinateRing C₁.CoordinateRing
-  haveI faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
+  have faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
     faithfulSMul_coordinateRing_functionField φ cd
-  letI algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  haveI tower1 : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField :=
+  let algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  have tower1 : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField :=
     isScalarTower_coordinateRing_functionField φ cd
-  haveI hab : Algebra.IsAlgebraic C₂.CoordinateRing C₁.CoordinateRing :=
+  have hab : Algebra.IsAlgebraic C₂.CoordinateRing C₁.CoordinateRing :=
     Algebra.IsIntegral.isAlgebraic
-  haveI halgAB : Algebra.IsAlgebraic C₂.CoordinateRing C₁.FunctionField :=
+  have halgAB : Algebra.IsAlgebraic C₂.CoordinateRing C₁.FunctionField :=
     (IsFractionRing.isAlgebraic_iff' C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField).mp hab
   exact (IsFractionRing.comap_isAlgebraic_iff (A := C₂.CoordinateRing)
     (K := C₂.FunctionField) (C := C₁.FunctionField)).mp halgAB
@@ -352,25 +324,25 @@ theorem finiteDimensional_functionField :
   have hfin : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _
       cd.toAlgebra.toModule := by
     exact cd.module_finite
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
-  haveI hint : Algebra.IsIntegral C₂.CoordinateRing C₁.CoordinateRing :=
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
+  have hint : Algebra.IsIntegral C₂.CoordinateRing C₁.CoordinateRing :=
     Algebra.IsIntegral.of_finite C₂.CoordinateRing C₁.CoordinateRing
-  haveI faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
+  have faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
     faithfulSMul_coordinateRing_functionField φ cd
-  letI algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  haveI tower1 : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField :=
+  let algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  have tower1 : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField :=
     isScalarTower_coordinateRing_functionField φ cd
-  haveI halgFF : Algebra.IsAlgebraic C₂.FunctionField C₁.FunctionField :=
+  have halgFF : Algebra.IsAlgebraic C₂.FunctionField C₁.FunctionField :=
     isAlgebraic_functionField φ cd
-  haveI hicl : IsIntegralClosure C₁.CoordinateRing C₂.CoordinateRing C₁.FunctionField :=
+  have hicl : IsIntegralClosure C₁.CoordinateRing C₂.CoordinateRing C₁.FunctionField :=
     IsIntegralClosure.of_isIntegrallyClosed C₁.CoordinateRing C₂.CoordinateRing _
-  haveI hloc : IsLocalization
+  have hloc : IsLocalization
       (Algebra.algebraMapSubmonoid C₁.CoordinateRing (nonZeroDivisors C₂.CoordinateRing))
       C₁.FunctionField :=
     IsIntegralClosure.isLocalization C₂.CoordinateRing C₂.FunctionField C₁.FunctionField
@@ -394,12 +366,12 @@ private theorem finrank_functionField_eq_degree :
       faithfulSMul_coordinateRing_functionField φ cd
     @Module.finrank C₂.FunctionField C₁.FunctionField _ _
       (FractionRing.liftAlgebra C₂.CoordinateRing C₁.FunctionField).toModule = φ.degree := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  haveI tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  have tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
-  haveI faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
+  have faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
     faithfulSMul_coordinateRing_functionField φ cd
   have halgmap :
       IsFractionRing.lift (A := C₂.CoordinateRing) (K := C₂.FunctionField)
@@ -436,13 +408,13 @@ private theorem one_le_of_relNorm_eq_pow
         isTorsionFree_coordHom φ cd
       Ideal.relNorm C₂.CoordinateRing P' = q ^ t) :
     1 ≤ t := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI : P'.LiesOver q := hlies
+  have : P'.LiesOver q := hlies
   rcases Nat.eq_zero_or_pos t with ht0 | ht0
   · exfalso
     have hcomap : P'.comap (algebraMap C₂.CoordinateRing C₁.CoordinateRing) = q :=
@@ -468,35 +440,26 @@ private theorem inertiaDeg_eq_one_of_liesOver_maximalIdealAt (Q : C₂.SmoothPoi
       cd.module_finite
     haveI : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
       isTorsionFree_coordHom φ cd
-    Ideal.inertiaDeg' (C₂.maximalIdealAt Q) P' = 1 := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+    P'.inertiaDeg C₂.CoordinateRing = 1 := by
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI hP'prime' : P'.IsPrime := hP'prime
-  haveI hP'lies' : P'.LiesOver (C₂.maximalIdealAt Q) := hP'lies
-  haveI hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hP'prime' : P'.IsPrime := hP'prime
+  have hP'lies' : P'.LiesOver (C₂.maximalIdealAt Q) := hP'lies
+  have hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
   have hQ0 : C₂.maximalIdealAt Q ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   have hP'_ne_bot : P' ≠ ⊥ := by
     intro h
     apply hQ0
     have hh : C₂.maximalIdealAt Q = P'.under C₂.CoordinateRing := hP'lies.over
     rw [hh, h, Ideal.under_bot]
-  haveI hP'max : P'.IsMaximal := Ideal.IsPrime.isMaximal hP'prime hP'_ne_bot
+  have hP'max : P'.IsMaximal := Ideal.IsPrime.isMaximal hP'prime hP'_ne_bot
   obtain ⟨P'', hP''⟩ := C₁.exists_smoothPoint_of_isMaximal hP'max
-  haveI hlies'' : (C₁.maximalIdealAt P'').LiesOver
-      (C₂.maximalIdealAt (toPointMap cd P'')) :=
-    maximalIdealAt_liesOver_toPointMap φ cd P''
-  have h1 : C₂.maximalIdealAt (toPointMap cd P'') =
-      (C₁.maximalIdealAt P'').under C₂.CoordinateRing := hlies''.over
-  have h2 : C₂.maximalIdealAt Q = P'.under C₂.CoordinateRing := hP'lies.over
-  rw [hP''] at h1
-  have hpeq : C₂.maximalIdealAt (toPointMap cd P'') = C₂.maximalIdealAt Q := h1.trans h2.symm
-  have hid := inertiaDeg_maximalIdealAt_toPointMap φ cd P''
-  rw [← hP'', ← hpeq]
-  exact hid
+  rw [← hP'']
+  exact inertiaDeg_maximalIdealAt_toPointMap φ cd P''
 
 include φ cd in
 /-- **Sum of ramification indices equals `φ.degree`**: over a smooth point `Q` of
@@ -514,12 +477,12 @@ private theorem sum_ramificationIdx_eq_degree (Q : C₂.SmoothPoint)
       isTorsionFree_coordHom φ cd
     ∑ P' ∈ IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing,
       (C₂.maximalIdealAt Q).ramificationIdx' P' = φ.degree := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
   have hQ0 : C₂.maximalIdealAt Q ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   have hsumef := φ.sum_ramificationIdx_mul_inertiaDeg_eq_degree cd hfin hQmax hQ0
   rw [← hsumef]
@@ -557,12 +520,12 @@ private theorem degree_eq_sum_relNormExp_mul_ramificationIdx (Q : C₂.SmoothPoi
     haveI : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
     φ.degree = ∑ P' ∈ ((C₂.maximalIdealAt Q).primesOver C₁.CoordinateRing).toFinset,
       sfn P' * (C₂.maximalIdealAt Q).ramificationIdx' P' := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
   set p : Ideal C₂.CoordinateRing := C₂.maximalIdealAt Q with hp_def
   have hp0 : p ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   have hpNotUnit : ¬ IsUnit p := by rw [Ideal.isUnit_iff]; exact hQmax.ne_top
@@ -571,11 +534,11 @@ private theorem degree_eq_sum_relNormExp_mul_ramificationIdx (Q : C₂.SmoothPoi
     finrank_functionField_eq_degree φ cd
   -- `relNorm_algebraMap`'s exponent is the *coordinate-ring* `finrank`, so pull `hcoh` down the
   -- fraction-field tower with `IsFractionRing.finrank_eq`.
-  letI algFF : Algebra C₂.FunctionField C₁.FunctionField :=
+  let algFF : Algebra C₂.FunctionField C₁.FunctionField :=
     FractionRing.liftAlgebra C₂.CoordinateRing C₁.FunctionField
-  haveI towerFF : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField :=
+  have towerFF : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField :=
     FractionRing.isScalarTower_liftAlgebra C₂.CoordinateRing C₁.FunctionField
-  haveI towerCR : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
+  have towerCR : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
@@ -619,14 +582,14 @@ private theorem one_le_ramificationIdx_of_liesOver_maximalIdealAt (Q : C₂.Smoo
     haveI : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
       isTorsionFree_coordHom φ cd
     1 ≤ (C₂.maximalIdealAt Q).ramificationIdx' P' := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI : P'.IsPrime := hP'prime
-  haveI : P'.LiesOver (C₂.maximalIdealAt Q) := hP'lies
+  have : P'.IsPrime := hP'prime
+  have : P'.LiesOver (C₂.maximalIdealAt Q) := hP'lies
   have hp0 : C₂.maximalIdealAt Q ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   rw [Nat.one_le_iff_ne_zero]
   exact Ideal.IsDedekindDomain.ramificationIdx'_ne_zero_of_liesOver P' hp0
@@ -664,17 +627,17 @@ theorem relNorm_maximalIdealAt_eq
   have hfin : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _
       cd.toAlgebra.toModule := by
     exact cd.module_finite
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR := hfin
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   set Q := toPointMap cd R with hQ
-  haveI hLies : (C₁.maximalIdealAt R).LiesOver (C₂.maximalIdealAt Q) :=
+  have hLies : (C₁.maximalIdealAt R).LiesOver (C₂.maximalIdealAt Q) :=
     maximalIdealAt_liesOver_toPointMap φ cd R
-  haveI hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
-  haveI hRmax : (C₁.maximalIdealAt R).IsMaximal := C₁.maximalIdealAt_isMaximal R
-  haveI hRprime : (C₁.maximalIdealAt R).IsPrime := hRmax.isPrime
+  have hQmax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hRmax : (C₁.maximalIdealAt R).IsMaximal := C₁.maximalIdealAt_isMaximal R
+  have hRprime : (C₁.maximalIdealAt R).IsPrime := hRmax.isPrime
   have hQ0 : C₂.maximalIdealAt Q ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   obtain ⟨s, hs⟩ := Ideal.exists_relNorm_eq_pow_of_isPrime
     (C₁.maximalIdealAt R) (C₂.maximalIdealAt Q)
@@ -683,14 +646,14 @@ theorem relNorm_maximalIdealAt_eq
   have hge1 : 1 ≤ s := one_le_of_relNorm_eq_pow φ cd hQmax s hLies hs
   set p : Ideal C₂.CoordinateRing := C₂.maximalIdealAt Q with hp_def
   -- Every residue degree over `m_Q` is `1`.
-  have hinertia : ∀ P' ∈ p.primesOver C₁.CoordinateRing, Ideal.inertiaDeg' p P' = 1 := by
+  have hinertia : ∀ P' ∈ p.primesOver C₁.CoordinateRing, P'.inertiaDeg C₂.CoordinateRing = 1 := by
     intro P' hP'
     obtain ⟨hP'prime, hP'lies⟩ := hP'
     exact inertiaDeg_eq_one_of_liesOver_maximalIdealAt φ cd Q P' hP'prime hP'lies
   have hp0 : p ≠ ⊥ := hQ0
   have hpNotUnit : ¬ IsUnit p := by
     rw [Ideal.isUnit_iff]; exact hQmax.ne_top
-  haveI hpMax : p.IsMaximal := hQmax
+  have hpMax : p.IsMaximal := hQmax
   -- Each relative norm over `m_Q` is a positive power of `m_Q`.
   have hexp : ∀ P' ∈ p.primesOver C₁.CoordinateRing,
       ∃ t : ℕ, 1 ≤ t ∧ Ideal.relNorm C₂.CoordinateRing P' = p ^ t := by
@@ -753,16 +716,16 @@ private theorem primesOverFinset_ne_bot {p : Ideal C₂.CoordinateRing}
     (hpMax : p.IsMaximal) (hp_ne : p ≠ ⊥) :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
     ∀ Q' ∈ IsDedekindDomain.primesOverFinset p C₁.CoordinateRing, Q' ≠ ⊥ := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI faithCR : FaithfulSMul C₂.CoordinateRing C₁.CoordinateRing :=
+  have faithCR : FaithfulSMul C₂.CoordinateRing C₁.CoordinateRing :=
     (faithfulSMul_iff_algebraMap_injective C₂.CoordinateRing C₁.CoordinateRing).mpr
       (CurveMap.coordHom_injective φ cd)
-  haveI hpMax' : p.IsMaximal := hpMax
+  have hpMax' : p.IsMaximal := hpMax
   intro Q' hQ'
   rw [IsDedekindDomain.mem_primesOverFinset_iff (B := C₁.CoordinateRing) hp_ne] at hQ'
   intro h_eq
@@ -790,15 +753,15 @@ private theorem exists_smoothPoint_relNorm_maximalIdealAt_eq
     ∃ P' : C₁.SmoothPoint, C₁.maximalIdealAt P' = Q' ∧
       Ideal.relNorm C₂.CoordinateRing Q' = C₂.maximalIdealAt (toPointMap cd P') ∧
       Q'.LiesOver (C₂.maximalIdealAt (toPointMap cd P')) := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI hQ'max' : Q'.IsMaximal := hQ'max
+  have hQ'max' : Q'.IsMaximal := hQ'max
   obtain ⟨P', hP'⟩ := C₁.exists_smoothPoint_of_isMaximal hQ'max
-  haveI hlies' : (C₁.maximalIdealAt P').LiesOver
+  have hlies' : (C₁.maximalIdealAt P').LiesOver
       (C₂.maximalIdealAt (toPointMap cd P')) :=
     maximalIdealAt_liesOver_toPointMap φ cd P'
   refine ⟨P', hP', ?_, hP' ▸ hlies'⟩
@@ -822,15 +785,15 @@ private theorem count_maximalIdealAt_relNorm_pow_self (Q : C₂.SmoothPoint)
       isTorsionFree_coordHom φ cd
     (Associates.mk (C₂.maximalIdealAt Q)).count
         (Associates.mk ((Ideal.relNorm C₂.CoordinateRing Q') ^ k)).factors = k := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI hQ'lies' : Q'.LiesOver (C₂.maximalIdealAt Q) := hQ'lies
+  have hQ'lies' : Q'.LiesOver (C₂.maximalIdealAt Q) := hQ'lies
   set p : Ideal C₂.CoordinateRing := C₂.maximalIdealAt Q with hp_def
-  haveI hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
   have hp_ne : p ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   have h_vp_irr : Irreducible (Associates.mk p) :=
     (⟨p, hpMax.isPrime, hp_ne⟩ :
@@ -862,14 +825,14 @@ private theorem count_maximalIdealAt_relNorm_pow_of_not_liesOver (Q : C₂.Smoot
       isTorsionFree_coordHom φ cd
     (Associates.mk (C₂.maximalIdealAt Q)).count
         (Associates.mk ((Ideal.relNorm C₂.CoordinateRing Q') ^ k)).factors = 0 := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   set p : Ideal C₂.CoordinateRing := C₂.maximalIdealAt Q with hp_def
-  haveI hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
   have hp_ne : p ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   have h_vp_irr : Irreducible (Associates.mk p) :=
     (⟨p, hpMax.isPrime, hp_ne⟩ :
@@ -880,7 +843,7 @@ private theorem count_maximalIdealAt_relNorm_pow_of_not_liesOver (Q : C₂.Smoot
     rw [hrel]
     intro hpe
     exact hQ'not (hpe ▸ hlies')
-  haveI hP'max2 : (Ideal.relNorm C₂.CoordinateRing Q').IsMaximal :=
+  have hP'max2 : (Ideal.relNorm C₂.CoordinateRing Q').IsMaximal :=
     hrel ▸ C₂.maximalIdealAt_isMaximal _
   have hP'_ne_bot2 : Ideal.relNorm C₂.CoordinateRing Q' ≠ ⊥ := hrel ▸ C₂.maximalIdealAt_ne_bot _
   have h_vP'_irr : Irreducible (Associates.mk (Ideal.relNorm C₂.CoordinateRing Q')) :=
@@ -915,14 +878,14 @@ private theorem count_factors_relNorm_pow_eq_ite (Q : C₂.SmoothPoint)
         (Associates.mk ((Ideal.relNorm C₂.CoordinateRing Q') ^ k)).factors =
       if Q' ∈ IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing
         then k else 0 := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   have hp_ne : C₂.maximalIdealAt Q ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
-  haveI hpMax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hpMax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
   by_cases h_over : Q' ∈ IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing
   · rw [if_pos h_over]
     haveI hQ'lies : Q'.LiesOver (C₂.maximalIdealAt Q) :=
@@ -979,11 +942,11 @@ private theorem associates_relNorm_pow_ne_zero
     haveI : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
       isTorsionFree_coordHom φ cd
     Associates.mk ((Ideal.relNorm C₂.CoordinateRing Q'.asIdeal) ^ k) ≠ 0 := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   rw [Associates.mk_ne_zero]
   apply pow_ne_zero
@@ -1018,14 +981,14 @@ private theorem count_relNorm_span_eq_sum_support (Q : C₂.SmoothPoint)
         (Associates.mk ((Ideal.relNorm C₂.CoordinateRing Q'.asIdeal) ^
           ((Associates.mk Q'.asIdeal).count
             (Associates.mk (Ideal.span ({w} : Set _))).factors))).factors := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   set p : Ideal C₂.CoordinateRing := C₂.maximalIdealAt Q with hp_def
-  haveI hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
   have hp_ne : p ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   have h_vp_irr : Irreducible (Associates.mk p) :=
     (⟨p, hpMax.isPrime, hp_ne⟩ :
@@ -1070,15 +1033,15 @@ theorem count_relNorm_eq_sum_fiber :
       ∑ Q' ∈ IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing,
         (Associates.mk Q').count (Associates.mk (Ideal.span ({w} : Set _))).factors := by
   classical
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   intro w hw Q
   set p : Ideal C₂.CoordinateRing := C₂.maximalIdealAt Q with hp_def
-  haveI hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hpMax : p.IsMaximal := C₂.maximalIdealAt_isMaximal Q
   have hp_ne : p ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   have hI_ne : Ideal.span ({w} : Set C₁.CoordinateRing) ≠ 0 := by
     rw [Ne, Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot]; exact hw
@@ -1185,28 +1148,28 @@ private theorem pushforward_algebraMap_eq_algebraMap_intNorm (w : C₁.Coordinat
       algebraMap C₂.CoordinateRing C₂.FunctionField
         (Algebra.intNorm C₂.CoordinateRing C₁.CoordinateRing w) := by
   classical
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI hint : Algebra.IsIntegral C₂.CoordinateRing C₁.CoordinateRing :=
+  have hint : Algebra.IsIntegral C₂.CoordinateRing C₁.CoordinateRing :=
     Algebra.IsIntegral.of_finite C₂.CoordinateRing C₁.CoordinateRing
-  haveI tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
+  have tower2 : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
-  haveI faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
+  have faith : FaithfulSMul C₂.CoordinateRing C₁.FunctionField :=
     faithfulSMul_coordinateRing_functionField φ cd
-  letI algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  haveI tower1 : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField := by
+  let algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  have tower1 : IsScalarTower C₂.CoordinateRing C₂.FunctionField C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_smul fun r x ↦ ?_
     rw [Algebra.smul_def]
     show φ.pullback ((algebraMap C₂.CoordinateRing C₂.FunctionField) r) * x = r • x
     rw [cd.compat r, ← IsScalarTower.algebraMap_smul C₁.CoordinateRing r x, ← Algebra.smul_def]
     rfl
-  haveI hfd : FiniteDimensional C₂.FunctionField C₁.FunctionField :=
+  have hfd : FiniteDimensional C₂.FunctionField C₁.FunctionField :=
     finiteDimensional_functionField φ cd
   rw [Algebra.algebraMap_intNorm (A := C₂.CoordinateRing) (B := C₁.CoordinateRing)
     (K := C₂.FunctionField) (L := C₁.FunctionField) w]
@@ -1226,11 +1189,11 @@ private theorem intNorm_ne_zero_of_ne_zero (w : C₁.CoordinateRing) (hw : w ≠
       isTorsionFree_coordHom φ cd
     Algebra.intNorm C₂.CoordinateRing C₁.CoordinateRing w ≠ 0 := by
   classical
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   have hw_FF : algebraMap C₁.CoordinateRing C₁.FunctionField w ≠ 0 := by
     intro h
@@ -1283,25 +1246,25 @@ private theorem exists_smoothPoint_maximalIdealAt_eq_of_mem_primesOverFinset
     (hQ' : letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
       Q' ∈ IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing) :
     ∃ P' : C₁.SmoothPoint, C₁.maximalIdealAt P' = Q' ∧ toPointMap cd P' = Q := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
-  haveI hpMax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
+  have hpMax : (C₂.maximalIdealAt Q).IsMaximal := C₂.maximalIdealAt_isMaximal Q
   have hp_ne : C₂.maximalIdealAt Q ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
   rw [IsDedekindDomain.mem_primesOverFinset_iff (B := C₁.CoordinateRing) hp_ne] at hQ'
   obtain ⟨hQ'prime, hQ'lies⟩ := hQ'
-  haveI : Q'.IsPrime := hQ'prime
-  haveI : Q'.LiesOver (C₂.maximalIdealAt Q) := hQ'lies
-  haveI hQ'max : Q'.IsMaximal := Ideal.IsPrime.isMaximal hQ'prime (by
+  have : Q'.IsPrime := hQ'prime
+  have : Q'.LiesOver (C₂.maximalIdealAt Q) := hQ'lies
+  have hQ'max : Q'.IsMaximal := Ideal.IsPrime.isMaximal hQ'prime (by
     intro h; apply hp_ne
     have : C₂.maximalIdealAt Q = Q'.under C₂.CoordinateRing := hQ'lies.over
     rw [this, h, Ideal.under_bot])
   obtain ⟨P', hP'⟩ := C₁.exists_smoothPoint_of_isMaximal hQ'max
   refine ⟨P', hP', ?_⟩
-  haveI hlies' : (C₁.maximalIdealAt P').LiesOver
+  have hlies' : (C₁.maximalIdealAt P').LiesOver
       (C₂.maximalIdealAt (toPointMap cd P')) :=
     maximalIdealAt_liesOver_toPointMap φ cd P'
   have h1 : C₂.maximalIdealAt (toPointMap cd P') =
@@ -1333,11 +1296,11 @@ private theorem affine_section_injOn_attach (Q : C₂.SmoothPoint) :
       (_ : b ∈ (IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing).attach),
       ProjectiveSmoothPoint.affine (g a.1 a.2) = ProjectiveSmoothPoint.affine (g b.1 b.2) →
         a = b := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   intro g hg a _ b _ hab
   simp only [ProjectiveSmoothPoint.affine.injEq] at hab
@@ -1373,11 +1336,11 @@ private theorem sum_filter_support_eq_sum_image_section
         ∑ x ∈ (IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q)
           C₁.CoordinateRing).attach.image
           (fun Q' ↦ ProjectiveSmoothPoint.affine (g Q'.1 Q'.2)), D x := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   intro g hg_ideal hg_Q
   have hp_ne : C₂.maximalIdealAt Q ≠ ⊥ := C₂.maximalIdealAt_ne_bot Q
@@ -1439,11 +1402,11 @@ private theorem sum_image_section_eq_sum_primesOver
           (C₁.projectiveDivisorOf (algebraMap C₁.CoordinateRing C₁.FunctionField w)) x) =
         ∑ Q' ∈ IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing,
           ((Associates.mk Q').count (Associates.mk (Ideal.span ({w} : Set _))).factors : ℤ) := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   intro g hg_ideal
   rw [Finset.sum_image (affine_section_injOn_attach φ cd Q g hg_ideal)]
@@ -1480,11 +1443,11 @@ private theorem pushforwardDivisorVal_projectiveDivisorOf_affine_eq_sum_fiber
       ∑ Q' ∈ IsDedekindDomain.primesOverFinset (C₂.maximalIdealAt Q) C₁.CoordinateRing,
         ((Associates.mk Q').count (Associates.mk (Ideal.span ({w} : Set _))).factors : ℤ) := by
   classical
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   -- Choose a section `pt` of the fibre `{primes over m_Q} → {smooth points P'}`, with
   -- `m_{pt Q'} = Q'` and `φ (pt Q') = Q`, via the existence lemma.
@@ -1532,11 +1495,11 @@ private theorem projectiveDivisorOf_pushforward_algebraMap_apply_affine
       φ.pushforwardDivisorVal cd (C₁.projectiveDivisorOf
         (algebraMap C₁.CoordinateRing C₁.FunctionField w)) (ProjectiveSmoothPoint.affine Q) := by
   classical
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing := cd.toAlgebra
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have hfin' : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     cd.module_finite
-  haveI htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
+  have htf : @Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing _ _ modCR :=
     isTorsionFree_coordHom φ cd
   have hnw : Algebra.intNorm C₂.CoordinateRing C₁.CoordinateRing w ≠ 0 :=
     intNorm_ne_zero_of_ne_zero φ cd w hw
@@ -1585,10 +1548,10 @@ theorem projectiveDivisorOf_pushforward_algebraMap_eq
 
 end NormConormSteps
 
-/-! ### NEW-1(ii): the `f = u/v` reduction
+/-! ### the `f = u/v` reduction
 
 The deep per-place arithmetic is the `algebraMap` case
-`projectiveDivisorOf_pushforward_algebraMap_eq` (in `section NormConormSteps`).
+`projectiveDivisorOf_pushforward_algebraMap_eq`.
 The norm–conorm identity for a *general* `f ∈ K(C₁)` follows from it by the
 multiplicativity of both sides together with `IsFractionRing.div_surjective`.
 The next four `private` helpers package that reduction:
@@ -1606,8 +1569,8 @@ private theorem projectiveDivisorOf_pushforward_zero (φ : CurveMap C₁ C₂)
       FiniteDimensional C₂.FunctionField C₁.FunctionField) :
     C₂.projectiveDivisorOf (φ.pushforward (0 : C₁.FunctionField)) =
       φ.pushforwardDivisorVal cd (C₁.projectiveDivisorOf (0 : C₁.FunctionField)) := by
-  letI algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  haveI : FiniteDimensional C₂.FunctionField C₁.FunctionField := hfd
+  let algFF : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  have : FiniteDimensional C₂.FunctionField C₁.FunctionField := hfd
   rw [show φ.pushforward (0 : C₁.FunctionField) = 0 from Algebra.norm_zero,
     C₂.projectiveDivisorOf_zero, C₁.projectiveDivisorOf_zero, map_zero]
 
@@ -1690,44 +1653,16 @@ private theorem projectiveDivisorOf_pushforward_eq_of_algebraMap (φ : CurveMap
       pushforwardDivisorVal_projectiveDivisorOf_eq_sub_of_mul_eq φ cd hf hav_ne hf_av,
       hau, hav, key u hu_ne, key v hv_ne]
 
-/-- **NEW-1(ii) — Silverman II.3.6, norm–conorm identity** `div(N_φ f) = φ_∗(div f)`.
-For a curve map `φ : C₁ → C₂` with coordinate-ring witness `cd` and a function
-`f ∈ K(C₁)`, the projective divisor of the conorm `N_φ f = φ.pushforward f`
-equals the valuation-theoretic pushforward of the projective divisor of `f`.
-
-The content is the per-place identity `ord_Q(N_φ f) = Σ_{P ↦ Q} f_{P/Q}·ord_P(f)`
-(with inertia degrees `f_{P/Q} = 1` over an algebraically closed field), proved
-via `Ideal.sum_ramification_inertia`, `Ideal.relNorm`, and `Algebra.intNorm`.
-
-The proof is the generalisation of the `F[X] → F[C]` machinery of
-`NormValuation.lean` (`count_relNorm_singleton_eq_sum_count_fiber`,
-`relNorm_maximalIdealAt`) to the *coordinate-ring extension* `F[C₂] → F[C₁]`
-induced by `cd`.  The instance-heavy steps are factored into the sub-lemmas of
-`section NormConormSteps`, each re-establishing the `cd`-induced algebra and its
-`Module.Finite`/`IsTorsionFree` structure internally (a `relNorm`/`intNorm`
-statement only typechecks with those instances in scope, so the sub-lemmas state
-them via `letI`/`haveI`-in-type and re-derive them in the body):
-* `finiteDimensional_functionField` — the finite extension `K(C₂) → K(C₁)`
-  (needed only for the `f = 0` branch below);
-* the degree/`finrank` coherence `finrank_{liftAlgebra} = φ.degree`
-  (`IsFractionRing.lift_unique` + `cd.compat`, so `relNorm_algebraMap` yields
-  `m_Q ^ φ.degree`), computed inside `relNorm_maximalIdealAt_eq`;
-* `relNorm_maximalIdealAt_eq` — the **`s = 1` core** `relNorm_{F[C₂]}(m_P) = m_{φP}`
-  from the global balance `relNorm(m_Q·F[C₁]) = m_Q^{φ.degree} =
-  ∏ relNorm(m_{P'})^{e_{P'}}` together with `Σ e_{P'}·f_{P'} = φ.degree`
-  (`sum_ramificationIdx_mul_inertiaDeg_eq_degree`) and `f_{P'} = 1`
-  (`inertiaDeg_maximalIdealAt_toPointMap`), forcing each exponent to 1;
-* `count_relNorm_eq_sum_fiber` — the affine count identity
-  `count_{m_Q}(relNorm (span{u})) = Σ_{Q' over m_Q} count_{Q'}(span{u})`;
-* `projectiveDivisorOf_pushforward_algebraMap_eq` — the `algebraMap` case, matching
-  the affine coefficients to the `mapDomain` fibre sum of `pushforwardDivisorVal`
-  via the bijection `{P : φP = Q} ≃ {primes over m_Q}` (`maximalIdealAt_toPointMap`
-  + `exists_smoothPoint_of_isMaximal`), the place at infinity being forced by
-  `projectiveDivisorOf_degree_eq_zero` (both projective divisors have degree `0`,
-  and `pushforwardDivisorVal` preserves degree).
-Here the general `f = u/v` reduces to `f = algebraMap u`, `u ∈ F[C₁]` nonzero, via
-`IsFractionRing.div_surjective` and the additivity of both sides
-(`projectiveDivisorOf_mul`, `pushforward_mul`, `pushforwardDivisorVal` a hom). -/
+/-- **Silverman II.3.6, norm–conorm identity** `div(N_φ f) = φ_∗(div f)`.
+
+For a curve map with a coordinate-ring witness over an algebraically closed field,
+the local norm formula is `ord_Q(N_φ f) = Σ_{P ↦ Q} ord_P(f)`, since the residue
+degrees are one. The prime-ideal norm formula follows from the fundamental
+ramification identity and the degree of the function-field extension. The
+bijection between points over `Q` and primes over its maximal ideal then gives
+the affine coefficients; degree zero determines the coefficient at infinity.
+The quotient representation `f = u/v` and additivity extend the identity from
+nonzero coordinate-ring elements to all rational functions. -/
 theorem projectiveDivisorOf_pushforward_eq_pushforwardDivisorVal [IsAlgClosed F]
     [IsDedekindDomain C₁.CoordinateRing] [IsDedekindDomain C₂.CoordinateRing]
     (φ : CurveMap C₁ C₂) (cd : φ.CoordHom)
@@ -1746,7 +1681,7 @@ theorem projectiveDivisorOf_pushforward_eq_pushforwardDivisorVal [IsAlgClosed F]
 
 end HasseWeil.Curves.CurveMap
 
-/-! ### NEW-1(iii)/(iv): compatibility + the gap `h_pres` -/
+/-! ### Compatibility of divisor pushforwards -/
 
 namespace HasseWeil.EC.Isogeny
 
@@ -1755,7 +1690,7 @@ open HasseWeil.Curves
 variable {F : Type*} [Field F] [DecidableEq F]
 variable {W₁ W₂ : Affine F} [W₁.IsElliptic] [W₂.IsElliptic]
 
-/-- **NEW-1(iii)**: the point-map pushforward `pushforwardProjectiveDivisor`
+/-- the point-map pushforward `pushforwardProjectiveDivisor`
 agrees with the valuation-theoretic pushforward `pushforwardDivisorVal`.  By
 construction the two `mapDomain` place-image maps coincide, so this is `rfl`
 up to the `(φP).toProjectiveSmoothPoint = affine (toPointMap cd P)` identity. -/
@@ -1776,10 +1711,10 @@ theorem pushforwardProjectiveDivisor_eq_pushforwardDivisorVal (φ : Isogeny W₁
     -- `affine (toPointMap cd P')` after the round-trip.
     rfl
 
-/-- **NEW-1(iv) / `h_pres` — Silverman II.3.7**: the pushforward `φ_∗` carries
+/-- **Silverman II.3.7**: the pushforward `φ_∗` carries
 principal projective divisors to principal ones.  This is the sole deep input to
 Silverman III.4.8 (`EC.Isogeny.addHomProperty`); it falls out of the norm–conorm
-identity (NEW-1 ii) since the pushforward of `div f` is `div(N_φ f)`, again
+identity since the pushforward of `div f` is `div(N_φ f)`, again
 principal. -/
 theorem pushforward_preserves_principal [IsAlgClosed F]
     [IsDedekindDomain (⟨W₁⟩ : SmoothPlaneCurve F).CoordinateRing]
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/AFConditional.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/AFConditional.lean
index d56788426..bb5320f8d 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/AFConditional.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/AFConditional.lean
@@ -3,27 +3,16 @@ import HasseWeil.Foundation.Curves.Ramification.PoleOrderParity
 import HasseWeil.Isogeny.GroupHom.PicZero
 
 /-!
-# AF unified package: conditional witnesses
+# Divisor witnesses for additivity of isogenies
 
-Wires together the shipped infrastructure into the witnesses needed by
-`AddHomProperty_of_picZero_witnesses` (the universal Silverman III.4.8
-theorem). Each witness is parametrized over the remaining outstanding
-hypotheses, making the dependency chain explicit:
+The reduction `D ~ (σD) − (O)` of degree-zero divisors identifies their
+Picard classes with point classes. Combined with the assertion that
+`(P) − (O)` is principal only when `P = O`, it shows that the sum of
+points vanishes on principal divisors. These two identities give the
+Picard-group witnesses for additivity of an isogeny whose pushforward
+preserves principal divisors.
 
-* `h_inj` (`κ ∘ σ̄ = id`): `DivZeroReduce W` (= reduction lemma).
-* `h_van` (`σ` vanishes on principal): `DivZeroReduce W` AND a
-  `point_minus_O_principal_eq_zero` over arbitrary functions
-  (= the no-finite-poles bridge).
-
-Once `DivZeroReduce W` lands (which itself decomposes into the
-list-induction `effective_sum_reduce` + Finsupp-to-list bridge for
-multiplicities) and the no-finite-poles bridge, the AF unified package
-fully closes B-4-003 conditionally on Miller (the geometric chord/tangent
-construction, the only remaining genuinely-new mathematical piece).
-
-## References
-
-* `T-PIC-AF-UNIFIED.md` for the full plan.
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.3–III.4.
 -/
 
 open WeierstrassCurve
@@ -33,19 +22,14 @@ namespace HasseWeil.Curves
 variable {F : Type*} [Field F] [DecidableEq F]
   (W : Affine F) [W.IsElliptic]
 
-/-- The full divisor reduction property: every degree-zero divisor is
-linearly equivalent to `(σD) − (O)`. The proof requires Miller (via
-`effective_sum_reduce`) plus a Finsupp-to-list bridge for handling
-multiplicities. Both are outstanding tickets. -/
+/-- Every degree-zero divisor is linearly equivalent to `(σD) − (O)`,
+where `σD` is its sum of points with multiplicity. -/
 def DivZeroReduce : Prop :=
   ∀ D : ProjectiveDivisor.degZero (⟨W⟩ : SmoothPlaneCurve F),
     SmoothPlaneCurve.ProjLinearlyEquiv (⟨W⟩ : SmoothPlaneCurve F)
       D.val (kappaDivisor W (projectiveDivisorSum W D.val))
 
-/-- The unconditional `point_minus_O` property: if `(P) − (O)` is
-principal, then `P = 0`. The proof requires the parity lemma (shipped
-as `point_minus_O_principal_eq_zero_of_coord`) plus the no-finite-poles
-→ CR-image bridge (outstanding ticket: `T-PIC-AF-UNIFIED.md` piece (a)). -/
+/-- A point divisor `(P) − (O)` is principal only when `P = O`. -/
 def PointMinusOPrincipalEqZero : Prop :=
   ∀ P : W.Point,
     SmoothPlaneCurve.ProjIsPrincipal (⟨W⟩ : SmoothPlaneCurve F)
@@ -112,11 +96,8 @@ theorem h_van_degZero_of_divZeroReduce_and_pointMinusO
 
 /-! ### `PointMinusOPrincipalEqZero` from the no-finite-poles bridge -/
 
-/-- The no-finite-poles → CR-image bridge: any nonzero function with all
-local orders nonneg lies in the coordinate-ring image. The proof composes
-worker-I's `pointValuation_algebraMap_le_one`, `smoothPointEquivMaxIdeal`
-(under `[IsAlgClosed F] [IsElliptic]`), and mathlib's
-`mem_coordinateRing_of_valuation_le_one`. ~80-150 LOC follow-up ticket. -/
+/-- A nonzero function with nonnegative order at every affine point
+lies in the image of the coordinate ring. -/
 def NoFinitePolesBridge : Prop :=
   ∀ (f : (⟨W⟩ : SmoothPlaneCurve F).FunctionField), f ≠ 0 →
     (∀ P : (⟨W⟩ : SmoothPlaneCurve F).SmoothPoint,
@@ -151,9 +132,9 @@ theorem pointMinusO_of_bridge
       ((ProjectiveSmoothPoint.infinity : ProjectiveSmoothPoint
         (⟨W⟩ : SmoothPlaneCurve F))) ≠ ProjectiveSmoothPoint.affine Q := by
     nofun
-  rw [if_neg h_inf_ne, sub_zero] at h_eq
+  rw [ite_eq_right h_inf_ne, sub_zero] at h_eq
   by_cases h_eq_pt : P.toProjectiveSmoothPoint = ProjectiveSmoothPoint.affine Q
-  · rw [if_pos h_eq_pt] at h_eq
+  · rw [ite_eq_left h_eq_pt] at h_eq
     cases h_top : (⟨W⟩ : SmoothPlaneCurve F).ord_P Q f with
     | top =>
       rw [h_top, WithTop.untopD_top] at h_eq
@@ -163,7 +144,7 @@ theorem pointMinusO_of_bridge
       have hn : n = 1 := h_eq.symm
       subst hn
       exact_mod_cast (by decide : (0 : ℤ) ≤ 1)
-  · rw [if_neg h_eq_pt] at h_eq
+  · rw [ite_eq_right h_eq_pt] at h_eq
     cases h_top : (⟨W⟩ : SmoothPlaneCurve F).ord_P Q f with
     | top =>
       exact absurd (((⟨W⟩ : SmoothPlaneCurve F).ord_P_eq_top_iff (P := Q) f).mp h_top) hf_ne
@@ -174,17 +155,10 @@ theorem pointMinusO_of_bridge
 
 /-! ### Bundled AFInputs structure -/
 
-/-- The three input Props needed for the AF unified package on a single
-elliptic curve `W`:
-
-* `miller`: chord/tangent geometric identity at the divisor level.
-* `divZeroReduce`: combinatorial reduction `D ~ (σD) − (O)` for D ∈ Div⁰.
-* `noFinitePolesBridge`: the algebraic bridge from "no finite poles" to
-  "in coordinate-ring image".
-
-Each is its own outstanding ticket; once all three are discharged for a
-specific `W`, both witnesses needed by `AddHomProperty_of_picZero_witnesses`
-(restricted to that curve's side of the isogeny) follow immediately. -/
+/-- The divisor identities needed to identify an elliptic curve with
+its degree-zero Picard group: the Miller relation, reduction of
+degree-zero divisors to point divisors, and the coordinate-ring
+characterization of functions with no finite poles. -/
 structure AFInputs (W : Affine F) [W.IsElliptic] where
   miller : MillerHypothesis W
   divZeroReduce : DivZeroReduce W
@@ -217,12 +191,9 @@ theorem h_van_degZero {W : Affine F} [W.IsElliptic] (a : AFInputs W)
 
 end AFInputs
 
-/-! ### Final B-4-003 wrapper -/
+/-! ### Additivity from divisor witnesses -/
 
-/-- The "principal divisors lie in degZero" predicate. Proved by
-worker-K's T-II-3-009 (`projectiveDivisorOf_degree_zero`) under
-`[IsAlgClosed F]`. We take it as a hypothesis here to keep the wrapper
-conditional. -/
+/-- Principal projective divisors have degree zero. -/
 def PrincipalImpliesDegZero (W : Affine F) [W.IsElliptic] : Prop :=
   ∀ D : ProjectiveDivisor (⟨W⟩ : SmoothPlaneCurve F),
     D ∈ (⟨W⟩ : SmoothPlaneCurve F).projPrincipalSubgroup →
@@ -230,8 +201,8 @@ def PrincipalImpliesDegZero (W : Affine F) [W.IsElliptic] : Prop :=
 
 namespace AFInputs
 
-/-- The full h_van witness, given `AFInputs` + the principal-implies-degZero
-bridge (worker-K's T-II-3-009). -/
+/-- The sum of points vanishes on principal divisors, given the divisor
+identities in `AFInputs` and the degree-zero property of principal divisors. -/
 theorem h_van {W : Affine F} [W.IsElliptic] (a : AFInputs W)
     (h_pdz : PrincipalImpliesDegZero W)
     (D : ProjectiveDivisor (⟨W⟩ : SmoothPlaneCurve F))
@@ -241,20 +212,9 @@ theorem h_van {W : Affine F} [W.IsElliptic] (a : AFInputs W)
 
 end AFInputs
 
-/-- **Final conditional B-4-003** for an isogeny: given `AFInputs` for
-both curves, the principal-degZero bridge for both curves, and the
-pushforward-preserves-principal hypothesis (T-PIC-C-003), the universal
-hom property holds.
-
-This closes B-4-003 over `[IsAlgClosed F]` modulo:
-- `MillerHypothesis` for both W₁ and W₂ (geometric chord/tangent — the BIG remaining piece)
-- `DivZeroReduce` for both W₁ and W₂ (Finsupp decomposition + miller corollaries)
-- `NoFinitePolesBridge` for both W₁ and W₂ (valuation bridge)
-- `PrincipalImpliesDegZero` for both W₁ and W₂ (worker-K's T-II-3-009)
-- `h_pres` (T-PIC-C-003 = norm-divisor identity)
-
-All the structural plumbing is here; the remaining work is to prove
-each of these Props for elliptic curves. -/
+/-- An isogeny is additive given the divisor identities on both curves,
+the degree-zero property of principal divisors, and preservation of
+principal divisors by its pushforward. See Silverman III.4.8. -/
 theorem AddHomProperty_of_AFInputs
     {W₁ W₂ : Affine F} [W₁.IsElliptic] [W₂.IsElliptic]
     (φ : HasseWeil.EC.Isogeny W₁ W₂)
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/GenericFiber.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/GenericFiber.lean
index 7a21963e5..91c992b49 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/GenericFiber.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Fiber/GenericFiber.lean
@@ -12,38 +12,13 @@ import Mathlib.RingTheory.DedekindDomain.Different
 import Mathlib.RingTheory.DedekindDomain.Factorization
 
 /-!
-# Generic fiber cardinality (T-II-2-009, Silverman II.2.6(b))
+For a coordinate-ring morphism, if a nonzero prime is unramified and all residue
+field degrees above it are one, its number of primes above equals the
+function-field degree. For a separable morphism this is the separable degree.
+The algebraic support also identifies finite ramification loci through the
+different ideal and constructs smooth points over algebraically closed fields.
 
-For a nonconstant morphism `φ : C₁ → C₂` of smooth curves, for almost all
-points `Q ∈ C₂`, the fiber `φ⁻¹(Q)` has cardinality equal to the
-separable degree `deg_s(φ)`.
-
-This file provides the **algebraic-geometric content** of T-II-2-009 at
-the Dedekind-domain level, building on the `CurveMap` + `CoordHom`
-infrastructure: if we can exhibit a maximal ideal `p ⊂ C₂.CoordinateRing`
-that is **unramified** and has **trivial residue-field degrees**, then
-the number of primes above it equals the function-field degree `deg(φ)`
-(which coincides with `deg_s(φ)` for separable `φ`).
-
-This is one side of Silverman's II.2.6(b): the fibre-cardinality statement
-in terms of the discrete-prime count. The other direction — **exhibiting**
-such an unramified prime `p` — requires the primitive-element-plus-
-discriminant construction over a base that sees "almost all" `Q` (i.e.
-the unramified locus is a dense open). Over arbitrary `F`, this is the
-main piece still missing; see the progress note at the end.
-
-## Main results
-
-* `CurveMap.primesOverFinset_card_eq_degree_of_unramified` — given an
-  unramified prime `p` with every residue-field degree `f_P = 1`, the
-  number of primes above `p` equals `φ.degree`.
-* `CurveMap.primesOverFinset_card_eq_sepDegree_of_separable_and_unramified`
-  — same conclusion in terms of `sepDegree` under the separability
-  hypothesis (which forces `deg = sepDeg`).
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], II.2.6(b).
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.2.6(b).
 -/
 
 namespace HasseWeil.Curves
@@ -52,12 +27,6 @@ namespace CurveMap
 
 variable {F : Type*} [Field F] {C₁ C₂ : SmoothPlaneCurve F}
 
-/-- **T-II-2-009, algebraic direction** (unramified + trivial residue
-degrees ⇒ fiber count = degree). Given a `CurveMap` with `CoordHom`
-witness and a maximal ideal `p ⊂ C₂.CoordinateRing` that is
-(i) non-zero, (ii) has every ramification index = 1 on the primes above it,
-and (iii) has every inertia (residue) degree = 1, the number of primes
-above `p` equals `φ.degree`. -/
 theorem primesOverFinset_card_eq_degree_of_unramified
     [IsIntegrallyClosed C₂.CoordinateRing]
     [IsIntegrallyClosed C₁.CoordinateRing]
@@ -67,10 +36,10 @@ theorem primesOverFinset_card_eq_degree_of_unramified
       letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
       ∀ P ∈ IsDedekindDomain.primesOverFinset p C₁.CoordinateRing,
         Ideal.ramificationIdx' p P *
-        Ideal.inertiaDeg' p P = 1) :
+        P.inertiaDeg C₂.CoordinateRing = 1) :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
     (IsDedekindDomain.primesOverFinset p C₁.CoordinateRing).card = φ.degree := by
-  letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
+  let : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
   have hsum := φ.sum_ramificationIdx_mul_inertiaDeg_eq_degree
     coordHom coordHom.module_finite hpMax hp0
   -- Σ_{P ∈ S} (e · f) = deg. With each e·f = 1, LHS = S.card.
@@ -80,10 +49,6 @@ theorem primesOverFinset_card_eq_degree_of_unramified
   rw [Finset.sum_const, Nat.smul_one_eq_cast] at hsum'
   exact_mod_cast hsum'
 
-/-- **T-II-2-009 (separable case)**: for a separable `CurveMap` with an
-unramified + trivial-residue-degree prime, the fiber count equals
-`sepDegree`. Uses the fact that `sepDegree = degree` for separable
-extensions (via `Field.finSepDegree_eq_finrank_iff`). -/
 theorem primesOverFinset_card_eq_sepDegree_of_separable_and_unramified
     [IsIntegrallyClosed C₂.CoordinateRing]
     [IsIntegrallyClosed C₁.CoordinateRing]
@@ -94,11 +59,11 @@ theorem primesOverFinset_card_eq_sepDegree_of_separable_and_unramified
       letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
       ∀ P ∈ IsDedekindDomain.primesOverFinset p C₁.CoordinateRing,
         Ideal.ramificationIdx' p P *
-        Ideal.inertiaDeg' p P = 1) :
+        P.inertiaDeg C₂.CoordinateRing = 1) :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
     (IsDedekindDomain.primesOverFinset p C₁.CoordinateRing).card = φ.separableDegree := by
   -- separable + finite-dim ⇒ sepDegree = degree.
-  letI : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  let : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
   have hsep_eq : φ.degree = φ.separableDegree := by
     change φ.degree = @Field.finSepDegree _ _ _ _ φ.toAlgebra
     have hinsep : φ.inseparableDegree = 1 := hsep
@@ -112,17 +77,6 @@ theorem primesOverFinset_card_eq_sepDegree_of_separable_and_unramified
   exact φ.primesOverFinset_card_eq_degree_of_unramified coordHom
     hpMax hp0 h_ef_one
 
-/-! ### Piece 1 — bad locus `{p : p ∣ differentIdeal}` is finite
-
-For a finite separable extension of Dedekind domains `A → B`, the set of
-maximal ideals `p ⊂ A` that ramify (i.e. divide `differentIdeal A B`)
-is **finite**. Direct consequence of `differentIdeal_ne_bot` +
-`Ideal.finite_factors`. -/
-
-/-- **T-II-2-009 Piece 1**: the set of height-one primes of `B` that
-divide `differentIdeal A B` is finite. These are the "ramified primes"
-in the `A → B` extension (primes above which at least one prime of `B`
-has `e_P ≥ 2`). Silverman II.2.6(b) "finitely many bad Q". -/
 theorem _root_.IsDedekindDomain.finite_ramified_primes
     {A : Type*} [CommRing A] [IsDedekindDomain A]
     {B : Type*} [CommRing B] [IsDedekindDomain B]
@@ -132,20 +86,13 @@ theorem _root_.IsDedekindDomain.finite_ramified_primes
       (FractionRing.liftAlgebra A (FractionRing B))) :
     {P : IsDedekindDomain.HeightOneSpectrum B |
       P.asIdeal ∣ differentIdeal A B}.Finite := by
-  letI : Algebra (FractionRing A) (FractionRing B) :=
+  let : Algebra (FractionRing A) (FractionRing B) :=
     FractionRing.liftAlgebra A (FractionRing B)
-  haveI : IsScalarTower A (FractionRing A) (FractionRing B) :=
+  have : IsScalarTower A (FractionRing A) (FractionRing B) :=
     FractionRing.isScalarTower_liftAlgebra A (FractionRing B)
-  haveI := hsep
+  have := hsep
   exact Ideal.finite_factors (differentIdeal_ne_bot (A := A) (B := B))
 
-/-! ### Piece 2 — `¬ P ∣ differentIdeal ⇒ IsUnramifiedAt A P`
-
-Direct corollary of mathlib's `not_dvd_differentIdeal_iff`. Packages the
-easier direction as a standalone lemma for use in Pieces 3–5. -/
-
-/-- **T-II-2-009 Piece 2**: a prime of `B` that does not divide the
-different ideal is unramified over `A`. -/
 theorem _root_.IsDedekindDomain.isUnramifiedAt_of_not_dvd_differentIdeal
     {A : Type*} [CommRing A] [IsDedekindDomain A]
     {B : Type*} [CommRing B] [IsDedekindDomain B]
@@ -155,50 +102,27 @@ theorem _root_.IsDedekindDomain.isUnramifiedAt_of_not_dvd_differentIdeal
       (FractionRing.liftAlgebra A (FractionRing B)))
     {P : Ideal B} [P.IsPrime] (hnd : ¬ P ∣ differentIdeal A B) :
     Algebra.IsUnramifiedAt A P := by
-  letI : Algebra (FractionRing A) (FractionRing B) :=
+  let : Algebra (FractionRing A) (FractionRing B) :=
     FractionRing.liftAlgebra A (FractionRing B)
-  haveI : IsScalarTower A (FractionRing A) (FractionRing B) :=
+  have : IsScalarTower A (FractionRing A) (FractionRing B) :=
     FractionRing.isScalarTower_liftAlgebra A (FractionRing B)
-  haveI := hsep
+  have := hsep
   exact not_dvd_differentIdeal_iff.mp hnd
 
-/-! ### Piece 3 — `IsUnramifiedAt P ⇒ e_{P|A} = 1`
-
-Corollary of mathlib's `Ideal.ramificationIdx_eq_one_of_isUnramifiedAt`.
-Packages the ramification-index side of unramifiedness; the inertia-degree
-side (`f_P = 1`) requires the residue field to be trivial over the base,
-which is **automatic over algebraically closed `F`** but must be supplied
-externally over arbitrary base fields. -/
-
-/-- **T-II-2-009 Piece 3 (ramification half)**: a prime `P` of `B` that is
-unramified over `A` has ramification index `1` (at its image in `A`).
-Direct corollary of mathlib's `Ideal.ramificationIdx_eq_one_of_isUnramifiedAt`. -/
 theorem _root_.IsDedekindDomain.ramificationIdx_eq_one_of_isUnramifiedAt_of_ne_bot
     {A : Type*} [CommRing A] [IsDomain A]
     {B : Type*} [CommRing B] [IsDomain B] [IsNoetherianRing B]
     [Algebra A B] [Algebra.EssFiniteType A B]
     {P : Ideal B} [P.IsPrime] [Algebra.IsUnramifiedAt A P] (hP : P ≠ ⊥) :
     Ideal.ramificationIdx' (P.under A) P = 1 := by
-  haveI : P.LiesOver (P.under A) := ⟨rfl⟩
-  letI := Localization.AtPrime.algebraOfLiesOver (P.under A) P
+  have : P.LiesOver (P.under A) := ⟨rfl⟩
+  let := Localization.AtPrime.algebraOfLiesOver (P.under A) P
   have hmap : (P.under A).map (algebraMap A (Localization.AtPrime P)) =
       IsLocalRing.maximalIdeal _ :=
     ((Algebra.isUnramifiedAt_iff_map_eq A (P.under A) P).mp ‹_›).2
   exact Ideal.ramificationIdx'_eq_one_of_map_localization Ideal.map_comap_le hP
     (Ideal.primeCompl_le_nonZeroDivisors P) hmap
 
-/-! ### Piece 4 — existence of an unramified prime
-
-Given Piece 1 (bad locus finite) and the assumption that `B` has
-infinitely many height-one primes (a genuine curve-theory input, since a
-Dedekind domain of infinite F-dimension has infinite spectrum), there
-exists a prime `P` of `B` outside the ramified locus. Piece 2 then turns
-"not in bad locus" into `IsUnramifiedAt A P`. -/
-
-/-- **T-II-2-009 Piece 4**: combining Piece 1 (bad locus finite) + Piece 2
-(not-in-bad-locus ⇒ IsUnramifiedAt), plus the hypothesis that `B` has
-infinitely many height-one primes, we extract a specific prime
-`P : HeightOneSpectrum B` that is unramified over `A`. -/
 theorem _root_.IsDedekindDomain.exists_unramified_prime
     {A : Type*} [CommRing A] [IsDedekindDomain A]
     {B : Type*} [CommRing B] [IsDedekindDomain B]
@@ -229,30 +153,9 @@ theorem _root_.IsDedekindDomain.exists_unramified_prime
     exact hfin_bad.subset this
   obtain ⟨P, hP⟩ := hgood_nonempty
   refine ⟨P, ?_⟩
-  haveI : P.asIdeal.IsPrime := P.isPrime
+  have : P.asIdeal.IsPrime := P.isPrime
   exact IsDedekindDomain.isUnramifiedAt_of_not_dvd_differentIdeal hsep hP
 
-/-! ### Piece 5 — composed theorem at the abstract Dedekind level
-
-Combining Pieces 1–4 yields: for a finite separable extension of Dedekind
-domains `A → B` with `B` having infinitely many height-one primes, there
-exists a specific prime `P` of `B` for which the ramification index is
-exactly `1`.
-
-Specialised to `CurveMap + CoordHom` + `[IsAlgClosed F]` (where residue
-field degrees are automatically `1`), this gives `(primesOverFinset).card
-= φ.degree` — the full Silverman II.2.6(b) content. -/
-
-/-- **T-II-2-009 Piece 5 (composed at A → B level)**: assembles Pieces
-1–4 into a single witness. For `A → B` Dedekind + finite separable +
-`B` infinite in primes, there exists a `P ∈ HeightOneSpectrum B`,
-nonzero, with `ramificationIdx' (P.under A → P) = 1`.
-
-The inertia degree `f_P = 1` is NOT concluded here — it depends on the
-residue-field extension being trivial, which is automatic over
-algebraically closed base (or for F-rational smooth points), and
-supplied as a separate hypothesis when needed (see
-`primesOverFinset_card_eq_degree_of_unramified`). -/
 theorem _root_.IsDedekindDomain.exists_unramifiedPrime_ramificationIdx_eq_one
     {A : Type*} [CommRing A] [IsDedekindDomain A]
     {B : Type*} [CommRing B] [IsDedekindDomain B]
@@ -265,31 +168,17 @@ theorem _root_.IsDedekindDomain.exists_unramifiedPrime_ramificationIdx_eq_one
     ∃ P : IsDedekindDomain.HeightOneSpectrum B,
       Ideal.ramificationIdx' (P.asIdeal.under A) P.asIdeal = 1 := by
   obtain ⟨P, hunram⟩ := IsDedekindDomain.exists_unramified_prime hsep hinf
-  haveI : P.asIdeal.IsPrime := P.isPrime
-  haveI := hunram
+  have : P.asIdeal.IsPrime := P.isPrime
+  have := hunram
   refine ⟨P, ?_⟩
   exact IsDedekindDomain.ramificationIdx_eq_one_of_isUnramifiedAt_of_ne_bot P.ne_bot
 
 end CurveMap
 
-/-! ### Piece 6 — HeightOneSpectrum infinite over alg-closed base
-
-Over an algebraically closed base `F` with `[IsElliptic]`, the coordinate
-ring `C.CoordinateRing` has infinitely many height-one primes. The proof
-goes via the `smoothPointEquivHeightOneSpectrum` bijection
-(in `HasseWeil/Curves/SmoothPointPrime.lean`), reducing to showing
-`C.SmoothPoint` is infinite. This follows because for each `x ∈ F`, the
-Weierstrass equation in `y` is a quadratic over an algebraically closed
-field and hence has a root — giving an injection `F → C.SmoothPoint`. -/
-
 namespace SmoothPlaneCurve
 
 variable {F : Type*} [Field F]
 
-/-- **T-II-2-009 Piece 6 (smooth points)**: for every `x ∈ F` over an
-algebraically closed field, there exists `y : F` with `(x, y)` a
-nonsingular point on the elliptic curve (any point on an elliptic curve
-is automatically nonsingular since `Δ ≠ 0`). -/
 theorem exists_smoothPoint_of_x
     [IsAlgClosed F] (C : SmoothPlaneCurve F) [C.toAffine.IsElliptic] (x : F) :
     ∃ P : C.SmoothPoint, P.x = x := by
@@ -320,9 +209,6 @@ theorem exists_smoothPoint_of_x
       (W := C.toAffine)).mp heq
   exact ⟨⟨x, y, hns⟩, rfl⟩
 
-/-- **T-II-2-009 Piece 6**: `C.SmoothPoint` is infinite over an
-algebraically closed base. Via the injection `x ↦ (smooth point with
-that x-coordinate)`. -/
 theorem smoothPoint_infinite
     [IsAlgClosed F] (C : SmoothPlaneCurve F) [C.toAffine.IsElliptic] :
     Infinite C.SmoothPoint := by
@@ -338,17 +224,13 @@ theorem smoothPoint_infinite
   rw [h₁, h₂] at this
   exact this
 
-/-- **T-II-2-009 Piece 6 (main)**: the height-one spectrum of
-`C.CoordinateRing` is infinite over an algebraically closed elliptic
-curve, via the bijection `SmoothPoint ≃ HeightOneSpectrum` from
-`HasseWeil/Curves/SmoothPointPrime.lean`. -/
 theorem heightOneSpectrum_infinite
     [IsAlgClosed F] (C : SmoothPlaneCurve F) [C.toAffine.IsElliptic]
     [IsIntegrallyClosed C.CoordinateRing] :
     Set.Infinite ({_P : IsDedekindDomain.HeightOneSpectrum C.CoordinateRing |
       True}) := by
-  haveI : Infinite C.SmoothPoint := C.smoothPoint_infinite
-  haveI : Infinite (IsDedekindDomain.HeightOneSpectrum C.CoordinateRing) :=
+  have : Infinite C.SmoothPoint := C.smoothPoint_infinite
+  have : Infinite (IsDedekindDomain.HeightOneSpectrum C.CoordinateRing) :=
     Infinite.of_injective C.smoothPointEquivHeightOneSpectrum
       C.smoothPointEquivHeightOneSpectrum.injective
   exact Set.infinite_univ
@@ -358,52 +240,11 @@ end SmoothPlaneCurve
 namespace CurveMap
 variable {F : Type*} [Field F] {C₁ C₂ : SmoothPlaneCurve F}
 
-/-! ### Piece 7 — residue fields at smooth points are `F`
-
-The residue-field-degree-one content (f_P = 1) of T-II-2-009 Piece 7.
-Rather than prove `inertiaDeg' = 1` directly — which hits a diamond
-between `Module.Free` from `Ideal.Quotient` vs `DivisionRing` paths —
-we package the residue-field F-rank: `finrank F (C.CoordinateRing /
-maximalIdealAt P) = 1`. Any consumer wanting `inertiaDeg' = 1` over an
-F-rational SmoothPoint combines this (for source + target) with the
-tower `finrank_mul_finrank`; the direct
-`inertiaDeg_maximalIdealAt` covers the specific `F[X] → F[C]` case. -/
-
-/-- **T-II-2-009 Piece 7 (residue field)**: the residue field of
-`C.CoordinateRing` at a smooth point `P` equals `F` (as `F`-module of
-rank 1). Direct re-export of `finrank_quotientMaximalIdealAt`
-for use in the Piece 8 chain. -/
 theorem finrank_quotientMaximalIdealAt_eq_one
     (C : SmoothPlaneCurve F) (P : C.SmoothPoint) :
     Module.finrank F (C.CoordinateRing ⧸ C.maximalIdealAt P) = 1 :=
   C.finrank_quotientMaximalIdealAt P
 
-/-! ### Piece 8 — final assembly: T-II-2-009 for CurveMap
-
-Combines Pieces 1–7: given a CurveMap with CoordHom, separable,
-and ([IsAlgClosed F] providing infinite spectrum + residue-degree=1),
-there's a maximal ideal of `C₂.CoordinateRing` whose fiber (primes above
-it in `C₁.CoordinateRing`) has cardinality equal to `φ.separableDegree`.
-
-The `f_P = 1` side of the inertia
-computation is supplied as a **witness hypothesis** (`h_inertia_one`)
-rather than proven in-line, due to the `Module.Free` diamond between
-`Ideal.Quotient.semiring` and `DivisionRing.toDivisionSemiring.toSemiring`
-paths. Under `[IsAlgClosed F]`, the hypothesis is always satisfied
-(both residue fields are `F`); the caller supplies it via
-`SmoothPlaneCurve.finrank_quotientMaximalIdealAt_eq_one` (Piece 7) for
-each prime in their specific setting. -/
-
-/-- **T-II-2-009 Piece 8 (full assembly, witness form)**: given the
-witness hypotheses for unramified existence + trivial residue degrees,
-produce a prime of `C₂.CoordinateRing` whose fiber in `C₁.CoordinateRing`
-has cardinality `φ.separableDegree`.
-
-The existence of an unramified Q (via Pieces 1–6 for alg-closed base) is
-bundled as `h_unramified_Q`. The trivial residue degree is bundled as
-`h_inertia_one` (follows from Piece 7 over [IsAlgClosed F] via the
-`maximalIdealAt` ↔ `HeightOneSpectrum` bijection from
-`smoothPointEquivHeightOneSpectrum`). -/
 theorem exists_heightOneSpectrum_fiber_card_eq_sepDegree
     [IsIntegrallyClosed C₂.CoordinateRing]
     [IsIntegrallyClosed C₁.CoordinateRing]
@@ -414,94 +255,15 @@ theorem exists_heightOneSpectrum_fiber_card_eq_sepDegree
       letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
       ∀ P ∈ IsDedekindDomain.primesOverFinset Q.asIdeal C₁.CoordinateRing,
         Ideal.ramificationIdx' Q.asIdeal P *
-        Ideal.inertiaDeg' Q.asIdeal P = 1) :
+        P.inertiaDeg C₂.CoordinateRing = 1) :
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
     (IsDedekindDomain.primesOverFinset Q.asIdeal C₁.CoordinateRing).card = φ.separableDegree := by
-  haveI : Q.asIdeal.IsPrime := Q.isPrime
-  haveI hQmax : Q.asIdeal.IsMaximal :=
+  have : Q.asIdeal.IsPrime := Q.isPrime
+  have hQmax : Q.asIdeal.IsMaximal :=
     Q.isPrime.isMaximal Q.ne_bot
   exact φ.primesOverFinset_card_eq_sepDegree_of_separable_and_unramified
     coordHom hsep hQmax Q.ne_bot h_ef_one_Q
 
-/-! ### Piece 9 — `inertiaDeg' = 1` at smooth points: diamond-blocked
-
-**Status**: blocked by the `Module.Free (C₂.CR/Q) (C₁.CR/P)` typeclass
-diamond. The attempted workaround — using `F` as outer ring in
-`Module.finrank_mul_finrank F _ _` — does **not** bypass the problem,
-because `finrank_mul_finrank` still requires `Module.Free` on the
-intermediate extension. Specifically:
-
-- `Module.Free.of_divisionRing` produces `Module.Free` via
-  `DivisionRing.toDivisionSemiring.toSemiring`.
-- The expected instance on `(C₂.CR/Q)`-module `(C₁.CR/P)` goes through
-  `Ideal.Quotient.semiring` and `Algebra.toModule`.
-- These two `Semiring` parent-paths do not unify — the same `Module.Free`
-  diamond between the two `Semiring` structures.
-
-The narrower witness form (`exists_heightOneSpectrum_fiber_card_eq_sepDegree`,
-Piece 8 above) takes the `e_P · f_P = 1` hypothesis as input and is the
-shippable deliverable. Closing `inertiaDeg' = 1` without the diamond
-requires either:
-
-1. A mathlib-level change to make `Ideal.Quotient.semiring` and
-   `Field`-derived `Semiring` agree definitionally, OR
-2. A residue-field algebra isomorphism `C.CR/(maximalIdealAt P) ≃ₐ[F] F`
-   (under `[IsAlgClosed F]`), expressed without passing through
-   `Module.Free` synthesis on the intermediate quotient, OR
-3. A direct computation of `Ideal.inertiaDeg'` from the algebra-map
-   surjectivity at smooth points on an algebraically closed base.
-
-The second route is the most promising — a dedicated file
-`HasseWeil/Curves/ResidueFieldAtSmoothPoint.lean` building an explicit
-`AlgEquiv` at each smooth point and then transporting `inertiaDeg'`
-through it would sidestep `finrank_mul_finrank` entirely. Estimated
-~100 LOC. Deferred. -/
-
-/-! ### Progress note — the remaining piece (existence of unramified p)
-
-The full T-II-2-009 statement **`∃ Q, #φ⁻¹(Q) = sepDeg(φ)`** reduces (via
-the theorems above) to exhibiting a maximal `p ⊂ C₂.CoordinateRing` with
-the ramification + inertia witnesses. Over an algebraically-closed base
-field `F`, such a `p` always exists because:
-
-1. **Primitive element**: the separable closure `L ⊂ C₁.FunctionField`
-   of `φ*(C₂.FunctionField)` is generated by a primitive element `α`
-   with minimal polynomial `m(T) ∈ (φ*C₂.FunctionField)[T]` of degree
-   `sepDeg(φ)`.
-
-2. **Discriminant**: `m` is separable (primitive-element condition), so
-   `discriminant(m)` is a nonzero element of `φ*C₂.FunctionField`.
-   Viewed in `C₂.FunctionField`, it has finitely many zeros (as an element
-   of a function field with a degree structure).
-
-3. **Dense open**: at any `p ⊂ C₂.CoordinateRing` outside the zero locus
-   of `discriminant(m)`, the specialisation `m̄ ∈ (C₂.CoordinateRing/p)[T]`
-   has `sepDeg(φ)` distinct roots in `C₁.CoordinateRing/P` for each `P`
-   above `p`. This gives the ramification + inertia witnesses.
-
-The formalisation gap:
-- Mathlib has `Field.exists_primitive_element` (step 1) and
-  `Polynomial.discriminant_ne_zero_of_separable` (step 2).
-- Step 3 requires the algebraic-geometric bridge between "discriminant
-  nonzero at `p`" and "unramified at `p` with trivial residue
-  extensions". For Dedekind extensions this is
-  `Algebra.IsUnramifiedAt ↔ ¬ p ∣ differentIdeal`, and over alg-closed
-  base the residue fields all equal `F`, making `inertiaDeg' = 1`
-  automatic.
-
-This last step is ~200 lines of algebraic-geometric infrastructure
-following Silverman + Neukirch (*Algebraic Number Theory*, III.2). The
-algebraic core (steps 1–2) is immediate in mathlib; step 3 is the
-substantive remaining content. Tracked under T-II-2-009's progress log.
-
-The **alternative path** `#ker β_pc = sepDeg β_pc` (for the specific
-β_pc = 1 − π) can be closed *without* the full T-II-2-009 once
-`AdditionPullback.lean` replaces the `isogOneSub` placeholder — that
-gives the real function-field pullback, from which `deg β_pc =
-[K(E) : φ*K(E)] = q + 1 − t` follows via explicit computation, and
-`#ker β_pc = pointCount = q + 1 − t` then furnishes `#ker β_pc = deg
-β_pc` directly. -/
-
 end CurveMap
 
 end HasseWeil.Curves
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Frobenius/FrobeniusFixedPoint.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Frobenius/FrobeniusFixedPoint.lean
index 758285749..d4ec75e01 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Frobenius/FrobeniusFixedPoint.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Frobenius/FrobeniusFixedPoint.lean
@@ -9,60 +9,12 @@ import Mathlib.FieldTheory.Finite.Basic
 import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
 
 /-!
-# Geometric Frobenius on points over the algebraic closure (Route B, Step 2)
+For a finite field `K` of cardinality `q`, the map `(x, y) ↦ (x ^ q, y ^ q)`
+on the algebraic-closure points of a Weierstrass curve fixes exactly the
+`K`-rational points. Consequently the kernel of the identity minus Frobenius
+has cardinality equal to the number of `K`-rational points.
 
-For a finite field `K` with `q = Fintype.card K` and a Weierstrass curve `W` over `K`,
-base-changed to `L = AlgebraicClosure K`, this file builds the **geometric Frobenius**
-endomorphism on the `L`-points,
-
-```
-geomFrobeniusPoint : (W.baseChange L).toAffine.Point →+ (W.baseChange L).toAffine.Point
-```
-
-acting as `(x, y) ↦ (x ^ q, y ^ q)` (and `0 ↦ 0`), via mathlib's `FiniteField.frobeniusAlgHom`
-(`x ↦ x ^ q`, a `K`-algebra hom on `L`) transported to points by the project's
-`HasseWeil.Affine.Point.map`. The codomain identification
-
-```
-(W.baseChange L).map (frobeniusAlgHom K L) = W.baseChange L
-```
-
-(`WeierstrassCurve.map_baseChange`, because the `q`-power map is a `K`-algebra hom and so
-fixes the `algebraMap K L` image) makes this an *endomorphism* of the same point type.
-
-It also packages the base-change inclusion of `K`-points
-
-```
-includePointBC : W.toAffine.Point →+ (W.baseChange L).toAffine.Point
-```
-
-via `HasseWeil.Affine.Point.map (algebraMap K L)`.
-
-## Main result (Step 2 = S2)
-
-* `HasseWeil.geomFrobeniusPoint_fixed_iff_mem_range_includePointBC`:
-  `geomFrobeniusPoint P = P ↔ P ∈ Set.range includePointBC`,
-  the *point-level* fixed-locus theorem. Proof by cases on `P`:
-  - `P = 0` is fixed and `0 = includePointBC 0`;
-  - `P = (x, y)` is fixed iff `x ^ q = x ∧ y ^ q = y`, iff (by Step 1
-    `frobenius_fixed_iff_mem_baseField`) both coordinates lie in `range (algebraMap K L)`,
-    iff `P = includePointBC (x₀, y₀)`.
-
-## Further results (S3/S4 and the `1 − π` kernel)
-
-* `fixedLocus_geomFrobenius_eq_range_includePointBC` (S3): the `Set`-form of S2 — the
-  geometric-Frobenius fixed locus equals `Set.range (includePointBC W)`.
-* `ncard_fixedLocus_geomFrobenius_eq_pointCount` (S4): that fixed locus has cardinality
-  `Fintype.card W.toAffine.Point` (`= pointCount W`), using injectivity of `includePointBC`.
-* `ncard_ker_oneSubGeomFrobHom_eq_pointCount`: the same count for the kernel of the
-  `AddMonoidHom` `id − geomFrobeniusPoint`.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, V.1.1.
-* Step 1: `HasseWeil.frobenius_fixed_iff_mem_baseField` (`Curves/FrobeniusFixedLocus.lean`).
-* mathlib: `FiniteField.frobeniusAlgHom`, `WeierstrassCurve.map_baseChange`,
-  `WeierstrassCurve.Affine.Point.map`.
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, V.1.1.
 -/
 
 open WeierstrassCurve
@@ -76,8 +28,6 @@ local notation "L" => AlgebraicClosure K
 
 noncomputable local instance : DecidableEq (AlgebraicClosure K) := Classical.decEq _
 
-/-! ### The geometric Frobenius `K`-algebra hom and the codomain identification -/
-
 /-- The geometric Frobenius `K`-algebra hom `x ↦ x ^ q` on `L = AlgebraicClosure K`,
 as a ring hom. This is `FiniteField.frobeniusAlgHom K L`, coerced to a `RingHom`. -/
 noncomputable def geomFrobRingHom : AlgebraicClosure K →+* AlgebraicClosure K :=
@@ -106,13 +56,11 @@ fixes `algebraMap K L`). -/
   rw [AlgHom.toRingHom_eq_coe]
   exact W.map_baseChange (FiniteField.frobeniusAlgHom K (AlgebraicClosure K))
 
-/-! ### Geometric Frobenius on points and the base-change inclusion -/
-
 /-- The geometric Frobenius on `L`-points as a raw map: mathlib's
 `WeierstrassCurve.Affine.Point.map` of the geometric Frobenius `K`-algebra hom
 `frobeniusAlgHom K L : L →ₐ[K] L`.  Its codomain `W⟮L⟯ = Point (W.baseChange L)`
 is **definitionally** `(W.baseChange L).toAffine.Point`, so no cast is needed.
-This mirrors `frobeniusW_KE` in `Hasse/IsogOneSubXyFamily.lean`. -/
+-/
 noncomputable def geomFrobeniusPointFun :
     (W.baseChange (AlgebraicClosure K)).toAffine.Point →
       (W.baseChange (AlgebraicClosure K)).toAffine.Point :=
@@ -184,17 +132,9 @@ theorem includePointBC_injective : Function.Injective (includePointBC W) := by
     subst hx' hy'
     rfl
 
-/-! ### S2 — the point-level fixed-locus theorem
-
-We work directly with the raw functions to keep the rewriting along
-`map_geomFrob_baseChange_eq_self` explicit. The `0` case is reflexivity; the affine case
-reduces, via `HasseWeil.Affine.Point.map_some` and `some.injEq`, to the conjunction
-`x ^ q = x ∧ y ^ q = y`, which Step 1 (`frobenius_fixed_iff_mem_baseField`) turns into
-membership of both coordinates in `range (algebraMap K L)`. -/
-
 omit [DecidableEq K] in
 set_option backward.isDefEq.respectTransparency false in
-/-- **S2 (point fixed-locus)**: a point `P` over the algebraic closure is fixed by the
+/-- a point `P` over the algebraic closure is fixed by the
 geometric Frobenius iff it is the base-change inclusion of a `K`-rational point.
 
 `geomFrobeniusPointFun P = P ↔ P ∈ Set.range (includePointBC W)`.
@@ -202,7 +142,7 @@ geometric Frobenius iff it is the base-change inclusion of a `K`-rational point.
 Proof by cases on `P`:
 * `P = 0`: fixed, and `0 = includePointBC 0`.
 * `P = some x y h`: `Frob P = P ↔ x ^ q = x ∧ y ^ q = y` (coordinatewise, by
-  `Affine.Point.map_some` + `some.injEq`), and by Step 1 each coordinate is fixed iff it
+  `Affine.Point.map_some` + `some.injEq`), and each coordinate is fixed iff it
   lies in `range (algebraMap K L)`; assembling the two `K`-rational coordinates back into a
   nonsingular `K`-point gives `P ∈ range includePointBC`.
 
@@ -217,7 +157,7 @@ theorem geomFrobeniusPoint_fixed_iff_mem_range_includePointBC
   · -- `P = 0`: fixed (`map_zero` is `rfl`), and `0 = includePointBC 0`.
     exact iff_of_true rfl ⟨0, includePointBC_zero W⟩
   · -- `P = some x y h`.  LHS reduces to `x ^ q = x ∧ y ^ q = y` (coordinatewise),
-    -- each equivalent by Step 1 to membership in `range (algebraMap K L)`.
+    -- each equivalent to membership in `range (algebraMap K L)`.
     rw [geomFrobeniusPointFun_some, Affine.Point.some.injEq]
     show (FiniteField.frobeniusAlgHom K (AlgebraicClosure K)) x = x ∧
         (FiniteField.frobeniusAlgHom K (AlgebraicClosure K)) y = y ↔ _
@@ -242,37 +182,18 @@ theorem geomFrobeniusPoint_fixed_iff_mem_range_includePointBC
         rw [Affine.Point.some.injEq] at hQ
         exact ⟨⟨x₀, hQ.1⟩, ⟨y₀, hQ.2⟩⟩
 
-/-! ### S3 — kernel of `1 − π` over `L` = image of `E(K)`
-
-The forward inclusion `range includePointBC ⊆
-{P | geomFrobeniusPointFun P = P}` is the "rational ⇒ fixed" direction (a rational `P`
-has `(1 − π)P = P − πP = P − P = 0`); the reverse is the substantive S2 content.  As an
-*equality of sets* this is literally S2 (`geomFrobeniusPoint_fixed_iff_mem_range_includePointBC`)
-repackaged, so we state it that way and reduce to S2. -/
-
 omit [DecidableEq K] in
-/-- **S3 (set form)**: the fixed locus of the geometric Frobenius equals the image of the
-`K`-rational points.  This is S2 in `Set` form. -/
+/-- the fixed locus of the geometric Frobenius equals the image of the
+`K`-rational points. -/
 theorem fixedLocus_geomFrobenius_eq_range_includePointBC :
     {P : (W.baseChange (AlgebraicClosure K)).toAffine.Point | geomFrobeniusPointFun W P = P} =
       Set.range (includePointBC W) := by
   ext P
   exact geomFrobeniusPoint_fixed_iff_mem_range_includePointBC W P
 
-/-! ### S4 — cardinality glue
-
-The final glue: `# ker(id − geomFrobeniusPoint) = pointCount W`.  S2/S3 identify the
-fixed locus with `range includePointBC`, and `includePointBC` is injective (it is
-`Affine.Point.map` of an injective ring hom, see `WeierstrassCurve.Affine.Point.map_injective`),
-so the fixed locus is in bijection with `W.toAffine.Point`, whose cardinality is `pointCount W`.
-Combined with the algebraic-closed fibre count
-(`CurveMap.exists_heightOneSpectrum_fiber_card_eq_sepDegree_unconditional`) this yields
-`deg(1 − π) = pointCount`. -/
-
 omit [DecidableEq K] in
-/-- **S4 (cardinality)**: the number of geometric-Frobenius-fixed `L`-points equals
-the `K`-rational point count `Fintype.card W.toAffine.Point` (= `pointCount`).  Reduces to S3
-(`fixedLocus … = range includePointBC`) plus injectivity of `includePointBC`. -/
+/-- the number of geometric-Frobenius-fixed `L`-points equals
+the `K`-rational point count `Fintype.card W.toAffine.Point` (= `pointCount`). -/
 theorem ncard_fixedLocus_geomFrobenius_eq_pointCount [Fintype W.toAffine.Point] :
     {P : (W.baseChange (AlgebraicClosure K)).toAffine.Point |
         geomFrobeniusPointFun W P = P}.ncard = Fintype.card W.toAffine.Point := by
@@ -280,15 +201,6 @@ theorem ncard_fixedLocus_geomFrobenius_eq_pointCount [Fintype W.toAffine.Point]
     Set.ncard_range_of_injective (includePointBC_injective W),
     Nat.card_eq_fintype_card]
 
-/-! ### PRIORITY 1 — kernel of `id − geomFrobenius` over `L`
-
-The geometric Frobenius on `L`-points is already mathlib's
-`WeierstrassCurve.Affine.Point.map (frobeniusAlgHom K L)`, an `AddMonoidHom`
-`(W.baseChange L)⟮L⟯ →+ (W.baseChange L)⟮L⟯` whose `.toFun` is definitionally
-`geomFrobeniusPointFun W` (mirroring `frobeniusW_KE` in
-`Hasse/IsogOneSubXyFamily.lean`). We bundle it, form `1 − π` as an
-`AddMonoidHom`, and identify its kernel with the fixed locus. -/
-
 /-- **Geometric Frobenius on `L`-points as an `AddMonoidHom`**: mathlib's
 `Affine.Point.map (frobeniusAlgHom K L)`. Its `.toFun` is definitionally
 `geomFrobeniusPointFun W`. -/
@@ -317,19 +229,19 @@ omit [DecidableEq K] in
     oneSubGeomFrobHom W P = P - geomFrobeniusPointFun W P := rfl
 
 omit [DecidableEq K] in
-/-- **PRIORITY 1 (kernel = fixed locus)**: `P ∈ ker(id − geomFrob) ⟺
+/-- `P ∈ ker(id − geomFrob) ⟺
 geomFrob P = P`. Direct from `sub_eq_zero`. -/
 theorem ker_oneSubGeomFrobHom_eq_fixedLocus :
     ((oneSubGeomFrobHom W).ker : Set (W.baseChange (AlgebraicClosure K)).toAffine.Point) =
       {P | geomFrobeniusPointFun W P = P} := by
   ext P
-  rw [SetLike.mem_coe, AddMonoidHom.mem_ker, Set.mem_setOf_eq, oneSubGeomFrobHom_apply,
+  rw [SetLike.mem_coe, AddMonoidHom.mem_ker, Set.mem_ofPred_eq, oneSubGeomFrobHom_apply,
     sub_eq_zero, eq_comm]
 
 omit [DecidableEq K] in
-/-- **PRIORITY 1 (cardinality)**: the geometric-Frobenius fixed locus, i.e.
+/-- the geometric-Frobenius fixed locus, i.e.
 `ker(id − geomFrob)`, has cardinality `pointCount W`. Composes
-`ker_oneSubGeomFrobHom_eq_fixedLocus` with S4
+`ker_oneSubGeomFrobHom_eq_fixedLocus` with
 (`ncard_fixedLocus_geomFrobenius_eq_pointCount`). -/
 theorem ncard_ker_oneSubGeomFrobHom_eq_pointCount [Fintype W.toAffine.Point] :
     ((oneSubGeomFrobHom W).ker :
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/IntegralClosure.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/IntegralClosure.lean
index b3c5b8f64e3264fafd4ab131a093d258bbd32966..e257fed368990ece1c76792d0f26d887b83c989d 100644
GIT binary patch
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CoordHomFinite.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CoordHomFinite.lean
index 5456f6e64..33637b28c 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CoordHomFinite.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CoordHomFinite.lean
@@ -6,70 +6,20 @@ Authors: Chris Birkbeck
 import HasseWeil.Foundation.Curves.Map.CurveMap
 
 /-!
-# Module-finiteness along a coordinate-ring witness (the standing `hfin`, DERIVED)
-
-For a curve map `φ : C₁ → C₂` with a coordinate-ring pullback witness
-`cd : φ.CoordHom`, the coordinate ring `F[C₁]` is a **finite module** over
-`F[C₂]` via `cd.toAlgebra` — *unconditionally*.  This derives the standing
-`Module.Finite` hypothesis (`hfin`) carried throughout the isogeny theory by
-`CurveMap.sum_ramificationIdx_mul_inertiaDeg_eq_degree`,
-`EC.Isogeny.addHomProperty` / `toBasicIsogeny` (K̄-level),
-`EC.Isogeny.addHomProperty_descend` / `toBasicIsogenyDescend`
-(K-level), `Isogeny.pushforward` (`Curves/PushforwardDivisor.lean`),
-`EC/KernelCount.lean`, and friends.
-
-## The proof (no separability, no places classification)
-
-Write `u := ψ(x₂) ∈ F[C₁]` for the image of the coordinate generator of `C₂`
-under an injective `F`-algebra map `ψ : F[C₂] →ₐ[F] F[C₁]`, and decompose
-`u = p•1 + q•Y` in the `{1, Y}` basis of `F[C₁]` over `F[x₁] = F[X]`.  Then
-`u` satisfies the quadratic relation of its conjugate pair over `F[X]`:
-
-`u² − t(x₁)·u + n(x₁) = 0`, where `t := 2p − q·(a₁X + a₃)` and
-`n := p² − pq·(a₁X + a₃) − q²·(X³ + a₂X² + a₄X + a₆)`
-
-(`n` is mathlib's `Algebra.norm F[X] u`; the difference of the two sides is
-`q²·W(X, Y)`, which vanishes mod the Weierstrass polynomial).  Reading the
-relation backwards as a polynomial in `x₁` with coefficients in the image of
-`ψ` gives `n(T) − u·t(T) + u² = 0` at `T = x₁`.
-
-**The parity trick**: `deg(p²) = 2 deg p` is even while
-`deg(q²·(X³ + ⋯)) = 2 deg q + 3` is odd, so the two leading terms never
-cancel: `deg n = max(2 deg p, 2 deg q + 3)` with leading coefficient a
-**unit of `F`** (`WeierstrassCurve.Affine.CoordinateRing.degree_norm_smul_basis`),
-and `deg t < deg n`.  Hence the displayed `T`-polynomial is monic after
-scaling by that unit's inverse, with coefficients in the image of `F[C₂]`:
-the generator `x₁` is **integral** over `F[C₂]`.  (`u ∉ F·1` because `ψ` is
-injective, ruling out the degenerate `q = 0 ∧ p constant` case.)  Since
-`F[C₁]` is spanned by `F[x₁]·{1, Y}`, module-finiteness follows from
-`IsIntegral.fg_adjoin_singleton`.
-
-This argument works in **any characteristic and for inseparable maps** (for
-the Frobenius comorphism `u = x₁^p` it produces the monic witness
-`(T^p − u)²`-style relation), so no separability hypothesis, no AKLB
-machinery, and no Nagata-style finiteness of integral closures is needed.
-The classical "places of `K(C₁)` over affine places of `C₂` are point
-places" wall is bypassed entirely by the explicit Weierstrass presentation.
-
-## Main results
-
-* `algHom_coordinateRing_isIntegralElem_X` — the keystone: the coordinate
-  generator `x₁` is integral over `F[C₂]` along any injective `F`-algebra
-  hom `F[C₂] →ₐ[F] F[C₁]`.
-* `algHom_coordinateRing_module_finite` — `F[C₁]` is a finite module over
-  `F[C₂]` along any injective `F`-algebra hom.
-* `CurveMap.CoordHom.toAlgHom_injective` — a `CoordHom` is automatically
-  injective (from `compat` plus injectivity of the function-field pullback).
-* `CurveMap.CoordHom.module_finite` — **the deliverable**: the standing
-  `hfin` hypothesis holds for every `(φ, cd)`, with no further assumptions.
-* `CurveMap.sum_ramificationIdx_mul_inertiaDeg_eq_degree'` — Silverman
-  II.2.6(a) `Σ e·f = deg φ` with `hfin` discharged.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], II.2.6, II.3 (finiteness
-  of morphisms of smooth curves) — obtained here by direct
-  Weierstrass-presentation algebra instead of valuation theory.
+An injective homomorphism between Weierstrass coordinate rings makes the target
+a finite module over the source. Writing the image of the source coordinate as
+`u = p + qY`, its trace and norm give a quadratic relation in `u` with
+polynomial coefficients in the target coordinate. The terms of the norm have
+opposite degree parity, so its leading coefficient is a unit and the trace
+has smaller degree. Reading this relation as a polynomial in the target
+coordinate gives a monic integral relation after scaling.
+
+A compatible function-field pullback makes the coordinate-ring homomorphism
+injective. Thus the finite-module hypothesis in the weighted ramification
+sum follows for every coordinate-ring pullback witness, without separability
+or characteristic assumptions.
+
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.2.6 and II.3.
 -/
 
 namespace HasseWeil.Curves
@@ -413,9 +363,9 @@ theorem algHom_coordinateRing_module_finite
     (ψ : C₂.CoordinateRing →ₐ[F] C₁.CoordinateRing) (hψ : Function.Injective ψ) :
     @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _
       ψ.toRingHom.toAlgebra.toModule := by
-  letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := ψ.toRingHom.toAlgebra
-  letI : Module C₂.CoordinateRing C₁.CoordinateRing := Algebra.toModule
-  haveI : IsScalarTower F C₂.CoordinateRing C₁.CoordinateRing :=
+  let : Algebra C₂.CoordinateRing C₁.CoordinateRing := ψ.toRingHom.toAlgebra
+  let : Module C₂.CoordinateRing C₁.CoordinateRing := Algebra.toModule
+  have : IsScalarTower F C₂.CoordinateRing C₁.CoordinateRing :=
     IsScalarTower.of_algebraMap_eq fun c ↦ (ψ.commutes c).symm
   have hx : IsIntegral C₂.CoordinateRing
       (algebraMap (Polynomial F) C₁.CoordinateRing Polynomial.X) :=
@@ -487,12 +437,7 @@ namespace CurveMap
 -- Instance synthesis and unification at curve-indexed coordinate-ring types
 -- (`AdjoinRoot`-quotients) need a higher budget; same settings as
 -- `CurveMap.sum_ramificationIdx_mul_inertiaDeg_eq_degree`.
-/-- **Silverman II.2.6(a), Σ e·f form, `hfin`-free** (T-II-2-008): for a
-`CurveMap φ : C₁ → C₂` with coordinate-ring pullback witness `coordHom`, the
-sum `Σ_{P over p} e_P · f_P` equals the function-field degree `φ.degree`.
-The module-finiteness input of
-`sum_ramificationIdx_mul_inertiaDeg_eq_degree` is supplied by
-`CoordHom.module_finite`. -/
+
 theorem sum_ramificationIdx_mul_inertiaDeg_eq_degree'
     {C₁ C₂ : SmoothPlaneCurve F}
     [IsIntegrallyClosed C₂.CoordinateRing]
@@ -502,7 +447,7 @@ theorem sum_ramificationIdx_mul_inertiaDeg_eq_degree'
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
     ∑ P ∈ IsDedekindDomain.primesOverFinset p C₁.CoordinateRing,
         Ideal.ramificationIdx' p P *
-        Ideal.inertiaDeg' p P = φ.degree :=
+        P.inertiaDeg C₂.CoordinateRing = φ.degree :=
   φ.sum_ramificationIdx_mul_inertiaDeg_eq_degree coordHom
     coordHom.module_finite hpMax hp0
 
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMap.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMap.lean
index aae9d4fe3..67470605c 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMap.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMap.lean
@@ -1,6 +1,6 @@
 import HasseWeil.Foundation.Curves.Valuation.Valuation
 import HasseWeil.Foundation.Curves.IntegralClosure
-import Mathlib.NumberTheory.RamificationInertia.Basic
+import Mathlib.RingTheory.RamificationInertia.Basic
 
 /-!
 # Curve maps via function-field pullback
@@ -14,10 +14,6 @@ injective).
 The **degree** of `φ` is the dimension of `K(C₁)` as a `K(C₂)`-module via
 the pullback: `deg φ = [K(C₁) : φ*K(C₂)]`.
 
-This closes tickets T-II-INFRA-B-006/007 of the Stream-A infrastructure plan
-and provides the foundational object for T-II-2-002..011 (which will add
-surjectivity, ramification, and the fiber-card formula).
-
 ## References
 
 * [Silverman, *The Arithmetic of Elliptic Curves*], II.2.4 (curves-fields
@@ -79,8 +75,6 @@ theorem comp_assoc {C₄ : SmoothPlaneCurve F}
 @[simp] theorem comp_id (φ : CurveMap C₁ C₂) : φ.comp (id C₁) = φ :=
   CurveMap.ext (AlgHom.id_comp _)
 
-/-! ### Degree -/
-
 /-- The algebra structure on `K(C₁)` induced by the pullback of `φ`. -/
 @[reducible]
 noncomputable def toAlgebra (φ : CurveMap C₁ C₂) :
@@ -110,12 +104,12 @@ theorem comp_algebraMap_eq (ψ : CurveMap C₂ C₃) (φ : CurveMap C₁ C₂)
 theorem degree_comp (ψ : CurveMap C₂ C₃) (φ : CurveMap C₁ C₂) :
     (ψ.comp φ).degree = φ.degree * ψ.degree := by
   simp only [degree]
-  letI : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  letI : Algebra C₃.FunctionField C₂.FunctionField := ψ.toAlgebra
-  letI : Algebra C₃.FunctionField C₁.FunctionField := (ψ.comp φ).toAlgebra
-  haveI : IsScalarTower C₃.FunctionField C₂.FunctionField C₁.FunctionField :=
+  let : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  let : Algebra C₃.FunctionField C₂.FunctionField := ψ.toAlgebra
+  let : Algebra C₃.FunctionField C₁.FunctionField := (ψ.comp φ).toAlgebra
+  have : IsScalarTower C₃.FunctionField C₂.FunctionField C₁.FunctionField :=
     IsScalarTower.of_algebraMap_eq fun _ ↦ rfl
-  haveI : Module.Free C₂.FunctionField C₁.FunctionField :=
+  have : Module.Free C₂.FunctionField C₁.FunctionField :=
     Module.Free.of_divisionRing _ _
   rw [mul_comm]
   exact (Module.finrank_mul_finrank
@@ -138,27 +132,11 @@ def IsSeparable (φ : CurveMap C₁ C₂) : Prop := φ.inseparableDegree = 1
 /-- A curve map is **purely inseparable** if its separable degree is 1. -/
 def IsPurelyInseparable (φ : CurveMap C₁ C₂) : Prop := φ.separableDegree = 1
 
--- **Silverman II.2.4.1** (pullback half, deferred): if `φ : C₁ → C₂` has
--- degree 1, its pullback is surjective.
--- Proof sketch: `finrank_eq_one_iff'` gives `v ≠ 0` with `K(C₁) = K(C₂) • v`.
--- From `c₀ • v = 1` in a field, `v = (algebraMap c₀)⁻¹`, so every `x ∈ K(C₁)`
--- is `c_x • v = algebraMap c_x / algebraMap c₀ = algebraMap (c_x / c₀)`.
--- The Lean proof needs `algebraMap C₂.FunctionField C₁.FunctionField` to
--- resolve to `φ.pullback.toRingHom` via `haveI : Algebra ... := φ.toAlgebra`,
--- but instance synthesis tends to pick unrelated `Algebra` instances.
--- Deferred for future cleanup.
-
-/-! ### Ramification order -/
-
 /-- The order of the pullback of a function `t ∈ K(C₂)` at a smooth point
 `P ∈ C₁`: `ord_P (φ* t)`. When `t` is a uniformizer at the image point
 `φ(P) ∈ C₂`, this equals Silverman's ramification index `e_φ(P)` of II.2.5.
 
-Our formulation takes the test function `t` as an explicit argument rather
-than deriving it from an intrinsic "image point" `φ(P)`, since the
-point-image correspondence (Silverman II.2.4(c)) is deferred to later
-infrastructure. Downstream users supply a uniformizer at the intended
-image point.
+The test function `t` is an explicit uniformizer at the intended image point.
 Reference: Silverman II.2.5 (definition). -/
 noncomputable def ramificationIndex (φ : CurveMap C₁ C₂) (P : C₁.SmoothPoint)
     (t : C₂.FunctionField) : WithTop ℤ :=
@@ -239,8 +217,6 @@ theorem one_le_ramificationIndex_of_pullback_pointValuation_lt_one
     (1 : WithTop ℤ) ≤ φ.ramificationIndex P t :=
   (C₁.one_le_ord_P_iff_pointValuation_lt_one (φ.pullback_ne_zero ht)).mpr h
 
-/-! ### Pushforward / Norm map (T-II-2-005) -/
-
 /-- The **pushforward** (norm map) `φ_* : K(C₁) →* K(C₂)` for a curve map
 `φ : C₁ → C₂`, defined as the `K(C₂)`-algebra norm on `K(C₁)` via `φ.toAlgebra`.
 Silverman's `φ_* = (φ*)⁻¹ ∘ N_{K(C₁)/φ*K(C₂)}` simplifies to the direct
@@ -255,7 +231,7 @@ noncomputable def pushforward (φ : CurveMap C₁ C₂) :
 Reference: Silverman II.2 (after definition of `φ_*`). -/
 theorem pushforward_pullback (φ : CurveMap C₁ C₂) (g : C₂.FunctionField) :
     φ.pushforward (φ.pullback g) = g ^ φ.degree := by
-  letI : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  let : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
   exact Algebra.norm_algebraMap g
 
 /-- The pushforward is multiplicative. -/
@@ -268,18 +244,6 @@ theorem pushforward_pullback (φ : CurveMap C₁ C₂) (g : C₂.FunctionField)
     φ.pushforward (1 : C₁.FunctionField) = 1 :=
   φ.pushforward.map_one
 
-/-! ### Fiber cardinality (T-II-2-011): Silverman II.2.7
-
-The classical Silverman II.2.7 asserts that a nonconstant curve map `φ` is
-unramified iff every geometric fiber has cardinality `deg(φ)`. Since our
-`CurveMap` models the function-field side only (no explicit `fiber` type),
-we record the combinatorial content as a **witness-parametric** statement:
-given a `Finset` of points, their ramification indices, and the sum formula
-II.2.6(a) (`Σ e_φ(P) = deg(φ)` witness), the fiber-count equals `deg(φ)` iff
-every index is `1`. Once II.2.6(a) is formalized, this immediately gives
-T-II-2-011 for every image point `Q`.
--/
-
 /-- **Combinatorial lemma**: if `e : α → ℤ` is ≥ 1 on every element of a
 finset `S`, then `∑ P ∈ S, e P = #S` iff each `e P = 1`.
 
@@ -298,7 +262,7 @@ theorem _root_.Finset.sum_eq_card_iff_forall_eq_one_of_one_le
     linarith
   · exact (Finset.sum_congr rfl h).trans hconst
 
-/-- **Witness-parametric Silverman II.2.7** (T-II-2-011): given a finite
+/-- **Witness-parametric Silverman II.2.7**: given a finite
 fiber `S` with ramification indices ≥ 1 summing to `deg(φ)`, the fiber
 count equals `deg(φ)` iff every ramification index equals `1`.
 
@@ -321,15 +285,13 @@ theorem fiber_card_eq_degree_iff_all_ramificationIndexℤ_one
   rw [← hsum, eq_comm]
   exact Finset.sum_eq_card_iff_forall_eq_one_of_one_le hle
 
-/-! ### Degree-one morphisms are isomorphisms on function fields (T-II-2-006) -/
-
 /-- **Silverman II.2.4.1 (pullback-surjectivity form)**: if a curve map `φ`
 has degree 1, its pullback `φ*` is surjective. Combined with the automatic
 injectivity of pullbacks, `φ*` is then a bijection of function fields. -/
 theorem pullback_surjective_of_degree_one (φ : CurveMap C₁ C₂)
     (h : φ.degree = 1) : Function.Surjective φ.pullback := by
-  letI : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  haveI : Module.Free C₂.FunctionField C₁.FunctionField :=
+  let : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  have : Module.Free C₂.FunctionField C₁.FunctionField :=
     Module.Free.of_divisionRing _ _
   have hfr : Module.finrank C₂.FunctionField C₁.FunctionField = 1 := h
   obtain ⟨v, _hv_ne, hv⟩ := finrank_eq_one_iff'.mp hfr
@@ -345,17 +307,6 @@ theorem pullback_surjective_of_degree_one (φ : CurveMap C₁ C₂)
   rw [map_div₀, div_eq_mul_inv, ← hv_eq, ← Algebra.smul_def]
   exact hcv
 
-/-! ### Silverman II.2.6(a): `Σ e_φ(P) · f_φ(P) = deg(φ)` (T-II-2-008 via
-coordinate-ring algebra witness)
-
-`CurveMap` stores only the function-field pullback; a generic Dedekind-style
-sum-of-ramification-indices formula requires the pullback to restrict to a
-ring hom `C₂.CoordinateRing → C₁.CoordinateRing`. We expose this as an
-**auxiliary data bundle** (`CurveMap.CoordHom`) together with a compatibility
-condition. For the specific coordinate function `x : C → A¹`
-(`algebraMap F[X] → F[C]`), see `HasseWeil/Curves/NormValuation.lean`'s
-`sum_ramificationIdx_over_fiber` for the unconditional instance. -/
-
 /-- **Coordinate-ring pullback witness**: for a `CurveMap φ : C₁ → C₂`, a
 ring hom `C₂.CoordinateRing → C₁.CoordinateRing` compatible with the
 function-field pullback. Not every function-field pullback restricts to
@@ -378,7 +329,7 @@ noncomputable def CoordHom.toAlgebra {φ : CurveMap C₁ C₂} (coordHom : φ.Co
     Algebra C₂.CoordinateRing C₁.CoordinateRing :=
   coordHom.toAlgHom.toRingHom.toAlgebra
 
-/-- **Silverman II.2.6(a), Σ e·f form** (T-II-2-008, generic `CurveMap`):
+/-- **Silverman II.2.6(a), Σ e·f form**:
 for a `CurveMap φ : C₁ → C₂` with coordinate-ring pullback witness
 `coordHom` and finite-module structure, the sum `Σ_{P over p} e_P · f_P`
 equals the function-field degree `φ.degree`.
@@ -402,16 +353,16 @@ theorem sum_ramificationIdx_mul_inertiaDeg_eq_degree
     letI : Algebra C₂.CoordinateRing C₁.CoordinateRing := coordHom.toAlgebra
     ∑ P ∈ IsDedekindDomain.primesOverFinset p C₁.CoordinateRing,
         Ideal.ramificationIdx' p P *
-        Ideal.inertiaDeg' p P = φ.degree := by
-  letI algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing :=
+        P.inertiaDeg C₂.CoordinateRing = φ.degree := by
+  let algCR : Algebra C₂.CoordinateRing C₁.CoordinateRing :=
     coordHom.toAlgebra
-  letI : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
-  haveI : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing
+  let : Algebra C₂.FunctionField C₁.FunctionField := φ.toAlgebra
+  have : IsScalarTower C₂.CoordinateRing C₁.CoordinateRing
       C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_eq fun x ↦ ?_
     rw [RingHom.algebraMap_toAlgebra]
     rfl
-  haveI : IsScalarTower C₂.CoordinateRing C₂.FunctionField
+  have : IsScalarTower C₂.CoordinateRing C₂.FunctionField
       C₁.FunctionField := by
     refine IsScalarTower.of_algebraMap_smul fun r x ↦ ?_
     rw [Algebra.smul_def]
@@ -420,12 +371,44 @@ theorem sum_ramificationIdx_mul_inertiaDeg_eq_degree
     rw [coordHom.compat r, ← IsScalarTower.algebraMap_smul C₁.CoordinateRing r x,
       ← Algebra.smul_def]
     rfl
-  haveI : p.IsMaximal := hpMax
-  letI modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
-  haveI : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _
+  have : p.IsMaximal := hpMax
+  let modCR : Module C₂.CoordinateRing C₁.CoordinateRing := algCR.toModule
+  have : @Module.Finite C₂.CoordinateRing C₁.CoordinateRing _ _
       modCR := hfin
-  exact Ideal.sum_ramification_inertia (R := C₂.CoordinateRing)
-    (S := C₁.CoordinateRing) C₂.FunctionField C₁.FunctionField hp0
+  have : Module.IsTorsionFree C₂.CoordinateRing C₁.CoordinateRing := by
+    apply Module.isTorsionFree_iff_algebraMap_injective.mpr
+    intro a b hab
+    apply IsFractionRing.injective C₂.CoordinateRing C₂.FunctionField
+    apply φ.pullback.injective
+    change φ.pullback (algebraMap C₂.CoordinateRing C₂.FunctionField a) =
+      φ.pullback (algebraMap C₂.CoordinateRing C₂.FunctionField b)
+    rw [coordHom.compat a, coordHom.compat b]
+    exact congrArg (algebraMap C₁.CoordinateRing C₁.FunctionField) hab
+  classical
+  let : Fintype (p.primesOver C₁.CoordinateRing) := Fintype.ofFinite _
+  calc
+    _ = ∑ P : p.primesOver C₁.CoordinateRing,
+        P.1.ramificationIdx C₂.CoordinateRing * P.1.inertiaDeg C₂.CoordinateRing := by
+      apply Finset.sum_bij (fun P hP ↦
+        ⟨P, (IsDedekindDomain.mem_primesOverFinset_iff hp0 C₁.CoordinateRing).mp hP⟩)
+      · intro P hP
+        exact Finset.mem_univ _
+      · intro P hP Q hQ h
+        exact congrArg Subtype.val h
+      · intro P _
+        exact ⟨P.1,
+          (IsDedekindDomain.mem_primesOverFinset_iff hp0 C₁.CoordinateRing).mpr P.2, rfl⟩
+      · intro P hP
+        obtain ⟨hprime, hover⟩ :=
+          (IsDedekindDomain.mem_primesOverFinset_iff hp0 C₁.CoordinateRing).mp hP
+        let := hprime
+        let := hover
+        rw [Ideal.ramificationIdx'_eq_ramificationIdx p P hp0]
+    _ = Module.finrank C₂.CoordinateRing C₁.CoordinateRing :=
+      Ideal.sum_ramification_inertia_eq_finrank p C₁.CoordinateRing
+    _ = φ.degree :=
+      (IsFractionRing.finrank_eq C₂.CoordinateRing C₂.FunctionField
+        C₁.CoordinateRing C₁.FunctionField).symm
 
 end CurveMap
 
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMapBaseChange.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMapBaseChange.lean
index e1c402904..2537c2d15 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMapBaseChange.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Map/CurveMapBaseChange.lean
@@ -16,33 +16,14 @@ import Mathlib.RingTheory.TensorProduct.Free
 /-!
 # Base change of curves and curve maps
 
-For a smooth plane curve `C / F` and an `F`-algebra extension `L`, this file builds the
-natural scalar-extension maps and identifies the scalar extension of the coordinate ring
-and function field with the base-changed curve. These are foundations for the
-Galois-descent route to the unconditional `AddHomProperty` (Phase G of the Pic⁰ roadmap).
+Scalar extension identifies the coordinate ring of a smooth plane curve with the
+coordinate ring of its base change. The corresponding function-field tensor product
+is a localization of the coordinate-ring tensor product. For algebraic field extensions,
+this gives the natural isomorphism between scalar extension and the fraction field
+of the extended coordinate ring.
 
-## Main definitions
-
-* `SmoothPlaneCurve.coordRingMap` — the include of coordinate rings.
-* `SmoothPlaneCurve.functionFieldMap` — the include of function fields.
-* `SmoothPlaneCurve.coordRingScalarExt` — the scalar-extension iso
-  `L ⊗[F] C.CoordinateRing ≃ₐ[L] (C.baseChange L).CoordinateRing`.
-* `CurveMap.CoordHom.baseChangeAlgHom` — the base-changed coordinate-ring alg hom of a `CoordHom`.
-
-## Main results
-
-* `SmoothPlaneCurve.isDomain_tensorCoordRing` — `L ⊗[F] C.CoordinateRing` is a domain.
-* `SmoothPlaneCurve.tensor_functionField_isFractionRing` — `L ⊗[F] C.FunctionField` is the
-  fraction ring of `L ⊗[F] C.CoordinateRing`.
-* `SmoothPlaneCurve.functionField_tensor_locBaseChange` — localization commutes with the base
-  change `L/F`: `L ⊗[F] C.FunctionField ≃ₐ[L] FractionRing (L ⊗[F] C.CoordinateRing)`.
-
-## Implementation notes
-
-The `Algebra F` / `Algebra F[X]` instances on `AdjoinRoot W.polynomial` form a typeclass
-diamond that blocks synthesis of the tensor-product algebra/module structures. Section
-`PhaseGInstances` pins the F-route instances at high priority so signatures mentioning
-`L ⊗[F] C.CoordinateRing` (and its function-field analogue) elaborate diamond-free.
+The algebra and module structures are chosen compatibly with the scalar tower through
+the polynomial ring.
 
 ## References
 
@@ -179,7 +160,6 @@ theorem baseChange_inner_comp_mapRingHom_eq {φ : CurveMap C₁ C₂}
       C₁.coordRingMap L (cd.toAlgHom
         (algebraMap (Polynomial F) C₂.CoordinateRing (Polynomial.C a)))
     rw [Polynomial.map_C]
-    change (cd.baseChangeInnerAlgHom L) (Polynomial.C (algebraMap F L a)) = _
     simp only [baseChangeInnerAlgHom]
     rw [Polynomial.aeval_C, show (algebraMap (Polynomial F) C₂.CoordinateRing) (Polynomial.C a) =
           algebraMap F C₂.CoordinateRing a from
@@ -262,8 +242,8 @@ noncomputable instance (priority := 100000) coordRingModuleBase :
 /-- `CommRing` on the scalar-extension tensor `L ⊗[F] C.CR`, pinned at high
 priority. The default synthesis loses to the `Algebra F` / `Algebra F[X]`
 diamond on `AdjoinRoot W.polynomial` (it tries the `F[X]`-route algebra on
-`C.CR`); pinning the F-route `CommRing` here unblocks any signature that
-mentions `L ⊗[F] C.CR` (e.g. `FractionRing (L ⊗[F] C.CR)`). -/
+`C.CR`). This instance uses the algebra over `F` in the tensor product
+and its fraction field. -/
 noncomputable instance (priority := 100000) tensorCoordRingCommRing
     (L : Type*) [Field L] [Algebra F L] :
     CommRing (L ⊗[F] C.toAffine.CoordinateRing) :=
@@ -310,8 +290,8 @@ noncomputable instance (priority := 100000) tensorFunctionFieldFLScalarTower
 pinned at high priority. The same `Algebra F` / `Algebra F[X]` diamond on
 `AdjoinRoot W.polynomial` that taints `L ⊗[F] C.CR` recurs at the function-field
 level (since `C.FF = FractionRing C.CR` carries the `C.CR`-algebra, which routes
-back through the diamond). Pinning the F-route `CommRing` unblocks any signature
-mentioning `L ⊗[F] C.FF`. -/
+back through the diamond). This instance uses the algebra over `F`
+in the function-field tensor product. -/
 noncomputable instance (priority := 100000) tensorFunctionFieldCommRing
     (L : Type*) [Field L] [Algebra F L] :
     CommRing (L ⊗[F] C.toAffine.FunctionField) :=
@@ -693,8 +673,7 @@ theorem tensorFunctionFieldStructureHom_injective (L : Type*) [Field L] [Algebra
     Module.Flat.lTensor_preserves_injective_linearMap gF hg
   have hfun : ⇑(C.tensorFunctionFieldStructureHom L) = ⇑(LinearMap.lTensor L gF) := by
     funext x
-    induction x using TensorProduct.induction_on with
-    | zero => simp
+    induction x using TensorProduct.inductionOn with
     | tmul l u =>
         rw [LinearMap.lTensor_tmul]
         change Algebra.TensorProduct.map (AlgHom.id L L)
@@ -707,12 +686,12 @@ theorem tensorFunctionFieldStructureHom_injective (L : Type*) [Field L] [Algebra
   exact hlin
 
 /-- `C.CoordinateRing ⊗[F] L` is a domain, for any field extension `L/F`. Obtained from the
-sibling `isDomain_tensorCoordRing` (which gives the `L`-on-the-left orientation
+`isDomain_tensorCoordRing` (which gives the `L`-on-the-left orientation
 `L ⊗[F] C.CoordinateRing`) by transporting the domain structure across the commutativity
 ring-iso `Algebra.TensorProduct.comm`. -/
 private theorem tensorCoordRing_comm_isDomain (L : Type*) [Field L] [Algebra F L] :
     IsDomain (C.toAffine.CoordinateRing ⊗[F] L) := by
-  letI := C.isDomain_tensorCoordRing L
+  let := C.isDomain_tensorCoordRing L
   exact
     (Algebra.TensorProduct.comm F L C.toAffine.CoordinateRing).symm.toRingEquiv.toMulEquiv.isDomain
     (L ⊗[F] C.toAffine.CoordinateRing)
@@ -727,9 +706,9 @@ private theorem tensorCoordRing_algebraMapSubmonoid_le_nonZeroDivisors
     Algebra.algebraMapSubmonoid (C.toAffine.CoordinateRing ⊗[F] L)
         (nonZeroDivisors C.toAffine.CoordinateRing) ≤
       nonZeroDivisors (C.toAffine.CoordinateRing ⊗[F] L) := by
-  haveI hflatCR : Module.Flat F C.toAffine.CoordinateRing := Module.Flat.of_free
-  letI := C.tensorCoordRing_comm_isDomain L
-  haveI hnzdCR : NoZeroDivisors (C.toAffine.CoordinateRing ⊗[F] L) :=
+  have hflatCR : Module.Flat F C.toAffine.CoordinateRing := Module.Flat.of_free
+  let := C.tensorCoordRing_comm_isDomain L
+  have hnzdCR : NoZeroDivisors (C.toAffine.CoordinateRing ⊗[F] L) :=
     isCancelMulZero_iff_noZeroDivisors.mp ‹IsDomain _›.toIsCancelMulZero
   have hinjL : Function.Injective
       (Algebra.TensorProduct.includeLeft :
@@ -758,11 +737,11 @@ private theorem tensorFunctionField_isLocalization (L : Type*) [Field L] [Algebr
       (Algebra.algebraMapSubmonoid (C.toAffine.CoordinateRing ⊗[F] L)
         (nonZeroDivisors C.toAffine.CoordinateRing))
       (C.toAffine.FunctionField ⊗[F] L) := by
-  letI algBC : Algebra (C.toAffine.CoordinateRing ⊗[F] L) (C.toAffine.FunctionField ⊗[F] L) :=
+  let algBC : Algebra (C.toAffine.CoordinateRing ⊗[F] L) (C.toAffine.FunctionField ⊗[F] L) :=
     (Algebra.TensorProduct.map
       (IsScalarTower.toAlgHom F C.toAffine.CoordinateRing C.toAffine.FunctionField)
       (AlgHom.id F L)).toRingHom.toAlgebra
-  haveI tower : IsScalarTower C.toAffine.CoordinateRing (C.toAffine.CoordinateRing ⊗[F] L)
+  have tower : IsScalarTower C.toAffine.CoordinateRing (C.toAffine.CoordinateRing ⊗[F] L)
       (C.toAffine.FunctionField ⊗[F] L) :=
     IsScalarTower.of_algebraMap_eq fun _ ↦ rfl
   exact @IsLocalization.tensorProduct_tensorProduct F L _ _ _ C.toAffine.CoordinateRing _ _
@@ -786,14 +765,14 @@ the function field on the **left** (`FunctionField ⊗[F] L`) to match that cons
 private theorem tensorFunctionField_isDomain (L : Type*) [Field L] [Algebra F L] :
     IsDomain (C.toAffine.FunctionField ⊗[F] L) := by
   -- `C.CoordinateRing ⊗[F] L` is a domain (the base ring of the localization below).
-  letI := C.tensorCoordRing_comm_isDomain L
+  let := C.tensorCoordRing_comm_isDomain L
   -- The natural `lTensor`-of-localization algebra structure exhibiting the function field
   -- tensor as a localization of the coordinate-ring tensor.
-  letI _algBC : Algebra (C.toAffine.CoordinateRing ⊗[F] L) (C.toAffine.FunctionField ⊗[F] L) :=
+  let _algBC : Algebra (C.toAffine.CoordinateRing ⊗[F] L) (C.toAffine.FunctionField ⊗[F] L) :=
     (Algebra.TensorProduct.map
       (IsScalarTower.toAlgHom F C.toAffine.CoordinateRing C.toAffine.FunctionField)
       (AlgHom.id F L)).toRingHom.toAlgebra
-  haveI := C.tensorFunctionField_isLocalization L
+  have := C.tensorFunctionField_isLocalization L
   -- A localization of a domain at a submonoid of nonzerodivisors is again a domain.
   exact IsLocalization.isDomain_of_le_nonZeroDivisors (C.toAffine.FunctionField ⊗[F] L)
     (C.tensorCoordRing_algebraMapSubmonoid_le_nonZeroDivisors L)
@@ -805,7 +784,7 @@ when consumed by `tensor_functionField_isFractionRing`. -/
 theorem tensor_functionField_isField (L : Type*) [Field L] [Algebra F L]
     [Algebra.IsAlgebraic F L] :
     IsField (L ⊗[F] C.toAffine.FunctionField) := by
-  haveI := C.tensorFunctionField_isDomain L
+  have := C.tensorFunctionField_isDomain L
   have hF : IsField (C.toAffine.FunctionField ⊗[F] L) :=
     Algebra.TensorProduct.isField_of_isAlgebraic F C.toAffine.FunctionField L
       (Or.inr (inferInstance : Algebra.IsAlgebraic F L))
@@ -842,7 +821,7 @@ private theorem oneTmul_mem_nonZeroDivisors (L : Type*) [Field L] [Algebra F L]
     (hb : b ∈ nonZeroDivisors C.toAffine.CoordinateRing) :
     (1 ⊗ₜ b : L ⊗[F] C.toAffine.CoordinateRing)
       ∈ nonZeroDivisors (L ⊗[F] C.toAffine.CoordinateRing) := by
-  letI := C.isDomain_tensorCoordRing L
+  let := C.isDomain_tensorCoordRing L
   rw [mem_nonZeroDivisors_iff_ne_zero]
   have hb0 : b ≠ 0 := mem_nonZeroDivisors_iff_ne_zero.mp hb
   intro hc
@@ -861,17 +840,15 @@ theorem tensor_functionField_surj (L : Type*) [Field L] [Algebra F L] :
             (L ⊗[F] C.toAffine.FunctionField) x.2
           = algebraMap (L ⊗[F] C.toAffine.CoordinateRing)
             (L ⊗[F] C.toAffine.FunctionField) x.1 := by
-  letI alg := C.tensorFunctionFieldAlgebra L
-  letI := C.isDomain_tensorCoordRing L
-  haveI hflatL : Module.Flat F L := Module.Flat.of_free
+  let alg := C.tensorFunctionFieldAlgebra L
+  let := C.isDomain_tensorCoordRing L
+  have hflatL : Module.Flat F L := Module.Flat.of_free
   have halg : ∀ (l : L) (u : C.toAffine.CoordinateRing),
       algebraMap (L ⊗[F] C.toAffine.CoordinateRing) (L ⊗[F] C.toAffine.FunctionField) (l ⊗ₜ u)
         = l ⊗ₜ algebraMap C.toAffine.CoordinateRing C.toAffine.FunctionField u :=
     fun l u ↦ rfl
   refine fun z ↦ ?_
-  induction z using TensorProduct.induction_on with
-  | zero =>
-      exact ⟨(0, 1), by rw [zero_mul, map_zero]⟩
+  induction z using TensorProduct.inductionOn with
   | tmul l f =>
       obtain ⟨a, b, hbmem, hab⟩ := IsFractionRing.div_surjective (A := C.toAffine.CoordinateRing) f
       refine ⟨(l ⊗ₜ a, ⟨1 ⊗ₜ b, C.oneTmul_mem_nonZeroDivisors L b hbmem⟩), ?_⟩
@@ -895,8 +872,8 @@ theorem tensor_functionField_isFractionRing (L : Type*) [Field L] [Algebra F L]
     [Algebra.IsAlgebraic F L] :
     @IsFractionRing (L ⊗[F] C.toAffine.CoordinateRing) _
       (L ⊗[F] C.toAffine.FunctionField) _ (C.tensorFunctionFieldAlgebra L) := by
-  letI alg := C.tensorFunctionFieldAlgebra L
-  letI := C.isDomain_tensorCoordRing L
+  let alg := C.tensorFunctionFieldAlgebra L
+  let := C.isDomain_tensorCoordRing L
   have hinj : Function.Injective
       (algebraMap (L ⊗[F] C.toAffine.CoordinateRing) (L ⊗[F] C.toAffine.FunctionField)) :=
     C.tensorFunctionFieldStructureHom_injective L
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/InseparableDegree.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/InseparableDegree.lean
index ebd777c8c..9132c544f 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/InseparableDegree.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/InseparableDegree.lean
@@ -8,29 +8,13 @@ import Mathlib.FieldTheory.PurelyInseparable.Basic
 import Mathlib.FieldTheory.SeparableClosure
 
 /-!
-# Inseparable degree API for isogenies (Silverman II.2.10-12)
+For an elliptic-curve isogeny, the separable degree times the inseparable degree
+equals the function-field degree. In positive characteristic the inseparable
+degree is a power of the characteristic. The separable closure of the pullback
+field gives the maximal intermediate field over which the extension is purely
+inseparable.
 
-This file builds the inseparable-degree / separable-closure API for
-elliptic-curve isogenies, building on mathlib's
-`Mathlib.FieldTheory.SeparableClosure` (`Field.sepDegree`,
-`Field.finInsepDegree`, `separableClosure`) and
-`Mathlib.FieldTheory.PurelyInseparable.Basic` (`IsPurelyInseparable`,
-`isPurelyInseparable_iff_pow_mem`, `IsPurelyInseparable.pow_mem`).
-
-## Main results
-
-* `Isogeny.inseparableDegree_dvd_degree` — the inseparable degree
-  divides the degree (consequence of `finSepDegree · finInsepDegree =
-  finrank`, mathlib `PurelyInseparable/Basic.lean:595`).
-* `Isogeny.inseparableDegree_eq_one_iff_separable` — direct from the
-  project's definition `IsSeparable := inseparableDegree = 1`.
-* `Isogeny.inseparableDegree_isPow_of_charP` — in characteristic `p`,
-  the inseparable degree is a power of `p`. Uses
-  `isPurelyInseparable_iff_pow_mem` (mathlib).
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], II.2.10-12.
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.2.10–12.
 -/
 
 open WeierstrassCurve
@@ -70,7 +54,7 @@ theorem separableDegree_mul_finInsepDegree (α : Isogeny W W) :
     α.separableDegree *
         Field.finInsepDegree W.FunctionField W.FunctionField =
       α.degree := by
-  letI : Algebra W.FunctionField W.FunctionField := α.toAlgebra
+  let : Algebra W.FunctionField W.FunctionField := α.toAlgebra
   rw [separableDegree_eq_finSepDegree, degree_eq_finrank]
   exact Field.finSepDegree_mul_finInsepDegree W.FunctionField W.FunctionField
 
@@ -83,7 +67,7 @@ theorem inseparableDegree_eq_finInsepDegree (α : Isogeny W W)
     letI : Algebra W.FunctionField W.FunctionField := α.toAlgebra
     α.inseparableDegree =
       Field.finInsepDegree W.FunctionField W.FunctionField := by
-  letI : Algebra W.FunctionField W.FunctionField := α.toAlgebra
+  let : Algebra W.FunctionField W.FunctionField := α.toAlgebra
   -- inseparableDegree = degree / separableDegree
   -- = (separableDegree * finInsepDegree) / separableDegree
   -- = finInsepDegree (since separableDegree > 0)
@@ -102,7 +86,7 @@ theorem inseparableDegree_dvd_degree (α : Isogeny W W) :
     α.inseparableDegree ∣ α.degree := by
   -- inseparableDegree = degree / separableDegree; degree = separableDegree * finInsepDegree.
   -- So separableDegree | degree, hence (degree / separableDegree) | degree.
-  letI : Algebra W.FunctionField W.FunctionField := α.toAlgebra
+  let : Algebra W.FunctionField W.FunctionField := α.toAlgebra
   have h_mul := separableDegree_mul_finInsepDegree α
   show α.degree / α.separableDegree ∣ α.degree
   rw [show α.degree = α.separableDegree *
@@ -130,15 +114,15 @@ theorem inseparableDegree_isPow_of_charP
     (α : Isogeny W.toAffine W.toAffine)
     (h_deg_pos : 0 < α.degree) :
     ∃ e : ℕ, α.inseparableDegree = p ^ e := by
-  letI alg : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  let alg : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
   -- CharP transports along the algebraMap (it's a ringHom into a field of char p)
-  haveI : CharP W.toAffine.FunctionField p :=
+  have : CharP W.toAffine.FunctionField p :=
     charP_of_injective_algebraMap
       (algebraMap K W.toAffine.FunctionField).injective p
   -- ExpChar from CharP + Prime
-  haveI : ExpChar W.toAffine.FunctionField p := ExpChar.prime Fact.out
+  have : ExpChar W.toAffine.FunctionField p := ExpChar.prime Fact.out
   -- From h_deg_pos: FiniteDimensional via the matching finrank
-  haveI hfin : @FiniteDimensional W.toAffine.FunctionField
+  have hfin : @FiniteDimensional W.toAffine.FunctionField
       W.toAffine.FunctionField _ _ alg.toModule :=
     @FiniteDimensional.of_finrank_pos _ _ _ _ alg.toModule h_deg_pos
   -- Pull down the mathlib lemma with explicit instances
@@ -155,7 +139,7 @@ theorem inseparableDegree_isPow_of_charP
   rw [inseparableDegree_eq_finInsepDegree α h_sep_pos]
   exact he
 
-/-! ### P0-B — `Isogeny.separableSubfield`
+/-! The separable subfield
 
 The maximal intermediate field over which `K(E)` is purely inseparable.
 Direct specialisation of `Mathlib.FieldTheory.separableClosure` to
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/PoleOrderParity.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/PoleOrderParity.lean
index 07d5e7569..e5fe1c773 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/PoleOrderParity.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/PoleOrderParity.lean
@@ -183,7 +183,7 @@ theorem point_minus_O_principal_eq_zero_of_coord
       have := congr_arg ProjectiveSmoothPoint.toAffinePoint h
       rwa [P.toProjectiveSmoothPoint_toAffinePoint,
         ProjectiveSmoothPoint.toAffinePoint_infinity] at this
-    rw [if_neg h_ne, if_pos rfl]
+    rw [ite_eq_right h_ne, ite_eq_left rfl]
     decide
   -- Use projectiveDivisorOf_apply_infinity to convert: -1 = (ordAtInfty f).untopD 0
   rw [SmoothPlaneCurve.projectiveDivisorOf_apply_infinity] at h_inf
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/RamificationAtInfinity.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/RamificationAtInfinity.lean
index c09705086..7142cc012 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/RamificationAtInfinity.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Ramification/RamificationAtInfinity.lean
@@ -3,82 +3,21 @@ Copyright (c) 2026 Chris Birkbeck. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Birkbeck
 -/
-import Mathlib.NumberTheory.RamificationInertia.Basic
+import Mathlib.RingTheory.RamificationInertia.Basic
+import Mathlib.LinearAlgebra.Dimension.Localization
+import Mathlib.NumberTheory.RamificationInertia.Ramification
 import Mathlib.RingTheory.DedekindDomain.IntegralClosure
 
 /-!
-# Abstract ramification at infinity (Worker A Action 1)
-
-For a smooth plane curve `C` over a field `k` and a nonconstant function
-`f ∈ K(C)`, this file establishes the **function-field-degree-equals-pole-
-divisor-degree identity**:
-
-```
-[K(C) : k(f)] = Σ_{P pole of f} (- ord_P f) · [κ(P) : k].
-```
-
-This is *not* Riemann–Roch. It is the degree formula for the finite morphism
-`f : C → ℙ¹`, scoped to the fibre at `∞`. The proof goes via Mathlib's
-`Ideal.sum_ramification_inertia` applied at the `(u)`-prime of `k[u]`, where
-`u = 1/f` is the uniformizer at infinity on `ℙ¹`:
-
-  * `e_P = ord_P(u) = -ord_P(f)` (ramification index),
-  * `f_P = [κ(P) : κ((u))] = [κ(P) : k]` (inertia degree).
-
-## Strategy
-
-The proof packages Mathlib's abstract ramification–inertia machinery: pick
-the integral closure `S` of `k[u]` in `K(C)` (via the algebra structure
-`u ↦ 1/f`), observe `S` is Dedekind, then `Ideal.sum_ramification_inertia`
-at the `(u)` prime gives the sum directly. The curve-specific notation
-(`poleSupport`, `ordAt`, `kappa`) is bookkeeping over the resulting finset.
-
-This is the deliverable for "Worker A — Action 1 abstract ramification-at-
-infinity setup". Worker B's closing application combines this with their
-Action 2 (translation valuation transport) plus Action 3 (a small closed-
-point lemma) to discharge the project's `pc_sepDeg_eq_pointCount` witness.
-
-## Main definitions
-
-* `polyToFieldOfInv f`, `ratFunToFieldOfInv hf` — the `k`-algebra maps
-  `Polynomial k →ₐ[k] L` and `FractionRing (Polynomial k) →ₐ[k] L` sending the
-  indeterminate to `f⁻¹` (the uniformizer `u = 1/f`).
-* `LinfAt f` — `L` as a type synonym carrying the `X ↦ f⁻¹` polynomial-algebra
-  structure, so it does not clash with the project's pre-existing `X ↦ coordX`
-  instance.
-* `Sinf f` — structural package of "an integral closure of `Polynomial k` in
-  `LinfAt f`": the carrier plus every typeclass instance the closing theorem
-  needs as an honest field.
-* `xIdeal`, `Sinf.ordAt`, `Sinf.kappa` — the prime `(X)` at infinity, the order
-  `-(ramificationIdx' …)`, and the residue field `carrier ⧸ P`.
-
-## Main results
-
-* `finrank_eq_sum_ramificationIdx_mul_inertiaDeg` — the abstract fundamental
-  identity `Σ e(P) · f(P) = [LinfAt f : k(f)]` from `Ideal.sum_ramification_inertia`.
-* `finrank_eq_weighted_poleDegree_of_nonconstant` — the closing corollary
-  `[L : k(f)] = Σ_P (−ord_P f) · inertiaDeg' P`.
-* `Sinf.inertiaDeg_eq_one_of_algebraMap_surjective` — `inertiaDeg' (X) P = 1`
-  when the residue map `k → κ(P)` is surjective.
-
-## Implementation notes
-
-`LinfAt` and `Sinf` exist to dodge two Lean-elaboration hazards. `LinfAt f`
-isolates the `X ↦ f⁻¹` algebra structure from the project's `X ↦ coordX` one;
-transcendence is carried as `Fact (Transcendental k f⁻¹)` (use sites declare
-`haveI : Fact (Transcendental k f⁻¹) := ⟨_⟩` once and the algebra instances
-resolve). `Sinf` bundles the integral-closure instances as record fields so they
-resolve by projection, sidestepping the `Meta.SynthInstance.tryResolve`
-negative-cache anomaly that the `↥(integralClosure …)` Subalgebra coercion
-triggers under `Ideal.sum_ramification_inertia` (commit 538ff64).
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, II.2 (degree of a
-  morphism between curves; formula II.2.6.a as the sum over preimages of
-  one point).
-* Mathlib's `Ideal.sum_ramification_inertia` provides the abstract
-  fundamental identity `Σ e · f = [Frac S : Frac R]`.
+# Ramification at infinity
+
+For a nonconstant function on a smooth curve, the degree of the function-field
+extension equals the degree of its pole divisor. Applying the fundamental
+ramification–inertia identity to the prime `(X)` of `k[X]`, embedded by
+`X ↦ f⁻¹`, identifies its ramification indices with pole orders and its
+inertia degrees with residue-field degrees.
+
+See Silverman, *The Arithmetic of Elliptic Curves*, II.2.
 -/
 
 namespace HasseWeil.Curves
@@ -144,11 +83,7 @@ theorem ratFunToFieldOfInv_injective {L : Type*} [Field L] [Algebra k L]
     Function.Injective (ratFunToFieldOfInv hf) :=
   (ratFunToFieldOfInv hf).toRingHom.injective
 
-/-- The function field viewed as a type-synonym, used to install a
-`Polynomial k`-algebra structure via `X ↦ f⁻¹` that does not conflict with
-the project's existing `X ↦ coordX` instance.
-The unused argument `_f` indexes the algebra structure intended for the
-synonym. -/
+
 @[nolint unusedArguments]
 def LinfAt {L : Type*} [Field L] [Algebra k L] (_f : L) : Type _ := L
 
@@ -238,12 +173,7 @@ instance isScalarTower_k_fractionRing
 
 end LinfAt
 
-/-- Structural package of "an integral closure of `Polynomial k` inside
-`LinfAt f`": carrier type plus every typeclass instance the closing
-`Ideal.sum_ramification_inertia` invocation requires, bundled as record
-fields so `letI` can install them at the use site. Sidesteps the Lean 4
-synthesis anomaly that affects the `↥(integralClosure …)` Subalgebra
-coercion (commit 538ff64). -/
+
 structure Sinf.{u} {L : Type*} [Field L] [Algebra k L] (f : L) where
   /-- Carrier type — canonically `↥(integralClosure (Polynomial k) (LinfAt f))`. -/
   carrier : Type u
@@ -262,18 +192,14 @@ structure Sinf.{u} {L : Type*} [Field L] [Algebra k L] (f : L) where
   [isFractionRing : IsFractionRing carrier (LinfAt (k := k) f)]
   /-- Compatible scalar tower with the ambient `Polynomial k → LinfAt f`. -/
   [isScalarTower : IsScalarTower (Polynomial k) carrier (LinfAt (k := k) f)]
-  /-- Finite as a `Polynomial k`-module. -/
+
   [moduleFinite : Module.Finite (Polynomial k) carrier]
-  /-- Torsion-free as a `Polynomial k`-module (algebraMap injective). -/
+
   [isTorsionFree : Module.IsTorsionFree (Polynomial k) carrier]
 
 section SinfConstruction
 
 
-/-- Canonical construction of `Sinf` from Mathlib's `integralClosure`
-Subalgebra. Given the ambient finite-separability hypotheses, all the
-required instances are synthesised once here; downstream code consumes
-them only by field projection. -/
 noncomputable def Sinf.ofIntegralClosure
     {L : Type*} [Field L] [Algebra k L] (f : L) [Fact (Transcendental k f⁻¹)]
     [Module.Finite (FractionRing (Polynomial k)) (LinfAt (k := k) f)]
@@ -288,13 +214,13 @@ noncomputable def Sinf.ofIntegralClosure
       (integralClosure (Polynomial k) (LinfAt (k := k) f)) :=
     IsIntegralClosure.finite (Polynomial k) (FractionRing (Polynomial k))
       (LinfAt (k := k) f) (integralClosure (Polynomial k) (LinfAt (k := k) f))
-  haveI : FaithfulSMul (Polynomial k) (LinfAt (k := k) f) := by
+  have : FaithfulSMul (Polynomial k) (LinfAt (k := k) f) := by
     rw [faithfulSMul_iff_algebraMap_injective]
     exact polyToFieldOfInv_injective_of_transcendental
       (Fact.out (p := Transcendental k f⁻¹))
-  haveI : Module.IsTorsionFree (Polynomial k) (LinfAt (k := k) f) :=
+  have : Module.IsTorsionFree (Polynomial k) (LinfAt (k := k) f) :=
     inferInstance
-  haveI : Module.IsTorsionFree (Polynomial k)
+  have : Module.IsTorsionFree (Polynomial k)
       (integralClosure (Polynomial k) (LinfAt (k := k) f)) :=
     Subalgebra.instIsTorsionFree (integralClosure (Polynomial k)
       (LinfAt (k := k) f))
@@ -316,13 +242,7 @@ theorem xIdeal_ne_bot : (xIdeal (k := k)) ≠ ⊥ := by
   rw [xIdeal, Ne, Ideal.span_singleton_eq_bot]
   exact Polynomial.X_ne_zero
 
-/-- **Abstract fundamental ramification–inertia identity at infinity.**
-For a function `f` realising `LinfAt f` as a finite separable extension of
-`FractionRing (Polynomial k) = k(f)` and an integral-closure package
-`data : Sinf k f`, the sum over primes of the carrier lying over `(X)` of
-`e(P) · f(P)` equals `[LinfAt f : k(f)]`. This is the abstract content of
-the `[K(C):k(f)] = Σ (pole degree)` identity, before specialisation to
-`LinfAt f = K(C)` for a smooth curve `C`. -/
+
 theorem finrank_eq_sum_ramificationIdx_mul_inertiaDeg
     {L : Type*} [Field L] [Algebra k L] (f : L) [Fact (Transcendental k f⁻¹)]
     [Module.Finite (FractionRing (Polynomial k)) (LinfAt (k := k) f)]
@@ -332,18 +252,42 @@ theorem finrank_eq_sum_ramificationIdx_mul_inertiaDeg
     letI := data.algPoly
     ∑ P ∈ IsDedekindDomain.primesOverFinset (xIdeal (k := k)) data.carrier,
         Ideal.ramificationIdx' (xIdeal (k := k)) P *
-        Ideal.inertiaDeg' (xIdeal (k := k)) P =
+        P.inertiaDeg (Polynomial k) =
       Module.finrank (FractionRing (Polynomial k)) (LinfAt (k := k) f) := by
-  letI := data.commRing
-  letI := data.isDomain
-  letI := data.isDedekindDomain
-  letI := data.algPoly
-  letI := data.algLinfAt
-  letI := data.isFractionRing
-  letI := data.isScalarTower
-  letI := data.moduleFinite
-  exact Ideal.sum_ramification_inertia (R := Polynomial k) data.carrier
-    (FractionRing (Polynomial k)) (LinfAt (k := k) f) xIdeal_ne_bot
+  let := data.commRing
+  let := data.isDomain
+  let := data.isDedekindDomain
+  let := data.algPoly
+  let := data.algLinfAt
+  let := data.isFractionRing
+  let := data.isScalarTower
+  let := data.moduleFinite
+  classical
+  let := data.isTorsionFree
+  let : Fintype ((xIdeal (k := k)).primesOver data.carrier) := Fintype.ofFinite _
+  calc
+    _ = ∑ P : (xIdeal (k := k)).primesOver data.carrier,
+        P.1.ramificationIdx (Polynomial k) * P.1.inertiaDeg (Polynomial k) := by
+      apply Finset.sum_bij (fun P hP ↦
+        ⟨P, (IsDedekindDomain.mem_primesOverFinset_iff xIdeal_ne_bot data.carrier).mp hP⟩)
+      · intro P hP
+        exact Finset.mem_univ _
+      · intro P hP Q hQ h
+        exact congrArg Subtype.val h
+      · intro P _
+        exact ⟨P.1,
+          (IsDedekindDomain.mem_primesOverFinset_iff xIdeal_ne_bot data.carrier).mpr P.2, rfl⟩
+      · intro P hP
+        obtain ⟨hprime, hover⟩ :=
+          (IsDedekindDomain.mem_primesOverFinset_iff xIdeal_ne_bot data.carrier).mp hP
+        let := hprime
+        let := hover
+        rw [Ideal.ramificationIdx'_eq_ramificationIdx (xIdeal (k := k)) P xIdeal_ne_bot]
+    _ = Module.finrank (Polynomial k) data.carrier :=
+      Ideal.sum_ramification_inertia_eq_finrank (xIdeal (k := k)) data.carrier
+    _ = Module.finrank (FractionRing (Polynomial k)) (LinfAt (k := k) f) :=
+      (IsFractionRing.finrank_eq (Polynomial k) (FractionRing (Polynomial k))
+        data.carrier (LinfAt (k := k) f)).symm
 
 /-- The `R[X] ⧸ (X) ≃ₐ[R] R`-style algebra equivalence specialised to
 `R = k`, given by evaluation at `0`. -/
@@ -374,8 +318,8 @@ theorem Sinf.toNat_neg_ordAt_eq_ramificationIdx
     ∀ P : Ideal data.carrier,
       (-(data.ordAt P)).toNat =
         Ideal.ramificationIdx' (xIdeal (k := k)) P := by
-  letI := data.commRing
-  letI := data.algPoly
+  let := data.commRing
+  let := data.algPoly
   intro P
   simp [Sinf.ordAt]
 
@@ -415,16 +359,34 @@ theorem Sinf.inertiaDeg_eq_finrank_kappa
     letI := data.commRing
     letI := data.algPoly
     ∀ (P : Ideal data.carrier) [P.LiesOver (xIdeal (k := k))],
-      Ideal.inertiaDeg' (xIdeal (k := k)) P =
-        Module.finrank (Polynomial k ⧸ xIdeal (k := k)) (data.kappa P) := by
-  letI := data.commRing
-  letI := data.algPoly
+      (if h : P.comap (algebraMap (Polynomial k) data.carrier) = xIdeal (k := k) then
+        letI : Algebra (Polynomial k ⧸ xIdeal (k := k)) (data.carrier ⧸ P) :=
+          Ideal.Quotient.algebraQuotientOfLEComap h.ge
+        Module.finrank (Polynomial k ⧸ xIdeal (k := k)) (data.kappa P)
+      else 0) = Module.finrank (Polynomial k ⧸ xIdeal (k := k)) (data.kappa P) := by
+  let := data.commRing
+  let := data.algPoly
   intro P _
-  exact Ideal.inertiaDeg'_algebraMap _ _
+  simp only [dite_eq_left (Ideal.over_def P (xIdeal (k := k))).symm]
+
+/-- For a prime above `(X)`, the inertia degree is its residue-field degree. -/
+theorem Sinf.prime_inertiaDeg_eq_finrank_kappa
+    {L : Type*} [Field L] [Algebra k L] {f : L}
+    (data : Sinf (k := k) f) :
+    letI := data.commRing
+    letI := data.algPoly
+    ∀ (P : Ideal data.carrier) [P.IsPrime] [P.LiesOver (xIdeal (k := k))],
+      P.inertiaDeg (Polynomial k) =
+        Module.finrank (Polynomial k ⧸ xIdeal (k := k)) (data.kappa P) := by
+  let := data.commRing
+  let := data.algPoly
+  let := data.moduleFinite
+  intro P _ _
+  let : P.IsMaximal := Ideal.isMaximal_of_isIntegral_of_isMaximal_under (R := Polynomial k) P (by
+    simpa only [← Ideal.over_def P (xIdeal (k := k))] using (xIdeal_isMaximal (k := k)))
+  exact Ideal.inertiaDeg_eq_of_isMaximal (xIdeal (k := k)) P
+
 
-/-- The `Polynomial k ⧸ xIdeal`-finrank of a module agrees with its
-`k`-finrank, given a compatible scalar tower `k → (Polynomial k ⧸ xIdeal)
-→ M`. Uses `quotientXAlgEquiv` plus the tower formula. -/
 theorem finrank_residue_eq_finrank_k
     {M : Type*} [AddCommGroup M] [Module (Polynomial k ⧸ xIdeal (k := k)) M]
     [Module k M] [IsScalarTower k (Polynomial k ⧸ xIdeal (k := k)) M]
@@ -439,7 +401,7 @@ theorem finrank_residue_eq_finrank_k
 /-- **Abstract inertia-degree-one criterion at `(X)`.** For an *arbitrary* prime `P` of the
 `Sinf` carrier lying over `xIdeal := (X)`, if the structure algebra map
 `(Polynomial k ⧸ (X)) → (carrier ⧸ P)` (`≅ k → κ(P)`) is surjective, then the inertia
-degree `inertiaDeg' (X) P` equals `1`. This is the field-agnostic, `CoordinateRing`-free,
+degree `P.inertiaDeg (Polynomial k)` equals `1`. This is the field-agnostic, `CoordinateRing`-free,
 `IsAlgClosed`-free criterion for any prime over `(X)`. -/
 theorem Sinf.inertiaDeg_eq_one_of_algebraMap_surjective
     {L : Type*} [Field L] [Algebra k L] {f : L}
@@ -455,31 +417,26 @@ theorem Sinf.inertiaDeg_eq_one_of_algebraMap_surjective
         (algebraMap (Polynomial k ⧸ xIdeal (k := k)) (data.carrier ⧸ P))) :
     letI := data.commRing
     letI := data.algPoly
-    Ideal.inertiaDeg' (xIdeal (k := k)) P = 1 := by
-  letI := data.commRing
-  letI := data.algPoly
-  letI := data.isDomain
-  letI := data.moduleFinite
-  haveI hmax : (xIdeal (k := k)).IsMaximal := xIdeal_isMaximal
-  letI : Algebra (Polynomial k ⧸ xIdeal (k := k)) (data.carrier ⧸ P) :=
+    P.inertiaDeg (Polynomial k) = 1 := by
+  let := data.commRing
+  let := data.algPoly
+  let := data.isDomain
+  let := data.moduleFinite
+  have hmax : (xIdeal (k := k)).IsMaximal := xIdeal_isMaximal
+  let : Algebra (Polynomial k ⧸ xIdeal (k := k)) (data.carrier ⧸ P) :=
     Ideal.Quotient.algebraQuotientOfLEComap
       (Ideal.LiesOver.over (p := xIdeal (k := k)) (P := P)).le
-  rw [Ideal.inertiaDeg'_algebraMap]
+  let : P.IsMaximal := Ideal.isMaximal_of_isIntegral_of_isMaximal_under (R := Polynomial k) P (by
+    simpa only [← Ideal.over_def P (xIdeal (k := k))] using (xIdeal_isMaximal (k := k)))
+  rw [Ideal.inertiaDeg_eq_of_isMaximal (xIdeal (k := k)) P]
   refine le_antisymm ?_ ?_
   · refine finrank_le_one (1 : data.carrier ⧸ P) fun w ↦ ?_
     obtain ⟨c, hc⟩ := h_surj w
     exact ⟨c, by rwa [Algebra.algebraMap_eq_smul_one] at hc⟩
-  · have hpos := Ideal.inertiaDeg'_pos (xIdeal (k := k)) P
-    rwa [Ideal.inertiaDeg'_algebraMap] at hpos
-
-/-- **Closing corollary (abstract form).** The function-field degree
-`[L : k(f)]` over the Sinf data equals the weighted pole-divisor degree
-`Σ_P (−ord_P f) · f_P` (with `f_P = inertiaDeg' xIdeal P`). This is the
-abstract form of the `[K(C) : k(f)] = Σ_{P pole of f} (−ord_P f) · [κ(P) :
-k]` identity; Worker B's PoleDivisorFallback specialisation substitutes
-`L = K(E)`, `f = γ*x_gen`, and converts `inertiaDeg'` to `[κ(P) : k]`
-using `inertiaDeg_eq_finrank_kappa` plus the `quotientXAlgEquiv`
-`Polynomial k ⧸ xIdeal ≃ k`. -/
+  · exact Nat.succ_le_of_lt (Module.finrank_pos :
+      0 < Module.finrank (Polynomial k ⧸ xIdeal (k := k)) (data.carrier ⧸ P))
+
+
 theorem finrank_eq_weighted_poleDegree_of_nonconstant
     {L : Type*} [Field L] [Algebra k L] (f : L) [Fact (Transcendental k f⁻¹)]
     [Module.Finite (FractionRing (Polynomial k)) (LinfAt (k := k) f)]
@@ -490,16 +447,16 @@ theorem finrank_eq_weighted_poleDegree_of_nonconstant
     Module.finrank (FractionRing (Polynomial k)) (LinfAt (k := k) f) =
     ∑ P ∈ IsDedekindDomain.primesOverFinset (xIdeal (k := k)) data.carrier,
       (-(data.ordAt P)).toNat *
-        Ideal.inertiaDeg' (xIdeal (k := k)) P := by
-  letI := data.commRing
-  letI := data.algPoly
-  letI := data.isDomain
-  letI := data.isDedekindDomain
-  letI := data.algLinfAt
-  letI := data.isFractionRing
-  letI := data.isScalarTower
-  letI := data.moduleFinite
-  letI := data.isTorsionFree
+        P.inertiaDeg (Polynomial k) := by
+  let := data.commRing
+  let := data.algPoly
+  let := data.isDomain
+  let := data.isDedekindDomain
+  let := data.algLinfAt
+  let := data.isFractionRing
+  let := data.isScalarTower
+  let := data.moduleFinite
+  let := data.isTorsionFree
   have h := finrank_eq_sum_ramificationIdx_mul_inertiaDeg
     (k := k) (L := L) f data
   rw [← h]
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Transcendence.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Transcendence.lean
index bdce413c5..a6490d166 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Transcendence.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Transcendence.lean
@@ -16,19 +16,6 @@ that a nonconstant curve map `φ : C₁ → C₂` induces a finite extension
 `K(C₁) / φ*(K(C₂))`: the range of `φ.pullback` is an intermediate ring
 containing the algebraic structure of `F[x]`, and transitivity does the rest.
 
-This closes the *first piece* of the infrastructure needed for
-T-II-INFRA-B-009 (Silverman II.2.4(a), `CurveMap.pullback_image_finite_codim`).
-The full finiteness still requires transcendence-degree theory, which
-mathlib provides via `IsTranscendenceBasis`/`trdeg` but whose bridge to our
-subalgebra setting (`Algebra.adjoin F {C.coordX}`) needs additional
-typeclass-plumbing. That bridge is tracked as a follow-up.
-
-## Main results
-
-- `SmoothPlaneCurve.isAlgebraic_polynomialX_functionField`: `F(C)` is
-  algebraic over `F[X]` (instance). Consequence of
-  `IsLocalization.isAlgebraic` + `Algebra.IsAlgebraic.of_finite` + transitivity.
-
 ## References
 
 * [Silverman, *The Arithmetic of Elliptic Curves*], II.1, II.2.4
@@ -49,29 +36,19 @@ algebraic over the base), `F(x) → F(C)` is algebraic (finite of rank 2), and
 algebraicness is transitive. -/
 instance isAlgebraic_polynomialX_functionField :
     Algebra.IsAlgebraic (Polynomial F) C.FunctionField := by
-  haveI : Algebra.IsAlgebraic (Polynomial F) (FractionRing (Polynomial F)) :=
+  have : Algebra.IsAlgebraic (Polynomial F) (FractionRing (Polynomial F)) :=
     IsLocalization.isAlgebraic _ (nonZeroDivisors _)
-  haveI : Algebra.IsAlgebraic (FractionRing (Polynomial F)) C.FunctionField :=
+  have : Algebra.IsAlgebraic (FractionRing (Polynomial F)) C.FunctionField :=
     Algebra.IsAlgebraic.of_finite _ _
   exact Algebra.IsAlgebraic.trans (R := Polynomial F)
     (S := FractionRing (Polynomial F)) (A := C.FunctionField)
 
-/-- Every element of `F(C)` is integral over `F[X]`: since `F[X]` is a field
-(well, an integral domain), integrality and algebraicness coincide over it
-only for the algebraic case when the ring is a field. Here, since `F[X]`
-is an ID and `F(C)` is a domain, we can state integrality directly.
-
-**Remark**: `F[X]` is NOT a field, so `IsAlgebraic` and `IsIntegral` over
-`F[X]` are distinct notions in general; however, an element of `F(C)` that
-is a root of any nonzero polynomial over `F[X]` is a root of SOME monic
-polynomial (divide by the leading coefficient after clearing denominators
-in the fraction field). We don't need this strengthening for the main
-application. -/
+/-- The function field is integral over the fraction field of the polynomial ring. -/
 instance isIntegral_fracPolynomialX_functionField :
     Algebra.IsIntegral (FractionRing (Polynomial F)) C.FunctionField := by
   exact (Algebra.IsAlgebraic.of_finite (FractionRing (Polynomial F)) C.FunctionField).isIntegral
 
-/-! ### T-II-1-005: Uniformizer is not a `p`-th power in char `p > 0` -/
+
 
 /-- **Silverman Exercise II.2.15 (⇐), char-`p` half**: if `t ∈ F(C)` is a
 uniformizer at some smooth point `P`, then `t` is not a `p`-th power in
@@ -103,7 +80,7 @@ theorem notMem_pthPowers_of_uniformizer {p : ℕ} (hp : 1 < p)
 `F(C)` is finite over `Frac(F[X]) = F(x)`, so trdeg = 1 by additivity of
 transcendence degree across the tower `F → F[X] → F(x) → F(C)`. -/
 
-/-- **trdeg F K(E) = 1 (axiom-clean)**: the function field of a smooth
+/-- **trdeg F K(E) = 1**: the function field of a smooth
 plane curve has transcendence degree 1 over the base field.
 
 Tower computation: `trdeg F (Polynomial F) = 1` (mathlib's
@@ -113,11 +90,11 @@ Tower computation: `trdeg F (Polynomial F) = 1` (mathlib's
 `FiniteOverKx`). Apply `trdeg_add_eq` twice. -/
 theorem functionField_trdeg_eq_one : Algebra.trdeg F C.FunctionField = 1 := by
   have h_fund : Algebra.trdeg F (FractionRing (Polynomial F)) = 1 := by
-    haveI : Algebra.IsAlgebraic (Polynomial F) (FractionRing (Polynomial F)) :=
+    have : Algebra.IsAlgebraic (Polynomial F) (FractionRing (Polynomial F)) :=
       IsLocalization.isAlgebraic _ (nonZeroDivisors _)
     rw [← trdeg_add_eq F (Polynomial F) (A := FractionRing (Polynomial F)),
       Polynomial.trdeg_of_isDomain, trdeg_eq_zero, add_zero]
-  haveI : Algebra.IsAlgebraic (FractionRing (Polynomial F)) C.FunctionField :=
+  have : Algebra.IsAlgebraic (FractionRing (Polynomial F)) C.FunctionField :=
     Algebra.IsAlgebraic.of_finite _ _
   rw [← trdeg_add_eq F (FractionRing (Polynomial F)) (A := C.FunctionField), h_fund,
     trdeg_eq_zero, add_zero]
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/AlgebraicNonNegOrd.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/AlgebraicNonNegOrd.lean
index c599cad55..ad80470ae 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/AlgebraicNonNegOrd.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/AlgebraicNonNegOrd.lean
@@ -84,7 +84,7 @@ theorem ord_P_nonneg_of_isAlgebraic
   · rw [hf, ord_P_zero]; exact le_top
   · have hv : C.pointValuation P f ≠ 0 := (C.pointValuation P).ne_zero_iff.mpr hf
     simp only [ord_P]
-    rw [dif_neg hv, show (0 : WithTop ℤ) = ((0 : ℤ) : WithTop ℤ) from rfl,
+    rw [dite_eq_right hv, show (0 : WithTop ℤ) = ((0 : ℤ) : WithTop ℤ) from rfl,
       WithTop.coe_le_coe]
     -- The order is `-toAdd (unzero hv)`, so `≤ 1` for the valuation becomes
     -- `toAdd (unzero hv) ≤ toAdd 1 = 0`, giving `0 ≤ -toAdd (unzero hv)`.
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Infinity.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Infinity.lean
index bf9827ace..e8e5e12b2 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Infinity.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Infinity.lean
@@ -22,25 +22,21 @@ with the classical values
 ordAtInfty(x) = -2,   ordAtInfty(y) = -3.
 ```
 
-Rather than building a local ring at infinity and extracting the valuation
-from its DVR structure (the approach outlined in the Phase-D ticket
-`T-II-INFRA-D-001`), this file defines `ordAtInfty` algebraically via the
-algebra norm `N : F(C) → F(x)`:
+The order at infinity is defined algebraically using the norm
+`N : F(C) → F(x)`:
 
 ```
 ordAtInfty(f) := - intDegree (N(f)) ∈ ℤ ∪ {∞}.
 ```
 
-The rationale is that for a degree-2 extension of `F(x)`, ramified only at
-the place at infinity of `ℙ¹_F` with ramification index 2 and residue
+The rationale is that when the unique place of a degree-2 extension of `F(x)` above
+the place at infinity of `ℙ¹_F` has ramification index 2 and residue
 degree 1, the valuation at the unique place of `F(C)` above infinity is
 related to the ℙ¹-valuation of the norm by the classical formula
 `v_K(N(f)) = f_w · w(f)` with `f_w = 1`. Concretely,
 `ordAtInfty(f) = ordAtInfty(N(f))` where the right-hand side is the order at
 infinity on `F(x)`, computed by `- intDegree`.
 
-This closes (partial form) ticket `T-II-INFRA-D-002`.
-
 ## References
 
 * [Silverman, *The Arithmetic of Elliptic Curves*], IV.1 (for
@@ -87,7 +83,7 @@ noncomputable def ordAtInfty (f : C.FunctionField) : WithTop ℤ :=
   if f = 0 then ⊤
   else ((- RatFunc.intDegree (C.normAsRatFunc f) : ℤ) : WithTop ℤ)
 
-@[simp] theorem ordAtInfty_zero : C.ordAtInfty 0 = ⊤ := if_pos rfl
+@[simp] theorem ordAtInfty_zero : C.ordAtInfty 0 = ⊤ := ite_eq_left rfl
 
 theorem ordAtInfty_eq_top_iff (f : C.FunctionField) :
     C.ordAtInfty f = ⊤ ↔ f = 0 := by
@@ -98,7 +94,7 @@ theorem ordAtInfty_eq_top_iff (f : C.FunctionField) :
 
 theorem ordAtInfty_of_ne {f : C.FunctionField} (hf : f ≠ 0) :
     C.ordAtInfty f = (- RatFunc.intDegree (C.normAsRatFunc f) : ℤ) :=
-  if_neg hf
+  ite_eq_right hf
 
 /-- Multiplicativity of `ordAtInfty`: for nonzero `f, g`,
 `ordAtInfty(f · g) = ordAtInfty(f) + ordAtInfty(g)`. -/
@@ -950,7 +946,7 @@ private theorem degree_sub_C_leadingCoeff_div_mul_lt {n d : Polynomial F}
   have h_deg : n.degree = (Polynomial.C lam * d).degree := by
     rw [Polynomial.degree_C_mul hlam_ne, Polynomial.degree_eq_natDegree hn,
       Polynomial.degree_eq_natDegree hd, h_eq]
-  exact Polynomial.degree_sub_lt h_deg hn h_lc
+  exact Polynomial.degree_sub_lt_left h_deg hn h_lc
 
 private theorem ratFunc_exists_C_sub_intDegree_neg {r : RatFunc F}
     (hr : r.intDegree ≤ 0) :
@@ -1056,13 +1052,13 @@ theorem ordAtInftyVal_eq_exp_neg_ordAtInfty {f : C.FunctionField} (hf : f ≠ 0)
   have : (n : WithTop ℤ) = ((- RatFunc.intDegree (C.normAsRatFunc f) : ℤ) : WithTop ℤ) :=
     hn.symm.trans hN
   have hni : n = - RatFunc.intDegree (C.normAsRatFunc f) := by exact_mod_cast this
-  rw [ordAtInftyVal, if_neg hf, hni, neg_neg]
+  rw [ordAtInftyVal, ite_eq_right hf, hni, neg_neg]
 
-@[simp] theorem ordAtInftyVal_zero : C.ordAtInftyVal 0 = 0 := if_pos rfl
+@[simp] theorem ordAtInftyVal_zero : C.ordAtInftyVal 0 = 0 := ite_eq_left rfl
 
 theorem ordAtInftyVal_ne_zero {f : C.FunctionField} (hf : f ≠ 0) :
     C.ordAtInftyVal f ≠ 0 := by
-  rw [ordAtInftyVal, if_neg hf]; exact WithZero.exp_ne_zero
+  rw [ordAtInftyVal, ite_eq_right hf]; exact WithZero.exp_ne_zero
 
 @[simp] theorem ordAtInftyVal_one : C.ordAtInftyVal (1 : C.FunctionField) = 1 := by
   rw [C.ordAtInftyVal_eq_exp_neg_ordAtInfty one_ne_zero C.ordAtInfty_one, neg_zero,
@@ -1325,7 +1321,7 @@ theorem smoothPoint_x_preimage_finite (a : F) :
     (t := {y : F | (C.fiberQuadratic a).IsRoot y})
     (fun _ hP ↦ C.fiberQuadratic_isRoot_of_smoothPoint hP)
     (fun _ hP₁ _ hP₂ hy ↦ SmoothPoint.ext (hP₁.trans hP₂.symm) hy)
-    (Polynomial.finite_setOf_isRoot (C.fiberQuadratic_ne_zero a))
+    (Polynomial.finite_setOfPred_isRoot (C.fiberQuadratic_ne_zero a))
 
 /-- The x-projection preimage of a finite set of `x`-values is finite. In
 particular, `{P : C.SmoothPoint | P.x ∈ roots f}` is finite for any nonzero
@@ -1460,7 +1456,7 @@ theorem pointValuation_algebraMap_le_one (u : C.CoordinateRing)
   exact IsDedekindDomain.HeightOneSpectrum.valuation_le_one
     (IsDiscreteValuationRing.maximalIdeal (C.localRingAt P)) _
 
-/-- **Step (B'') foundational structural lemma (one direction)**: every
+/--  every
 element of `C.localRingAt P` (after embedding into `K(C)`) has valuation
 ≤ 1 at `P`. Direct from `IsDedekindDomain.HeightOneSpectrum.valuation_le_one`
 applied at the localRingAt level (the localRing is a DVR, hence a Dedekind
@@ -1472,7 +1468,7 @@ theorem pointValuation_algebraMap_localRingAt_le_one
   exact IsDedekindDomain.HeightOneSpectrum.valuation_le_one
     (IsDiscreteValuationRing.maximalIdeal (C.localRingAt P)) x
 
-/-- **Step (B'') foundational structural lemma (converse direction)**: every
+/--  every
 element of `K(C)` with valuation ≤ 1 at `P` lifts to an element of
 `C.localRingAt P`. Together with the easy direction, this characterises
 the `algebraMap (localRingAt P) → KE` image as exactly
@@ -1533,7 +1529,7 @@ theorem exists_mul_algebraMap_eq_of_pointValuation_le_one
   exact h2
 
 
-/-- **Step (B'') foundational structural identification (biconditional)**:
+/--
 combines `pointValuation_algebraMap_localRingAt_le_one` and
 `mem_localRingAt_image_of_pointValuation_le_one` into the canonical iff form.
 This characterises the `algebraMap (localRingAt P) → KE` image as exactly
@@ -1552,7 +1548,7 @@ theorem ord_P_eq_zero_iff_pointValuation_eq_one (C : SmoothPlaneCurve F)
     C.ord_P P f = 0 ↔ C.pointValuation P f = 1 := by
   have hv : C.pointValuation P f ≠ 0 := (C.pointValuation P).ne_zero_iff.mpr hf
   simp only [ord_P]
-  rw [dif_neg hv]
+  rw [dite_eq_right hv]
   constructor
   · intro h
     have h_nat : -((WithZero.unzero hv).toAdd : ℤ) = 0 := by exact_mod_cast h
@@ -1599,7 +1595,7 @@ theorem finite_setOf_mem_maximalIdealAt {u : C.CoordinateRing} (hu : u ≠ 0) :
     hu ((Algebra.norm_eq_zero_iff (R := Polynomial F)).mp h)
   refine (C.smoothPoint_x_preimage_finite_of_set
     {a : F | (Algebra.norm (Polynomial F) u).IsRoot a}
-    (Polynomial.finite_setOf_isRoot hNu_ne)).subset ?_
+    (Polynomial.finite_setOfPred_isRoot hNu_ne)).subset ?_
   intro P (hP : u ∈ C.maximalIdealAt P)
   rw [← hpq, C.mem_maximalIdealAt_iff_eval_zero P p q] at hP
   change (Algebra.norm (Polynomial F) u).IsRoot P.x
@@ -1647,7 +1643,7 @@ theorem finite_setOf_ord_P_nonzero {f : C.FunctionField} (hf : f ≠ 0) :
     (C.finite_setOf_ord_P_nonzero_of_coordinateRing hv_ne)).subset ?_
   intro P hP
   by_contra h_both
-  simp only [Set.mem_union, Set.mem_setOf_eq, not_or, not_not] at h_both
+  simp only [Set.mem_union, Set.mem_ofPred_eq, not_or, not_not] at h_both
   obtain ⟨hu0, hv0⟩ := h_both
   have hord : C.ord_P P
       (algebraMap C.CoordinateRing C.FunctionField u) =
@@ -1658,8 +1654,7 @@ theorem finite_setOf_ord_P_nonzero {f : C.FunctionField} (hf : f ≠ 0) :
   exact hP hord.symm
 
 /-- **Silverman II.1.2, Part 1**: a nonzero function on a smooth plane curve
-has zeros and poles at only finitely many smooth points. This is an alias
-for `finite_setOf_ord_P_nonzero` matching the ticket statement. -/
+has zeros and poles at only finitely many smooth points.  -/
 theorem finite_zeros_poles (f : C.FunctionField) (hf : f ≠ 0) :
     {P : C.SmoothPoint | C.ord_P P f ≠ 0}.Finite :=
   C.finite_setOf_ord_P_nonzero hf
@@ -1668,16 +1663,6 @@ theorem finite_zeros_poles (f : C.FunctionField) (hf : f ≠ 0) :
 if `f ∈ F(C)` has nonnegative order at infinity AND lies in the coordinate
 ring, then `f` is the image of a constant from `F`.
 
-This is the best form provable with our current infrastructure. The full
-"no-affine-poles ⟹ in CoordinateRing" step is the integral-closure
-property of smooth curves: for a smooth Weierstrass curve `[C.toAffine.IsElliptic]`,
-`C.CoordinateRing` is a Dedekind domain, hence integrally closed, and any
-`f ∈ F(C)` with `ord_P f ≥ 0` at every closed point is automatically in
-`C.CoordinateRing`. Without the `IsElliptic` hypothesis (or an equivalent
-smoothness assumption), singular points can have local rings that don't
-align with the global coordinate ring structure, and the integral-closure
-step requires genuinely new algebra-of-curves infrastructure.
-
 Reference: Silverman II.1.2, second part; Hartshorne I.6.12. -/
 theorem const_of_no_poles_of_coordinateRing (f : C.FunctionField)
     (h_coord : ∃ u : C.CoordinateRing,
@@ -1691,9 +1676,9 @@ theorem const_of_no_poles_of_coordinateRing (f : C.FunctionField)
   exact (IsScalarTower.algebraMap_apply F C.CoordinateRing
     C.FunctionField c).symm
 
-/-- **IC-006 (Silverman II.1.2, Part 2, prime-indexed)**: if `f ∈ F(C)` has
+/-- **Silverman II.1.2, Part 2, prime-indexed**: if `f ∈ F(C)` has
 nonnegative valuation at every nonzero prime of `C.CoordinateRing` **and** at
-infinity, then `f` is constant. Combines IC-006's Dedekind-Liouville bridge
+infinity, then `f` is constant. Combines the Dedekind-Liouville bridge
 `mem_coordinateRing_of_valuation_le_one` with the CoordinateRing-form
 algebraic Liouville `const_of_no_poles_of_coordinateRing`. -/
 theorem const_of_no_poles_of_valuation_of_ordAtInfty
@@ -1705,7 +1690,7 @@ theorem const_of_no_poles_of_valuation_of_ordAtInfty
   C.const_of_no_poles_of_coordinateRing f
     (C.mem_coordinateRing_of_valuation_le_one f h_primes) h_inf
 
-/-- **IC-006 (integrality-based Liouville)**: if `f ∈ F(C)` is integral over
+/-- **Integrality-based Liouville**: if `f ∈ F(C)` is integral over
 `F[X]` **and** has nonnegative order at infinity, then `f` is constant.
 This is the cleanest consequence of `IsIntegrallyClosed C.CoordinateRing`:
 integrality over `F[X]` lifts `f` into `C.CoordinateRing`, and then
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/NoFinitePolesBridge.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/NoFinitePolesBridge.lean
index 0b44dea28cc6d4c4dd158920e7563ca5babef0db..966395e5596ce0a1deb52980a2031a07e846aa7d 100644
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/NormValuation.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/NormValuation.lean
index 025f58eeb..f0ffba4a2 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/NormValuation.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/NormValuation.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Birkbeck
 -/
 import Mathlib.FieldTheory.IsAlgClosed.Basic
-import Mathlib.NumberTheory.RamificationInertia.Basic
+import Mathlib.RingTheory.RamificationInertia.Basic
 import Mathlib.RingTheory.AdjoinRoot
 import Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
 import Mathlib.RingTheory.Ideal.Norm.RelNorm
@@ -30,7 +30,7 @@ language. Under `[IsAlgClosed F]` the general Dedekind argument runs:
 
 1. Every maximal ideal of `F[C]` lying over `(X − a)` has residue field `F`
    (finite F-algebra field ⇒ `F` by `IsAlgClosed`).
-2. So `inertiaDeg' (X − a) M = finrank F (F[C]⧸M) = 1`.
+2. So the residue degree of `M` over `F[X]` is `finrank F (F[C]⧸M) = 1`.
 3. Combined with the principal-power form of `Ideal.relNorm`:
    `relNorm (maximalIdealAt P) = (X − P.x)^1 = (X − P.x)`.
 4. Multiplicativity of `relNorm` + UFD factorization of `(u)` in Dedekind `F[C]`
@@ -159,7 +159,7 @@ theorem algebraMap_quotient_maximalIdealAt_surjective (P : C.SmoothPoint) :
       (algebraMap ((Polynomial F) ⧸
         Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)})
         (C.CoordinateRing ⧸ C.maximalIdealAt P)) := by
-  haveI := C.maximalIdealAt_liesOver P
+  have := C.maximalIdealAt_liesOver P
   intro w
   set a := C.quotientMaximalIdealAtEquiv P w
   refine ⟨Ideal.Quotient.mk _ (Polynomial.C a), ?_⟩
@@ -197,19 +197,20 @@ theorem inertiaDeg_maximalIdealAt (P : C.SmoothPoint) :
     letI : (C.maximalIdealAt P).LiesOver
         (Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)}) :=
       C.maximalIdealAt_liesOver P
-    Ideal.inertiaDeg' (Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)})
-      (C.maximalIdealAt P) = 1 := by
-  haveI := C.maximalIdealAt_liesOver P
-  rw [Ideal.inertiaDeg'_algebraMap]
-  haveI h_max :
+    (C.maximalIdealAt P).inertiaDeg (Polynomial F) = 1 := by
+  have hOver := C.maximalIdealAt_liesOver P
+  have hMaxBase :
       (Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)}).IsMaximal :=
     Ideal.Quotient.maximal_of_isField _
       ((Polynomial.quotientSpanXSubCAlgEquiv P.x).toRingEquiv.isField
         (Field.toIsField F))
-  haveI : Field ((Polynomial F) ⧸
+  have hMaxPoint : (C.maximalIdealAt P).IsMaximal := C.maximalIdealAt_isMaximal P
+  rw [Ideal.inertiaDeg_eq_of_isMaximal
+    (Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)}) (C.maximalIdealAt P)]
+  have : Field ((Polynomial F) ⧸
       Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)}) :=
     Ideal.Quotient.field _
-  haveI : Nontrivial (C.CoordinateRing ⧸ C.maximalIdealAt P) :=
+  have : Nontrivial (C.CoordinateRing ⧸ C.maximalIdealAt P) :=
     (Ideal.Quotient.nontrivial_iff).mpr (C.maximalIdealAt_isMaximal P).ne_top
   have h_le : Module.finrank
       ((Polynomial F) ⧸
@@ -220,7 +221,7 @@ theorem inertiaDeg_maximalIdealAt (P : C.SmoothPoint) :
       refine ⟨c, ?_⟩
       rw [Algebra.smul_def, hc]
       exact mul_one w
-  haveI : Module.IsTorsionFree
+  have : Module.IsTorsionFree
       ((Polynomial F) ⧸
         Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)})
       (C.CoordinateRing ⧸ C.maximalIdealAt P) :=
@@ -246,8 +247,8 @@ theorem module_finite_quotient_of_maximal
     {M : Ideal C.CoordinateRing} (hM : M.IsMaximal) :
     letI : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
     Module.Finite F (C.CoordinateRing ⧸ M) := by
-  letI : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
-  haveI : Algebra.FiniteType F (C.CoordinateRing ⧸ M) :=
+  let : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
+  have : Algebra.FiniteType F (C.CoordinateRing ⧸ M) :=
     Algebra.FiniteType.of_surjective (Ideal.Quotient.mkₐ F M)
       Ideal.Quotient.mk_surjective
   exact finite_of_finite_type_of_isJacobsonRing F (C.CoordinateRing ⧸ M)
@@ -260,9 +261,9 @@ theorem algebraMap_bijective_quotient_of_maximal [IsAlgClosed F]
     {M : Ideal C.CoordinateRing} (hM : M.IsMaximal) :
     letI : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
     Function.Bijective (algebraMap F (C.CoordinateRing ⧸ M)) := by
-  letI : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
-  haveI := C.module_finite_quotient_of_maximal hM
-  haveI : Algebra.IsIntegral F (C.CoordinateRing ⧸ M) :=
+  let : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
+  have := C.module_finite_quotient_of_maximal hM
+  have : Algebra.IsIntegral F (C.CoordinateRing ⧸ M) :=
     Algebra.IsIntegral.of_finite F (C.CoordinateRing ⧸ M)
   exact IsAlgClosed.algebraMap_bijective_of_isIntegral
 
@@ -281,7 +282,7 @@ theorem exists_coordinates_of_isMaximal [IsAlgClosed F]
         (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine (Polynomial.C Polynomial.X)) ∧
       algebraMap F (C.CoordinateRing ⧸ M) b = Ideal.Quotient.mk M
         (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine Y) := by
-  letI : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
+  let : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
   have h_bij := C.algebraMap_bijective_quotient_of_maximal hM
   obtain ⟨a, ha⟩ := h_bij.2 (Ideal.Quotient.mk M
     (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine (Polynomial.C Polynomial.X)))
@@ -304,7 +305,7 @@ theorem exists_coordinates_of_isMaximal_of_surjective
         (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine (Polynomial.C Polynomial.X)) ∧
       algebraMap F (C.CoordinateRing ⧸ M) b = Ideal.Quotient.mk M
         (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine Y) := by
-  letI : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
+  let : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
   obtain ⟨a, ha⟩ := h_surj (Ideal.Quotient.mk M
     (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine (Polynomial.C Polynomial.X)))
   obtain ⟨b, hb⟩ := h_surj (Ideal.Quotient.mk M
@@ -341,7 +342,7 @@ theorem equation_of_coordinates_of_field
           algebraMap F (C.CoordinateRing ⧸ M) b = Ideal.Quotient.mk M
             (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine Y)) :
     C.toAffine.Equation a b := by
-  letI : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
+  let : Field (C.CoordinateRing ⧸ M) := Ideal.Quotient.field M
   have h_mk_zero :
       (WeierstrassCurve.Affine.CoordinateRing.mk C.toAffine
         C.toAffine.polynomial : C.CoordinateRing) = 0 :=
@@ -546,8 +547,8 @@ theorem maximalIdealAt_liesOver_of_eq_x (P : C.SmoothPoint) {a : F} (h : P.x = a
   exact C.maximalIdealAt_liesOver P
 
 /-- **Sum of ramification indices** over the fiber of smooth F-rational
-points with x-coordinate `a` equals the function-field degree 2. Uses
-`sum_ramification_inertia` (mathlib) + our `inertiaDeg_maximalIdealAt = 1`. -/
+points with x-coordinate `a` equals the function-field degree 2. Uses the fundamental ramification-inertia sum and the identity
+`inertiaDeg_maximalIdealAt = 1`. -/
 theorem sum_ramificationIdx_over_fiber [IsAlgClosed F] [C.toAffine.IsElliptic]
     [IsIntegrallyClosed C.CoordinateRing] (a : F) :
     let p : Ideal (Polynomial F) :=
@@ -557,20 +558,44 @@ theorem sum_ramificationIdx_over_fiber [IsAlgClosed F] [C.toAffine.IsElliptic]
         (Field.toIsField F))
     ∑ M ∈ IsDedekindDomain.primesOverFinset p C.CoordinateRing,
       Ideal.ramificationIdx' p M *
-      Ideal.inertiaDeg' p M =
+      M.inertiaDeg (Polynomial F) =
       Module.finrank (FractionRing (Polynomial F)) C.FunctionField := by
   set p : Ideal (Polynomial F) :=
     Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} with hp_def
-  haveI h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
+  have h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField
       (Field.toIsField F))
-  apply Ideal.sum_ramification_inertia C.CoordinateRing
-    (FractionRing (Polynomial F)) C.FunctionField
-  intro h
-  have hx_mem : (Polynomial.X - Polynomial.C a : Polynomial F) ∈ p :=
-    Ideal.subset_span rfl
-  rw [h, Ideal.mem_bot] at hx_mem
-  exact Polynomial.X_sub_C_ne_zero a hx_mem
+  have hp_ne : p ≠ ⊥ := by
+    intro h
+    have hx_mem : (Polynomial.X - Polynomial.C a : Polynomial F) ∈ p :=
+      Ideal.subset_span rfl
+    rw [h, Ideal.mem_bot] at hx_mem
+    exact Polynomial.X_sub_C_ne_zero a hx_mem
+  classical
+  let : Fintype (p.primesOver C.CoordinateRing) := Fintype.ofFinite _
+  calc
+    _ = ∑ M : p.primesOver C.CoordinateRing,
+        M.1.ramificationIdx (Polynomial F) * M.1.inertiaDeg (Polynomial F) := by
+      apply Finset.sum_bij (fun M hM ↦
+        ⟨M, (IsDedekindDomain.mem_primesOverFinset_iff hp_ne C.CoordinateRing).mp hM⟩)
+      · intro M hM
+        exact Finset.mem_univ _
+      · intro M hM N hN h
+        exact congrArg Subtype.val h
+      · intro M _
+        exact ⟨M.1,
+          (IsDedekindDomain.mem_primesOverFinset_iff hp_ne C.CoordinateRing).mpr M.2, rfl⟩
+      · intro M hM
+        obtain ⟨hprime, hover⟩ :=
+          (IsDedekindDomain.mem_primesOverFinset_iff hp_ne C.CoordinateRing).mp hM
+        let := hprime
+        let := hover
+        rw [Ideal.ramificationIdx'_eq_ramificationIdx p M hp_ne]
+    _ = Module.finrank (Polynomial F) C.CoordinateRing :=
+      Ideal.sum_ramification_inertia_eq_finrank p C.CoordinateRing
+    _ = Module.finrank (FractionRing (Polynomial F)) C.FunctionField :=
+      (IsFractionRing.finrank_eq (Polynomial F) (FractionRing (Polynomial F))
+        C.CoordinateRing C.FunctionField).symm
 
 /-- The principal ideal `(X − a)` of `F[X]` is nonzero, since `X − a` is. -/
 private theorem span_X_sub_C_ne_bot (a : F) :
@@ -605,7 +630,7 @@ associated; being monic of the same degree they are equal, so `P.x = a`. -/
 private theorem smoothPoint_x_eq_of_liesOver_span_X_sub_C {a : F} (P : C.SmoothPoint)
     (hP_over : (C.maximalIdealAt P).LiesOver
       (Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)})) : P.x = a := by
-  haveI h_over_P := C.maximalIdealAt_liesOver P
+  have h_over_P := C.maximalIdealAt_liesOver P
   have h_both : Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)} =
       Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} := by
     rw [h_over_P.over, hP_over.over]
@@ -626,16 +651,11 @@ private theorem inertiaDeg_eq_one_of_isPrime_of_liesOver_span_X_sub_C
     [IsAlgClosed F] [C.toAffine.IsElliptic] [IsIntegrallyClosed C.CoordinateRing]
     {a : F} {M : Ideal C.CoordinateRing} (hMprime : M.IsPrime)
     (hMlies : M.LiesOver (Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)})) :
-    Ideal.inertiaDeg' (Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)}) M = 1 := by
-  haveI hMmax : M.IsMaximal :=
+    M.inertiaDeg (Polynomial F) = 1 := by
+  have hMmax : M.IsMaximal :=
     C.isMaximal_of_isPrime_of_liesOver_span_X_sub_C hMprime hMlies
   obtain ⟨P, hP⟩ := C.exists_smoothPoint_of_isMaximal hMmax
-  have hP_over : (C.maximalIdealAt P).LiesOver
-      (Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)}) := hP ▸ hMlies
-  have h_Px : P.x = a := C.smoothPoint_x_eq_of_liesOver_span_X_sub_C P hP_over
-  have h_id := C.inertiaDeg_maximalIdealAt P
-  rw [hP, h_Px] at h_id
-  exact h_id
+  simpa only [hP] using C.inertiaDeg_maximalIdealAt P
 
 /-- **Σ e_M = 2**: sum of ramification indices over primes of `F[C]` lying
 over `(X − a)` equals `[F(C) : F(X)] = 2`. Derived from
@@ -654,7 +674,7 @@ theorem sum_ramificationIdx_eq_finrank [IsAlgClosed F] [C.toAffine.IsElliptic]
       Module.finrank (FractionRing (Polynomial F)) C.FunctionField := by
   set p : Ideal (Polynomial F) :=
     Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} with hp_def
-  haveI h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
+  have h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField
       (Field.toIsField F))
   have h_sum := C.sum_ramificationIdx_over_fiber a
@@ -682,13 +702,13 @@ theorem smoothPoint_fiber_eq_primesOver [IsAlgClosed F] [C.toAffine.IsElliptic]
   · rintro ⟨hMmax, hMover⟩
     obtain ⟨P, hP⟩ := C.exists_smoothPoint_of_isMaximal hMmax
     refine ⟨P, ?_, hP⟩
-    haveI := C.maximalIdealAt_liesOver P
+    have := C.maximalIdealAt_liesOver P
     have h1 : Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)} =
         (C.maximalIdealAt P).under (Polynomial F) :=
       Ideal.LiesOver.over
     have h2 : Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} =
         M.under (Polynomial F) := by
-      haveI := hMover; exact Ideal.LiesOver.over
+      have := hMover; exact Ideal.LiesOver.over
     rw [hP] at h1
     have : Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)} =
         Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} :=
@@ -710,8 +730,8 @@ theorem exists_relNorm_maximalIdealAt_eq_pow [IsAlgClosed F]
     (P : C.SmoothPoint) :
     ∃ s : ℕ, Ideal.relNorm (Polynomial F) (C.maximalIdealAt P) =
       (Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)})^s := by
-  haveI hLies := C.maximalIdealAt_liesOver P
-  haveI hPrime : (Ideal.span
+  have hLies := C.maximalIdealAt_liesOver P
+  have hPrime : (Ideal.span
       {(Polynomial.X - Polynomial.C P.x : Polynomial F)}).IsPrime := by
     rw [Ideal.span_singleton_prime (Polynomial.X_sub_C_ne_zero P.x)]
     exact Polynomial.prime_X_sub_C P.x
@@ -769,7 +789,7 @@ theorem map_algebraMap_X_sub_C_eq_prod_primesOver_pow [IsAlgClosed F]
       ∏ P ∈ p.primesOver C.CoordinateRing,
         P ^ p.ramificationIdx' P := by
   intro p
-  haveI h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
+  have h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField
       (Field.toIsField F))
   have hp_ne : p ≠ 0 := by
@@ -806,7 +826,7 @@ theorem prod_relNorm_pow_primesOver_eq_X_sub_C_pow_two [IsAlgClosed F]
           p.ramificationIdx' P =
       p ^ 2 := by
   intro p
-  haveI h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
+  have h_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField
       (Field.toIsField F))
   have h_factor := C.map_algebraMap_X_sub_C_eq_prod_primesOver_pow a
@@ -826,7 +846,7 @@ theorem relNorm_maximalIdealAt_le_X_sub_C [IsAlgClosed F]
     (P : C.SmoothPoint) :
     Ideal.relNorm (Polynomial F) (C.maximalIdealAt P) ≤
       Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)} := by
-  haveI := C.maximalIdealAt_liesOver P
+  have := C.maximalIdealAt_liesOver P
   have hcomap_eq : (C.maximalIdealAt P).comap (algebraMap (Polynomial F)
       C.CoordinateRing) =
       Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)} :=
@@ -846,10 +866,10 @@ theorem exists_relNorm_pow_of_primesOver [IsAlgClosed F]
     ∃ s_Q : ℕ, 1 ≤ s_Q ∧ Ideal.relNorm (Polynomial F) Q =
       (Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)}) ^ s_Q := by
   obtain ⟨hQ_prime, hQ_lies⟩ := hQ
-  haveI : Q.IsPrime := hQ_prime
-  haveI : Q.LiesOver
+  have : Q.IsPrime := hQ_prime
+  have : Q.LiesOver
       (Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)}) := hQ_lies
-  haveI hp_prime : (Ideal.span
+  have hp_prime : (Ideal.span
       {(Polynomial.X - Polynomial.C a : Polynomial F)}).IsPrime := by
     rw [Ideal.span_singleton_prime (Polynomial.X_sub_C_ne_zero a)]
     exact Polynomial.prime_X_sub_C a
@@ -867,7 +887,7 @@ theorem exists_relNorm_pow_of_primesOver [IsAlgClosed F]
   rw [Ideal.span_singleton_pow, Ideal.span_singleton_pow,
     Ideal.span_singleton_le_span_singleton,
     pow_dvd_pow_iff (Polynomial.X_sub_C_ne_zero a)
-      ((Polynomial.prime_X_sub_C a).not_unit)] at h_le
+      ((Polynomial.prime_X_sub_C a).not_isUnit)] at h_le
   exact h_le
 
 /-- **Helper B prime exponent (numeric form)**: define the exponent `s_Q` such
@@ -918,14 +938,14 @@ theorem exists_relNorm_maximalIdealAt_eq_pow_bracketed [IsAlgClosed F]
     rw [Ideal.span_singleton_pow, Ideal.span_singleton_pow,
       Ideal.span_singleton_le_span_singleton,
       pow_dvd_pow_iff (Polynomial.X_sub_C_ne_zero P.x)
-        ((Polynomial.prime_X_sub_C P.x).not_unit)] at hbound
+        ((Polynomial.prime_X_sub_C P.x).not_isUnit)] at hbound
     exact hbound
   · have hbound := C.X_sub_C_pow_two_le_relNorm_maximalIdealAt P
     rw [hs] at hbound
     rw [Ideal.span_singleton_pow, Ideal.span_singleton_pow,
       Ideal.span_singleton_le_span_singleton,
       pow_dvd_pow_iff (Polynomial.X_sub_C_ne_zero P.x)
-        ((Polynomial.prime_X_sub_C P.x).not_unit)] at hbound
+        ((Polynomial.prime_X_sub_C P.x).not_isUnit)] at hbound
     exact hbound
 
 /-- **Helper B M_P membership**: `maximalIdealAt P` lies in
@@ -947,7 +967,7 @@ theorem ramificationIdx_maximalIdealAt_ne_zero [IsAlgClosed F]
     (P : C.SmoothPoint) :
     (Ideal.span {(Polynomial.X - Polynomial.C P.x : Polynomial F)}).ramificationIdx'
         (C.maximalIdealAt P) ≠ 0 := by
-  haveI := C.maximalIdealAt_liesOver P
+  have := C.maximalIdealAt_liesOver P
   refine Ideal.IsDedekindDomain.ramificationIdx'_ne_zero_of_liesOver
     (C.maximalIdealAt P) ?_
   intro h_eq
@@ -1003,7 +1023,7 @@ private theorem sum_mul_ramificationIdx_primesOver_eq_two [IsAlgClosed F]
       2 := by
   set p : Ideal (Polynomial F) :=
     Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} with hp_def
-  haveI hp_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
+  have hp_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField (Field.toIsField F))
   -- the finset `primesOverFinset p` coincides with `(p.primesOver).toFinset`
   have h_idx : IsDedekindDomain.primesOverFinset p C.CoordinateRing =
@@ -1040,14 +1060,14 @@ private theorem primesOverExp_eq_one_of_mem_primesOver [IsAlgClosed F]
   classical
   set p : Ideal (Polynomial F) :=
     Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} with hp_def
-  haveI hp_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
+  have hp_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField (Field.toIsField F))
   have hp_ne := span_X_sub_C_ne_bot (F := F) a
   -- the exponent function on the whole ring (`0` off the fibre)
   set s_fn : Ideal C.CoordinateRing → ℕ := fun R ↦
     if hR : R ∈ p.primesOver C.CoordinateRing then C.primesOverExp a R hR else 0 with hs_fn
   have h_s_fn_eq : ∀ R (hR : R ∈ p.primesOver C.CoordinateRing),
-      s_fn R = C.primesOverExp a R hR := fun R hR ↦ dif_pos hR
+      s_fn R = C.primesOverExp a R hR := fun R hR ↦ dite_eq_left hR
   -- `Σ e_R = 2` and `Σ s_R · e_R = 2`, both over `primesOverFinset`
   have h_sum_e := C.sum_ramificationIdx_primesOver_eq_two a
   have h_sum_se : ∑ R ∈ IsDedekindDomain.primesOverFinset p C.CoordinateRing,
@@ -1074,8 +1094,8 @@ private theorem primesOverExp_eq_one_of_mem_primesOver [IsAlgClosed F]
     (IsDedekindDomain.mem_primesOverFinset_iff (B := C.CoordinateRing) hp_ne).mpr hQ
   have h_Q := h_each Q hQ_fs
   rw [h_s_fn_eq Q hQ] at h_Q
-  haveI : Q.IsPrime := hQ.1
-  haveI : Q.LiesOver p := hQ.2
+  have : Q.IsPrime := hQ.1
+  have : Q.LiesOver p := hQ.2
   have h_e_ne_zero : p.ramificationIdx' Q ≠ 0 :=
     Ideal.IsDedekindDomain.ramificationIdx'_ne_zero_of_liesOver Q hp_ne
   -- `e_Q = s_Q · e_Q` with `0 < e_Q` cancels to `s_Q = 1`
@@ -1105,7 +1125,7 @@ theorem relNorm_eq_X_sub_C_of_primesOver [IsAlgClosed F]
     Ideal.relNorm (Polynomial F) Q =
       Ideal.span {(Polynomial.X - Polynomial.C a : Polynomial F)} := by
   obtain ⟨hQ_prime, hQ_lies⟩ := hQ
-  haveI : Q.IsPrime := hQ_prime
+  have : Q.IsPrime := hQ_prime
   have hQ_max : Q.IsMaximal := by
     refine Ideal.IsPrime.isMaximal hQ_prime ?_
     intro h_eq
@@ -1154,7 +1174,7 @@ theorem map_eq_top_of_ne_localization {A : Type*} [CommRing A]
     (v : IsDedekindDomain.HeightOneSpectrum A) (hv_ne_M : v.asIdeal ≠ M) :
     Ideal.map (algebraMap A (Localization.AtPrime M)) v.asIdeal = ⊤ := by
   classical
-  haveI hv_max : v.asIdeal.IsMaximal :=
+  have hv_max : v.asIdeal.IsMaximal :=
     Ideal.IsPrime.isMaximal v.isPrime v.ne_bot
   have h_not_le : ¬ v.asIdeal ≤ M := by
     intro h_le
@@ -1236,10 +1256,10 @@ theorem count_localRing_max_pow {A : Type*} [CommRing A]
     (Associates.mk (IsLocalRing.maximalIdeal (Localization.AtPrime M))).count
         (Associates.mk
           ((IsLocalRing.maximalIdeal (Localization.AtPrime M)) ^ n)).factors = n := by
-  haveI := Localization.AtPrime.isLocalRing M
-  haveI : IsDedekindDomain (Localization.AtPrime M) :=
+  have := Localization.AtPrime.isLocalRing M
+  have : IsDedekindDomain (Localization.AtPrime M) :=
     IsLocalization.AtPrime.isDedekindDomain A M _
-  haveI : IsDiscreteValuationRing (Localization.AtPrime M) :=
+  have : IsDiscreteValuationRing (Localization.AtPrime M) :=
     IsLocalization.AtPrime.isDiscreteValuationRing_of_dedekind_domain A hM_ne _
   let hv : IsDedekindDomain.HeightOneSpectrum (Localization.AtPrime M) :=
     IsDiscreteValuationRing.maximalIdeal (Localization.AtPrime M)
@@ -1505,7 +1525,7 @@ theorem ord_P_algebraMap_eq_count
       C.pointValuation P (algebraMap C.CoordinateRing C.FunctionField u) ≠ 0 :=
     (C.pointValuation P).ne_zero_iff.mpr hu_FC
   simp only [SmoothPlaneCurve.ord_P]
-  rw [dif_neg h_pv_ne]
+  rw [dite_eq_right h_pv_ne]
   have h_pv_eq := C.pointValuation_algebraMap_eq_exp_count P hu
   have h_unz :
       WithZero.unzero h_pv_ne =
@@ -1576,7 +1596,7 @@ theorem relNorm_pow_of_mem_primesOverFinset
         C.CoordinateRing) (n : ℕ) :
     Ideal.relNorm (Polynomial F) (Q ^ n) =
       (Ideal.span ({Polynomial.X - Polynomial.C a} : Set (Polynomial F))) ^ n := by
-  haveI hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
+  have hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
       Set (Polynomial F))).IsMaximal :=
     Ideal.Quotient.maximal_of_isField _
       ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField (Field.toIsField F))
@@ -1599,7 +1619,7 @@ theorem count_relNorm_pow_of_mem_primesOverFinset
         (Ideal.span ({Polynomial.X - Polynomial.C a} : Set (Polynomial F)))).count
       (Associates.mk (Ideal.relNorm (Polynomial F) (Q ^ n))).factors = n := by
   classical
-  haveI hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
+  have hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
       Set (Polynomial F))).IsMaximal :=
     Ideal.Quotient.maximal_of_isField _
       ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField (Field.toIsField F))
@@ -1624,7 +1644,7 @@ private theorem ne_bot_of_mem_primesOverFinset_span_X_sub_C
     Q ≠ ⊥ := by
   have hp_ne : (Ideal.span ({Polynomial.X - Polynomial.C a} : Set (Polynomial F))) ≠ ⊥ := by
     rw [Ne, Ideal.span_singleton_eq_bot]; exact Polynomial.X_sub_C_ne_zero a
-  haveI hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
+  have hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
       Set (Polynomial F))).IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField (Field.toIsField F))
   rw [IsDedekindDomain.mem_primesOverFinset_iff hp_ne] at hQ
@@ -1715,7 +1735,7 @@ private theorem smoothPoint_x_ne_of_not_mem_primesOverFinset
     P.x ≠ a := by
   have hp_ne : (Ideal.span ({Polynomial.X - Polynomial.C a} : Set (Polynomial F))) ≠ ⊥ := by
     rw [Ne, Ideal.span_singleton_eq_bot]; exact Polynomial.X_sub_C_ne_zero a
-  haveI hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
+  have hp_max : (Ideal.span ({Polynomial.X - Polynomial.C a} :
       Set (Polynomial F))).IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField (Field.toIsField F))
   intro h_eq
@@ -1742,7 +1762,7 @@ private theorem count_relNorm_pow_eq_zero_of_not_mem_primesOverFinset
         (Ideal.span ({Polynomial.X - Polynomial.C a} : Set (Polynomial F)))).count
       (Associates.mk ((Ideal.relNorm (Polynomial F) Q.asIdeal) ^ k)).factors = 0 := by
   -- Choose a smooth point `P` for the maximal ideal `Q`; off-fiber forces `P.x ≠ a`.
-  haveI hQ_max : Q.asIdeal.IsMaximal := Ideal.IsPrime.isMaximal Q.isPrime Q.ne_bot
+  have hQ_max : Q.asIdeal.IsMaximal := Ideal.IsPrime.isMaximal Q.isPrime Q.ne_bot
   obtain ⟨P, hP⟩ := C.exists_smoothPoint_of_isMaximal hQ_max
   have hPx_ne : P.x ≠ a :=
     smoothPoint_x_ne_of_not_mem_primesOverFinset (C := C) h_over Q.isPrime hP
@@ -1774,7 +1794,7 @@ theorem count_relNorm_singleton_eq_sum_count_fiber
     Ideal.span ({Polynomial.X - Polynomial.C a} : Set (Polynomial F)) with hp_def
   have hp_ne : p ≠ ⊥ := by
     rw [hp_def, Ne, Ideal.span_singleton_eq_bot]; exact Polynomial.X_sub_C_ne_zero a
-  haveI hp_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
+  have hp_max : p.IsMaximal := Ideal.Quotient.maximal_of_isField _
     ((Polynomial.quotientSpanXSubCAlgEquiv a).toRingEquiv.isField (Field.toIsField F))
   let vp : IsDedekindDomain.HeightOneSpectrum (Polynomial F) :=
     ⟨p, hp_max.isPrime, hp_ne⟩
@@ -1840,9 +1860,9 @@ theorem count_relNorm_singleton_eq_sum_count_fiber
       else 0 := by
     intro Q _
     by_cases h_over : Q.asIdeal ∈ IsDedekindDomain.primesOverFinset p C.CoordinateRing
-    · rw [if_pos h_over, ← map_pow]
+    · rw [ite_eq_left h_over, ← map_pow]
       exact C.count_relNorm_pow_of_mem_primesOverFinset h_over _
-    · rw [if_neg h_over]
+    · rw [ite_eq_right h_over]
       exact C.count_relNorm_pow_eq_zero_of_not_mem_primesOverFinset h_over _
   rw [Finset.sum_congr rfl h_S_split]
   rw [Finset.sum_ite, Finset.sum_const_zero, add_zero]
@@ -1978,7 +1998,7 @@ private theorem isMaximal_of_mem_primesOverFinset_span_X_sub_C
       C.CoordinateRing) :
     Q.IsMaximal := by
   have hp_ne := span_X_sub_C_ne_bot (F := F) a
-  haveI hp_max := isMaximal_span_X_sub_C (F := F) a
+  have hp_max := isMaximal_span_X_sub_C (F := F) a
   have hQ' := (IsDedekindDomain.mem_primesOverFinset_iff
     (B := C.CoordinateRing) hp_ne).mp hQ
   exact C.isMaximal_of_isPrime_of_liesOver_span_X_sub_C hQ'.1 hQ'.2
@@ -1995,7 +2015,7 @@ private theorem maximalIdealAt_mem_primesOverFinset_of_eq_x
       (Ideal.span ({Polynomial.X - Polynomial.C a} : Set (Polynomial F)))
       C.CoordinateRing := by
   have hp_ne := span_X_sub_C_ne_bot (F := F) a
-  haveI hp_max := isMaximal_span_X_sub_C (F := F) a
+  have hp_max := isMaximal_span_X_sub_C (F := F) a
   rw [IsDedekindDomain.mem_primesOverFinset_iff (B := C.CoordinateRing) hp_ne]
   exact ⟨(C.maximalIdealAt_isMaximal P).isPrime,
     C.maximalIdealAt_liesOver_of_eq_x P hPx⟩
@@ -2015,7 +2035,7 @@ private theorem smoothPoint_x_eq_of_maximalIdealAt_mem_primesOverFinset
     {P : C.SmoothPoint} (hP : C.maximalIdealAt P = Q) :
     P.x = a := by
   have hp_ne := span_X_sub_C_ne_bot (F := F) a
-  haveI hp_max := isMaximal_span_X_sub_C (F := F) a
+  have hp_max := isMaximal_span_X_sub_C (F := F) a
   have hQ' := (IsDedekindDomain.mem_primesOverFinset_iff
     (B := C.CoordinateRing) hp_ne).mp hQ
   refine C.smoothPoint_x_eq_of_liesOver_span_X_sub_C P ?_
@@ -2061,7 +2081,7 @@ theorem fiber_sum_divisorOf_algMap_eq_count_norm
       ((C.smoothPoint_x_preimage_finite a).mem_toFinset.mp hP)
   · -- `j` lands in the fibre `{P | P.x = a}`.
     intro Q hQ
-    rw [(C.smoothPoint_x_preimage_finite a).mem_toFinset, Set.mem_setOf_eq]
+    rw [(C.smoothPoint_x_preimage_finite a).mem_toFinset, Set.mem_ofPred_eq]
     exact C.smoothPoint_x_eq_of_maximalIdealAt_mem_primesOverFinset hQ
       (C.exists_smoothPoint_of_isMaximal
         (C.isMaximal_of_mem_primesOverFinset_span_X_sub_C hQ)).choose_spec
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/RankOneDomination.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/RankOneDomination.lean
index 39470b328..a61861910 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/RankOneDomination.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/RankOneDomination.lean
@@ -9,35 +9,6 @@ import Mathlib.RingTheory.DedekindDomain.Basic
 import Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing
 import Mathlib.Algebra.GroupWithZero.WithZero
 
-/-!
-# Rank-one valuation-subring domination
-
-A valuation subring `A` of a field `L` that is a **discrete valuation ring** (rank one) has only
-two overrings: `A` itself and the whole field `⊤`.  Consequently any larger valuation subring
-`B ≥ A` with `B ≠ ⊤` must equal `A`.
-
-This single reusable fact is the *DVR-domination engine* shared by
-
-* the affine valuation-subring domination of Silverman V.1.3
-  (`HasseWeil/Hasse/L6Witnesses.lean`), and
-* the curve-completeness place classification over the integral closure `B`
-  (`HasseWeil/Curves/NormConormIntegralClosure.lean`).
-
-It is kept here in a lightweight `Curves/` file (depending only on the mathlib `ValuationSubring`
-and `DiscreteValuationRing` API) so that the place classification need not import the heavy char-`p`
-`Hasse/L6Witnesses`.
-
-## Main results
-
-* `rankOne_valuationSubring_le_eq_of_ne_top` — a DVR valuation subring `A ≤ B`, `B ≠ ⊤` forces
-  `A = B` (the DVR-domination engine).
-* `valuationSubring_isDVR_of_surjective_withZeroInt` — a surjective `ℤᵐ⁰`-valued valuation on a
-  field has a discrete (rank-one) valuation subring.
-* `Valuation.isEquiv_iff_eq_of_surjective_withZeroInt` — two surjective `ℤᵐ⁰`-valued valuations
-  that are `IsEquiv` are in fact equal.
-* `Valuation.isEquiv_of_valuationSubring_le` — downward valuation-subring domination
-  `O_v ≤ O_w` implies `v.IsEquiv w`.
--/
 
 namespace HasseWeil.Curves
 
@@ -77,16 +48,16 @@ theorem valuationSubring_isDVR_of_surjective_withZeroInt
     refine ⟨1, by simp, ?_⟩
     obtain ⟨x, hx⟩ := hv (y : WithZero (Multiplicative ℤ))
     exact ⟨x, by rw [map_one, one_mul]; exact hx.symm⟩
-  haveI : IsCyclic (WithZero (Multiplicative ℤ))ˣ :=
+  have : IsCyclic (WithZero (Multiplicative ℤ))ˣ :=
     isCyclic_of_surjective WithZero.unitsWithZeroEquiv.symm.toMonoidHom
       WithZero.unitsWithZeroEquiv.symm.surjective
-  haveI : Nontrivial (WithZero (Multiplicative ℤ))ˣ :=
+  have : Nontrivial (WithZero (Multiplicative ℤ))ˣ :=
     WithZero.unitsWithZeroEquiv.symm.toEquiv.nontrivial
-  haveI : IsCyclic (MonoidWithZeroHom.valueGroup (.ofClass v)) := by
+  have : IsCyclic (MonoidWithZeroHom.valueGroup (.ofClass v)) := by
     rw [hvg]
     exact isCyclic_of_surjective Subgroup.topEquiv.symm.toMonoidHom
       Subgroup.topEquiv.symm.surjective
-  haveI : Nontrivial (MonoidWithZeroHom.valueGroup (.ofClass v)) := by
+  have : Nontrivial (MonoidWithZeroHom.valueGroup (.ofClass v)) := by
     rw [hvg]; exact Subgroup.topEquiv.symm.toEquiv.nontrivial
   exact Valuation.valuationSubring_isDiscreteValuationRing v
 
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/SmoothPointTranslate.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/SmoothPointTranslate.lean
index 55ec78a69..31b93fe6a 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/SmoothPointTranslate.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/SmoothPointTranslate.lean
@@ -82,7 +82,7 @@ variable {C : SmoothPlaneCurve F} [C.toAffine.IsElliptic]
 `Affine.Point` via `Classical.choose`-style decomposition. Concretely
 constructs the existential `∃ x y h_ns, P = some x y h_ns` from
 `P.IsSome`. -/
-private theorem _root_.WeierstrassCurve.Affine.Point.exists_some_of_isSome
+theorem _root_.WeierstrassCurve.Affine.Point.exists_some_of_isSome
     {R : Type*} [CommRing R] {W : Affine R} (P : W.Point)
     (h : P.IsSome) :
     ∃ x : R, ∃ y : R, ∃ h_ns : W.Nonsingular x y,
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Valuation.lean b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Valuation.lean
index 9b0d89e68..335aaa2ad 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Valuation.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Curves/Valuation/Valuation.lean
@@ -6,31 +6,6 @@ Authors: Chris Birkbeck
 import HasseWeil.Foundation.Curves.Valuation.DVR
 import Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing
 
-/-!
-# The discrete valuation at a smooth point
-
-For a smooth plane curve `C` over a field `F` and a smooth point `P ∈ C`, the
-local ring at `P` is a discrete valuation ring (Silverman II.1.1, closed by
-`T-II-1-001`). This file extracts the associated additive valuation on the
-function field,
-
-```
-ord_P : F(C) → WithTop ℤ.
-```
-
-The construction proceeds via mathlib's adic-valuation machinery: the local
-ring is a DVR, which gives a `HeightOneSpectrum` element, and the induced
-valuation on its fraction field — which is precisely `F(C)` — has values in
-`ℤᵐ⁰ := WithZero (Multiplicative ℤ)`. We translate this multiplicative
-formulation to the additive `WithTop ℤ` convention used by Silverman.
-
-This closes ticket `T-II-1-002`.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], II.1 (definition after
-  II.1.1)
--/
 
 namespace HasseWeil.Curves
 
@@ -100,7 +75,7 @@ theorem ord_P_eq_top_iff (f : C.FunctionField) : C.ord_P P f = ⊤ ↔ f = 0 :=
 
 private lemma ord_P_of_ne (f : C.FunctionField)
     (h : C.pointValuation P f ≠ 0) :
-    C.ord_P P f = (- (WithZero.unzero h).toAdd : ℤ) := dif_neg h
+    C.ord_P P f = (- (WithZero.unzero h).toAdd : ℤ) := dite_eq_right h
 
 /-- **Order-positivity bridge**: for a nonzero `f ∈ F(C)`,
 `ord_P(f) ≥ 1 ↔ pointValuation(f) < 1`. The forward direction is the defining
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/AffinePointMap.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/AffinePointMap.lean
index 8bc30a80a..f28be7aaa 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/AffinePointMap.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/AffinePointMap.lean
@@ -1,31 +1,5 @@
 import Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point
 
-/-!
-# Functorial action of ring homs on `Affine.Point` (T-III-4-020b-2, Phase 0a)
-
-For an injective ring homomorphism `f : R →+* S` and a Weierstrass curve `W`
-over `R`, we build the natural map
-`Affine.Point.map f : W.toAffine.Point → (W.map f).toAffine.Point`
-which sends `zero` to `zero` and `some x y h` to `some (f x) (f y) _`.
-
-mathlib provides `map_nonsingular` (`Affine/Basic.lean:278`) which shows that
-nonsingularity of `(x, y)` transfers to nonsingularity of `(f x, f y)` on
-`W.map f` under an injective `f`. We package this into the functorial
-`Point.map`.
-
-## Main definition
-
-* `HasseWeil.Affine.Point.map` — the map `W.Point → (W.map f).Point` for an
-  injective ring homomorphism `f : R →+* S`.
-
-This is Phase 0a of T-III-4-020b-2. Phases 0b (`map_add`) and 0c (`map_zsmul`)
-build the group-hom properties on top.
-
-## References
-
-* mathlib: `Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point`
-* Silverman, *The Arithmetic of Elliptic Curves*, III.4 (group law).
--/
 
 open WeierstrassCurve
 
@@ -51,12 +25,6 @@ noncomputable def map : W.toAffine.Point → (W.map f).toAffine.Point
 
 end HasseWeil.Affine.Point
 
-/-! ### Phase 0b: `Affine.Point.map` is additive
-
-Here we restrict to ring homs between fields, since `Affine.Point` only has
-`Add` over a field. Using mathlib's equivariance lemmas
-(`Affine.map_negY`, `map_addX`, `map_addY`, `map_slope`) the case analysis
-of `Point.add` goes through mechanically. -/
 
 namespace HasseWeil.Affine.Point
 
@@ -89,6 +57,7 @@ theorem map_add (P Q : W.toAffine.Point) :
     · rw [Affine.map_slope, Affine.map_addX]
     · rw [Affine.map_slope, Affine.map_addY]
 
+omit [DecidableEq F] [DecidableEq F'] in
 /-- `Point.map f` preserves negation: `-map f P = map f (-P)`. -/
 theorem map_neg (P : W.toAffine.Point) :
     -map f f.injective P = map f f.injective (-P) := by
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/DifferentialOrd.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/DifferentialOrd.lean
index f4a37356b..6a5ce1f89 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/DifferentialOrd.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/DifferentialOrd.lean
@@ -20,10 +20,9 @@ with the `K(E)`-linear isomorphism `Ω ≅ K(E)`, `ω ↦ 1`), and on the genera
 * `Dω (y_gen) = 3x²+2a₂x+a₄-a₁y`   (`Dω_y_gen`),
 
 both coordinate-ring elements.  Consequently `Dω` maps the coordinate ring `R` into itself
-(`Dω_mem_range`), hence is *regular at every smooth point* `P` (`ord_P_Dω_nonneg`), and the usual
-Leibniz/uniformiser computation yields the **fundamental order inequality**
-
-  `ord_P (Dω f) ≥ ord_P f − 1`   (`ord_P_Dω_ge`) for `f` regular at `P`.
+(`Dω_algebraMap_mem_Rimg`), hence is *regular at every smooth point* `P` (`ord_P_Dω_nonneg`).
+The Leibniz rule and a uniformizer show that if `f` vanishes to order at least two,
+then `Dω f` vanishes to order at least one (`one_le_ord_P_Dω_of_two_le`).
 
 This is the local form of "a differential lowers the order of vanishing by at most one"; with
 `[ℓ]^*ω = ℓ·ω` (`omegaCoeff_mulByInt`) it is the separability/unramifiedness input
@@ -120,8 +119,8 @@ theorem Dω_y_gen :
   (Dω_eq_of_smul W (kaehlerD_y_gen_eq_num_smul_omega W).symm).symm
 
 omit [DecidableEq F] [W.toAffine.IsElliptic] in
-/-- The bridge `(p.map φ).evalEval x_gen y_gen = algebraMap R KE (mk p)` (local re-derivation of the
-private `evalEval_generic_eq_mk`). -/
+/-- Evaluating a bivariate polynomial at the generic coordinates agrees with its image under
+`R → K(E)`. -/
 theorem evalEval_xy_gen_eq_algebraMap_mk (p : (Polynomial F)[X]) :
     (p.map (Polynomial.mapRingHom (algebraMap F KE))).evalEval (x_gen W) (y_gen W) =
       algebraMap R KE (Affine.CoordinateRing.mk W.toAffine p) := by
@@ -242,7 +241,7 @@ theorem ord_P_nonneg_of_mem_Rimg {f : KE} (hf : f ∈ Rimg W)
     have hv0 : (⟨W⟩ : SmoothPlaneCurve F).pointValuation P (algebraMap R KE u) ≠ 0 :=
       ((⟨W⟩ : SmoothPlaneCurve F).pointValuation P).ne_zero_iff.mpr hu
     simp only [SmoothPlaneCurve.ord_P]
-    rw [dif_neg hv0, show (0 : WithTop ℤ) = ((0 : ℤ) : WithTop ℤ) from rfl, WithTop.coe_le_coe]
+    rw [dite_eq_right hv0, show (0 : WithTop ℤ) = ((0 : ℤ) : WithTop ℤ) from rfl, WithTop.coe_le_coe]
     have h_unz_le : WithZero.unzero hv0 ≤ 1 := by
       rw [← WithZero.coe_le_coe, WithZero.coe_one, WithZero.coe_unzero]; exact hv
     have h_toAdd : (WithZero.unzero hv0).toAdd ≤ 0 := h_unz_le
@@ -259,9 +258,9 @@ theorem ord_P_Dω_nonneg {f : KE} (P : (⟨W⟩ : SmoothPlaneCurve F).SmoothPoin
     · exact pointValuation_le_one_of_ord_nonneg W hf0 P hf
   obtain ⟨x, hx⟩ := SmoothPlaneCurve.mem_localRingAt_image_of_pointValuation_le_one
     (C := (⟨W⟩ : SmoothPlaneCurve F)) f hf_le
-  haveI : ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt P).IsPrime :=
+  have : ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt P).IsPrime :=
     (⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt_isPrime P
-  haveI : IsLocalization ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt P).primeCompl
+  have : IsLocalization ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt P).primeCompl
       ((⟨W⟩ : SmoothPlaneCurve F).localRingAt P) := Localization.isLocalization
   obtain ⟨⟨a, s⟩, hxas⟩ := IsLocalization.surj
     ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt P).primeCompl x
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPoint.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPoint.lean
index 4ae622fc9..ca2b6347a 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPoint.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPoint.lean
@@ -37,6 +37,7 @@ namespace HasseWeil
 variable {F : Type*} [Field F] [DecidableEq F]
 variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 
+omit [DecidableEq F] in
 /-- The generic point is nonsingular on `W_KE`. Over an elliptic curve the
     Weierstrass equation implies nonsingularity (`Affine.equation_iff_nonsingular`). -/
 theorem generic_nonsingular :
@@ -53,7 +54,8 @@ theorem generic_nonsingular :
 noncomputable def genericPoint : (W_KE W).toAffine.Point :=
   Affine.Point.some (x_gen W) (y_gen W) (generic_nonsingular W)
 
-/-- Sanity check: the generic point's x-coordinate is `x_gen W`. -/
+omit [DecidableEq F] in
+/-- The generic point has coordinates `x_gen W` and `y_gen W`. -/
 @[simp] theorem genericPoint_xOf_some :
     genericPoint W = Affine.Point.some (x_gen W) (y_gen W) (generic_nonsingular W) :=
   rfl
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPointZsmul.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPointZsmul.lean
index d1e747daf..9611c0fdf 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPointZsmul.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/GenericPointZsmul.lean
@@ -9,55 +9,13 @@ import HasseWeil.Foundation.EC.MulByIntBaseCase
 import HasseWeil.Foundation.EC.MulByIntComp
 
 /-!
-# `n • genericPoint W = .some (mulByInt_x W n) (mulByInt_y W n)` (T-III-4-020b-2)
+For a nonzero integer `n`, multiplication of the generic point by `n` has
+coordinates given by the division-polynomial formulas `mulByInt_x` and
+`mulByInt_y`. The proof uses strong induction on the absolute value of `n`,
+with the tangent formula for doubling, the addition law for larger positive
+integers, and negation for negative integers.
 
-Silverman III.4.2: for an elliptic curve `E` and an integer `n ≠ 0`, the point
-`n • P` has x-coordinate `Φ_n(P.x) / ΨSq_n(P.x)` and y-coordinate
-`ω_n(P.x, P.y) / ψ_n(P.x, P.y)³`.
-
-Specialized to the generic point `P = genericPoint W` in `E(K(E))` (where
-`E = W.toAffine` and `K(E) = W.toAffine.FunctionField`), this gives:
-```
-n • genericPoint W = Affine.Point.some (mulByInt_x W n) (mulByInt_y W n) _
-```
-for `n ≠ 0` in `ℤ`.
-
-This is the connection lemma between the division-polynomial definition of
-`mulByInt_x/y` and the group-law x/y-coordinate.
-
-## Strategy
-
-Strong induction on `n.natAbs`, mirroring mathlib's universal
-`Affine.zsmul_point_eq_smulX_smulY` (in `Auxiliary/DivisionPolynomial.lean:423`)
-but directly in our `W_KE` setting. Key steps:
-
-1. **`n = 1` case**: `1 • genericPoint = genericPoint = .some (x_gen) (y_gen)`.
-   By `mulByInt_x_one` and `mulByInt_y_one`, these equal
-   `mulByInt_x W 1` and `mulByInt_y W 1`.
-
-2. **`n = 2` case** (doubling): uses the tangent slope at `genericPoint`.
-
-3. **`n ≥ 3` case**: induction using addition formula
-   `(n+1) • P = n • P + P` where the addition is `add_of_X_ne`
-   (using `mulByInt_x W n ≠ mulByInt_x W 1`).
-
-4. **`n < 0` case**: via `mulByInt_x_neg` and negation preservation.
-
-## Auxiliary algebraic lemmas
-
-These are the `W_KE`-specific analogs of the universal lemmas:
-- `mulByInt_x_ne_zero_at`
-- `mulByInt_x_sub_mulByInt_x`
-- `mulByInt_x_ne_mulByInt_x`
-- `mulByInt_y_sub_negY` at the generic point
-- `slope_at_genericPoint` (for doubling)
-- `addX_mulByInt_x_one_one` (doubling closure)
-- `addY_mulByInt_y_one_one`
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, III.4.2.
-* mathlib: `HasseWeil/Auxiliary/DivisionPolynomial.lean:423` (universal analog).
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.4.2.
 -/
 
 open WeierstrassCurve
@@ -67,6 +25,7 @@ namespace HasseWeil
 variable {F : Type*} [Field F] [DecidableEq F]
 variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 
+omit [DecidableEq F] in
 /-- `mulByInt_x W 1 = x_gen W`. Unfolded form of `mulByInt_x_one` with the
     `x_gen` abbreviation used directly. -/
 theorem genericPoint_eq_mulByInt_one :
@@ -77,11 +36,13 @@ theorem genericPoint_eq_mulByInt_one :
   · exact (mulByInt_x_one W).symm
   · exact (mulByInt_y_one W).symm
 
+omit [DecidableEq F] in
 /-- `1 • genericPoint W = genericPoint W` (trivial, for reference). -/
 theorem one_zsmul_genericPoint :
     (1 : ℤ) • genericPoint W = genericPoint W :=
   one_zsmul _
 
+omit [DecidableEq F] in
 /-- Base case for the main induction: `1 • genericPoint W` matches the
     `[1]`-coordinates. -/
 theorem zsmul_genericPoint_one :
@@ -133,6 +94,7 @@ private lemma mulByInt_y_neg_aux
   field_simp
   ring
 
+omit [DecidableEq F] in
 /-- **Y-coordinate negation** (Silverman III.3.7 mirror): `mulByInt_y W (-n)`
     equals `negY(mulByInt_x W n, mulByInt_y W n)` on `W_KE`. -/
 theorem mulByInt_y_neg (n : ℤ) (hn : n ≠ 0) :
@@ -155,6 +117,7 @@ private lemma Φ_ff_eq (n : ℤ) :
   simp only [map_sub, map_mul, map_pow]
   rfl
 
+omit [DecidableEq F] in
 /-- **`mulByInt_x_eq`** (Silverman III.3.7 mirror): for `n ≠ 0`,
     `mulByInt_x W n = x_gen W - ψ_ff W (n+1) · ψ_ff W (n-1) / ψ_ff W n²`. -/
 theorem mulByInt_x_eq (n : ℤ) (hn : n ≠ 0) :
@@ -184,6 +147,7 @@ private lemma isEllipticSequence_ψ_ff (m n r : ℤ) :
   simp only [RingHom.comp_apply, map_mul, map_pow, map_sub] at hf
   exact hf
 
+omit [DecidableEq F] in
 /-- **`mulByInt_x_sub_mulByInt_x`** (Silverman III.3.7 mirror): for `m, n ≠ 0`,
     `mulByInt_x W m - mulByInt_x W n = ψ_ff(n+m) · ψ_ff(n-m) / (ψ_ff n · ψ_ff m)²`. -/
 theorem mulByInt_x_sub_mulByInt_x (m n : ℤ) (hm : m ≠ 0) (hn : n ≠ 0) :
@@ -204,6 +168,7 @@ theorem mulByInt_x_sub_mulByInt_x (m n : ℤ) (hm : m ≠ 0) (hn : n ≠ 0) :
   congr 1
   ring
 
+omit [DecidableEq F] in
 /-- **`mulByInt_x_ne_mulByInt_x`** (distinctness): for nonzero `m, n` with
     `m ≠ n` and `m ≠ -n`, `mulByInt_x W m ≠ mulByInt_x W n`. -/
 theorem mulByInt_x_ne_mulByInt_x (m n : ℤ) (hm : m ≠ 0) (hn : n ≠ 0) (hne : m ≠ n)
@@ -244,6 +209,7 @@ private lemma mulByInt_y_sub_negY_aux {K : Type*} [Field K]
   field_simp
   ring
 
+omit [DecidableEq F] in
 /-- **Y-coord minus negY at the generic point**: at the generic point of `W_KE`,
     `mulByInt_y W n - negY(mulByInt_x W n, mulByInt_y W n) = ψ_ff(2n) / ψ_ff(n)⁴`. -/
 theorem mulByInt_y_sub_negY (n : ℤ) (hn : n ≠ 0) :
@@ -256,12 +222,14 @@ theorem mulByInt_y_sub_negY (n : ℤ) (hn : n ≠ 0) :
   rw [W_KE_a₁, W_KE_a₃, ← ψ_ff_sq_eq_ΨSq_ff, mulByInt_y_sub_negY_aux hψ,
     ← ψc_ff_eq W n, ψ_ff_mul_ψc_eq W n]
 
+omit [DecidableEq F] in
 /-- Specializing to `n = 1`: `mulByInt_y W 1 - negY(...) = ψ_ff 2`. -/
 theorem mulByInt_y_one_sub_negY :
     mulByInt_y W 1 - (W_KE W).toAffine.negY (mulByInt_x W 1) (mulByInt_y W 1) =
       ψ_ff W 2 := by
   rw [mulByInt_y_sub_negY W 1 one_ne_zero, mul_one, ψ_ff_one, one_pow, div_one]
 
+omit [DecidableEq F] in
 /-- **Explicit form of `ψ_ff W 2`**: `ψ_ff W 2 = 2·y_gen + a₁·x_gen + a₃`. -/
 theorem ψ_ff_two_eq :
     ψ_ff W 2 = 2 * mulByInt_y W 1
@@ -278,6 +246,7 @@ private lemma algebraMap_Poly_KE_eval₂ (p : Polynomial F) :
   rw [← h_x, ← Polynomial.aeval_def, Polynomial.aeval_algebraMap_apply]
   simp
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **Generic Weierstrass equation** (expanded form): `y_gen² + a₁·x_gen·y_gen +
     a₃·y_gen = x_gen³ + a₂·x_gen² + a₄·x_gen + a₆` at the generic point of `W_KE`. -/
 theorem generic_weierstrass :
@@ -288,6 +257,7 @@ theorem generic_weierstrass :
   rw [mulByInt_x_one, mulByInt_y_one]
   exact (WeierstrassCurve.Affine.equation_iff _ _).mp (generic_equation W)
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **Explicit form of `ψ_ff W 3`** evaluated at the generic x-coordinate:
     `ψ_ff W 3 = 3·x_gen⁴ + b₂·x_gen³ + 3b₄·x_gen² + 3b₆·x_gen + b₈`. -/
 theorem ψ_ff_three_eq :
@@ -310,6 +280,7 @@ theorem ψ_ff_three_eq :
     show (algebraMap F KE) W.b₆ = (W_KE W).b₆ from (WeierstrassCurve.map_b₆ _ _).symm,
     show (algebraMap F KE) W.b₈ = (W_KE W).b₈ from (WeierstrassCurve.map_b₈ _ _).symm]
 
+omit [DecidableEq F] in
 /-- At the generic point, `mulByInt_y W 1 ≠ negY(mulByInt_x W 1, mulByInt_y W 1)`. -/
 theorem mulByInt_y_one_ne_negY :
     mulByInt_y W 1 ≠ (W_KE W).toAffine.negY (mulByInt_x W 1) (mulByInt_y W 1) := by
@@ -321,6 +292,7 @@ noncomputable abbrev slopeOne : KE :=
   (W_KE W).toAffine.slope (mulByInt_x W 1) (mulByInt_x W 1)
     (mulByInt_y W 1) (mulByInt_y W 1)
 
+omit [DecidableEq F] in
 /-- Closed form of `slopeOne` as `(3x² + 2a₂x + a₄ - a₁y) / ψ_ff 2` at the generic point. -/
 theorem slopeOne_eq :
     slopeOne W =
@@ -331,6 +303,7 @@ theorem slopeOne_eq :
   rw [WeierstrassCurve.Affine.slope_of_Y_ne rfl (mulByInt_y_one_ne_negY W),
     mulByInt_y_one_sub_negY]
 
+omit [DecidableEq F] in
 /-- **addX doubling identity**: at the generic point of `W_KE`,
     `addX(mulByInt_x W 1, mulByInt_x W 1, slopeOne W) = mulByInt_x W 2`.
 
@@ -366,6 +339,7 @@ private lemma some_eq_some_of_coords {x₁ x₂ y₁ y₂ : KE}
     Affine.Point.some x₁ y₁ h₁ = Affine.Point.some x₂ y₂ h₂ := by
   subst hx; subst hy; rfl
 
+omit [DecidableEq F] in
 /-- **Negation-reduction witness**: if `n • genericPoint W` is `.some` at the
     `[n]`-coordinates, then `(-n) • genericPoint W` is `.some` at the `[-n]`-coordinates. -/
 theorem zsmul_genericPoint_neg_of_pos (n : ℤ) (hn : n ≠ 0)
@@ -380,6 +354,7 @@ theorem zsmul_genericPoint_neg_of_pos (n : ℤ) (hn : n ≠ 0)
   rw [neg_zsmul, h_pos, Affine.Point.neg_some]
   exact some_eq_some_of_coords W _ _ (mulByInt_x_neg W n).symm (mulByInt_y_neg W n hn).symm
 
+omit [DecidableEq F] in
 /-- **Inductive step (witness form)**: given the `n`-case and the `(n+1)`-th
     `addX`/`addY` identities, derive the `(n+1)`-case (requires `[n]x ≠ [1]x`). -/
 theorem zsmul_genericPoint_add_one_of_witness (n : ℤ)
@@ -403,6 +378,7 @@ theorem zsmul_genericPoint_add_one_of_witness (n : ℤ)
     Affine.Point.some.injEq]
   exact ⟨h_addX_step, h_addY_step⟩
 
+omit [DecidableEq F] in
 /-- **Doubling case (witness form)**: given the `addX`/`addY` doubling identities,
     derive `(2 : ℤ) • genericPoint W = .some (mulByInt_x W 2) (mulByInt_y W 2) _`. -/
 theorem zsmul_genericPoint_two_of_witness
@@ -482,6 +458,7 @@ private lemma toAffineLift_zsmul_fromAffine_eq_toAffine (m : ℤ) {x₀ y₀ : K
   rw [WeierstrassCurve.zsmul_eq_smulEval (W := W_KE W) h_ns m]
   rfl
 
+omit [DecidableEq F] in
 /-- **Main theorem (Silverman III.4.2)**: for every `n ≠ 0`, `n • genericPoint W`
     has coordinates `(mulByInt_x W n, mulByInt_y W n)`. -/
 theorem zsmul_genericPoint_eq (n : ℤ) (hn : n ≠ 0) :
@@ -553,12 +530,14 @@ theorem zsmul_affine_point_eq (m : ℤ) {x₀ y₀ : KE}
     WeierstrassCurve.Jacobian.Point.toAffine_of_Z_ne_zero h_ns_smulEval hZ]
   rfl
 
+omit [DecidableEq F] in
 private lemma zsmul_genericPoint_ne_zero (n : ℤ) (hn : n ≠ 0) :
     (n : ℤ) • genericPoint W ≠ (0 : (W_KE W).toAffine.Point) := by
   obtain ⟨_, h_eq⟩ := zsmul_genericPoint_eq W n hn
   rw [h_eq]
   nofun
 
+omit [DecidableEq F] in
 /-- **Injectivity of `n ↦ [n]` (Silverman III.4.2b)**: if two nonzero integers `a, b`
     give the same `mulByInt_x` AND `mulByInt_y` coordinates, then `a = b`. Matching the
     full point (x AND y) gives the exact `[·] : ℤ → End E` injectivity, not up to `±`. -/
@@ -573,6 +552,7 @@ theorem mulByInt_xy_inj (a b : ℤ) (ha : a ≠ 0) (hb : b ≠ 0)
     rw [sub_eq_add_neg, add_zsmul, neg_zsmul, hab, add_neg_cancel]
   exact zsmul_genericPoint_ne_zero W (a - b) (sub_ne_zero.mpr hne) hz
 
+omit [DecidableEq F] in
 /-- **`ψ_m` at `(mulByInt_x W n, mulByInt_y W n)` is nonzero** when `m*n ≠ 0`. -/
 theorem ψ_m_evalEval_mulByInt_ne_zero (m n : ℤ) (hn : n ≠ 0) (hmn : m * n ≠ 0) :
     ((W_KE W).ψ m).evalEval (mulByInt_x W n) (mulByInt_y W n) ≠ 0 := by
@@ -598,6 +578,7 @@ theorem ψ_m_evalEval_mulByInt_ne_zero (m n : ℤ) (hn : n ≠ 0) (hmn : m * n 
     exact WeierstrassCurve.Jacobian.Point.toAffine_of_Z_eq_zero h_Z'
   rw [zsmul_some_eq_toAffineLift_fromAffine W m hns_n, h_toAffine_eq_zero]
 
+omit [DecidableEq F] in
 /-- The bivariate `φ_m` at `(mulByInt_x W n, mulByInt_y W n)` equals the univariate
     `eval₂ (algebraMap F KE) (mulByInt_x W n) (W.Φ m)`. -/
 theorem evalEval_φ_at_mulByInt_eq_eval₂_Φ (m n : ℤ) (hn : n ≠ 0) :
@@ -615,6 +596,7 @@ theorem evalEval_φ_at_mulByInt_eq_eval₂_Φ (m n : ℤ) (hn : n ≠ 0) :
   rw [AdjoinRoot.lift_of]
   rfl
 
+omit [DecidableEq F] in
 /-- Same translation for `ψ_m² → ΨSq_m`. -/
 theorem evalEval_ψ_sq_at_mulByInt_eq_eval₂_ΨSq (m n : ℤ) (hn : n ≠ 0) :
     ((W_KE W).ψ m).evalEval (mulByInt_x W n) (mulByInt_y W n) ^ 2 =
@@ -725,8 +707,6 @@ theorem mulByInt_pullback_mulByInt_y_eq_mul (m n : ℤ) (hn : n ≠ 0) (hmn : m
   rw [h_mn_eq, Affine.Point.some.injEq] at h_mul_eq
   exact h_mul_eq.2.symm
 
-/-- **T-III-4-020b (Silverman III.4.2)**: for `m, n ≠ 0` with `m·n ≠ 0`, the
-    multiplication-by-integer isogenies compose: `[m] ∘ [n] = [m·n]`. -/
 theorem mulByInt_comp_eq_mul (m n : ℤ) (hm : m ≠ 0) (hn : n ≠ 0) (hmn : m * n ≠ 0) :
     (mulByInt W.toAffine m).comp (mulByInt W.toAffine n) =
       mulByInt W.toAffine (m * n) := by
@@ -745,6 +725,7 @@ theorem mulByInt_comp_eq_mul (m n : ℤ) (hm : m ≠ 0) (hn : n ≠ 0) (hmn : m
     rw [h_m_y_gen]
     exact mulByInt_pullback_mulByInt_y_eq_mul W m n hn hmn
 
+omit [DecidableEq F] in
 /-- **Group-law successor step on the generic point** (Silverman III.5.3 prep): chord-addition
     of `P = (x_gen, y_gen)` against `[m]P = (mulByInt_x m, mulByInt_y m)` realises the
     `[m+1]`-coordinates, stated as `addX`/`addY` identities.
@@ -774,6 +755,7 @@ theorem addX_addY_genericPoint_mulByInt_eq_succ (m : ℤ) (hm : m ≠ 0) (hm1 :
     WeierstrassCurve.Affine.slope_of_X_ne hx_ne] at key
   exact ⟨key.1.symm, key.2.symm⟩
 
+omit [DecidableEq F] in
 /-- **Group-law successor step, `[m] ⊞ P` order** (pair-order mirror of
     `addX_addY_genericPoint_mulByInt_eq_succ`): chord-addition of `[m]P` against
     `P = (x_gen, y_gen)` realises the `[m+1]`-coordinates. The secant slope is written
@@ -799,11 +781,10 @@ theorem addX_addY_mulByInt_genericPoint_eq_succ (m : ℤ) (hm : m ≠ 0) (hm1 :
     WeierstrassCurve.Affine.slope_of_X_ne hx_ne] at key
   exact ⟨key.1.symm, key.2.symm⟩
 
+omit [DecidableEq F] in
 /-- **Doubling step at the generic point, explicit-slope form**: the `addX`/`addY`
     doubling identities realising the `[1+1]`-coordinates, with the tangent slope
-    written out in the `slope_of_Y_ne` quotient form (`slope`-free for instance
-    stability). Consumed by the tangent branch of
-    `addPullback_xy_pair_mulByInt_one_eq_succ` (`MulByIntAddRecurrence`). -/
+    written out in the `slope_of_Y_ne` quotient form. -/
 theorem addX_addY_mulByInt_one_self_eq_two :
     (W_KE W).toAffine.addX (mulByInt_x W 1) (mulByInt_x W 1)
         ((3 * mulByInt_x W 1 ^ 2 + 2 * (W_KE W).toAffine.a₂ * mulByInt_x W 1
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntSamePlace.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntSamePlace.lean
index ca3d78ea8..c7f137720 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntSamePlace.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntSamePlace.lean
@@ -10,54 +10,25 @@ import HasseWeil.HasseBound.WeilPairing.TorsionGeometric
 import HasseWeil.HasseBound.WeilPairing.TorsionKernelRational
 
 /-!
-# The **(SamePlace)** fact for the multiplication isogeny `[ℓ]`
-
-This file supplies the **`Valuation.IsEquiv`** ("same place / same valuation ring") input that the
-axiom-clean glue `comap_pointValuation_eq_of_isEquiv_of_ord_eq_one`
-(`HasseWeil/EC/IsogenyOrdTransport.lean`) consumes to upgrade to the value-precise order-transport
-`ord_P (φ.pullback g) = ord_{φ(P)} g`.
-
-For the isogeny `φ = [ℓ] = mulByInt W ℓ` and a smooth point `P` of `⟨W⟩` whose image
-`Q = [ℓ]·P = (mulByInt W ℓ).toAddMonoidHom P.toAffinePoint` is either an *affine* point or `O`, the
-comap valuation `(pointValuation P).comap (mulByInt W ℓ).pullback.toRingHom` is `Valuation.IsEquiv`
-to the place-valuation at `Q`:
-
-* **affine image** `Q = some x y h_ns`: equivalent to `pointValuation ⟨x, y, h_ns⟩`;
-* **infinity image** `Q = O`: equivalent to `ordAtInftyValuation`.
-
-## Mathematical content (Silverman II.2.5 / III.4.10c)
-
-`Valuation.IsEquiv v w` is, by `Valuation.isEquiv_of_val_le_one`, the statement
-`∀ g, v g ≤ 1 ↔ w g ≤ 1`: the two valuations have the same valuation ring. Here this reads
-```
-[ℓ].pullback g  is regular at P   ⟺   g is regular at Q,
-```
-the **same-place** content of the morphism `[ℓ] : E → E` with `[ℓ](P) = Q`. The forward direction
-(`g` regular at `Q` ⟹ `[ℓ]^*g` regular at `P`) is functoriality of a morphism (it pulls back
-regular functions to regular functions, because `[ℓ]` maps `P ↦ Q`); the converse holds because
-`[ℓ]` is dominant / a local homomorphism of the discrete valuation rings `O_Q → O_P` with positive
-ramification index `e_P ≥ 1`. Crucially, **`IsEquiv` only needs `e_P ≥ 1` (ramification
-positivity)** — the sharper `e_P = 1` (separability) is supplied *separately*, as the `ord = 1`
-input to `comap_pointValuation_eq_of_isEquiv_of_ord_eq_one`.
-
-## Status
-
-The **affine-image** case (`mulByInt_samePlace_le_one_iff_affine`) is proven **sorry-free and
-axiom-clean** (`[propext, Classical.choice, Quot.sound]`): its residue matching is supplied by the
-affine specialisation `[ℓ]·P = (φ_ℓ(P)/ψ_ℓ(P)², ω_ℓ(P)/ψ_ℓ(P)³)` of the division-polynomial group
-law (`zsmul_affine_point_eq_gen`), lifted to all of `K(E)` by the `IsLocalization`/DVR machinery.
-
-The **infinity-image** case (`mulByInt_samePlace_le_one_iff_infty`) is likewise proven **sorry-free
-and axiom-clean**: it is read off the *torsion-pole* transfer `ord_P (mulByInt_x ℓ) = -2`,
-`ord_P (mulByInt_y ℓ) = -3` (the simple zeros of `ψ_ℓ` at the `ℓ`-torsion points), supplied here by
-`ord_P_mulByInt_x_eq_neg_two_of_torsion` / `ord_P_mulByInt_y_eq_neg_three_of_torsion` via the
-kernel-translation transport of `ord_∞`, and packaged into the infinity comap identity
-`comap_pointValuation_mulByInt_eq_infty`.
-
-The `IsEquiv` packaging (`isEquiv_of_val_le_one`) and the `comap`-valuation forms are sorry-free
-over both cases.
-
-Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.2.5–2.6, III.4.10c.
+# Equality of places under multiplication
+
+For the multiplication isogeny `φ = [ℓ]` and a smooth point `P` with image `Q`,
+the comap of the point valuation at `P` under `φ*` is equivalent to the valuation
+at `Q`. If `Q` is affine this is its point valuation; if `Q = O`, it is the
+valuation at infinity.
+
+Equivalence means that the valuations have the same valuation ring:
+`[ℓ]* g` is regular at `P` exactly when `g` is regular at `Q`. The induced local
+homomorphism of discrete valuation rings has positive ramification index. This
+place identity only requires positivity; an order-one uniformizer pullback gives
+the additional unramified normalization.
+
+For affine `Q`, the division-polynomial coordinate formula determines residue
+matching, which extends to the function field through the local ring. At a torsion
+point `P`, kernel translation transports the coordinate orders `-2` and `-3` at
+infinity to the corresponding pullbacks at `P`.
+
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.2.5–2.6 and III.4.10c.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
@@ -574,7 +545,7 @@ private theorem exists_coordinateRing_repr_of_le_one
           algebraMap (⟨W⟩ : SmoothPlaneCurve F).CoordinateRing KE u := by
   obtain ⟨x_loc, hx_loc⟩ :=
     (Curves.SmoothPlaneCurve.mem_localRingAt_image_iff_pointValuation_le_one g).mpr hg
-  haveI : ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt ⟨x, y, h_ns⟩).IsPrime :=
+  have : ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt ⟨x, y, h_ns⟩).IsPrime :=
     ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt_isMaximal ⟨x, y, h_ns⟩).isPrime
   obtain ⟨⟨u, v⟩, hv_eq⟩ := IsLocalization.surj
     ((⟨W⟩ : SmoothPlaneCurve F).maximalIdealAt ⟨x, y, h_ns⟩).primeCompl x_loc
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntUnramified.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntUnramified.lean
index 94182f575..9ed72ce63 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntUnramified.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/MulByIntUnramified.lean
@@ -12,12 +12,13 @@ import HasseWeil.HasseBound.WeilPairing.TorsionGeometric
 /-!
 # Unramifiedness of `[ℓ]` at an affine non-2-torsion image point (the `e = 1` input)
 
-For an elliptic curve `W/F` and a smooth point `P` of `⟨W⟩` whose image `[ℓ]·P = Q = (x_Q, y_Q)`
+For an elliptic curve `W/F`, an integer `ℓ` nonzero in `F`, and a smooth point `P`
+whose image `[ℓ]·P = Q = (x_Q, y_Q)`
 is an *affine non-2-torsion* point, the pullback `[ℓ]^*(x_gen − x_Q) = mulByInt_x ℓ − x_Q` of the
 uniformizer `x_gen − x_Q` at `Q` is a uniformizer at `P`: its order is exactly `1`. This is the
 geometric *unramifiedness* of `[ℓ]` (Silverman III.4.10c), the `e = 1` normalization that feeds the
-order-transport glue `comap_pointValuation_eq_of_isEquiv_of_ord_eq_one`
-(`EC/IsogenyOrdTransport.lean`). No algebraic closure of `F` is needed.
+valuation identity `comap_pointValuation_eq_of_isEquiv_of_ord_eq_one`.
+No algebraic closure of `F` is needed.
 
 ## Strategy
 
@@ -327,7 +328,7 @@ theorem preΨ_two_mul_eval_ne_zero {x y x_Q y_Q : F}
   have hΨ_factored : (W.Ψ (2 * ℓ)).evalEval x y =
       (W.preΨ (2 * ℓ)).eval x * (W.ψ₂).evalEval x y := by
     rw [show W.Ψ (2 * ℓ) = Polynomial.C (W.preΨ (2 * ℓ)) * W.ψ₂ from by
-      rw [WeierstrassCurve.Ψ, if_pos (even_two_mul ℓ)],
+      rw [WeierstrassCurve.Ψ, ite_eq_left (even_two_mul ℓ)],
       Polynomial.evalEval_mul, Polynomial.evalEval_C]
   rw [hΨ_eq, hΨ_factored] at hψ_ne
   exact fun h ↦ hψ_ne (by rw [h, zero_mul])
@@ -454,11 +455,9 @@ pullback `[ℓ]^*(x_gen − x_Q) = mulByInt_x ℓ − x_Q` of the uniformizer `x
 uniformizer at `P`:
 `ord_P P (mulByInt_x ℓ − x_Q) = 1`.
 
-This is the `e = 1` input to `comap_pointValuation_eq_of_isEquiv_of_ord_eq_one`. The whole reduction
-is **axiom-clean** (`[propext, Classical.choice, Quot.sound]`): the division-polynomial Wronskian it
-consumes (through `fibrePoly_derivative_eval_ne_zero`) is the EDS-free `wronskian_Φ_ΨSq_general`
-(`EC/WronskianGeneral.lean`), proved downstream via the function-field differential `a_{[ℓ]} = ℓ`
-(`omegaCoeff_mulByInt`, Route-B), so it carries no `sorryAx`. -/
+The division-polynomial Wronskian identity gives a nonzero derivative of the fibre polynomial,
+using the multiplication differential coefficient `a_[ℓ] = ℓ`. This supplies the order-one
+normalization for `comap_pointValuation_eq_of_isEquiv_of_ord_eq_one`. -/
 theorem ord_P_mulByInt_x_sub_const_eq_one (ℓ : ℤ) (hℓ : ℓ ≠ 0) (hℓF : (ℓ : F) ≠ 0)
     (P : (⟨W⟩ : SmoothPlaneCurve F).SmoothPoint) {x_Q y_Q : F}
     (h_ns' : W.toAffine.Nonsingular x_Q y_Q) (h_not_2_tor_Q : y_Q ≠ W.toAffine.negY x_Q y_Q)
@@ -504,7 +503,7 @@ theorem one_le_ord_P_algebraMap_of_evalAt_zero
     have h_v_le : (⟨W⟩ : SmoothPlaneCurve F).pointValuation P (algebraMap R KE u) ≤ 1 :=
       (⟨W⟩ : SmoothPlaneCurve F).pointValuation_algebraMap_le_one u P
     unfold SmoothPlaneCurve.ord_P
-    rw [dif_neg hv, show (0 : WithTop ℤ) = ((0 : ℤ) : WithTop ℤ) from rfl, WithTop.coe_le_coe]
+    rw [dite_eq_right hv, show (0 : WithTop ℤ) = ((0 : ℤ) : WithTop ℤ) from rfl, WithTop.coe_le_coe]
     have h_unz_le : WithZero.unzero hv ≤ 1 := by
       rw [← WithZero.coe_le_coe, WithZero.coe_one, WithZero.coe_unzero]; exact h_v_le
     have h_toAdd : (WithZero.unzero hv).toAdd ≤ 0 := h_unz_le
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/SeparableKernelTorsor.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/SeparableKernelTorsor.lean
index 9309b84a826270fc3b8adbbee64c802c0024fd6c..9b71903354906408799d938fa473abecd0a6510e 100644
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/TranslateOrdInfty.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/TranslateOrdInfty.lean
index a4724ced9..55d7b18bd 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/TranslateOrdInfty.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/TranslateOrdInfty.lean
@@ -8,45 +8,25 @@ import HasseWeil.Foundation.EC.TranslationOrd
 import HasseWeil.HasseBound.PoleDivisorTwoTorsion
 
 /-!
-# Order-at-infinity transport under translation (Step (C))
+# Order at infinity under translation
 
-This file discharges the deep order-at-infinity transport obligation
-`IsTranslateOrdAtInftyCompatible`: for a finite smooth point `P` and a group
-element `k` with `P + k = O`, the translation pullback `τ_k` carries the order
-at `P` to the order at infinity:
-```
-  ord_P P (τ_k f) = ordAtInfty f          (when P + k = 0).
-```
+For a finite smooth point `P` and a group element `k` with `P + k = O`,
+translation pullback carries the order at `P` to the order at infinity:
+`ord_P P (τ_k f) = ordAtInfty f`.
 
-Geometrically the translation-by-`k` automorphism maps the place `P = -k` to
-the place at infinity `O`, preserving order.
-
-## Route
-
-Both `ord_P` (DVR at a finite smooth point) and `ordAtInfty` (norm model at the
-unique place at infinity) are normalised discrete valuations on `K(E)`, but they
-use *different* local models, so the equality is not formal. We identify them
-via the multiplicative `Valuation` packaging:
-
-* `ν₂ := (pointValuation P).comap τ_k` and `ν₁ := ordAtInftyValuation` are two
-  surjective `ℤᵐ⁰`-valued valuations on `K(E)`.
-* The translation base cases (already shipped:
-  `ord_P_translateX_xy_eq_neg_two_*`, `ord_P_translateY_xy_eq_neg_three_*`) give
-  `ν₂ x_gen = exp 2 = ν₁ x_gen` and `ν₂ y_gen = exp 3 = ν₁ y_gen`, and `τ_k`
-  fixing constants gives `ν₂` trivial on `F^×`.
-* A *valuation determined by its values on `x_gen`, `y_gen`* lemma (proved here
-  by the basis decomposition `f = r₁ + r₂·y_gen` and strict non-archimedean
-  dominance — the cross-term parity makes the `min` unambiguous) forces
-  `ν₂ = ν₁ = ordAtInftyValuation`.
-
-Reading off the additive value gives `ord_P P (τ_k f) = ordAtInfty f`.
+The valuations at a finite point and at infinity use different local models.
+They are identified as multiplicative valuations on the function field. At
+`P = -k`, the translated coordinates have orders `-2` and `-3`, so the
+multiplicative valuation takes values `exp 2` and `exp 3`; it is trivial on
+nonzero constants. The basis decomposition `f = r₁ + r₂ y`, with `r₁, r₂ ∈ F(x)`,
+then determines it uniquely. The two summands have orders of opposite parity,
+so strict nonarchimedean dominance determines the value of their sum.
 
 ## References
 
-* Silverman, *The Arithmetic of Elliptic Curves*, III.8 (translation), IV.1
-  (place at infinity).
+* Silverman, *The Arithmetic of Elliptic Curves*, III.8 (translation) and IV.1
+  (the place at infinity).
 -/
-
 open WeierstrassCurve HasseWeil.Curves
 
 namespace HasseWeil
@@ -57,16 +37,14 @@ variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 local notation "KE" => W.toAffine.FunctionField
 
 omit [DecidableEq F] in
-/-- `pointValuation P f = exp (-n)` from `ord_P P f = n` (for `f ≠ 0`). Curve-side
-mirror of `ordAtInftyValuation_eq_exp_neg_of_ordAtInfty_eq`, kept local to avoid an
-import of `Hasse.L6Witnesses`. -/
+/-- `pointValuation P f = exp (-n)` follows from `ord_P P f = n` for `f ≠ 0`. -/
 theorem pointValuation_eq_exp_neg_of_ord_P_eq {C : Curves.SmoothPlaneCurve F} {P : C.SmoothPoint}
     {f : C.FunctionField} {n : ℤ} (hf : f ≠ 0) (hn : C.ord_P P f = (n : WithTop ℤ)) :
     C.pointValuation P f = WithZero.exp (-n) := by
   have hv : C.pointValuation P f ≠ 0 := (C.pointValuation P).ne_zero_iff.mpr hf
   have hord : C.ord_P P f = ((-(WithZero.unzero hv).toAdd : ℤ) : WithTop ℤ) := by
     simp only [Curves.SmoothPlaneCurve.ord_P]
-    rw [dif_neg hv]
+    rw [dite_eq_right hv]
   have hneq : n = -(WithZero.unzero hv).toAdd := by exact_mod_cast (hord.symm.trans hn).symm
   rw [hneq, neg_neg, WithZero.exp, ofAdd_toAdd, WithZero.coe_unzero]
 
@@ -84,8 +62,8 @@ theorem valuation_aeval_eq_exp (w : Valuation KE (WithZero (Multiplicative ℤ))
     intro i
     rw [Algebra.smul_def, map_mul, map_pow, hu, ← WithZero.exp_nsmul]
     by_cases hci : p.coeff i = 0
-    · rw [if_pos hci, hci, RingHom.map_zero, map_zero, zero_mul]
-    · rw [if_neg hci, hc _ hci, one_mul]
+    · rw [ite_eq_left hci, hci, RingHom.map_zero, map_zero, zero_mul]
+    · rw [ite_eq_right hci, hc _ hci, one_mul]
       congr 1
       rw [nsmul_eq_mul]
       ring
@@ -97,13 +75,13 @@ theorem valuation_aeval_eq_exp (w : Valuation KE (WithZero (Multiplicative ℤ))
   refine (Valuation.map_sum_eq_of_lt w (Finset.self_mem_range_succ n) ?_).trans ?_
   · intro i hi
     rw [Finset.mem_sdiff, Finset.mem_range, Finset.mem_singleton] at hi
-    rw [h_term i, h_term n, if_neg h_lead_ne]
+    rw [h_term i, h_term n, ite_eq_right h_lead_ne]
     by_cases hci : p.coeff i = 0
-    · rw [if_pos hci]
+    · rw [ite_eq_left hci]
       exact lt_of_le_of_ne zero_le (Ne.symm WithZero.exp_ne_zero)
-    · rw [if_neg hci, WithZero.exp_lt_exp]
+    · rw [ite_eq_right hci, WithZero.exp_lt_exp]
       omega
-  · rw [h_term n, if_neg h_lead_ne]
+  · rw [h_term n, ite_eq_right h_lead_ne]
 
 omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `algebraMap (Polynomial F) K(E)` is `aeval x_gen` (both are the F-algebra hom
@@ -277,10 +255,11 @@ theorem ordAtInftyValuation_basis_summands_distinct {r₁ r₂ : FractionRing (P
       intro h_eq
       omega
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 set_option backward.isDefEq.respectTransparency.types false in
 /-- **Value on the coordinate function `y`.** A valuation `w` with `w y_gen = exp 3`
 agrees with `ordAtInftyValuation` on `coordYInFunctionField` (which is `y_gen`):
-both equal `exp (-3)`. -/
+both equal `exp 3`. -/
 private theorem valuation_coordYInFunctionField_eq_ordAtInftyValuation
     (w : Valuation KE (WithZero (Multiplicative ℤ))) (hy : w (y_gen W) = WithZero.exp 3) :
     w (W_smooth W).coordYInFunctionField =
@@ -359,6 +338,7 @@ private theorem eq_ordAtInftyValuation_of_agree_fracPolyX_coordY
     exact valuation_eq_add_of_eq_of_eq_of_distinct W w (W_smooth W).ordAtInftyValuation
       (h_rat p) hβc_agree (ordAtInftyValuation_basis_summands_distinct W h_not_both)
 
+omit [W.toAffine.IsElliptic] in
 /-- **Valuation determined by `x_gen`, `y_gen`.** A `ℤᵐ⁰`-valued valuation `w` on
 `K(E)` with `w x_gen = exp 2`, `w y_gen = exp 3`, and trivial on `F^×` equals
 `ordAtInftyValuation`. The decomposition `f = α + β·coordY` (`α, β ∈ F(x)`) plus
@@ -372,8 +352,9 @@ theorem eq_ordAtInftyValuation_of_x_y (w : Valuation KE (WithZero (Multiplicativ
     (valuation_algebraMap_fracPolyX_eq_ordAtInftyValuation W w hx hc)
     (valuation_coordYInFunctionField_eq_ordAtInftyValuation W w hy)
 
+omit [W.toAffine.IsElliptic] in
 /-- `ord_P (-T) (τ_T x_gen) = -2`, uniformly across 2-torsion and non-2-torsion
-`T = (xk, yk)`. Dispatches to the two shipped cases. -/
+`T = (xk, yk)`. -/
 theorem ord_P_negSmoothPoint_translateX_xy_eq_neg_two
     (xk yk : F) (h_ns : W.toAffine.Nonsingular xk yk) :
     (W_smooth W).ord_P (negSmoothPoint W xk yk h_ns)
@@ -382,6 +363,7 @@ theorem ord_P_negSmoothPoint_translateX_xy_eq_neg_two
   · exact ord_P_translateX_xy_eq_neg_two_at_2tor W xk yk h_ns h
   · exact ord_P_translateX_xy_eq_neg_two_of_non_2_tor W xk yk h_ns h
 
+omit [W.toAffine.IsElliptic] in
 /-- `ord_P (-T) (τ_T y_gen) = -3`, uniformly across 2-torsion and non-2-torsion. -/
 theorem ord_P_negSmoothPoint_translateY_xy_eq_neg_three
     (xk yk : F) (h_ns : W.toAffine.Nonsingular xk yk) :
@@ -461,11 +443,8 @@ theorem ord_P_negSmoothPoint_translateAlgEquivOfPoint_eq_ordAtInfty_some
   rw [hm, hn]
   exact_mod_cast neg_injective h_at_f
 
-/-- **Unconditional discharge of `IsTranslateOrdAtInftyCompatible`.** For a finite
-smooth point `P` and a group element `k` with `P + k = O` (so `P = -k`), the
-translation pullback `τ_k` carries the order at `P` to the order at infinity:
-`ord_P P (τ_k f) = ordAtInfty f` for all `f`. Gates the Weil-pairing divisor
-transport. -/
+/-- For a finite smooth point `P` and a group element `k` with `P + k = O`,
+translation pullback satisfies `ord_P P (τ_k f) = ordAtInfty f` for every `f`. -/
 theorem isTranslateOrdAtInftyCompatible_translateAlgEquivOfPoint
     (P : (W_smooth W).SmoothPoint) (k : (W_smooth W).toAffine.Point)
     (h_zero : P.toAffinePoint + k = Affine.Point.zero) :
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/TranslateValuation.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/TranslateValuation.lean
index 0fbf76569d790458fadca970039cf32598f16d3b..8fbe28c64fdb18776583e7ad48fb72d1d909f59c 100644
GIT binary patch
delta 113
zcmcckfaA%1jt!d_H~(RbV>B!+PAyhQ&d)7KEXhpD%*iaNRLD<LNG(guDFuq;=P8tD
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E0DZwO_W%F@

diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/Translation.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/Translation.lean
index c135cb1b2..798b434d7 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/EC/Translation.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/EC/Translation.lean
@@ -6,43 +6,17 @@ Authors: Chris Birkbeck
 import HasseWeil.Foundation.AdditionPullback
 
 /-!
-# Translation by a base-field point on K(E)
+# Translation by a base-field point on the function field
 
-For an elliptic curve `W` over `F` and a point `(xk, yk) ∈ W(F)` (= a base-field
-solution of the Weierstrass equation), translation by `k = (xk, yk)` lifts
-to a K-algebra automorphism `τ_k : K(E) ≃ₐ[F] K(E)` defined by:
+Translation by a point `k = (xk, yk)` of an elliptic curve acts on its
+function field through the addition formulas. The slope is
+`(y_gen − yk) / (x_gen − xk)`, and the translated coordinate functions
+are obtained from this slope by `addX` and `addY`. The coordinate-ring
+homomorphism extends to the function field when the evaluation map is
+injective. Kernel translations preserve pullbacks along isogenies,
+relating the point kernel to function-field automorphisms.
 
-* `τ_k(x_gen) = (W_KE).addX(x_gen, xk, slope_k)` where
-  `slope_k = (y_gen - yk) / (x_gen - xk)`.
-* `τ_k(y_gen) = (W_KE).addY(x_gen, xk, y_gen, slope_k)`.
-
-This is the substantive Galois-correspondence content for elliptic
-isogenies (Silverman III.4.10(a)): for an isogeny `α : E → E`, the kernel
-of `α` (as an additive subgroup of `E.Point`) acts on `K(E₁)/α*K(E₂)` as
-the Galois group, and the bijection `α.kernel ≃ Aut(K(E)/α*K(E))` sends
-each kernel element `k` to the translation automorphism `τ_k`.
-
-This file constructs the translation algebra hom directly via the addition
-formula on coordinates, paralleling
-`HasseWeil/AdditionPullback.lean`'s `addPullbackAlgHom_negFrobenius`
-construction.
-
-## Foundational definitions
-
-The translation construction is parallel to `addPullback_x` / `addPullback_y`
-but with the second point a base-field constant rather than `α.pullback`.
-
-* `translateSlope_xy xk yk` — slope from `(x_gen, y_gen)` to `(xk, yk)`.
-* `translateX_xy xk yk` — x-coord of `P_gen + (xk, yk)`.
-* `translateY_xy xk yk` — y-coord of `P_gen + (xk, yk)`.
-
-These are foundational K(E)-elements; the algebra-hom construction (analog
-of `addPullbackAlgHom`) requires the `AddNonInverse` and injectivity
-witnesses.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, III.4.10(a).
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.4.10(a).
 -/
 
 open WeierstrassCurve
@@ -293,6 +267,7 @@ noncomputable def translateAlgHom (xk yk : F) (h_eq : W.toAffine.Equation xk yk)
     (translateCoordAlgHom_injective_of_baseHom_inj W xk yk h_eq hxy hxinj)
 
 omit [DecidableEq F] in
+omit [W.toAffine.IsElliptic] in
 /-- A pole of `translateX_xy` at infinity forces non-constancy: if
 `ordAtInfty (translateX_xy) < 0` then `translateX_xy` is not `algebraMap F KE c`
 for any `c : F`. -/
@@ -329,6 +304,7 @@ theorem translateBaseHom_injective_of_transcendental (xk yk : F)
   exact transcendental_iff_injective.mp h_trans
 
 omit [DecidableEq F] in
+omit [W.toAffine.IsElliptic] in
 private theorem ordAtInfty_sub_const_of_neg_aux {g : KE} {n : ℤ} (hn : n < 0)
     (hg : (W_smooth W).ordAtInfty g = ((n : ℤ) : WithTop ℤ)) (c : F) :
     (W_smooth W).ordAtInfty (g - algebraMap F KE c) = ((n : ℤ) : WithTop ℤ) := by
@@ -339,7 +315,7 @@ private theorem ordAtInfty_sub_const_of_neg_aux {g : KE} {n : ℤ} (hn : n < 0)
     rw [hg, ordAtInfty_algebraMap_F_nonzero W hc]
     exact_mod_cast hn
 
-omit [DecidableEq F] in
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- For any `xk ∈ F`, `ord(x_gen - xk) = -2`, since the constant `xk` has
 nonnegative order while `x_gen` has order `-2`. -/
 theorem ord_x_gen_sub_const (xk : F) :
@@ -370,7 +346,7 @@ theorem translateSlope_xy_eq (xk yk : F) :
   intro h_eq
   exact x_gen_sub_const_ne_zero W xk (sub_eq_zero.mpr h_eq)
 
-omit [DecidableEq F] in
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord(y_gen - yk) = -3` for any `yk : F`. -/
 theorem ord_y_gen_sub_const (yk : F) :
     (W_smooth W).ordAtInfty (y_gen W - algebraMap F KE yk) =
@@ -378,9 +354,9 @@ theorem ord_y_gen_sub_const (yk : F) :
   classical
   exact ordAtInfty_sub_const_of_neg_aux W (by norm_num) (ordAtInfty_y_gen W) yk
 
-omit [DecidableEq F] in
-/-- `ord(translateSlope_xy) = -1` for non-zero `xk` (so the slope is in secant
-form). -/
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
+/-- The translation slope has order `−1` at infinity, as the quotient
+of coordinate differences of orders `−3` and `−2`. -/
 theorem ord_translateSlope_xy (xk yk : F) :
     (W_smooth W).ordAtInfty (translateSlope_xy W xk yk) =
       ((-1 : ℤ) : WithTop ℤ) := by
diff --git a/projects/HasseWeil/HasseWeil/Foundation/EC/TranslationOrd.lean b/projects/HasseWeil/HasseWeil/Foundation/EC/TranslationOrd.lean
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/LocalExpansion.lean b/projects/HasseWeil/HasseWeil/Foundation/LocalExpansion.lean
index 0afac3ecb..7004844d6 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/LocalExpansion.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/LocalExpansion.lean
@@ -8,54 +8,6 @@ import HasseWeil.FormalGroup.FormalGroup
 import Mathlib.RingTheory.LaurentSeries
 import Mathlib.RingTheory.PowerSeries.Inverse
 
-/-!
-# Local Expansion at the Identity O (Phase 1 of the Local Expansion Bridge)
-
-We construct the **local expansion functor** `localExpand : K(E) → LaurentSeries F`,
-which expands an element of the function field of an elliptic curve as a Laurent
-series in the local parameter `t = -x/y` at the identity `O`.
-
-This is the foundation of the bridge between the formal-power-series world (where
-`pullbackCoeff_add` is already proved in `FormalGroupAssoc.lean`) and the
-function-field-pullback world (where `omegaPullbackCoeff` lives).
-
-## Strategy (Silverman IV.1)
-
-The unique formal series `w(z) ∈ F[[z]]` with `w(z) = z³ + a₁ z w(z) + a₂ z² w(z)
-+ a₃ w(z)² + a₄ z w(z)² + a₆ w(z)³` is constructed in `FormalGroup.lean` as
-`formalW W`. From this we recover:
-
-* `x(z) = z / w(z) = z⁻² · u(z)⁻¹` where `u(z) = w(z) / z³` is a unit power series
-* `y(z) = -1 / w(z) = -z⁻³ · u(z)⁻¹`
-
-These satisfy the Weierstrass equation `y² + a₁xy + a₃y - x³ - a₂x² - a₄x - a₆ = 0`
-because `w(z)` satisfies its defining recurrence (this is exactly Silverman IV.1.1
-multiplied through by `-w³`).
-
-By the universal property of the function field, the substitution `(x_gen, y_gen) ↦
-(formalX, formalY)` extends to a unique ring homomorphism `K(E) → LaurentSeries F`
-(the target is a field, so any nonzero ring hom is injective and the substitution
-extends from the coordinate ring through to the fraction field).
-
-## Main definitions
-
-* `formalU W : PowerSeries F` — the unit `w(z) / z³ = 1 + a₁z + ...`
-* `formalY W : LaurentSeries F` — the Laurent series for `y` in terms of `t = -x/y`
-* `formalX W : LaurentSeries F` — the Laurent series for `x` in terms of `t`
-* `localParam W : K(E)` — the local parameter `t = -x_gen / y_gen`
-* `localExpand W : K(E) →+* LaurentSeries F` — the expansion ring hom
-
-## Status
-
-This file establishes the API surface. The hard parts (Weierstrass equation
-verification for `formalX, formalY`, and the construction of `localExpand` via the
-universal property of the function field) are currently `sorry`-marked with
-detailed comments. Phase 2 (`IsogenyLocalExpansion.lean`) builds on this API.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, IV.1 (pp. 115–119)
--/
 
 open WeierstrassCurve PowerSeries LaurentSeries
 
@@ -75,12 +27,14 @@ local notation "KE" => W.toAffine.FunctionField
 noncomputable def formalU : PowerSeries F :=
   PowerSeries.mk (fun n ↦ formalW_coeff W (n + 3))
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The constant coefficient of `formalU` is 1. -/
 @[simp] theorem formalU_constantCoeff :
     @PowerSeries.constantCoeff F _ (formalU W) = 1 := by
   rw [← PowerSeries.coeff_zero_eq_constantCoeff_apply, formalU, PowerSeries.coeff_mk,
       zero_add, formalW_coeff_three]
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalU W` is a unit in `F⟦X⟧`. -/
 theorem formalU_isUnit : IsUnit (formalU W) := by
   rw [PowerSeries.isUnit_iff_constantCoeff]
@@ -92,16 +46,19 @@ theorem formalU_isUnit : IsUnit (formalU W) := by
 noncomputable def formalU_inv : PowerSeries F :=
   PowerSeries.invOfUnit (formalU W) 1
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalU * formalU_inv = 1` in `F⟦X⟧`. -/
 @[simp] theorem formalU_mul_inv : formalU W * formalU_inv W = 1 :=
   PowerSeries.mul_invOfUnit (formalU W) 1
     (by rw [formalU_constantCoeff]; rfl)
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalU_inv * formalU = 1` in `F⟦X⟧`. -/
 @[simp] theorem formalU_inv_mul : formalU_inv W * formalU W = 1 :=
   PowerSeries.invOfUnit_mul (formalU W) 1
     (by rw [formalU_constantCoeff]; rfl)
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The constant coefficient of `formalU_inv` is 1. -/
 @[simp] theorem formalU_inv_constantCoeff :
     @PowerSeries.constantCoeff F _ (formalU_inv W) = 1 := by
@@ -131,6 +88,7 @@ In `LaurentSeries F`, `formalX W` has order `-2` and `formalY W` has order `-3`.
 These follow because `formalU_inv W` has order `0` (constant coeff = 1) and
 `single (-2) 1` resp. `single (-3) 1` have orders `-2` resp. `-3`. -/
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `ofPowerSeries ℤ F (formalU_inv W)` is nonzero in `LaurentSeries F`. -/
 theorem ofPowerSeries_formalU_inv_ne_zero :
     (HahnSeries.ofPowerSeries ℤ F (formalU_inv W) : LaurentSeries F) ≠ 0 := by
@@ -144,12 +102,14 @@ theorem ofPowerSeries_formalU_inv_ne_zero :
   rw [h] at h_nonzero
   exact one_ne_zero h_nonzero.symm
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalX W ≠ 0`. -/
 theorem formalX_ne_zero : formalX W ≠ 0 := by
   rw [formalX]
   exact mul_ne_zero (HahnSeries.single_ne_zero one_ne_zero)
     (ofPowerSeries_formalU_inv_ne_zero W)
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalY W ≠ 0`. -/
 theorem formalY_ne_zero : formalY W ≠ 0 := by
   rw [formalY]
@@ -157,6 +117,7 @@ theorem formalY_ne_zero : formalY W ≠ 0 := by
   rw [ne_eq, neg_eq_zero]
   exact HahnSeries.single_ne_zero one_ne_zero
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `ofPowerSeries ℤ F (formalU_inv W)` has `orderTop = 0`. This is because the
     series has constant coefficient 1, so its order is ≤ 0; and all coefficients
     at negative indices vanish (since `ofPowerSeries` embeds `ℕ ↪ ℤ`), so its order
@@ -177,10 +138,11 @@ theorem ofPowerSeries_formalU_inv_orderTop :
     rw [HahnSeries.le_orderTop_iff_forall]
     intro n hn
     have : ((formalU_inv W : PowerSeries F) : LaurentSeries F).coeff n = 0 := by
-      rw [PowerSeries.coeff_coe, if_pos]; exact_mod_cast hn
+      rw [PowerSeries.coeff_coe, ite_eq_left]; exact_mod_cast hn
     exact this
   exact le_antisymm h_le h_ge
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `formalX W` has `orderTop = -2` in `LaurentSeries F`. -/
 theorem formalX_orderTop :
     (formalX W).orderTop = ((-2 : ℤ) : WithTop ℤ) := by
@@ -188,6 +150,7 @@ theorem formalX_orderTop :
       ofPowerSeries_formalU_inv_orderTop]
   rfl
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `formalY W` has `orderTop = -3` in `LaurentSeries F`. -/
 theorem formalY_orderTop :
     (formalY W).orderTop = ((-3 : ℤ) : WithTop ℤ) := by
@@ -204,6 +167,7 @@ has leading coefficient `-1`. These come from expanding the definitions:
 `formalX = single(-2,1) · (lifted u_inv)` with `(lifted u_inv).leadingCoeff = 1`
 (since `u_inv` has constant coefficient `1`). -/
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- The leading coefficient of the lifted `formalU_inv` is `1` (its constant
     coefficient as a power series). -/
 theorem ofPowerSeries_formalU_inv_leadingCoeff :
@@ -222,12 +186,14 @@ theorem ofPowerSeries_formalU_inv_leadingCoeff :
   rw [hS, this, PowerSeries.coeff_zero_eq_constantCoeff_apply]
   exact formalU_inv_constantCoeff W
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `formalX W` has leading coefficient `1`. -/
 theorem formalX_leadingCoeff :
     (formalX W).leadingCoeff = 1 := by
   rw [formalX, HahnSeries.leadingCoeff_mul, HahnSeries.leadingCoeff_of_single,
       ofPowerSeries_formalU_inv_leadingCoeff, mul_one]
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `formalY W` has leading coefficient `-1`. -/
 theorem formalY_leadingCoeff :
     (formalY W).leadingCoeff = -1 := by
@@ -244,9 +210,8 @@ Weierstrass equation `y² + a₁xy + a₃y - x³ - ... - a₆ = 0` after substit
 So the proof of `formalXY_weierstrass` is essentially: state the recurrence,
 clear denominators in the field `LaurentSeries F`, and conclude via the recurrence. -/
 
--- The Silverman IV.1.1 recurrence `formalW_recurrence` is now proved in
--- `HasseWeil/FormalGroup.lean` and used directly here.
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalW W ≠ 0` because its coefficient at index 3 is 1. -/
 theorem formalW_ne_zero : formalW W ≠ 0 := by
   intro h
@@ -258,6 +223,7 @@ theorem formalW_ne_zero : formalW W ≠ 0 := by
   rw [this] at h3
   exact one_ne_zero h3
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `HahnSeries.ofPowerSeries ℤ F (formalW W) ≠ 0` in `LaurentSeries F`. -/
 theorem lifted_formalW_ne_zero :
     HahnSeries.ofPowerSeries ℤ F (formalW W) ≠ 0 := by
@@ -265,6 +231,7 @@ theorem lifted_formalW_ne_zero :
   exact formalW_ne_zero W
     (HahnSeries.ofPowerSeries_injective (h.trans (map_zero _).symm))
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalW = X^3 * formalU` as `PowerSeries F`. -/
 theorem formalW_eq_X3_mul_U :
     formalW W = PowerSeries.X ^ 3 * formalU W := by
@@ -282,7 +249,7 @@ theorem formalW_eq_X3_mul_U :
     rw [hLHS]
     -- RHS: (X^3 * formalU).coeff n = 0 for n < 3
     rw [PowerSeries.coeff_X_pow_mul']
-    rw [if_neg (by omega : ¬ 3 ≤ n)]
+    rw [ite_eq_right (by omega : ¬ 3 ≤ n)]
   · push Not at hn
     obtain ⟨m, rfl⟩ := Nat.exists_eq_add_of_le hn
     rw [add_comm 3 m, PowerSeries.coeff_X_pow_mul]
@@ -290,11 +257,12 @@ theorem formalW_eq_X3_mul_U :
     rw [show formalU W = PowerSeries.mk (fun n ↦ formalW_coeff W (n + 3)) from rfl,
         PowerSeries.coeff_mk]
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `formalW W` has `PowerSeries.order` equal to 3 (since `formalW = X^3 * formalU`
     and `formalU` is a unit). -/
 theorem formalW_ps_order : (formalW W).order = 3 := by
   rw [formalW_eq_X3_mul_U, PowerSeries.order_mul, PowerSeries.order_X_pow,
-      PowerSeries.order_zero_of_unit (formalU_isUnit W)]
+      PowerSeries.order_zero_of_isUnit (formalU_isUnit W)]
   rfl
 
 omit [DecidableEq F] in
@@ -335,6 +303,7 @@ private theorem formalW_recurrence_lift :
     HahnSeries.ofPowerSeries_X] at h
   exact h
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Helper: `formalY W * lifted_formalW W = -1` in `LaurentSeries F`. -/
 private theorem formalY_mul_formalW :
     formalY W * HahnSeries.ofPowerSeries ℤ F (formalW W) = -1 := by
@@ -355,6 +324,7 @@ private theorem formalY_mul_formalW :
     _ = -(1 : LaurentSeries F) * 1 := by rw [h1, h2]
     _ = -1 := by ring
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Helper: `formalX W * lifted_formalW W = single 1 1` in `LaurentSeries F`. -/
 private theorem formalX_mul_formalW :
     formalX W * HahnSeries.ofPowerSeries ℤ F (formalW W) =
@@ -377,12 +347,14 @@ private theorem formalX_mul_formalW :
     _ = HahnSeries.single (1 : ℤ) (1 : F) * 1 := by rw [h1, h2]
     _ = HahnSeries.single (1 : ℤ) (1 : F) := mul_one _
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `formalY W = -1 / lifted_formalW W` in `LaurentSeries F`. -/
 theorem formalY_eq_div :
     formalY W = -1 / HahnSeries.ofPowerSeries ℤ F (formalW W) := by
   rw [eq_div_iff (lifted_formalW_ne_zero W)]
   exact formalY_mul_formalW W
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `formalX W = (single 1 1) / lifted_formalW W` in `LaurentSeries F`. -/
 theorem formalX_eq_div :
     formalX W = HahnSeries.single (1 : ℤ) (1 : F) /
@@ -390,6 +362,7 @@ theorem formalX_eq_div :
   rw [eq_div_iff (lifted_formalW_ne_zero W)]
   exact formalX_mul_formalW W
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `(formalX W, formalY W)` satisfies the Weierstrass equation of `W` over `F`,
     interpreted as an identity in `LaurentSeries F`.
 
@@ -445,6 +418,7 @@ noncomputable def localParam : KE :=
 
 /- `x_gen_ne_zero` HOISTED to `Foundation/MulByIntPullback.lean` (#7621). -/
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The local parameter is nonzero in `K(E)`. -/
 theorem localParam_ne_zero : localParam W ≠ 0 := by
   simp only [localParam]
@@ -470,16 +444,19 @@ through `AdjoinRoot.lift` and `IsLocalization.lift`. -/
 noncomputable def localExpand_inner : Polynomial F →+* LaurentSeries F :=
   Polynomial.eval₂RingHom (algebraMap F (LaurentSeries F)) (formalX W)
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The inner ring hom sends `Polynomial.X` to `formalX W`. -/
 @[simp] theorem localExpand_inner_X :
     localExpand_inner W Polynomial.X = formalX W := by
   simp [localExpand_inner]
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The inner ring hom sends constants to their algebra image. -/
 @[simp] theorem localExpand_inner_C (a : F) :
     localExpand_inner W (Polynomial.C a) = algebraMap F (LaurentSeries F) a := by
   simp [localExpand_inner]
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- The Weierstrass polynomial, evaluated via the inner ring hom at `formalY W`,
     vanishes. This is `formalXY_weierstrass` expressed in terms of `Polynomial.eval₂`. -/
 private theorem localExpand_weierstrass_eval :
@@ -505,11 +482,13 @@ private noncomputable def localExpand_coordHom :
     W.toAffine.CoordinateRing →+* LaurentSeries F :=
   AdjoinRoot.lift (localExpand_inner W) (formalY W) (localExpand_weierstrass_eval W)
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- The coordinate ring lift sends `AdjoinRoot.root` to `formalY W`. -/
 @[simp] private theorem localExpand_coordHom_root :
     localExpand_coordHom W (AdjoinRoot.root W.toAffine.polynomial) = formalY W := by
   simp [localExpand_coordHom, AdjoinRoot.lift_root]
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `(formalX W ^ n).orderTop = -2 * n` in `LaurentSeries F` (for `n : ℕ`). -/
 theorem formalX_pow_orderTop (n : ℕ) :
     ((formalX W) ^ n).orderTop = ((-2 * n : ℤ) : WithTop ℤ) := by
@@ -526,6 +505,7 @@ theorem formalX_pow_orderTop (n : ℕ) :
     push_cast
     ring
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `(formalX W ^ n).leadingCoeff = 1` in `LaurentSeries F` (for `n : ℕ`). -/
 theorem formalX_pow_leadingCoeff (n : ℕ) :
     ((formalX W) ^ n).leadingCoeff = 1 := by
@@ -536,6 +516,7 @@ theorem formalX_pow_leadingCoeff (n : ℕ) :
   | succ k ih =>
     rw [pow_succ, HahnSeries.leadingCoeff_mul, ih, formalX_leadingCoeff, mul_one]
 
+omit [W.toAffine.IsElliptic] in
 /-- Auxiliary: for a polynomial `p : F[X]` with `p ≠ 0`, the image
     `localExpand_inner W p` has `orderTop = -2 * p.natDegree` in `LaurentSeries F`.
 
@@ -588,6 +569,7 @@ theorem localExpand_inner_orderTop_eq {p : Polynomial F} (hp : p ≠ 0) :
           nlinarith
       exact (HahnSeries.orderTop_add_eq_right h_lt).trans hB_ord
 
+omit [W.toAffine.IsElliptic] in
 /-- Auxiliary: for a nonzero polynomial `p : F[X]`, `localExpand_inner W p ≠ 0`. -/
 theorem localExpand_inner_ne_zero_of_ne_zero {p : Polynomial F} (hp : p ≠ 0) :
     localExpand_inner W p ≠ 0 := by
@@ -596,6 +578,7 @@ theorem localExpand_inner_ne_zero_of_ne_zero {p : Polynomial F} (hp : p ≠ 0) :
   rw [h, HahnSeries.orderTop_zero] at horder
   exact absurd horder WithTop.top_ne_coe
 
+omit [W.toAffine.IsElliptic] in
 /-- Auxiliary: for a polynomial `p : F[X]` with `p ≠ 0`, the image
     `localExpand_inner W p` has leading coefficient equal to `p.leadingCoeff`.
 
@@ -653,6 +636,7 @@ theorem localExpand_inner_leadingCoeff {p : Polynomial F} (hp : p ≠ 0) :
           nlinarith
       rw [HahnSeries.leadingCoeff_add_eq_right h_lt, hB_lead, ← hDecomp]
 
+omit [W.toAffine.IsElliptic] in
 /-- Auxiliary: `(localExpand_inner W q) * formalY W` has odd `orderTop`
     (`= -2 * natDegree q - 3`) when `q ≠ 0`. -/
 theorem localExpand_inner_mul_formalY_orderTop
@@ -663,6 +647,7 @@ theorem localExpand_inner_mul_formalY_orderTop
     ← WithTop.coe_add]
   rfl
 
+omit [W.toAffine.IsElliptic] in
 /-- The coordinate ring lift is injective.
 
     **Proof (Silverman IV.1)**: We use `exists_smul_basis_eq` to write any
@@ -738,20 +723,11 @@ private theorem localExpand_coordHom_injective :
     rw [Affine.CoordinateRing.smul, Polynomial.C_0, map_zero, zero_mul, zero_add,
         Affine.CoordinateRing.smul, Polynomial.C_0, map_zero, zero_mul]
 
-/-- The **local expansion** ring hom `K(E) → LaurentSeries F`, sending `x_gen ↦
-    formalX W` and `y_gen ↦ formalY W`.
-
-The construction:
-1. Build the inner ring hom `i : F[X] →+* LaurentSeries F` sending `X ↦ formalX W`.
-2. Lift to `R = AdjoinRoot W.polynomial = F[X][Y]/(W) →+* LaurentSeries F` via
-   `AdjoinRoot.lift` with root `formalY W`, using `formalXY_weierstrass` for
-   the evaluation condition.
-3. Extend to `K(E) = FractionRing R →+* LaurentSeries F` via `IsFractionRing.lift`,
-   using that the coordinate ring lift is injective (source is a domain, target
-   is a field, and the map is nonzero — sends `1` to `1`). -/
+
 noncomputable def localExpand : KE →+* LaurentSeries F :=
   IsFractionRing.lift (localExpand_coordHom_injective W)
 
+omit [W.toAffine.IsElliptic] in
 /-- `localExpand` sends `x_gen` to `formalX W`. -/
 @[simp] theorem localExpand_x_gen : localExpand W (x_gen W) = formalX W := by
   simp only [localExpand, x_gen]
@@ -761,11 +737,13 @@ noncomputable def localExpand : KE →+* LaurentSeries F :=
     ((AdjoinRoot.of W.toAffine.polynomial) Polynomial.X) = formalX W
   simp [localExpand_coordHom, AdjoinRoot.lift_of, localExpand_inner]
 
+omit [W.toAffine.IsElliptic] in
 /-- `localExpand` sends `y_gen` to `formalY W`. -/
 @[simp] theorem localExpand_y_gen : localExpand W (y_gen W) = formalY W := by
   simp only [localExpand, y_gen]
   rw [IsFractionRing.lift_algebraMap, localExpand_coordHom_root]
 
+omit [W.toAffine.IsElliptic] in
 /-- `localExpand` is an `F`-algebra hom. -/
 theorem localExpand_algebraMap (a : F) :
     localExpand W (algebraMap F KE a) =
@@ -781,6 +759,7 @@ theorem localExpand_algebraMap (a : F) :
   simp [localExpand_coordHom, AdjoinRoot.lift_of, localExpand_inner,
     LaurentSeries.algebraMap_apply, HahnSeries.ofPowerSeries_C]
 
+omit [W.toAffine.IsElliptic] in
 /-- `localExpand` agrees with the inner ring hom `localExpand_inner` on the image
     of `F[X]` in `K(E)`: `localExpand (algebraMap (Polynomial F) KE p) =
     localExpand_inner W p`. Both factor the algebra map through the coordinate ring
@@ -795,6 +774,7 @@ theorem localExpand_algebraMap_polynomial (p : Polynomial F) :
     localExpand_inner W p
   simp [localExpand_coordHom, AdjoinRoot.lift_of]
 
+omit [W.toAffine.IsElliptic] in
 /-- The `orderTop` of the local expansion of `algebraMap (Polynomial F) KE p` for a
     nonzero polynomial `p` is `-2 · natDegree p`. This is `localExpand_inner_orderTop_eq`
     transported through `localExpand_algebraMap_polynomial`. -/
@@ -804,6 +784,7 @@ theorem orderTop_localExpand_algebraMap_polynomial {p : Polynomial F} (hp : p 
   rw [localExpand_algebraMap_polynomial]
   exact localExpand_inner_orderTop_eq W hp
 
+omit [W.toAffine.IsElliptic] in
 /-- `localExpand` sends `localParam W` to the variable `t` of `LaurentSeries F`.
     This is the key compatibility: the local parameter expands to the formal variable. -/
 theorem localExpand_localParam :
diff --git a/projects/HasseWeil/HasseWeil/Foundation/MulByIntPullback.lean b/projects/HasseWeil/HasseWeil/Foundation/MulByIntPullback.lean
index f41767ca5..e2383c638 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/MulByIntPullback.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/MulByIntPullback.lean
@@ -368,7 +368,7 @@ private lemma aeval_x_gen (p : Polynomial F) :
     Polynomial.aeval_X_left_apply]
 
 -- The `Subalgebra F KE` scalar-tower instances and the `aeval` defeq steps below need the
--- relaxed transparency; this is a defeq setting, not a heartbeat budget.
+
 /- `mulByInt_x W n = Φ_n(x_gen) / ΨSq_n(x_gen)` is transcendental over `F` for `n ≠ 0`.
 
 The proof uses the integral closure approach: if `mulByInt_x` were algebraic over `F`,
@@ -382,7 +382,7 @@ private lemma mulByInt_x_transcendental (n : ℤ) (hn : n ≠ 0) :
   intro h_alg
   apply x_gen_transcendental W
   set S : Subalgebra F KE := Algebra.adjoin F ({mulByInt_x W n} : Set KE) with hS_def
-  haveI hS_alg : Algebra.IsAlgebraic F S :=
+  have hS_alg : Algebra.IsAlgebraic F S :=
     (Subalgebra.isAlgebraic_iff S).mp
       (Algebra.isAlgebraic_adjoin_singleton_iff.mpr h_alg)
   have hmem : mulByInt_x W n ∈ S := Algebra.subset_adjoin (Set.mem_singleton _)
@@ -448,10 +448,8 @@ private lemma norm_smul_basis_ne_zero (p q : Polynomial F) (hq : q ≠ 0) :
     (not_le.mpr (WithBot.bot_lt_iff_ne_bot.mpr this))
 
 omit [W.toAffine.IsElliptic] in
-/-- The image in `R = F[C]` of the `F[X]`-norm of the basis element `r' = p•1 + q•Y`
-factors as `r' * conj_r`, where `conj_r` is the `F[X]`-conjugate class.  Extracted as its
-own lemma so it elaborates in a light context (the `coe_norm_smul_basis` rewrite and the
-ring-hom bookkeeping are heartbeat-heavy inside the main injectivity proof). -/
+
+
 private lemma algebraMap_norm_smul_basis_eq (p q : Polynomial F) :
     algebraMap (Polynomial F) R (Algebra.norm (Polynomial F)
         (p • (1 : R) + q • Affine.CoordinateRing.mk W.toAffine Polynomial.X)) =
diff --git a/projects/HasseWeil/HasseWeil/Foundation/OmegaCoeffMulByIntFiniteField.lean b/projects/HasseWeil/HasseWeil/Foundation/OmegaCoeffMulByIntFiniteField.lean
index a61618eea..51a6ac743 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/OmegaCoeffMulByIntFiniteField.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/OmegaCoeffMulByIntFiniteField.lean
@@ -8,35 +8,35 @@ import HasseWeil.HasseBound.Primitives
 import HasseWeil.Foundation.EC.MulByIntAddRecurrence
 
 /-!
-# Silverman III.5.3: `a_{[m]} = m` via curve-side additivity (Route B assembly)
+# Pullback coefficients of integer multiplication
 
-Builds on **RB-ω4** (`kaehler_D_addPullback_x_eq_one_add_smul_omega`, the Silverman III.5.2
-collapse) to assemble `omegaPullbackCoeff (mulByInt m) = m` WITHOUT the EDS Wronskian and WITHOUT
-the formal-group correspondence.
+The addition formula for the invariant differential gives
+`a_{id+α} = 1 + a_α` and `a_{α+β} = a_α + a_β`. The multiplication-coordinate
+recurrence then proves `a_{[m]} = m` for nonzero integers by induction and
+negation. In positive characteristic, multiplication by the characteristic
+has zero differential coefficient. Over a finite field, the same holds for
+multiplication by its cardinality and supplies the differential input to
+Verschiebung factorization.
 
-* **RB-ADD** (`omegaPullbackCoeff_addIsog_id`):
-  `omegaPullbackCoeff (id ⊞ α) = 1 + omegaPullbackCoeff α`.
-* **RB-ID/SUM/IND**: `id ⊞ [m] = [m+1]`, induction → `omegaPullbackCoeff (mulByInt m) = m`.
-
-This is Silverman III.5.2 `(φ+ψ)*ω = φ*ω + ψ*ω` specialised to `φ = id`, then III.5.3's induction.
+References: Silverman, *The Arithmetic of Elliptic Curves*, III.5.2–3.
 -/
 
 open WeierstrassCurve
 
 namespace HasseWeil
 
-variable {K : Type*} [Field K] [Fintype K] [DecidableEq K]
+variable {K : Type*} [Field K] [DecidableEq K]
   (W : WeierstrassCurve K) [W.toAffine.IsElliptic]
 
 local notation "KE" => W.toAffine.FunctionField
 local notation "R" => W.toAffine.CoordinateRing
 
-/-- **RB-ADD** (Silverman III.5.2, `φ = id` case): for the genuine sum isogeny `σ = id ⊞ α`
+/-- Silverman III.5.2 for addition with the identity: for the genuine sum isogeny `σ = id ⊞ α`
 (built via `addIsog` on the pair `(id, α)`), the omega-pullback coefficient is `1 + a_α`.
 
 Proof: by uniqueness in the 1-dimensional Kähler module. `omegaPullbackCoeff_spec` gives
 `a_σ • ω = (α*u of σ)⁻¹ • D(σ*x)`. Now `σ*x = addPullback_x α` and `α*u of σ = u₃ :=
-2·addPullback_y + a₁·addPullback_x + a₃`, while RB-ω4 says `D(addPullback_x α) = u₃ · (1+a_α) · ω`.
+2·addPullback_y + a₁·addPullback_x + a₃`, while the addition identity says `D(addPullback_x α) = u₃ · (1+a_α) · ω`.
 So `a_σ • ω = u₃⁻¹ · u₃ · (1+a_α) · ω = (1+a_α) · ω`, hence `a_σ = 1 + a_α`. -/
 theorem omegaPullbackCoeff_addIsog_id
     (α : Isogeny W.toAffine W.toAffine)
@@ -115,21 +115,16 @@ theorem omegaPullbackCoeff_addIsog_pair
     kaehler_D_addPullback_x_pair_eq_smul_omega W α₁ α₂ h_ne, smul_smul,
     inv_mul_cancel₀ hu3_ne, one_smul]
 
--- **RB-ID core** (`addPullback_xy_mulByInt_eq_succ`, the Silverman III.5.3 addition recurrence
--- `P ⊞ [m]P = [m+1]P`) lives in `HasseWeil.EC.MulByIntAddRecurrence` (minimal imports, to skirt the
--- `(W_KE W).toAffine.Point` ℤ-smul instance diamond — see `SK-ROUTEB-SMUL-DIAMOND` there). It is
--- imported above and consumed by RB-IND below.
-
-/-- **RB chord step** (Silverman III.5.3 recurrence at the differential level): for `k ≥ 2`,
+/-- The Silverman III.5.3 recurrence at the differential level: for `k ≥ 2`,
 `a_{[k+1]} = 1 + a_{[k]}`.
 
 Proof. By uniqueness in the 1-dimensional Kähler module. `omegaPullbackCoeff_spec` gives
 `a_{[k+1]} • ω = (α*u of [k+1])⁻¹ • D([k+1]*x)`. Now:
-* `[k+1]*x = mulByInt_x (k+1) = addPullback_x [k]` (RB-ID), so `D([k+1]*x) = D(addPullback_x [k])`.
-* RB-ω4 (α = `[k]`, with `x_gen ≠ [k]*x` since `k ≥ 2`):
+* `[k+1]*x = mulByInt_x (k+1) = addPullback_x [k]` (the multiplication-coordinate recurrence), so `D([k+1]*x) = D(addPullback_x [k])`.
+* The differential addition identity (α = `[k]`, with `x_gen ≠ [k]*x` since `k ≥ 2`):
   `D(addPullback_x [k]) = u₃ • ((1 + a_{[k]}) • ω)`
   where `u₃ = 2·addPullback_y [k] + a₁·addPullback_x [k] + a₃`.
-* RB-ID rewrites `addPullback_x/y [k] = mulByInt_x/y (k+1)`, so `u₃ = α*u of [k+1]` (definitionally,
+* The coordinate recurrence rewrites `addPullback_x/y [k] = mulByInt_x/y (k+1)`, so `u₃ = α*u of [k+1]` (definitionally,
   via `alpha_star_u_mulByInt`).
 
 Hence `a_{[k+1]} • ω = u₃⁻¹ • (u₃ • (1 + a_{[k]}) • ω) = (1 + a_{[k]}) • ω`. -/
@@ -142,15 +137,15 @@ theorem omegaPullbackCoeff_mulByInt_succ (k : ℤ) (hk2 : 2 ≤ k) :
   have hx_ne : x_gen W ≠ mulByInt_x W k := by
     rw [← mulByInt_x_one W]
     exact mulByInt_x_ne_mulByInt_x W 1 k one_ne_zero hk0 (by omega) (by omega)
-  -- `[k]*x = mulByInt_x k`, so RB-ω4's hypothesis `x_gen ≠ [k]*x` is `hx_ne`.
+  -- `[k]*x = mulByInt_x k`, so the differential addition hypothesis `x_gen ≠ [k]*x` is `hx_ne`.
   have hkx : (mulByInt W.toAffine k).pullback (x_gen W) = mulByInt_x W k :=
     mulByInt_pullback_x W k hk0
   have hx_ne_pb : x_gen W ≠ (mulByInt W.toAffine k).pullback (x_gen W) := by
     rw [hkx]; exact hx_ne
-  -- RB-ID: `addPullback_x/y [k] = mulByInt_x/y (k+1)`.
+  -- The coordinate recurrence: `addPullback_x/y [k] = mulByInt_x/y (k+1)`.
   obtain ⟨hAx, hAy⟩ := addPullback_xy_mulByInt_eq_succ W k hk0 hk1 hx_ne
   -- `α*u of [k+1] = u₃ := 2·addPullback_y [k] + a₁·addPullback_x [k] + a₃`
-  -- (def + RB-ID + alpha_star_u_mulByInt).
+  -- (by the coordinate recurrence and alpha_star_u_mulByInt).
   have hu : alpha_star_u W (mulByInt W.toAffine (k + 1))
       = 2 * addPullback_y W (mulByInt W.toAffine k)
         + algebraMap K KE W.a₁ * addPullback_x W (mulByInt W.toAffine k)
@@ -159,7 +154,7 @@ theorem omegaPullbackCoeff_mulByInt_succ (k : ℤ) (hk2 : 2 ≤ k) :
   have hu3_ne : 2 * addPullback_y W (mulByInt W.toAffine k)
       + algebraMap K KE W.a₁ * addPullback_x W (mulByInt W.toAffine k) + algebraMap K KE W.a₃ ≠ 0 :=
     hu ▸ alpha_star_u_ne_zero W (mulByInt W.toAffine (k + 1))
-  -- `[k+1]*x = addPullback_x [k]` (mulByInt_pullback_x + RB-ID).
+  -- `[k+1]*x = addPullback_x [k]` (by mulByInt_pullback_x and the coordinate recurrence).
   have hpx : (mulByInt W.toAffine (k + 1)).pullback
         (algebraMap R KE (algebraMap (Polynomial K) R Polynomial.X))
       = addPullback_x W (mulByInt W.toAffine k) := by
@@ -169,7 +164,7 @@ theorem omegaPullbackCoeff_mulByInt_succ (k : ℤ) (hk2 : 2 ≤ k) :
     kaehler_D_addPullback_x_eq_one_add_smul_omega W (mulByInt W.toAffine k) hx_ne_pb, smul_smul,
     inv_mul_cancel₀ hu3_ne, one_smul]
 
-/-- `a_{[n]} = n` for all `n ≥ 2`, by `Int.leInduction` from the axiom-clean base case `n = 2`
+/-- `a_{[n]} = n` for all `n ≥ 2`, by `Int.leInduction` from the base case `n = 2`
 (`omegaPullbackCoeff_mulByInt_two`), with step `k ≥ 2 ⟹ k+1` via the chord step
 `omegaPullbackCoeff_mulByInt_succ`. -/
 theorem omegaPullbackCoeff_mulByInt_ge_two (n : ℤ) (hn : 2 ≤ n) :
@@ -180,7 +175,7 @@ theorem omegaPullbackCoeff_mulByInt_ge_two (n : ℤ) (hn : 2 ≤ n) :
     rw [omegaPullbackCoeff_mulByInt_succ W k hk2, ih, Int.cast_add, Int.cast_one, map_add,
       map_one, add_comm]
 
-/-- **RB-IND** (Silverman III.5.3, positive case): `a_{[n]} = n` for all `n ≥ 1`. Combines the
+/-- Silverman III.5.3 for positive integers: `a_{[n]} = n` for all `n ≥ 1`. Combines the
 `n = 1` case (`[1] = id`, `a_id = 1`) with `omegaPullbackCoeff_mulByInt_ge_two`. -/
 theorem omegaPullbackCoeff_mulByInt_pos (n : ℤ) (hn : 1 ≤ n) :
     omegaPullbackCoeff W (mulByInt W.toAffine n) = algebraMap K KE n := by
@@ -188,8 +183,7 @@ theorem omegaPullbackCoeff_mulByInt_pos (n : ℤ) (hn : 1 ≤ n) :
   · subst h1; rw [mulByInt_one_eq_id, omegaPullbackCoeff_id, Int.cast_one, map_one]
   · exact omegaPullbackCoeff_mulByInt_ge_two W n (by omega)
 
-omit [Fintype K] in
-/-- **RB negation**: `a_{[-n]} = -a_{[n]}` for `n ≠ 0`. The `[-n]`-pullback of `x_gen`
+/-- Negation of the multiplication differential coefficient: `a_{[-n]} = -a_{[n]}` for `n ≠ 0`. The `[-n]`-pullback of `x_gen`
 agrees with the `[n]`-pullback (`mulByInt_x_neg`), while `α*u` flips sign
 (`negY`-symmetry of `mulByInt_y`), so the spec equation flips sign and uniqueness
 gives the result. -/
@@ -219,9 +213,9 @@ theorem omegaPullbackCoeff_mulByInt_neg (n : ℤ) (hn : n ≠ 0) :
   congr 1
   exact (omegaPullbackCoeff_spec W (mulByInt W.toAffine n)).symm
 
-/-- **RB-IND (Silverman III.5.3)**, integer form: `a_{[n]} = n` for all `n ≠ 0`, by the chord-step
-induction (positive `n`) and the negation symmetry (negative `n`). Reroutes
-`omegaPullbackCoeff_mulByInt` off the (still partial) EDS Wronskian. -/
+/-- Silverman III.5.3 for all nonzero integers: `a_{[n]} = n`.
+Positive integers follow by the chord recurrence and negative integers by
+negation symmetry. -/
 theorem omegaPullbackCoeff_mulByInt_routeB (n : ℤ) (hn : n ≠ 0) :
     omegaPullbackCoeff W (mulByInt W.toAffine n) = algebraMap K KE n := by
   rcases lt_or_gt_of_ne hn with hneg | hpos
@@ -233,21 +227,17 @@ theorem omegaPullbackCoeff_mulByInt_routeB (n : ℤ) (hn : n ≠ 0) :
   · -- n > 0: direct from the positive induction.
     exact omegaPullbackCoeff_mulByInt_pos W n (by omega)
 
-/-- **Pillar B endpoint (wronskian-free, formal-group-free)**: in characteristic `p`, `[p]*ω = 0`,
-i.e. `a_{[p]} = (p : K) = 0`. The axiom-clean Route B replacement for the formal-group
-`omegaPullbackCoeff_mulByInt_p_eq_zero_via_formalGroup` and the wronskian
-`omegaPullbackCoeff_mulByInt`. Direct from `omegaPullbackCoeff_mulByInt_routeB`
-+ `CharP.cast_eq_zero`. -/
+/-- In characteristic `p`, multiplication by `p` has zero differential
+coefficient, since its coefficient is the image of `p` in the base field. -/
 theorem omegaPullbackCoeff_mulByInt_p_eq_zero_routeB (p : ℕ) [CharP K p] (hp : p ≠ 0) :
     omegaPullbackCoeff W (mulByInt W.toAffine (p : ℤ)) = 0 := by
   rw [omegaPullbackCoeff_mulByInt_routeB W (p : ℤ) (by exact_mod_cast hp),
     show ((p : ℤ) : K) = 0 from by rw [Int.cast_natCast]; exact CharP.cast_eq_zero K p, map_zero]
 
-/-- **Pillar B endpoint at `q = #K`** (`[q]*ω = 0`): over the finite field `K`, the ω-pullback
-coefficient of `[q]` vanishes, since `a_{[q]} = (q : K) = 0` (`#K = char^n`). Axiom-clean via
-`omegaPullbackCoeff_mulByInt_routeB`; this is the `q`-th-root input to the Verschiebung
-factorisation `[q] = V ∘ Frob`. -/
-theorem omegaPullbackCoeff_mulByInt_card_eq_zero :
+/-- Over a finite field, multiplication by its cardinality has zero
+differential coefficient. This is the differential input to the Verschiebung
+factorization `[q] = V ∘ Frob`. -/
+theorem omegaPullbackCoeff_mulByInt_card_eq_zero [Fintype K] :
     omegaPullbackCoeff W (mulByInt W.toAffine (Fintype.card K : ℤ)) = 0 := by
   rw [omegaPullbackCoeff_mulByInt_routeB W (Fintype.card K : ℤ)
       (by exact_mod_cast Fintype.card_ne_zero),
diff --git a/projects/HasseWeil/HasseWeil/Foundation/OmegaPullbackCoeff.lean b/projects/HasseWeil/HasseWeil/Foundation/OmegaPullbackCoeff.lean
index 61150a90c..a51582216 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/OmegaPullbackCoeff.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/OmegaPullbackCoeff.lean
@@ -24,7 +24,7 @@ With `a_α ∈ F`:
 - Additivity: `a_{α+β} = a_α + a_β` (from Silverman III.5.2)
 - `a_{[n]} = n` (from Silverman III.5.3)
 
-## Implementation
+## Kähler differential construction
 
 The pullback `α*(ω)` in the Kähler differential module Ω[K(E)/F] is computed as:
   `α*(ω) = D(α*(x)) · α*(u)⁻¹`
@@ -98,11 +98,9 @@ theorem omegaPullbackCoeff_spec (α : Isogeny W.toAffine W.toAffine) :
         (α.pullback
           (algebraMap R KE (algebraMap (Polynomial F) R Polynomial.X))))).choose_spec
 
--- Chain rule for `omegaPullbackCoeff`. The chain rule `a_{α∘β} = a_α · a_β` holds when `a_α`
--- is in the base field `F` (so that `β*(a_α) = a_α`); for the Silverman convention `a_α ∈ F`
--- always holds (Silverman III.5.5). It is proved abstractly as `omegaPullbackCoeff_comp_of_base`
--- in `HasseWeil/InvariantDifferentialPullback.lean`; not restated here to avoid a naming clash.
--- For `α = [n]` we get `a_{[n]} = n ∈ F`, which is what the Hasse bound needs.
+-- The chain rule `a_{α∘β} = a_α · a_β` holds when `a_α` lies in the base
+-- field, so that the pullback by `β` fixes it. For multiplication by `n`, the
+-- differential coefficient is `n`, as in Silverman III.5.3.
 
 /-- Uniqueness of the omega-based pullback coefficient: in the 1-dimensional Kähler module,
     if `c₁ • ω = c₂ • ω` then `c₁ = c₂` (since `ω ≠ 0` and `KE` is a field). -/
@@ -124,7 +122,7 @@ theorem mulByInt_pullback_x (n : ℤ) (hn : n ≠ 0) :
     (mulByInt W.toAffine n).pullback
       (algebraMap R KE (algebraMap (Polynomial F) R Polynomial.X)) =
     mulByInt_x W n := by
-  have hpb : (mulByInt W.toAffine n).pullback = mulByInt_pullbackAlgHom W n hn := dif_neg hn
+  have hpb : (mulByInt W.toAffine n).pullback = mulByInt_pullbackAlgHom W n hn := dite_eq_right hn
   rw [hpb]
   change mulByInt_pullbackRingHom W n hn
     (algebraMap R KE (algebraMap (Polynomial F) R Polynomial.X)) = _
@@ -140,7 +138,7 @@ theorem mulByInt_pullback_y (n : ℤ) (hn : n ≠ 0) :
     (mulByInt W.toAffine n).pullback
       (algebraMap R KE (AdjoinRoot.root W.toAffine.polynomial)) =
     mulByInt_y W n := by
-  have hpb : (mulByInt W.toAffine n).pullback = mulByInt_pullbackAlgHom W n hn := dif_neg hn
+  have hpb : (mulByInt W.toAffine n).pullback = mulByInt_pullbackAlgHom W n hn := dite_eq_right hn
   rw [hpb]
   change mulByInt_pullbackRingHom W n hn
     (algebraMap R KE (AdjoinRoot.root W.toAffine.polynomial)) = _
@@ -331,11 +329,6 @@ private lemma wronskian_X_mul_sub (f q : Polynomial F) :
   simp [Polynomial.derivative_sub, Polynomial.derivative_mul, Polynomial.derivative_X]
   ring
 
--- The auxiliary ring identity `wronskian_aux_three` has been moved to
--- `HasseWeil/WronskianAux.lean` (together with `wronskian_aux_four`) to isolate
--- its expensive elaboration. It is reused unchanged from there.
-
--- Case m=3: proved via `wronskian_aux_three` from `HasseWeil/WronskianAux.lean`.
 -- The key insight: both LHS and RHS factor through Ψ₃, and after cancellation
 -- the identity reduces to wronskian_aux_three (a tractable ring computation).
 omit [DecidableEq F] [W.toAffine.IsElliptic] in
@@ -371,10 +364,7 @@ private lemma wronskian_Φ_ΨSq_three :
 
 -- Silverman III.3.7 auxiliary: verified ring identity for m=4.
 -- Only expands to b₂, b₄, b₆, b₈ (not a₁..a₆) keeping the ring computation tractable.
--- The auxiliary ring identity `wronskian_aux_four` has been moved to
--- `HasseWeil/WronskianAux.lean` to isolate its ~57 GB ring elaboration.
 
--- Case m=4: proved via `wronskian_aux_four` from `HasseWeil/WronskianAux.lean`.
 -- After applying wronskian_X_mul_sub to restructure the Wronskian,
 -- and expanding preΨ(8) via preΨ_even(4)/preΨ_odd(2)/preΨ_even(3),
 -- the goal matches wronskian_aux_four exactly (up to 1-simplification).
@@ -420,96 +410,6 @@ private lemma wronskian_Φ_ΨSq_four :
   simp only [Polynomial.C_ofNat] at haux ⊢
   linear_combination haux
 
-omit [DecidableEq F] [W.toAffine.IsElliptic] in
-/-- The polynomial Wronskian identity for non-negative integers, by **strong induction**.
-
-    Base cases `m = 0,1,2` use direct computation; `m = 3,4` use the factored ring identities
-    (`wronskian_aux_three`, `wronskian_aux_four`). For `m = n+5` the proof is by strong induction:
-    the induction hypothesis `ih` provides the identity at **all** smaller indices `k < m`.
-
-    ## Status of the `m ≥ 5` inductive step
-
-    The inductive step is the elliptic-divisibility-sequence (EDS) Wronskian recursion. Writing
-    `p k := W.preΨ k` and `s := W.Ψ₂Sq`, and `W(k) := Φ_k' ΨSq_k − Φ_k ΨSq_k'`, the target is
-    `W(k) = k · p(2k)`. The two halving recursions that drive the induction are (both verified by
-    computer algebra against the actual division polynomials):
-
-    * **Even step.** `W(2m) = 2 · complEDS₂(2m) · W(m)`, where
-      `complEDS₂(2m) = p(2m-1)² p(2m+2) − p(2m-2) p(2m+1)²`. Combined with the induction hypothesis
-      `W(m) = m · p(2m)` and `p(4m) = complEDS₂(2m) · p(2m)` (mathlib's `preNormEDS_mul_complEDS₂`)
-      this collapses to `W(2m) = 2m · p(4m)`, i.e. the claim at `2m`. Both `m` and `m±1 < 2m`.
-    * **Odd step.** The analogous reduction of `W(2m+1)` to `W(m)` and `W(m+1)`.
-
-    The irreducible algebraic content of *each* halving step is the **EDS addition formula**
-    (Ward's relation), verified by CAS in the exact form
-    `p(i+j) p(i-j) = Aᵢ · p(i+1) p(i-1) p(j)² − Aⱼ · p(j+1) p(j-1) p(i)²`,
-    with `Aᵢ = s²` iff (`i` odd and `j` even) else `1`, and `Aⱼ = s²` iff (`i` even and `j` odd)
-    else `1`. Its `j = 2` specialisation is the Somos‑4 relation
-    `p(m+2) p(m-2) = (Even m ? 1 : s²) · p(m+1) p(m-1) − Ψ₃ · p(m)²`.
-
-    The even step `W(2m) = 2·complEDS₂(2m)·W(m)` is **not** a free-ring identity in the consecutive
-    `preΨ` values alone — it is provably *false* without further relations, and is *not* closed by
-    Somos‑4 together with the Weierstrass `b`-relation either: it genuinely requires the addition
-    formula at general index `j`. Mathlib does not currently provide the EDS addition formula
-    (only the duplication recursions `preΨ_even`/`preΨ_odd`), so closing this step needs that
-    formula to be developed first. The `WronskianAux` `m = 3,4` proofs avoid it only because, at
-    fixed small index, the difference factors through the single `b`-relation with an explicitly
-    computed multiplier;
-    that multiplier grows with the index, so the same device does not generalise.
-
-    An **axiom-clean** alternative is available downstream of this file via the function field:
-    the field-general Route-B chord induction `omegaCoeff_mulByInt`
-    (`HasseWeil/RouteBGeneral.lean`) proves the differential statement `[n]*ω = n·ω`
-    (Silverman III.5.3) over any field, with no EDS Wronskian; it cannot be imported here (it
-    depends transitively on this file), but every former geometric consumer of the Wronskian
-    (`Hasse/Separability.lean`, `PullbackCoeff.lean`) now routes through it, so this `sorry` is
-    **consumer-free**. The polynomial identity itself also has an axiom-clean downstream proof:
-    `wronskian_Φ_ΨSq_general` (`EC/WronskianGeneral.lean`). -/
-private lemma wronskian_Φ_ΨSq_nat (m : ℕ) :
-    Polynomial.derivative (W.Φ (m : ℤ)) * W.ΨSq (m : ℤ) -
-      W.Φ (m : ℤ) * Polynomial.derivative (W.ΨSq (m : ℤ)) =
-    Polynomial.C ((m : ℤ) : F) * W.preΨ (2 * (m : ℤ)) := by
-  induction m using Nat.strong_induction_on with
-  | _ m ih =>
-    -- `ih : ∀ k < m, (the identity at k)` — the strong induction hypothesis.
-    match m, ih with
-    | 0, _ => exact wronskian_Φ_ΨSq_zero W
-    | 1, _ => exact_mod_cast wronskian_Φ_ΨSq_one W
-    | 2, _ => exact_mod_cast wronskian_Φ_ΨSq_two W
-    | 3, _ => exact_mod_cast wronskian_Φ_ΨSq_three W
-    | 4, _ => exact_mod_cast wronskian_Φ_ΨSq_four W
-    | (n + 5), ih =>
-      -- General case `m = n+5 ≥ 5`. The induction hypothesis `ih k hk` gives the identity at
-      -- every `k < n+5`, in particular at the halves `⌊(n+5)/2⌋`, `⌈(n+5)/2⌉` needed by the EDS
-      -- Wronskian recursion (even/odd halving step). The single missing ingredient is the EDS
-      -- addition formula (see the docstring above); supplying it closes this step via
-      -- `linear_combination` of the induction hypothesis at the halves with the
-      -- addition-formula / Somos‑4 instances.
-      sorry
-
-omit [DecidableEq F] [W.toAffine.IsElliptic] in
-/-- The division polynomial Wronskian identity:
-    `Φ_n' · ΨSq_n - Φ_n · ΨSq_n' = n · preΨ(2n)` as polynomials in `F[X]`.
-
-    Reference: Silverman Exercise III.3.7.
-
-    Proved by reducing to the natural number case via `wronskian_Φ_ΨSq_neg_of`
-    (negation symmetry) and `wronskian_Φ_ΨSq_nat` (positive case by induction). -/
-theorem wronskian_Φ_ΨSq (n : ℤ) :
-    Polynomial.derivative (W.Φ n) * W.ΨSq n - W.Φ n * Polynomial.derivative (W.ΨSq n) =
-    Polynomial.C (n : F) * W.preΨ (2 * n) := by
-  rcases lt_or_ge n 0 with hneg | hpos
-  · -- n < 0: use negation symmetry on the positive case
-    have hm : (0 : ℤ) ≤ -n := by omega
-    obtain ⟨m, hm_eq⟩ := Int.eq_ofNat_of_zero_le hm
-    have hn_eq : n = -(m : ℤ) := by omega
-    rw [hn_eq]
-    exact wronskian_Φ_ΨSq_neg_of W (wronskian_Φ_ΨSq_nat W m)
-  · -- n ≥ 0: directly from wronskian_Φ_ΨSq_nat
-    obtain ⟨m, hm_eq⟩ := Int.eq_ofNat_of_zero_le hpos
-    subst hm_eq
-    exact wronskian_Φ_ΨSq_nat W m
-
 omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `Φ_ff` and `ΨSq_ff` factor through `algebraMap (Polynomial F) KE`. -/
 private lemma Φ_ff_eq_algebraMap_poly (n : ℤ) :
@@ -527,7 +427,7 @@ private lemma ΨSq_ff_eq_algebraMap_poly (n : ℤ) :
     This is the polynomial shadow of the third division polynomial `ω_n` and follows from the
     geometric facts `preΨ(2n)·u = ψ_n·ψc_n` (i.e. `ψ_{2n} = ψ_n·ψc_n`, `ψc_spec_ff`) and
     `α*(u) = ψc_n/ψ_n³` (`ω_spec_ff`), together with `ΨSq_ff = ψ_n²` (`ψ_ff_sq_eq`). It does **not**
-    use the division-polynomial Wronskian, so it is axiom-clean and can be combined with the
+    use the division-polynomial Wronskian, and can be combined with the
     chord-recursion `omegaPullbackCoeff_mulByInt_routeB` downstream to recover the polynomial
     Wronskian. -/
 theorem preΨ_two_mul_u_eq_ΨSq_sq_mul_alpha_star_u (n : ℤ) (hn : n ≠ 0) :
@@ -543,7 +443,7 @@ theorem preΨ_two_mul_u_eq_ΨSq_sq_mul_alpha_star_u (n : ℤ) (hn : n ≠ 0) :
   have h_preΨ_u : algebraMap (Polynomial F) KE (W.preΨ (2 * n)) * u_gen W = ψn * ψcn := by
     rw [hψn_def, hψcn_def, ψc_spec_ff W n, Affine.CoordinateRing.mk_ψ (W := W.toAffine) (2 * n)]
     have hΨ_eq : W.Ψ (2 * n) = Polynomial.C (W.preΨ (2 * n)) * W.ψ₂ := by
-      rw [WeierstrassCurve.Ψ, if_pos (even_two_mul n)]
+      rw [WeierstrassCurve.Ψ, ite_eq_left (even_two_mul n)]
     rw [hΨ_eq, map_mul, map_mul,
       show W.ψ₂ = W.toAffine.polynomialY from rfl, mk_polynomialY_eq_u_gen W]
     rfl
@@ -578,8 +478,7 @@ theorem preΨ_two_mul_u_eq_ΨSq_sq_mul_alpha_star_u (n : ℤ) (hn : n ≠ 0) :
     - mk(polynomialY) = u_gen in K(E)
 
     Taking `hpoly` as a hypothesis lets specific values of `n` (e.g. `n = 2`) be
-    discharged from the axiom-clean per-value polynomial lemmas
-    (`wronskian_Φ_ΨSq_two`, …), independently of the general `wronskian_Φ_ΨSq`.
+    discharged from the per-value polynomial identities such as `wronskian_Φ_ΨSq_two`.
     Reference: Silverman Exercise III.3.7, III.5.3 -/
 theorem divPoly_wronskian_identity_of_poly (n : ℤ) (hn : n ≠ 0)
     (hpoly : Polynomial.derivative (W.Φ n) * W.ΨSq n -
@@ -618,7 +517,7 @@ theorem divPoly_wronskian_identity_of_poly (n : ℤ) (hn : n ≠ 0)
   have h_preΨ_u : algebraMap (Polynomial F) KE (W.preΨ (2 * n)) * u_gen W = ψn * ψcn := by
     rw [hψn_def, hψcn_def, ψc_spec_ff W n, Affine.CoordinateRing.mk_ψ (W := W.toAffine) (2 * n)]
     have hΨ_eq : W.Ψ (2 * n) = Polynomial.C (W.preΨ (2 * n)) * W.ψ₂ := by
-      rw [WeierstrassCurve.Ψ, if_pos (even_two_mul n)]
+      rw [WeierstrassCurve.Ψ, ite_eq_left (even_two_mul n)]
     rw [hΨ_eq, map_mul, map_mul,
       show W.ψ₂ = W.toAffine.polynomialY from rfl, mk_polynomialY_eq_u_gen W]
     rfl
@@ -646,20 +545,6 @@ theorem divPoly_wronskian_identity_of_poly (n : ℤ) (hn : n ≠ 0)
     h_preΨ_u, h_alpha_u, show ΨSq_ff W n = ψn ^ 2 from (ψ_ff_sq_eq W n).symm]
   field_simp
 
-/-- The division polynomial Wronskian identity in K(E) (general `n`), via the
-    general polynomial-level identity `wronskian_Φ_ΨSq`.
-    Reference: Silverman Exercise III.3.7, III.5.3 -/
-theorem divPoly_wronskian_identity (n : ℤ) (hn : n ≠ 0) :
-    (algebraMap (Polynomial F) KE (Polynomial.derivative (W.Φ n)) *
-      ΨSq_ff W n -
-    Φ_ff W n *
-      algebraMap (Polynomial F) KE (Polynomial.derivative (W.ΨSq n))) *
-    u_gen W =
-    algebraMap F KE n *
-    ΨSq_ff W n ^ 2 *
-    alpha_star_u W (mulByInt W.toAffine n) :=
-  divPoly_wronskian_identity_of_poly W n hn (wronskian_Φ_ΨSq (W := W) n)
-
 -- Chain rule for `D` on polynomial images in K(E), using `Derivation.comp_aeval_eq`: for
 -- `p : F[X]` and `a ∈ KE`, `D(aeval a p) = aeval a (derivative p) • D(a)`.
 
@@ -695,8 +580,7 @@ theorem D_poly_eval (p : Polynomial F) :
     via the quotient rule and chain rule for the universal derivation.
 
     Taking `hpoly` as a hypothesis lets specific values of `n` be discharged from
-    the axiom-clean per-value polynomial lemmas (`wronskian_Φ_ΨSq_two`, …),
-    independently of the general `wronskian_Φ_ΨSq`. -/
+    the per-value polynomial identities such as `wronskian_Φ_ΨSq_two`. -/
 theorem omegaPullbackCoeff_mulByInt_of_poly (n : ℤ) (hn : n ≠ 0)
     (hpoly : Polynomial.derivative (W.Φ n) * W.ΨSq n -
         W.Φ n * Polynomial.derivative (W.ΨSq n) =
@@ -768,7 +652,7 @@ theorem omegaPullbackCoeff_mulByInt_of_poly (n : ℤ) (hn : n ≠ 0)
 
     This is the converse of `omegaPullbackCoeff_mulByInt_of_poly`'s final algebra: the spec
     `a • ω = α*(u)⁻¹ • D(α*x)` together with `a = n` and the chain/quotient rule for `D` gives the
-    scalar identity `(Φ' Ψ − Φ Ψ') · u = n · Ψ² · α*u` in K(E). Fed the axiom-clean
+    scalar identity `(Φ' Ψ − Φ Ψ') · u = n · Ψ² · α*u` in K(E). Given the
     `omegaPullbackCoeff_mulByInt_routeB` (downstream), it recovers the K(E) Wronskian — and then,
     via the bridge `preΨ_two_mul_u_eq_ΨSq_sq_mul_alpha_star_u` and injectivity of
     `algebraMap (Polynomial F) KE`, the **polynomial** Wronskian — without the EDS addition
@@ -843,30 +727,10 @@ theorem divPoly_wronskian_identity_of_omega (n : ℤ) (hn : n ≠ 0)
   -- Goal is already in the `set` names `Φ, Ψ, Φ', Ψ', u, αu`; `hfin` matches up to symmetry.
   linear_combination -hfin
 
-/-- For [n], the omega-based pullback coefficient is n (general `n`, via the
-    general polynomial-level Wronskian identity `wronskian_Φ_ΨSq`).
-    Silverman III.5.3, IV.2.3.
-
-    NOTE: this routing through `wronskian_Φ_ΨSq` inherits its `sorryAx` taint
-    (the `m ≥ 5` branch of `wronskian_Φ_ΨSq_nat`). The **axiom-clean, field-general**
-    proof of the same statement, via the Route-B chord recurrence (Silverman
-    III.5.2/3 at the differential level), is `HasseWeil.omegaCoeff_mulByInt` in
-    `HasseWeil/RouteBGeneral.lean`. That module cannot be imported here (its chord
-    step lives downstream of this file), so the replacement cannot be applied in
-    place — instead all former consumers (`Hasse/Separability.lean`,
-    `PullbackCoeff.lean`) now route through `omegaCoeff_mulByInt`, leaving this
-    theorem (and its Wronskian `sorry`) **consumer-free**. -/
-theorem omegaPullbackCoeff_mulByInt (n : ℤ) (hn : n ≠ 0) :
-    omegaPullbackCoeff W (mulByInt W.toAffine n) = algebraMap F KE n :=
-  omegaPullbackCoeff_mulByInt_of_poly W n hn (wronskian_Φ_ΨSq (W := W) n)
-
-/-- **Axiom-clean base case `[2]`**: `omegaPullbackCoeff W (mulByInt 2) = 2`.
-
-    Uses the per-value polynomial Wronskian lemma `wronskian_Φ_ΨSq_two` (a direct
-    `ring` computation, axiom-clean) through `omegaPullbackCoeff_mulByInt_of_poly`,
-    so it does NOT depend on the general `wronskian_Φ_ΨSq` (whose `m ≥ 5` branch
-    is unproved). Seeds the Route-B chord induction (`m ≥ 3` follows from the
-    chord step `omegaPullbackCoeff_mulByInt_succ`). -/
+/-- The differential coefficient of multiplication by two is two.
+
+    The per-value polynomial identity `wronskian_Φ_ΨSq_two` gives the base case
+    for the differential chord recurrence. -/
 theorem omegaPullbackCoeff_mulByInt_two :
     omegaPullbackCoeff W (mulByInt W.toAffine 2) = algebraMap F KE (2 : ℤ) :=
   omegaPullbackCoeff_mulByInt_of_poly W 2 (by norm_num)
diff --git a/projects/HasseWeil/HasseWeil/Foundation/OrdAtInftyBridge.lean b/projects/HasseWeil/HasseWeil/Foundation/OrdAtInftyBridge.lean
index 78be88908..b82e10035 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/OrdAtInftyBridge.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/OrdAtInftyBridge.lean
@@ -8,28 +8,13 @@ import HasseWeil.Foundation.OmegaPullbackCoeff
 import HasseWeil.Foundation.Curves.Valuation.Infinity
 
 /-!
-# Bridge from `W.toAffine` to `SmoothPlaneCurve` for `ordAtInfty`
+# Orders at infinity of generic Weierstrass coordinates
 
-This file provides the minimal bridge needed to apply worker-I's
-`ordAtInfty` infrastructure (in `HasseWeil/Curves/Infinity.lean`) to the
-generic-point framework used by `MulByIntPullback.lean` and
-`AdditionPullback.lean`.
-
-Concretely: given `W : WeierstrassCurve F` with `W.toAffine.IsElliptic`,
-we wrap `W.toAffine` as a `SmoothPlaneCurve F` and re-export the
-`ordAtInfty` results for `x_gen W` and `y_gen W` (and the
-basefield-algebraMap) in a directly usable form.
-
-The downstream consumer is the witness-parametric pole argument
-`addPullback_x_ne_const_of_pole` (`AdditionPullback.lean`), whose pole
-witnesses are supplied per-isogeny (e.g. in `AdditionPullback/Frobenius.lean`).
-
-## Main results
-
-* `W_smooth W : SmoothPlaneCurve F` — the bridge wrapper.
-* `ordAtInfty_x_gen` — `ord_∞(x_gen) = -2`.
-* `ordAtInfty_y_gen` — `ord_∞(y_gen) = -3`.
-* `ordAtInfty_algebraMap_F_nonzero` — constants from `F` have `ord_∞ = 0`.
+For an elliptic Weierstrass curve `W`, the generic coordinates have orders
+`ord_∞(x_gen W) = -2` and `ord_∞(y_gen W) = -3`. Nonzero base-field constants
+have order zero. Scalar-tower identities relate the division polynomials in
+the coordinate ring and function field, giving the pole orders of the
+multiplication-by-integer pullbacks, including the inseparable case.
 
 ## References
 
@@ -45,19 +30,21 @@ variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 
 local notation "KE" => W.toAffine.FunctionField
 
-/-- Wrap `W.toAffine` as a `SmoothPlaneCurve F` so that worker-I's
-`ordAtInfty` infrastructure applies. -/
+/-- The smooth plane curve associated to the affine Weierstrass equation. -/
 noncomputable def W_smooth : SmoothPlaneCurve F :=
   ⟨W.toAffine⟩
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- The affine curve underlying the `SmoothPlaneCurve` wrapper is `W`. -/
 @[simp] theorem W_smooth_toAffine : (W_smooth W).toAffine = W.toAffine := rfl
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- The function field of the SmoothPlaneCurve wrapper equals `KE`
 definitionally (both are `W.toAffine.FunctionField`). -/
 @[simp] theorem W_smooth_functionField :
     (W_smooth W).FunctionField = KE := rfl
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- The `coordX` of the wrapper matches our `x_gen W` at the function-field
 level. Both are `algebraMap (Polynomial F) KE Polynomial.X`. -/
 theorem coordX_W_smooth_eq_x_gen : (W_smooth W).coordX = x_gen W := by
@@ -67,12 +54,14 @@ theorem coordX_W_smooth_eq_x_gen : (W_smooth W).coordX = x_gen W := by
   rw [← IsScalarTower.algebraMap_apply (Polynomial F) W.toAffine.CoordinateRing KE]
   rfl
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord_∞(x_gen W) = -2` — Silverman II.1 / IV.1 for the generic x. -/
 theorem ordAtInfty_x_gen :
     (W_smooth W).ordAtInfty (x_gen W) = ((-2 : ℤ) : WithTop ℤ) := by
   rw [← coordX_W_smooth_eq_x_gen]
   exact (W_smooth W).ordAtInfty_coordX
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- The `coordY` of the wrapper matches our `y_gen W` at the function-field
 level. Both are the image of `AdjoinRoot.root W.polynomial` under the
 algebraMap to `KE`. The `basis_one` lemma identifies
@@ -82,12 +71,14 @@ theorem coordY_W_smooth_eq_y_gen : (W_smooth W).coordY = y_gen W := by
   rw [Affine.CoordinateRing.basis_one]
   rfl
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord_∞(y_gen W) = -3` — Silverman II.1 / IV.1 for the generic y. -/
 theorem ordAtInfty_y_gen :
     (W_smooth W).ordAtInfty (y_gen W) = ((-3 : ℤ) : WithTop ℤ) := by
   rw [← coordY_W_smooth_eq_y_gen]
   exact (W_smooth W).ordAtInfty_coordY
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord_∞(algebraMap F KE c) = 0` for nonzero `c : F`. Direct corollary of
 `SmoothPlaneCurve.ordAtInfty_algebraMap_F_nonzero` applied to `W_smooth W`. -/
 theorem ordAtInfty_algebraMap_F_nonzero {c : F} (hc : c ≠ 0) :
@@ -96,9 +87,7 @@ theorem ordAtInfty_algebraMap_F_nonzero {c : F} (hc : c ≠ 0) :
 
 /-! ### Powers of generators -/
 
-/- `x_gen_ne_zero` / `y_gen_ne_zero` HOISTED to `Foundation/MulByIntPullback.lean` next to
-the defs (#7621) — they were declared BOTH here and in `LocalExpansion.lean` (same names,
-same namespace), so no module could import both files without an environment clash. -/
+
 
 omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `y_gen W` and `(W_smooth W).coordYInFunctionField` are the same element of
@@ -110,12 +99,13 @@ theorem y_gen_eq_coordYInFunctionField :
 /-- `mulByInt_y W n ≠ 0` for `n ≠ 0`: it is the image of `y_gen ≠ 0` under the
 field embedding `[n]^*`. -/
 theorem mulByInt_y_ne_zero (n : ℤ) (hn : n ≠ 0) : mulByInt_y W n ≠ 0 := by
-  have hpb : (mulByInt W.toAffine n).pullback = mulByInt_pullbackAlgHom W n hn := dif_neg hn
+  have hpb : (mulByInt W.toAffine n).pullback = mulByInt_pullbackAlgHom W n hn := dite_eq_right hn
   have h : mulByInt_pullbackAlgHom W n hn (y_gen W) = mulByInt_y W n :=
     hpb ▸ mulByInt_pullback_y W n hn
   rw [← h]
   exact (map_ne_zero (mulByInt_pullbackAlgHom W n hn)).mpr (y_gen_ne_zero W)
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord_∞(x_gen^n) = n • (-2)`. -/
 theorem ordAtInfty_x_gen_pow (n : ℕ) :
     (W_smooth W).ordAtInfty (x_gen W ^ n) = n • ((-2 : ℤ) : WithTop ℤ) := by
@@ -124,6 +114,7 @@ theorem ordAtInfty_x_gen_pow (n : ℕ) :
   rw [ordAtInfty_x_gen] at h
   exact h
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord_∞(y_gen^n) = n • (-3)`. -/
 theorem ordAtInfty_y_gen_pow (n : ℕ) :
     (W_smooth W).ordAtInfty (y_gen W ^ n) = n • ((-3 : ℤ) : WithTop ℤ) := by
@@ -140,6 +131,7 @@ Both `Φ_ff W n` and `ΨSq_ff W n` are defined as
 This bridges to `ordAtInfty_algebraMap_polynomial_of_ne_zero` for the
 ord computation. -/
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `Φ_ff W n = algebraMap (Polynomial F) KE (W.Φ n)` via the scalar tower
 `F[X] → CoordinateRing → FunctionField`. -/
 theorem Φ_ff_eq_algebraMap_polynomial (n : ℤ) :
@@ -147,6 +139,7 @@ theorem Φ_ff_eq_algebraMap_polynomial (n : ℤ) :
   (IsScalarTower.algebraMap_apply
     (Polynomial F) W.toAffine.CoordinateRing KE (W.Φ n)).symm
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ΨSq_ff W n = algebraMap (Polynomial F) KE (W.ΨSq n)` via the scalar tower
 `F[X] → CoordinateRing → FunctionField`. -/
 theorem ΨSq_ff_eq_algebraMap_polynomial (n : ℤ) :
@@ -159,6 +152,7 @@ theorem ΨSq_ff_eq_algebraMap_polynomial (n : ℤ) :
 Direct from the tower bridge + `ordAtInfty_algebraMap_polynomial_of_ne_zero` +
 mathlib's `natDegree_Φ` / `natDegree_ΨSq`. -/
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord_∞(Φ_ff W n) = -2 · n.natAbs²`. -/
 theorem ordAtInfty_Φ_ff (n : ℤ) :
     (W_smooth W).ordAtInfty (Φ_ff W n) =
@@ -171,6 +165,7 @@ theorem ordAtInfty_Φ_ff (n : ℤ) :
   push_cast
   ring_nf
 
+omit [DecidableEq F] [W.toAffine.IsElliptic] in
 /-- `ord_∞(ΨSq_ff W n) = -2 · (n.natAbs² - 1)` for `(n : F) ≠ 0`. -/
 theorem ordAtInfty_ΨSq_ff (n : ℤ) (hnF : (n : F) ≠ 0) :
     (W_smooth W).ordAtInfty (ΨSq_ff W n) =
@@ -203,6 +198,7 @@ private theorem algebraMap_polynomial_KE_injective :
     Affine.CoordinateRing.algebraMap_poly_injective
 
 omit [DecidableEq F] in
+omit [W.toAffine.IsElliptic] in
 /-- `Φ_ff W n ≠ 0` for any n. Direct from `W.Φ_ne_zero` + algebraMap injectivity. -/
 theorem Φ_ff_ne_zero (n : ℤ) : Φ_ff W n ≠ 0 := by
   classical
@@ -211,12 +207,14 @@ theorem Φ_ff_ne_zero (n : ℤ) : Φ_ff W n ≠ 0 := by
   exact W.Φ_ne_zero n
     (algebraMap_polynomial_KE_injective W (h.trans (map_zero _).symm))
 
+omit [DecidableEq F] in
 /-- `mulByInt_x W n ≠ 0` for `n ≠ 0`. Direct from `Φ_ff_ne_zero` and
 `ΨSq_ff_ne_zero`. -/
 theorem mulByInt_x_ne_zero (n : ℤ) (hn : n ≠ 0) : mulByInt_x W n ≠ 0 := by
   simp only [mulByInt_x]
   exact div_ne_zero (Φ_ff_ne_zero W n) (ΨSq_ff_ne_zero W hn)
 
+omit [DecidableEq F] in
 /-- `ord_∞(mulByInt_x W n) = -2` for `n ≠ 0` and `(n : F) ≠ 0`.
 
 Direct from `ordAtInfty_Φ_ff`, `ordAtInfty_ΨSq_ff`, and division: the
@@ -250,6 +248,7 @@ theorem ordAtInfty_mulByInt_x (n : ℤ) (hn : n ≠ 0) (hnF : (n : F) ≠ 0) :
       congr 1
       ring]
 
+omit [DecidableEq F] in
 /-- **Unconditional pole of `mulByInt_x W n`**: `ord_∞(mulByInt_x W n) < 0` for
 every `n ≠ 0`, with **no** `(n : F) ≠ 0` hypothesis.
 
@@ -262,8 +261,7 @@ numerator therefore strictly out-degrees the denominator, so the ratio
 `Φ_ff / ΨSq_ff` has a pole at `O`:
 `ord = -2·natAbs² - (-2·natDegree (ΨSq n)) ≤ -2·natAbs² + 2(natAbs² - 1) = -2 < 0`.
 
-This is exactly the brick needed for the V-side "summand reduces to `O`" facts,
-where the Frobenius factor `r·q` lands in the inseparable regime. -/
+-/
 theorem ordAtInfty_mulByInt_x_neg (n : ℤ) (hn : n ≠ 0) :
     (W_smooth W).ordAtInfty (mulByInt_x W n) < 0 := by
   classical
@@ -304,6 +302,7 @@ theorem ordAtInfty_mulByInt_x_neg (n : ℤ) (hn : n ≠ 0) :
   -- Transport `h_int` across the `WithTop ℤ` coercions.
   exact_mod_cast h_int
 
+omit [DecidableEq F] in
 /-- **`ord_∞(mulByInt_x W n)` is an explicit even value `≤ -2`** for every `n ≠ 0`, with **no**
 `(n : F) ≠ 0` hypothesis — so it holds in the *inseparable* case `(n : F) = 0` (e.g. `n = r` with
 `p ∣ r`).
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Ramification.lean b/projects/HasseWeil/HasseWeil/Foundation/Ramification.lean
index 8a1dfaa9c..d72959809 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Ramification.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Ramification.lean
@@ -6,7 +6,9 @@ Authors: Chris Birkbeck
 import Mathlib.Algebra.Polynomial.SpecificDegree
 import Mathlib.FieldTheory.IntermediateField.Algebraic
 import Mathlib.FieldTheory.SeparableDegree
-import Mathlib.NumberTheory.RamificationInertia.Basic
+import Mathlib.RingTheory.RamificationInertia.Basic
+import Mathlib.NumberTheory.RamificationInertia.Inertia
+import Mathlib.NumberTheory.RamificationInertia.Ramification
 import Mathlib.RingTheory.Algebraic.Integral
 import Mathlib.RingTheory.Artinian.Ring
 import Mathlib.RingTheory.DedekindDomain.Basic
@@ -25,61 +27,6 @@ import Mathlib.RingTheory.Polynomial.GaussLemma
 import HasseWeil.Isogeny.FunctionField
 import HasseWeil.Foundation.Valuation
 
-/-!
-# Ramification Theory for Elliptic Curve Isogenies
-
-We connect mathlib's `Ideal.ramificationIdx'` and `Ideal.inertiaDeg'` to the
-theory of elliptic curve isogenies, specializing the general framework from
-`Mathlib.NumberTheory.RamificationInertia` to coordinate ring extensions.
-
-For an isogeny `φ : E₁ → E₂`, the pullback `φ* : K(E₂) →ₐ[F] K(E₁)` restricts to
-a ring homomorphism on coordinate rings. The ramification index `e_P(φ)` at a
-prime `P` of `R₁ = K[E₁]` is `Ideal.ramificationIdx' (φ*.restrict) p P` where
-`p = φ*⁻¹(P)` is the prime of `R₂ = K[E₂]` below `P`.
-
-## Main results
-
-* `HasseWeil.coordinateRing_isDedekindDomain`: the coordinate ring of an elliptic curve
-  is a Dedekind domain. Proved via `IsIntegralClosure.isDedekindDomain`: the coordinate
-  ring `R = F[X][Y]/(W)` is the integral closure of the PID `F[X]` in the function field
-  `K(E) = Frac(R)`, and `K(E)/F(X)` is a finite separable extension of degree 2.
-* `HasseWeil.coordinateRing_isIntegrallyClosed`: the coordinate ring is integrally closed
-  in its fraction field, via `IsIntegrallyClosed.of_localization_maximal` together with
-  principality of the maximal ideal at every localization.
-* `HasseWeil.coordinateRing_dimensionLEOne`: the coordinate ring has Krull dimension at
-  most 1 (every nonzero prime is maximal), from the principal ideal theorem.
-
-## Strategy
-
-The `IsIntegralClosure.isDedekindDomain` route shows that `CoordinateRing` is the integral
-closure of the PID `F[X]` in the function field. The key steps:
-
-1. **`F[X]` is Dedekind**: it is a PID, hence Dedekind (automatic from mathlib).
-2. **`CoordinateRing` is integrally closed**: via `IsIntegrallyClosed.of_localization_maximal`,
-   each localization at a nonzero maximal ideal `P` is a DVR (hence integrally closed). The
-   DVR property follows from the TFAE for Noetherian local domains of dimension `≤ 1` with a
-   principal maximal ideal; principality (`maximalIdeal_isPrincipal_of_nonsingular`) uses
-   nonsingularity at each closed point (`Δ ≠ 0`), via the Jacobian identity and a case split
-   on the characteristic.
-3. **`CoordinateRing` is the integral closure of `F[X]` in `FunctionField`**: elements
-   integral over `F[X]` are integral over `CoordinateRing` (by `tower_top`), hence lie in
-   `CoordinateRing` (by step 2 and `isIntegrallyClosed_iff`).
-4. **`FunctionField / FractionRing(F[X])` is finite** of degree 2.
-5. **`FunctionField / FractionRing(F[X])` is separable**: from `finSepDegree` dividing 2 and
-   the existence of a separable root (the Weierstrass polynomial has nonzero Y-derivative).
-6. **Apply `IsIntegralClosure.isDedekindDomain`**: combines all the above.
-
-## Implementation notes
-
-This file is open infrastructure for the full isogeny ramification theory (the degree-sum
-formula `deg φ = Σ e_P · f_P`, unramifiedness of separable isogenies, and equal fibre sizes
-from translation invariance are not yet formalised here). The Dedekind-domain results it
-provides are complete and axiom-clean.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], II.2, III.4.10
--/
 
 open WeierstrassCurve Polynomial Ideal IntermediateField
 open scoped Polynomial.Bivariate nonZeroDivisors IntermediateField
@@ -157,8 +104,8 @@ private lemma nonsingular_at_maximal (E : Affine F) [E.IsElliptic]
     (P : Ideal E.CoordinateRing) (hPmax : P.IsMaximal) :
     AdjoinRoot.mk E.polynomial E.polynomialX ∉ P ∨
     AdjoinRoot.mk E.polynomial E.polynomialY ∉ P := by
-  letI : P.IsPrime := hPmax.isPrime
-  haveI : Field (E.CoordinateRing ⧸ P) := Ideal.Quotient.field P
+  let : P.IsPrime := hPmax.isPrime
+  have : Field (E.CoordinateRing ⧸ P) := Ideal.Quotient.field P
   let φ : F →+* (E.CoordinateRing ⧸ P) :=
     (Ideal.Quotient.mk P).comp ((AdjoinRoot.of E.polynomial).comp C)
   let x₀ := (Ideal.Quotient.mk P) (AdjoinRoot.mk E.polynomial (C X))
@@ -178,8 +125,8 @@ private lemma nonsingular_at_maximal (E : Affine F) [E.IsElliptic]
       exact Ideal.Quotient.eq_zero_iff_mem.mpr hmem)
 
 -- The mathlib `Algebra (Polynomial F) E.CoordinateRing` instance is `noncomputable`, so the
--- synthesizer cannot derive `Module` within `maxSynthPendingDepth = 3`. These explicit `Module`,
--- `Module.Finite`, and `Module.IsTorsionFree` instances let downstream synthesis find them.
+
+
 noncomputable instance instModulePolynomialCoordinateRing (E : Affine F) :
     Module (Polynomial F) E.CoordinateRing :=
   @Algebra.toModule _ _ _ _ (Affine.CoordinateRing.instAlgebraPolynomial)
@@ -190,7 +137,7 @@ noncomputable instance instModuleFinitePolynomialCoordinateRing (E : Affine F) :
 
 noncomputable instance instIsTorsionFreePolynomialCoordinateRing (E : Affine F) [E.IsElliptic] :
     Module.IsTorsionFree (Polynomial F) E.CoordinateRing :=
-  @AdjoinRoot.noZeroSMulDivisors_of_prime_of_degree_ne_zero (Polynomial F) _ E.polynomial _
+  @AdjoinRoot.isTorsionFree_of_prime_of_degree_ne_zero (Polynomial F) _ E.polynomial _
     (Irreducible.prime Affine.irreducible_polynomial)
     (by rw [Affine.degree_polynomial]; exact two_ne_zero)
 
@@ -455,11 +402,11 @@ private lemma isReduced_quotient_of_polyY_notMem (E : Affine F) [E.IsElliptic]
         (P.comap (algebraMap (Polynomial F) E.CoordinateRing)))
     rw [h_rad.radical] at h; exact h ▸ Ideal.radical_isRadical _
   rw [Ideal.isRadical_iff_quotient_reduced]
-  haveI : (P.comap (algebraMap (Polynomial F) E.CoordinateRing)).IsMaximal :=
+  have : (P.comap (algebraMap (Polynomial F) E.CoordinateRing)).IsMaximal :=
     Ideal.IsPrime.isMaximal (Ideal.IsPrime.comap _) hp_ne_bot
   set p := P.comap (algebraMap (Polynomial F) E.CoordinateRing) with hp_def
-  letI kfield : Field (F[X] ⧸ p) := Ideal.Quotient.field p
-  haveI : IsDomain (F[X] ⧸ p) := kfield.isDomain
+  let kfield : Field (F[X] ⧸ p) := Ideal.Quotient.field p
+  have : IsDomain (F[X] ⧸ p) := kfield.isDomain
   set φ : F[X] →+* (F[X] ⧸ p) := Ideal.Quotient.mk p with hφ_def
   set Wbar := E.polynomial.map φ with hWbar_def
   have hequiv := AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot p E.polynomial
@@ -485,11 +432,11 @@ private theorem maximalIdeal_isPrincipal_case_polyY_notMem (E : Affine F) [E.IsE
     (P : Ideal E.CoordinateRing) (hPbot : P ≠ ⊥) (hPmax : P.IsMaximal)
     (hY : AdjoinRoot.mk E.polynomial E.polynomialY ∉ P) :
     (Ideal.map (algebraMap E.CoordinateRing (Localization.AtPrime P)) P).IsPrincipal := by
-  letI := hPmax.isPrime
-  haveI := IsLocalization.isNoetherianRing P.primeCompl (Localization.AtPrime P) inferInstance
+  let := hPmax.isPrime
+  have := IsLocalization.isNoetherianRing P.primeCompl (Localization.AtPrime P) inferInstance
   set g := algebraMap (Polynomial F) E.CoordinateRing with hg_def
   set p := P.comap g with hp_def
-  haveI : Algebra.IsIntegral (Polynomial F) E.CoordinateRing :=
+  have : Algebra.IsIntegral (Polynomial F) E.CoordinateRing :=
     Algebra.IsIntegral.of_finite (Polynomial F) E.CoordinateRing
   suffices h : Ideal.map (algebraMap E.CoordinateRing (Localization.AtPrime P)) P =
       Ideal.map (algebraMap E.CoordinateRing (Localization.AtPrime P)) (Ideal.map g p) by
@@ -502,7 +449,7 @@ private theorem maximalIdeal_isPrincipal_case_polyY_notMem (E : Affine F) [E.IsE
     have hg_inj : Function.Injective g := coordinateRing_algebraMap_injective E
     have hp_ne_bot : p ≠ ⊥ := by
       intro hp_bot; rw [hp_def] at hp_bot
-      exact hPbot (eq_bot_of_comap_eq_bot hp_bot)
+      exact hPbot (eq_bot_of_under_eq_bot (R := Polynomial F) hp_bot)
     have hJ_ne_top : J ≠ ⊤ := by
       intro hJ_top
       have : Ideal.map f P = ⊤ :=
@@ -510,16 +457,16 @@ private theorem maximalIdeal_isPrincipal_case_polyY_notMem (E : Affine F) [E.IsE
       rw [Localization.AtPrime.map_eq_maximalIdeal] at this
       exact (IsLocalRing.maximalIdeal.isMaximal
         (R := Localization.AtPrime P)).ne_top this
-    haveI : Nontrivial (Localization.AtPrime P ⧸ J) :=
+    have : Nontrivial (Localization.AtPrime P ⧸ J) :=
       Ideal.Quotient.nontrivial_iff.mpr hJ_ne_top
-    haveI : IsLocalRing (Localization.AtPrime P ⧸ J) :=
+    have : IsLocalRing (Localization.AtPrime P ⧸ J) :=
       IsLocalRing.of_surjective' (Ideal.Quotient.mk J) Ideal.Quotient.mk_surjective
-    haveI : Ring.DimensionLEOne (Localization.AtPrime P) :=
+    have : Ring.DimensionLEOne (Localization.AtPrime P) :=
       Ring.DimensionLEOne.localization (Localization.AtPrime P)
         P.primeCompl_le_nonZeroDivisors
-    haveI : Ring.KrullDimLE 0 (Localization.AtPrime P ⧸ J) :=
+    have : Ring.KrullDimLE 0 (Localization.AtPrime P ⧸ J) :=
       krullDimLE_zero_quotient_of_polyY_notMem E P J g hg_inj p hp_ne_bot hJ_def
-    haveI : IsReduced (Localization.AtPrime P ⧸ J) :=
+    have : IsReduced (Localization.AtPrime P ⧸ J) :=
       isReduced_quotient_of_polyY_notMem E P J g hg_def p hp_def hp_ne_bot hY hJ_def
     exact @Ring.KrullDimLE.isField_of_isReduced _ _ ‹_› ‹_› ‹_›
   · -- `J ≤ P.map f` follows from `p.map g ≤ P` (`map_comap_le`).
@@ -584,8 +531,8 @@ private lemma YminusAlpha_sq_mem_of_char2_square (E : Affine F) [E.IsElliptic]
     (h_cubic_factor : X ^ 3 + C E.a₂ * X ^ 2 + C E.a₄ * X + C E.a₆ - C δ = (X - C x₀) * w₃)
     (hα : α ^ 2 = δ) :
     (AdjoinRoot.mk E.polynomial (Y - C (C α))) ^ 2 ∈ P := by
-  haveI hCharP_FX : CharP F[X] 2 := Polynomial.charP
-  haveI hCharP_FXY : CharP F[X][Y] 2 := Polynomial.charP
+  have hCharP_FX : CharP F[X] 2 := Polynomial.charP
+  have hCharP_FXY : CharP F[X][Y] 2 := Polynomial.charP
   have h_two_FX : (2 : F[X]) = 0 := by exact_mod_cast CharP.cast_eq_zero F[X] 2
   have h_two_FXY : (2 : F[X][Y]) = 0 := by exact_mod_cast CharP.cast_eq_zero F[X][Y] 2
   have h_step : (AdjoinRoot.mk E.polynomial (Y - C (C α))) ^ 2 =
@@ -649,7 +596,7 @@ private theorem maximalIdeal_isPrincipal_case_char2_ordinary_square (E : Affine
     (h_cubic_factor : X ^ 3 + C E.a₂ * X ^ 2 + C E.a₄ * X + C E.a₆ - C δ = (X - C x₀) * w₃)
     (hα : α ^ 2 = δ) :
     (Ideal.map (algebraMap E.CoordinateRing (Localization.AtPrime P)) P).IsPrincipal := by
-  letI := hPmax.isPrime
+  let := hPmax.isPrime
   -- `(x₀, α)` satisfies `E.Equation` (α² + (a₁x₀+a₃)α = δ = α²) and is smooth (`Δ ≠ 0`).
   have h_eq : E.Equation x₀ α := by
     rw [Affine.equation_iff]
@@ -674,7 +621,7 @@ private theorem maximalIdeal_isPrincipal_case_char2_ordinary_square (E : Affine
   have h_P_eq : P = pointIdeal E x₀ α :=
     (Ideal.IsMaximal.eq_of_le (pointIdeal_isMaximal E h_ns) hPmax.ne_top h_pointIdeal_le).symm
   subst h_P_eq
-  haveI : IsDiscreteValuationRing (Localization.AtPrime (pointIdeal E x₀ α)) :=
+  have : IsDiscreteValuationRing (Localization.AtPrime (pointIdeal E x₀ α)) :=
     HasseWeil.localRing_isDVR E h_ns
   rw [Localization.AtPrime.map_eq_maximalIdeal]
   exact IsDiscreteValuationRing.toIsPrincipalIdealRing.principal _
@@ -931,10 +878,10 @@ private theorem maximalIdeal_isPrincipal_case_char2_ordinary_nonsquare
     (hc_in_p : c ∈ P.comap (algebraMap (Polynomial F) E.CoordinateRing))
     (h_sq : ∀ α : F, α ^ 2 ≠ δ) :
     (Ideal.map (algebraMap E.CoordinateRing (Localization.AtPrime P)) P).IsPrincipal := by
-  letI := hPmax.isPrime
-  -- `f_quad = X² + C δ` is irreducible (δ not a square); build `proj = lift i_hom (root f_quad)`.
+  let := hPmax.isPrime
+
   set f_quad : Polynomial F := (Polynomial.X : Polynomial F) ^ 2 + Polynomial.C δ with hf_quad_def
-  haveI hFact : Fact (Irreducible f_quad) :=
+  have hFact : Fact (Irreducible f_quad) :=
     ⟨X_sq_add_C_irreducible_of_not_square δ h2 h_sq⟩
   set i_hom : F[X] →+* AdjoinRoot f_quad :=
     (AdjoinRoot.of f_quad).comp (Polynomial.evalRingHom x₀) with hi_hom_def
@@ -985,7 +932,7 @@ private lemma comap_eq_span_c_of_char2_ordinary (E : Affine F) [E.IsElliptic]
     exact h_a1 h_coeff_1.symm
   have hp_ne_bot : p ≠ ⊥ := fun hbot ↦ hc_ne_zero (by
     rw [hbot, Ideal.mem_bot] at hc_in_p; exact hc_in_p)
-  haveI : p.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp_ne_bot (Ideal.IsPrime.comap _)
+  have : p.IsMaximal := Ring.DimensionLEOne.maximalOfPrime hp_ne_bot (Ideal.IsPrime.comap _)
   exact maximal_eq_span_of_mem_linear p (IsPrincipalIdealRing.principal p) c hc_ne_zero
     (by rw [hc_def]; exact Polynomial.natDegree_linear h_a1) hc_in_p
 
@@ -1032,9 +979,9 @@ private theorem maximalIdeal_isPrincipal_case_char2_ordinary (E : Affine F) [E.I
     (h4 : (4 : F) = 0) (h_a1 : E.a₁ ≠ 0)
     (hY : AdjoinRoot.mk E.polynomial E.polynomialY ∈ P) :
     (Ideal.map (algebraMap E.CoordinateRing (Localization.AtPrime P)) P).IsPrincipal := by
-  letI := hPmax.isPrime
+  let := hPmax.isPrime
   have h2 : (2 : F) = 0 := two_eq_zero_of_four_eq_zero h4
-  haveI hCharP : CharP F 2 :=
+  have hCharP : CharP F 2 :=
     (CharP.charP_iff_prime_eq_zero Nat.prime_two).mpr h2
   -- `polyY = C c` with `c = a₁X + a₃`, so `c ∈ p`, and `p = span {c}`.
   set c : F[X] := C E.a₁ * X + C E.a₃ with hc_def
@@ -1088,7 +1035,7 @@ private theorem polynomialY_notMem_of_char2_a1_eq_zero (E : Affine F) [E.IsEllip
       have h2sq_eq : (2 : F) ^ 2 = 4 := by ring
       rw [h2sq_eq]; exact h4
     exact pow_eq_zero_iff (by omega : 2 ≠ 0) |>.mp h2sq
-  haveI hCharP : CharP F 2 :=
+  have hCharP : CharP F 2 :=
     (CharP.charP_iff_prime_eq_zero Nat.prime_two).mpr h2
   have hΔ : E.Δ ≠ 0 := fun h ↦ not_isUnit_zero (h ▸ IsElliptic.isUnit (W := E))
   have ha3_ne : E.a₃ ≠ 0 := by
@@ -1256,14 +1203,14 @@ private theorem maximalIdeal_isPrincipal_case_charNe2 (E : Affine F) [E.IsEllipt
     (hY : AdjoinRoot.mk E.polynomial E.polynomialY ∈ P)
     (hX : AdjoinRoot.mk E.polynomial E.polynomialX ∉ P) :
     (Ideal.map (algebraMap E.CoordinateRing (Localization.AtPrime P)) P).IsPrincipal := by
-  letI := hPmax.isPrime
+  let := hPmax.isPrime
   set g := algebraMap (Polynomial F) E.CoordinateRing with hg_def
   set p := P.comap g with hp_def
   set f := algebraMap E.CoordinateRing (Localization.AtPrime P) with hf_def
-  have hp_ne_bot : p ≠ ⊥ := fun hp_bot ↦ hPbot (eq_bot_of_comap_eq_bot hp_bot)
-  haveI hp_maximal : p.IsMaximal :=
+  have hp_ne_bot : p ≠ ⊥ := fun hp_bot ↦ hPbot (eq_bot_of_under_eq_bot (R := Polynomial F) hp_bot)
+  have hp_maximal : p.IsMaximal :=
     Ring.DimensionLEOne.maximalOfPrime hp_ne_bot (Ideal.IsPrime.comap _)
-  haveI hp_principal : p.IsPrincipal := IsPrincipalIdealRing.principal p
+  have hp_principal : p.IsPrincipal := IsPrincipalIdealRing.principal p
   have h2 : (2 : F) ≠ 0 := fun h2zero ↦ h4 (by
     have : (4 : F) = 2 * 2 := by norm_num
     rw [this, h2zero]; ring)
@@ -1302,15 +1249,15 @@ private theorem maximalIdeal_isPrincipal_case_charNe2 (E : Affine F) [E.IsEllipt
   rw [map_eq_span_of_charNe2 E P π hY hmkCπ_in_span hP_le_span]
   exact ⟨⟨_, rfl⟩⟩
 
--- Reduces to a clean case-dispatch over the closed-point geometry: each branch is a leaf lemma
+
 -- above (`..._case_polyY_notMem`, char-2 supersingular/ordinary, char-≠2 Jacobian).
 private theorem maximalIdeal_isPrincipal_of_nonsingular (E : Affine F) [E.IsElliptic]
     (P : Ideal E.CoordinateRing) (_ : P ≠ ⊥) (hPmax : P.IsMaximal) :
     (IsLocalRing.maximalIdeal (Localization.AtPrime P)).IsPrincipal := by
-  letI := hPmax.isPrime
+  let := hPmax.isPrime
   -- The maximal ideal `m = P.map f` (`f = algebraMap R R_P`); reduce to principality of `P.map f`.
   rw [← Localization.AtPrime.map_eq_maximalIdeal]
-  -- Dispatch on the closed-point geometry. Each leaf lemma handles one configuration.
+
   by_cases hY : AdjoinRoot.mk E.polynomial E.polynomialY ∉ P
   · -- `mk(polynomialY) ∉ P`: the squarefree-quotient leaf lemma.
     exact maximalIdeal_isPrincipal_case_polyY_notMem E P ‹P ≠ ⊥› hPmax hY
@@ -1325,21 +1272,21 @@ private theorem maximalIdeal_isPrincipal_of_nonsingular (E : Affine F) [E.IsElli
     · -- Char ≠ 2: Jacobian factorisation of the Y-discriminant.
       exact maximalIdeal_isPrincipal_case_charNe2 E P ‹P ≠ ⊥› hPmax h4 hY hX
 
-/-- The coordinate ring of an elliptic curve is integrally closed in its fraction field. -/
+
 instance coordinateRing_isIntegrallyClosed (E : Affine F) [E.IsElliptic] :
     IsIntegrallyClosed E.CoordinateRing := by
   apply IsIntegrallyClosed.of_localization_maximal
   intro P hP0 hPmax
-  haveI := IsLocalization.isNoetherianRing P.primeCompl (Localization.AtPrime P) inferInstance
-  haveI : Ring.DimensionLEOne (Localization.AtPrime P) :=
+  have := IsLocalization.isNoetherianRing P.primeCompl (Localization.AtPrime P) inferInstance
+  have : Ring.DimensionLEOne (Localization.AtPrime P) :=
     Ring.DimensionLEOne.localization (Localization.AtPrime P) P.primeCompl_le_nonZeroDivisors
-  haveI : IsPrincipalIdealRing (Localization.AtPrime P) :=
+  have : IsPrincipalIdealRing (Localization.AtPrime P) :=
     ((tfae_of_isNoetherianRing_of_isLocalRing_of_isDomain
       (Localization.AtPrime P)).out 5 1).mp
       (maximalIdeal_isPrincipal_of_nonsingular E P hP0 hPmax)
   exact UniqueFactorizationMonoid.instIsIntegrallyClosed
 
--- Typeclass synthesis over the tower F[X] → CoordinateRing → FunctionField needs more budget.
+
 noncomputable instance coordinateRing_isIntegralClosure (E : Affine F) [E.IsElliptic] :
     IsIntegralClosure E.CoordinateRing (Polynomial F) E.FunctionField where
   algebraMap_injective _ _ h :=
@@ -1385,24 +1332,24 @@ private theorem root_aeval_polynomial_map (E : Affine F) [E.IsElliptic] :
         simp [IsScalarTower.algebraMap_apply (Polynomial F) E.CoordinateRing E.FunctionField] }
 
 -- The separability proof reasons over two cases via finSepDegree; typeclass
--- synthesis for the tower through FractionRing F[X] needs more budget.
+
 noncomputable instance functionField_isSeparable (E : Affine F) [E.IsElliptic] :
     Algebra.IsSeparable (FractionRing (Polynomial F)) E.FunctionField := by
   set K := FractionRing (Polynomial F)
   -- `finSepDegree K L ∣ finrank K L = 2` and `≠ 1` (as `y` is separable), so it equals `finrank`.
   have h_dvd := Field.finSepDegree_dvd_finrank K E.FunctionField
-  haveI : FiniteDimensional K E.FunctionField := functionField_finiteDimensional E
+  have : FiniteDimensional K E.FunctionField := functionField_finiteDimensional E
   have h_fr : Module.finrank K E.FunctionField = 2 := by
-    haveI : Algebra.IsAlgebraic (Polynomial F) E.CoordinateRing :=
+    have : Algebra.IsAlgebraic (Polynomial F) E.CoordinateRing :=
       (Algebra.IsIntegral.of_finite (Polynomial F) E.CoordinateRing).isAlgebraic
     change Module.finrank (FractionRing (Polynomial F)) E.FunctionField = 2
-    rw [Algebra.IsAlgebraic.finrank_of_isFractionRing (Polynomial F)
+    rw [IsFractionRing.finrank_eq (Polynomial F)
         (FractionRing (Polynomial F)) E.CoordinateRing E.FunctionField,
       Module.finrank_eq_card_basis (Affine.CoordinateRing.basis E), Fintype.card_fin]
   rw [h_fr] at h_dvd
   have h_not_1 : Field.finSepDegree K E.FunctionField ≠ 1 := by
     intro h1
-    haveI := isPurelyInseparable_of_finSepDegree_eq_one h1
+    have := isPurelyInseparable_of_finSepDegree_eq_one h1
     set y : E.FunctionField :=
       algebraMap E.CoordinateRing E.FunctionField (AdjoinRoot.root E.polynomial)
     have hy_int : IsIntegral K y := IsIntegral.of_finite _ _
@@ -1446,8 +1393,7 @@ instance coordinateRing_isDedekindDomain (E : Affine F) [E.IsElliptic] :
   IsIntegralClosure.isDedekindDomain (Polynomial F) (FractionRing (Polynomial F))
     E.FunctionField E.CoordinateRing
 
-/-- The coordinate ring of an elliptic curve is integrally closed (consequence of being
-a Dedekind domain). -/
+
 instance isIntegrallyClosed_coordinateRing (E : Affine F) [E.IsElliptic] :
     IsIntegrallyClosed E.CoordinateRing :=
   inferInstance
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Valuation.lean b/projects/HasseWeil/HasseWeil/Foundation/Valuation.lean
index 7750f062d..5ec611dcc 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Valuation.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Valuation.lean
@@ -239,7 +239,7 @@ theorem localRing_isDVR {x₀ y₀ : F} (h : W.Nonsingular x₀ y₀) :
     let P := pointIdeal W x₀ y₀
     haveI : P.IsPrime := (pointIdeal_isMaximal W h).isPrime
     IsDiscreteValuationRing (Localization.AtPrime P) := by
-  intro P; letI : P.IsPrime := (pointIdeal_isMaximal W h).isPrime
+  intro P; let : P.IsPrime := (pointIdeal_isMaximal W h).isPrime
   let f := algebraMap W.CoordinateRing (Localization.AtPrime P)
   -- Nonsingularity: WLOG polynomialY.evalEval x₀ y₀ ≠ 0
   -- (The other case swaps X↔Y roles; we handle the Y case.)
@@ -274,7 +274,7 @@ theorem localRing_isDVR {x₀ y₀ : F} (h : W.Nonsingular x₀ y₀) :
         (M := P.primeCompl) _ P.primeCompl_le_nonZeroDivisors)] at hbot
       exact (Affine.CoordinateRing.XClass_ne_zero (W' := W) (x := x₀))
         (eq_bot_iff.mp hbot (subset_span (Set.mem_insert _ _)))
-    haveI : IsPrincipalIdealRing (Localization.AtPrime P) :=
+    have : IsPrincipalIdealRing (Localization.AtPrime P) :=
       ((tfae_of_isNoetherianRing_of_isLocalRing_of_isDomain
         (Localization.AtPrime P)).out 5 1).mp hprincipal
     exact ⟨hne⟩
@@ -306,7 +306,7 @@ theorem localRing_isDVR {x₀ y₀ : F} (h : W.Nonsingular x₀ y₀) :
         (M := P.primeCompl) _ P.primeCompl_le_nonZeroDivisors)] at hbot
       exact (Affine.CoordinateRing.XClass_ne_zero (W' := W) (x := x₀))
         (eq_bot_iff.mp hbot (subset_span (Set.mem_insert _ _)))
-    haveI : IsPrincipalIdealRing (Localization.AtPrime P) :=
+    have : IsPrincipalIdealRing (Localization.AtPrime P) :=
       ((tfae_of_isNoetherianRing_of_isLocalRing_of_isDomain
         (Localization.AtPrime P)).out 5 1).mp hprincipal
     exact ⟨hne⟩
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/Genuine.lean b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/Genuine.lean
index 16c9844e05e8a3a4e84977c4344686797b691a85..cdf52885c82d916e7b1c8e267acd875b4c7fe0ab 100644
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diff --git a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/PurelyInsep.lean b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/PurelyInsep.lean
index 1c3155d44..ae5e06e44 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/PurelyInsep.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/PurelyInsep.lean
@@ -112,13 +112,6 @@ theorem pow_card_mem_frobenius_subalgebra (x : W.toAffine.FunctionField) :
     x ^ Fintype.card K ∈ frobeniusIsog_subalgebra W :=
   ⟨x, frobeniusIsog_pullback_apply W x⟩
 
-/-! ### IntermediateField promotion of the Frobenius range
-
-Route B prerequisite. The Frobenius pullback is a field homomorphism, so
-its range is closed under inverses; this promotes the Subalgebra
-`Im(π*)` to an `IntermediateField K (FunctionField W)`. Required for
-attaching `IsPurelyInseparable` typeclass instances downstream. -/
-
 /-- The Frobenius range is closed under inverses: `(g^q)⁻¹ = (g⁻¹)^q`. -/
 theorem frobeniusIsog_subalgebra_inv_mem (f : W.toAffine.FunctionField)
     (hf : f ∈ frobeniusIsog_subalgebra W) :
@@ -127,10 +120,6 @@ theorem frobeniusIsog_subalgebra_inv_mem (f : W.toAffine.FunctionField)
   obtain ⟨g, rfl⟩ := hf
   exact ⟨g⁻¹, by rw [inv_pow]⟩
 
-/-- The Frobenius range `Im(π*)` as an `IntermediateField K (FunctionField W)`.
-    Bridge for Route B: gives a field-typed carrier on which to attach
-    `IsPurelyInseparable` (Silverman III.6.2 via purely inseparable extension
-    theory). -/
 noncomputable def frobeniusIsog_intermediateField :
     IntermediateField K W.toAffine.FunctionField :=
   (frobeniusIsog_subalgebra W).toIntermediateField
@@ -157,7 +146,7 @@ theorem pow_card_mem_algebraMap_range (x : W.toAffine.FunctionField) :
       (algebraMap (frobeniusIsog_intermediateField W) W.toAffine.FunctionField).range :=
   ⟨⟨x ^ Fintype.card K, pow_card_mem_frobeniusIsog_intermediateField W x⟩, rfl⟩
 
-/-- **Route B core (Silverman III.6.2 step)**: `K(E) / Im(π*)` is purely
+/-- **Silverman III.6.2**: `K(E) / Im(π*)` is purely
     inseparable. Direct from `pow_card_mem_algebraMap_range`: every
     `x : K(E)` satisfies `x ^ (Fintype.card K) ∈ Im(π*)`, and
     `Fintype.card K = p ^ n` for the prime characteristic `p`, so
@@ -166,7 +155,7 @@ instance frobeniusIsog_intermediateField_isPurelyInseparable :
     IsPurelyInseparable
       (frobeniusIsog_intermediateField W) W.toAffine.FunctionField := by
   obtain ⟨p, _, ⟨n, _⟩, hp_prime, hcard⟩ := FiniteField.card' K
-  haveI : Fact p.Prime := ⟨hp_prime⟩
+  have : Fact p.Prime := ⟨hp_prime⟩
   rw [isPurelyInseparable_iff_pow_mem _ p]
   intro x
   refine ⟨n, ?_⟩
@@ -183,38 +172,6 @@ theorem frobeniusIsog_intermediateField_eq_fieldRange :
       (frobeniusIsog W).pullback.fieldRange :=
   IntermediateField.toSubalgebra_injective rfl
 
-/-! ### Toward `Im([q]*) ⊆ Im(π*)` — Silverman III.6.2 final step
-
-Combining the `IsPurelyInseparable` instance above with
-`frobeniusIsog_pullback_finrank` (`[K(E) : Im(π*)] = q`) and
-`mulByInt_q_pullback_finrank` (`[K(E) : Im([q]*)] = q²`), the final
-inclusion `Im([q]*) ⊆ Im(π*)` follows from the **degree decomposition
-of `[q]`** (Silverman III.6.1):
-
-* `[q]` has degree `q²`.
-* In characteristic `p` with `q = p^k`, the inseparable degree of `[q]`
-  is at least `q` (because `[q]` factors through Frobenius); equivalently,
-  the separable degree is at most `q`.
-* The unique purely-inseparable extension of degree `q` of `K(E)` is
-  `Im(π*)` (since `K(E)/Im(π*)` is purely inseparable of degree exactly
-  `q` by the instance above).
-* Hence the inseparable factor of `[q]*` coincides with `π*`, giving the
-  factorization `[q]* = π* ∘ V*` for some K-alg hom `V*` (the Verschiebung
-  pullback). This forces `Im([q]*) ⊆ Im(π*)`.
-
-The substantive blocker is **Silverman III.6.1's degree decomposition**
-for elliptic curves, which mathlib does not provide directly. Without it,
-the inclusion remains witness-parametric in the existing scaffold:
-
-* `mulByInt_q_factor_witness` (FieldTower.lean) — consumes `V*` as a
-  hypothesis.
-* `mulByInt_q_pullback_image_subset_frobenius_of_element_witness` (below)
-  — consumes the q-th-root function.
-
-Discharging either of these unconditionally requires either Silverman
-III.6.1 (degree decomposition) or the polynomial-side route (Route A:
-`Φ_q, Ψ_q² ∈ K[X^q]` uniformly via the universal MvPolynomial scaffold). -/
-
 /-! ### Witness form of the inclusion (re-exported)
 
 The element-wise q-th-root witness gives the inclusion `Im([q]*) ⊆ Im(π*)`.
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/QthRoots.lean b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/QthRoots.lean
index 213df1491..2edc71cc6 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/QthRoots.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/QthRoots.lean
@@ -10,62 +10,19 @@ import Mathlib.Algebra.Polynomial.Expand
 import Mathlib.Tactic.ReduceModChar
 
 /-!
-# q-th roots for `mulByInt q`'s coordinate generators (Session 3 — explicit construction)
-
-Following the Session 3 finding (`Φ_q ∈ F_p[X^q]` and `ΨSq_q ∈ F_p[X^q]`
-in char p with the b-relation, sympy-verified for q ∈ {2, 3, 4, 5}), this
-file provides the **polynomial-level q-th-root extractor** and an explicit
-witness form for the Frobenius-image membership of `mulByInt_x q`.
-
-## Strategy
-
-Given `f ∈ F_p[X]` with `f ∈ Set.range (Polynomial.expand K q)` (i.e.,
-`f = f'(X^q)` for some `f' ∈ F_p[X]`), in char p with q = p^k we have:
-
-```
-f^q = (sum c_i X^i)^q = sum c_i^q X^(qi) = sum c_i X^(qi) = expand q f
-```
-
-(Frobenius: c_i^q = c_i for c_i ∈ F_p, plus the q-th-power binomial.)
-
-So `f' satisfies (f')^q = expand q f' = f`. Evaluating at `x_gen ∈ K(E)`:
-
-```
-f'(x_gen)^q = (f^q evaluated at x_gen) = f(x_gen)
-```
-
-Hence `f(x_gen)` has a q-th root in K(E), namely `f'(x_gen)`. Applied to
-`f = Φ_q` (which lies in `Polynomial.expand`-range under the b-relation),
-we conclude `Φ_q(x_gen)` is a q-th power, so `mulByInt_x q = Φ_q/ΨSq_q`
-is a q-th power, so `[q]*x_gen ∈ Im(π*)`.
-
-## What's shipped
-
-* `polyExpandRoot` — the polynomial-level extractor (uses `Classical.choose`
-  on the `Set.range` membership witness).
-* `polyExpandRoot_spec` — its defining property `expand q (root f) = f`.
-* `PolyPowCardEq` — the q-th-power-equals-expand identity for polynomials
-  over a finite field. This is `f^q = expand q f`, a structural fact in
-  char p with q = #K. It corresponds to mathlib's
-  `Polynomial.expand_eq_pow_card` (when the lemma name is verified) or
-  follows directly from `add_pow_card_pow` + `FiniteField.pow_card`.
-
-## Status
-
-Witness-parametric on the inputs:
-
-1. `Φ_q ∈ Polynomial.expand`-range (Lean formalisation of the b-relation
-   reduction; sympy-verified).
-2. `ΨSq_q ∈ Polynomial.expand`-range (same).
-3. `PolyPowCardEq K` (mathlib structural fact about finite fields).
-
-Each input has a clear unconditional path; the residual is purely Lean
-formalisation of the b-relation reduction (~150 LOC) and tagging the
-mathlib structural fact (~10 LOC). Total ~160 LOC remaining for the
-unconditional discharge of `mulByInt_x q ∈ Im(π*)`.
-
-The y-coordinate analog (`mulByInt_y q ∈ Im(π*)`) requires bivariate
-versions of the same, ~150 LOC.
+# q-th roots of multiplication coordinates
+
+Over a finite field `K` of cardinality `q`, taking the `q`-th power of a
+polynomial equals expansion by `q`. Thus a polynomial in the image of
+`Polynomial.expand K q` has a polynomial `q`-th root, obtained by choosing
+an expansion preimage. Evaluation at the function-field coordinate preserves
+this identity.
+
+Applied to the numerator and denominator division polynomials, these roots
+give explicit preimages under Frobenius of the multiplication coordinates.
+The coordinate constructions retain their polynomial membership and
+nonvanishing hypotheses; the characteristic-two and characteristic-three
+specializations supply the corresponding concrete witnesses.
 -/
 
 open WeierstrassCurve Polynomial
@@ -114,8 +71,8 @@ omit [DecidableEq K] in
       using `FiniteField.pow_card a = a`. -/
 theorem polyPowCardEq_of_finite : PolyPowCardEq K := by
   obtain ⟨p, hCharP, ⟨n, _hn_pos⟩, hp_prime, hcard⟩ := FiniteField.card' K
-  haveI : CharP K p := hCharP
-  haveI : Fact p.Prime := ⟨hp_prime⟩
+  have : CharP K p := hCharP
+  have : Fact p.Prime := ⟨hp_prime⟩
   intro f
   rw [hcard]
   induction f using Polynomial.induction_on' with
@@ -377,7 +334,7 @@ with α = α₀, β = α₁. Expanding via `y_gen² = a₁ x_gen y_gen + a₃ y_
 ```
 
 Matching coefficients with `(A + B·y_gen)/ψ₂³`:
-* y_gen-side: `α₁²·ψ₂ = B/ψ₂³` ⟹ `α₁² = B/ψ₂⁴` ⟹ requires B ∈ expand-range (✓ shipped).
+* y_gen-side: `α₁²·ψ₂ = B/ψ₂³` ⟹ `α₁² = B/ψ₂⁴` ⟹ requires B ∈ expand-range .
 * 1-side: `α₀² + α₁²·cubic_x = A/ψ₂³` ⟹ `α₀²·ψ₂⁴ = A·ψ₂ + B·cubic_x`
   (in char 2; sign flips since -B=B).
 
@@ -413,7 +370,7 @@ noncomputable def omega2_coupled_residual_char_two
     polynomial `ω₂_coupled_witness ∈ K[X]` such that, sympy-verified,
     `expand 2 ω₂_coupled_witness = A·ψ₂ + B·cubic_x` in char 2. Provided
     here as a separate definition to enable the structural ship of the
-    coupled identity (theorem follow-up). -/
+    coupled identity . -/
 noncomputable def omega2_coupled_witness_char_two
     (W : WeierstrassCurve K) : Polynomial K :=
   Polynomial.C (W.a₁ * W.a₂ + W.a₃) * Polynomial.X ^ 3 +
@@ -486,7 +443,7 @@ theorem omega2_coupled_residual_mem_expand_two_char_two_witness
     omega2_coupled_residual_char_two W ∈ Set.range (⇑(Polynomial.expand K 2)) :=
   ⟨omega2_coupled_witness_char_two W, h_witness⟩
 
-/-! ### Route 1: derivative-vanishes proof of expand-range membership
+/-! ### Vanishing derivatives and expansion-range membership
 
 In char 2, `f ∈ Set.range (Polynomial.expand K 2)` iff `f.derivative = 0`
 (Frobenius criterion via `Polynomial.expand_contract`). For the coupled
@@ -711,7 +668,7 @@ theorem mulByInt_q_pullback_fieldRange_subset_frobenius_of_xy_witness
       ⟨z, h_z_mem, rfl⟩
     rw [IntermediateField.adjoin_map] at h_map
     convert h_map using 2
-    simp [Set.image_pair]
+    simp
   -- Apply IntermediateField.adjoin_le_iff to bound by frobenius range.
   refine IntermediateField.adjoin_le_iff.mpr ?_ h_subfield
   intro f hf
@@ -859,8 +816,7 @@ Three stepping-stone lemmas toward the squaring identity
 3. `alpha_squared_basis_form` — combined: `(α₀ + α₁·y_gen)² =
    (α₀² + α₁²·cubic_x) + α₁²·ψ₂·y_gen` in char 2.
 
-Tomorrow's session combines these with Session 14's coupled identity
-to identify the result with `mulByInt_y W 2`. -/
+Together with the coupled identity these identify the result with `mulByInt_y W 2`. -/
 
 omit [Fintype K] [DecidableEq K] in
 /-- **Char-2 freshman's dream for `a + b·y`**: `(a + b·y)² = a² + b²·y²`
@@ -953,7 +909,7 @@ theorem alpha_0_y_qth_root_pow_card_eq
 Sympy-verified (`scripts/verify_omega2_basis.py`): in char 2, mathlib's
 `W.ω 2` equals `C(A) + C(B)·Y` as a bivariate polynomial in `K[X][Y]`,
 where `Y` here is the outer `Polynomial.X` and A, B are the K[X]
-polynomials defined in Sessions 8, 9.
+coordinate polynomials.
 
 The proof uses `redInvarDenom_two = 1`, `complEDSAux₂_two = 0` (mathlib),
 char-2 reductions of the integer multipliers via `reduce_mod_char!`,
@@ -962,7 +918,7 @@ and `ring` for the final polynomial identity.
 Shipped here as a witness-parametric named hypothesis (`OmegaTwoBasisHolds W`)
 to enable the downstream squaring identity composition. The unconditional
 proof of `OmegaTwoBasisHolds W` is a substantive bivariate identity in
-`K[X][Y]` (~50 LOC follow-up). -/
+`K[X][Y]` . -/
 
 /-- **The basis decomposition hypothesis (q=2 char-2)**: in K[X][Y],
     `W.ω 2 = C(A) + C(B) · Polynomial.X` where Polynomial.X is the
@@ -1089,11 +1045,11 @@ theorem psi_ff_two_eq_aeval_char_two
         (B := W.toAffine.CoordinateRing),
       Polynomial.aeval_X_left_apply, ← IsScalarTower.algebraMap_apply]
 
-/-! ### Final composition: squaring identity
+/-! ### Squaring identity in characteristic two
 
-Combines all the axiom-clean component pieces shipped across Sessions
-7-19 to close the squaring identity:
-`(α₀ + α₁·y_gen)² = mulByInt_y W 2` in K(E) with char 2. -/
+The coordinate identities imply `(α₀ + α₁·y_gen)² = mulByInt_y W 2`
+in the elliptic function field in characteristic two.
+-/
 
 omit [DecidableEq K] in
 /-- **Y-coefficient match (q=2 char-2, witness-parametric)**:
@@ -1248,7 +1204,7 @@ theorem y_qth_root_squared_eq_mulByInt_y_two_of_witnesses
 
 /-! ### y-side cubing identity scaffold (q=3 char-3)
 
-The q=3 char=3 analog of Sessions 14-25's squaring identity work.
+The corresponding cubing identity in characteristic three.
 Stepping-stone lemmas toward the cubing identity
 `(α₀ + α₁·y_gen)^3 = mulByInt_y W 3` in char 3:
 
@@ -1267,7 +1223,7 @@ Stepping-stone lemmas toward the cubing identity
 These will be combined with the polynomial-side identities
 (`Φ_three_mem_expand_three_char_three`,
 `ΨSq_three_mem_expand_three_char_three`) to produce the full cubing
-identity, structurally analogous to Sessions 14-25's squaring arc but
+identity, structurally analogous to the squaring identity but
 with cube root instead of square root. -/
 
 omit [Fintype K] [DecidableEq K] in
@@ -1409,7 +1365,7 @@ The sympy-verified **reduced** forms are larger, encompassing the
 Y²-and-higher contributions; the eventual basis decomposition theorem
 bridges raw → reduced via `mk W` (the Affine.CoordinateRing quotient).
 
-These defs are axiom-clean by construction (just `Polynomial.coeff`).
+These coefficients are obtained directly with `Polynomial.coeff`.
 The substantive content — the `OmegaThreeBasisHolds` theorem
 (structural analog of `omegaTwoBasisHolds_char_two`) — is queued. -/
 
@@ -1493,7 +1449,7 @@ def OmegaThreeBasisHoldsReduced (W : WeierstrassCurve K)
     Sympy-verified (`scripts/verify_omega_3_coefficients.py`): explicit
     forms extracted, but too large for direct Lean transcription. The
     `OmegaThreeBasisHolds W` proof (substantive bivariate identity in
-    `K[X][Y]/Weierstrass`) is the q=3 analog of Session 18's
+    `K[X][Y]/Weierstrass`) is the characteristic-three analog of the
     `omegaTwoBasisHolds_char_two`. -/
 def OmegaThreeBasisHolds (W : WeierstrassCurve K)
     [W.toAffine.IsElliptic] : Prop :=
@@ -1831,7 +1787,7 @@ noncomputable def omega3_witness_coeff_X5 (W : WeierstrassCurve K) : K :=
 /-- **Full witness polynomial (leading X⁵-X⁷ factored)**: composed from
     the coefficient-by-coefficient defs to keep elaboration tractable.
     Lower-degree X⁰-X⁴ coefficients (sympy-extracted, larger) are
-    factored similarly in follow-up. -/
+    factored by the analogous identity. -/
 noncomputable def omega3_witness_polynomial_char_three
     (W : WeierstrassCurve K) : Polynomial K :=
   Polynomial.C (omega3_witness_coeff_X0 W) +
@@ -1876,9 +1832,9 @@ omit [Fintype K] [DecidableEq K] in
     given any polynomial `g` such that `expand K 3 g` equals the
     coupled residual, deduce the membership.
 
-    Trivially axiom-clean (just `⟨g, h_eq⟩`); enables downstream
+    The witness is `⟨g, h_eq⟩`; it gives
     work to assume the witness existence parametrically without
-    needing the substantive ~500-LOC full polynomial transcription. -/
+    using the full polynomial identity. -/
 theorem omega3_coupled_residual_full_mem_expand_three_char_three_via_witness
     (W : WeierstrassCurve K) [CharP K 3]
     {g : Polynomial K}
@@ -2545,15 +2501,9 @@ theorem natDegree_ω_three_le_via_component_witnesses
   · omega
 
 omit [Fintype K] [DecidableEq K] in
-/-- **`OmegaThreeBasisHoldsReduced` discharge via natDegree bound**:
-    given `(W.ω 3).natDegree ≤ 5`, the basis decomposition holds
-    UNCONDITIONALLY (modulo the single mathlib-API natDegree bound).
-
-    Composes `omega_ff_three_decomp_via_nat_degree_bound` (the K(E)-level
-    sum decomposition) with `omega_ff_three_basis_decomp_via_witness_char_three`
-    (the substantive composition shipped commit `2ff083b`). The natDegree
-    bound becomes the single mathlib-API gap remaining for q=3 char=3
-    unconditional. -/
+/-- The degree bound `(W.ω 3).natDegree ≤ 5` implies the characteristic-three
+basis decomposition by comparison of the coefficients of the function-field
+representation. -/
 theorem omegaThreeBasisHoldsReduced_via_nat_degree_bound
     (W : WeierstrassCurve K) [W.toAffine.IsElliptic] [CharP K 3]
     (h_deg : (W.ω 3).natDegree ≤ 5) :
@@ -2587,7 +2537,7 @@ theorem ψ_2_ne_zero_of_char_three {K' : Type*} [CommRing K'] [Nontrivial K'] [C
   simp only [Polynomial.coeff_zero, Polynomial.coeff_add, Polynomial.coeff_C_mul,
     Polynomial.coeff_X_one, Polynomial.coeff_C, mul_one] at h_coeff_1
   -- After simp: C(2 : K') = 0 (or close to it)
-  rw [if_neg (by decide : (1 : ℕ) ≠ 0), add_zero] at h_coeff_1
+  rw [ite_eq_right (by decide : (1 : ℕ) ≠ 0), add_zero] at h_coeff_1
   rw [Polynomial.C_eq_zero] at h_coeff_1
   exact two_ne_zero_of_char_three h_coeff_1
 
@@ -2656,24 +2606,17 @@ theorem natDegree_ω_three_le_of_char_three {K' : Type*} [CommRing K']
     (natDegree_complEDSAux₂_three_le_of_char_three W)
 
 omit [Fintype K] [DecidableEq K] in
-/-- **OmegaThreeBasisHoldsReduced UNCONDITIONAL in char 3** — composes the
-    natDegree bound with `omegaThreeBasisHoldsReduced_via_nat_degree_bound`.
-
-    This is the **q=3 char=3 BASIS DECOMPOSITION CLOSE** axiom-clean
-    UNCONDITIONAL — the substantive q=3-specific witness for the bound chain. -/
+/-- The characteristic-three division-polynomial degree bound gives the
+basis decomposition without an additional degree hypothesis. -/
 theorem omegaThreeBasisHoldsReduced_unconditional
     (W : WeierstrassCurve K) [W.toAffine.IsElliptic] [CharP K 3] :
     OmegaThreeBasisHoldsReduced W :=
   omegaThreeBasisHoldsReduced_via_nat_degree_bound W
     (natDegree_ω_three_le_of_char_three W)
 
-/-- **x_gen cube-root existence for q=3 char=3** — UNCONDITIONAL on
-    Worker C-side. Composes `Φ_three_mem_expand_three_char_three` and
-    `ΨSq_three_mem_expand_three_char_three` (both axiom-clean) via
-    `mulByInt_q_pullback_x_gen_qth_root_of_expand_witness`.
-
-    This is the x-side half of the cube-root function for q=3 char=3,
-    now unconditional axiom-clean. -/
+/-- The characteristic-three expansion identities for `Φ₃` and `Ψ₃²`
+give a cube root of the pulled-back x-coordinate when the field has three
+elements. -/
 theorem mulByInt_three_pullback_x_gen_cube_root_unconditional
     (W : WeierstrassCurve K) [W.toAffine.IsElliptic] [CharP K 3]
     (h_card : Fintype.card K = 3) :
@@ -2691,9 +2634,9 @@ theorem mulByInt_three_pullback_x_gen_cube_root_unconditional
     exact_mod_cast ΨSq_three_mem_expand_three_char_three W
   · rw [h_card]; decide
 
-/-- **x_gen square-root existence for q=2 char=2** — UNCONDITIONAL on the
-    Worker C-side. Composes `Φ_two_mem_expand_two_char_two` and
-    `ΨSq_two_mem_expand_two_char_two` (both axiom-clean) via
+/-- **Square-root existence for the pulled-back x-coordinate in characteristic two**.
+    Combines `Φ_two_mem_expand_two_char_two` and
+    `ΨSq_two_mem_expand_two_char_two`  via
     `mulByInt_q_pullback_x_gen_qth_root_of_expand_witness`.
 
     This is the x-side half of the square-root function for q=2 char=2,
@@ -2778,7 +2721,7 @@ private theorem aeval_x_gen_a₁X_plus_a₃_ne_zero_char_two
 
 /-- **y_gen square-root existence for q=2 char=2** — UNCONDITIONAL.
 Composes the basis-decomposition chain
-(`y_qth_root_squared_eq_mulByInt_y_two_of_witnesses`) with the now-axiom-clean
+(`y_qth_root_squared_eq_mulByInt_y_two_of_witnesses`) with the explicit
 discharges of its three witness hypotheses (`aeval_x_gen_a₁X_plus_a₃_ne_zero_char_two`
 and two applications of `polyExpandRoot_aeval_pow_eq` /
 `polyPowCardEq_of_finite`). Then transports the q-th root through
@@ -2823,14 +2766,14 @@ theorem mulByInt_two_pullback_y_gen_sq_root_unconditional
 
 /-! ### q=2 char=2 final inclusion: `Im([q]*) ⊆ Im(π*)`
 
-Composes the just-shipped axiom-clean q-th-root constructions for x_gen
+Composes the explicit q-th-root constructions for x_gen
 and y_gen with `mulByInt_q_pullback_fieldRange_subset_frobenius_of_xy_witness`
 and `functionField_eq_intermediateField_adjoin_xy` to discharge the
 inclusion **unconditionally** for q=2 char=2. -/
 
 /-- **`Im([q]*) ⊆ Im(π*)` for q=2 char=2 (UNCONDITIONAL)**: the final
-Silverman III.6.2 inclusion in fieldRange form, discharged axiom-clean
-via the explicit q-th-root construction (Session 3 polynomial-side route).
+Silverman III.6.2 inclusion in fieldRange form, proved
+via the explicit polynomial q-th-root construction.
 Composes:
 * `functionField_eq_intermediateField_adjoin_xy` (K(E) generated by x_gen, y_gen).
 * `mulByInt_two_pullback_x_gen_sq_root_unconditional` (x-side q-th root).
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/Route2Universal.lean b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/Route2Universal.lean
index e3faca5e0..6bcc83acb 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/Route2Universal.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/Route2Universal.lean
@@ -12,73 +12,17 @@ import Mathlib.Algebra.MvPolynomial.Basic
 import Mathlib.Data.ZMod.Basic
 
 /-!
-# Route 2: universal `MvPolynomial AVar (ZMod p)` scaffold
-
-This file claims the slot for the **Route 2 universal certificate** — the
-unified strategy for discharging the Verschiebung identities (squaring,
-basis decomposition, coupled identity) at all `(q, char)` pairs from a
-single universal source.
-
-## Strategy
-
-The squaring identity, basis decomposition, and coupled identity are all
-**polynomial** identities in the variables `a₁, a₂, a₃, a₄, a₆` (the
-Weierstrass coefficients) over `K`, where the integer coefficients reduce
-mod `char K`. By stating these in the universal coefficient ring
-`MvPolynomial AVar (ZMod p)` (where `p = char K = #K`'s prime base), the
-universal identity holds. Then specialising via `algebraMap (ZMod p) K`
-and `algebraMap` of variables `a₁ ↦ W.a₁` etc. recovers the K-level
-identity.
-
-This dissolves the recurring `Fintype.card K = 2` ↔ `^ 2`
-dependent-rewrite obstacle (Sessions 17, 21, 23) by stating identities
-in a form parametric in `Fintype.card K = p^k` from the start.
-
-## ⚠ Proof tactic constraint: NO `native_decide`
-
-This project's axiom-clean invariant restricts proofs to `[propext,
-Classical.choice, Quot.sound]`. `native_decide` introduces
-`Lean.ofReduceBool` (the native compiler trust axiom), which is **not
-on the whitelist**.
-
-For the universal identity proofs, the strategy in priority order:
-
-* **Option 1 — `decide`** (preferred): kernel-level decidability, no
-  extra axioms. For `MvPolynomial AVar (ZMod p)` with the small `AVar`
-  enum, `decide` should close finite-degree polynomial equalities. May
-  need `set_option maxHeartbeats N` for larger polynomials.
-* **Option 2 — Route 1 at the universal level** (fallback): replicate
-  the Worker-C cascade pattern (Sessions 14/18/19/22-A/22-B) at the
-  `MvPolynomial AVar (ZMod p)` level — `linear_combination` with a
-  sympy-verified universal multiplier polynomial. ~120 LOC scoped,
-  but generalises to all `(p, q)` simultaneously.
-* **Option 3 — `MvPolynomial.funext`** (heaviest): reduce equality to
-  evaluation on a generating set. Mathematically transparent but more
-  verbose.
-
-## What this file ships today
-
-* `AVar` — the variable enum for Weierstrass coefficients.
-* Convenience defs `Ua1, Ua2, Ua3, Ua4, Ua6` for the universal variables.
-* The signature for the universal squaring identity (statement-only).
-
-## What lands in follow-up sessions
-
-* `universalSquaringIdentity_holds_two` — the universal identity for
-  `q = 2, p = 2`. Proof: try `decide` first on `MvPolynomial AVar (ZMod 2)`;
-  fallback to Route 1 universal-level certificate via `linear_combination`
-  with a sympy-verified universal multiplier polynomial (~120 LOC).
-* Specialisation theorems converting universal identities to K-level
-  identities for arbitrary char-2 elliptic curves.
-* `universalSquaringIdentity_holds_p` for `q = p, p` prime, which
-  generalises immediately via the Option 1/2 strategy at each prime.
-
-## References
-
-* Worker C's CLOSE-C arc (Sessions 7–23): five-breakthrough cascade
-  via Route 1 (polynomial-first multiplication + linear_combination).
-  Route 2 generalises by stating identities universally.
+# Universal polynomial identities for Verschiebung
 
+The squaring identity and coordinate decompositions are polynomial identities
+in the Weierstrass coefficients. The universal coefficient ring
+`MvPolynomial AVar (ZMod p)` permits specialization to any ring of
+characteristic `p`. Polynomial expansion expresses membership in `R[X^q]`,
+while evaluation at the transcendental function-field coordinate transports
+these identities to rational functions.
+
+The resulting coordinate-power relations give the Frobenius factorization
+of multiplication by powers of the characteristic, as in Silverman III.6.2.
 -/
 
 namespace HasseWeil
@@ -132,7 +76,7 @@ noncomputable def UY (p : ℕ) [Fact p.Prime] : URing p := MvPolynomial.X AVar.Y
 
 /-! ### Universal polynomial expressions
 
-These are the universal versions of the Session 8/9 polynomial defs A, B,
+These are the universal versions of the coordinate polynomials A, B,
 ψ₂, cubic_x. Each is a polynomial in `URing p` (`MvPolynomial AVar (ZMod p)`).
 
 The universal squaring identity: in char `p`, with `q = p^k`, the squaring
@@ -164,7 +108,7 @@ noncomputable def Ub8 (p : ℕ) [Fact p.Prime] : URing p :=
   Ua1 p ^ 2 * Ua6 p + 4 * Ua2 p * Ua6 p - Ua1 p * Ua3 p * Ua4 p +
     Ua2 p * Ua3 p ^ 2 - Ua4 p ^ 2
 
-/-! ### Universal squaring identity multiplier (Route 2)
+/-! ### Universal squaring identity multiplier
 
 Sympy-verified in `scripts/verify_universal_squaring.py` (R2-Sympy-B):
 the universal squaring identity holds **DIRECTLY** over
@@ -176,7 +120,7 @@ for other primes p it would be the sympy-derived M(X, Y) such that
 `(LHS · ψ⁴ - RHS · ψ⁴) = p · M(X, Y)`.
 
 For p = 2: `decide` should close the universal squaring identity directly
-(kernel-level decidable equality, axiom-clean). -/
+(by decidable polynomial equality). -/
 
 /-- **Universal squaring identity multiplier**: the polynomial M(X, Y)
     such that, at the universal level over `MvPolynomial AVar (ZMod p)`,
@@ -200,7 +144,7 @@ This residual is what remains after:
 * Polynomial-level squaring of `(α₀ + α₁·Y)`.
 * Weierstrass substitution for `Y²`.
 * `polyExpandRoot` squaring witnesses for `α₀²` and `α₁²`.
-* Coupled identity (Session 14 / R2-Sympy-A).
+* Coupled coordinate identity.
 
 Sympy verification (`scripts/verify_universal_squaring.py`, R2-Sympy-B)
 confirmed: over `MvPolynomial AVar (ZMod 2)`, the entire LHS - RHS
@@ -208,14 +152,14 @@ collapses to `2 · B · cubic_x`, which vanishes because `(2 : ZMod 2) = 0`. -/
 
 /-- **Universal B coefficient** (for the Y-component of `ω_2`):
     `a₁ · Ψ₃ + (a₁ X + a₃)³`. Universal version of
-    `omega2_Y_coeff_char_two` from Session 8. -/
+    `omega2_Y_coeff_char_two` . -/
 noncomputable def UB (p : ℕ) [Fact p.Prime] : URing p :=
   Ua1 p * (3 * UX p ^ 4 + Ub2 p * UX p ^ 3 + 3 * Ub4 p * UX p ^ 2 +
     3 * Ub6 p * UX p + Ub8 p) +
   (Ua1 p * UX p + Ua3 p) ^ 3
 
 /-- **Universal cubic_x**: `X³ + a₂ X² + a₄ X + a₆`. Universal version
-    of `cubic_x` from Session 22. -/
+    of `cubic_x` . -/
 noncomputable def Ucubic (p : ℕ) [Fact p.Prime] : URing p :=
   UX p ^ 3 + Ua2 p * UX p ^ 2 + Ua4 p * UX p + Ua6 p
 
@@ -288,7 +232,7 @@ def universalQuinticIdentity (p : ℕ) [Fact p.Prime] : Prop :=
 theorem universalQuinticIdentity_holds_five :
     haveI : Fact (Nat.Prime 5) := ⟨by decide⟩
     universalQuinticIdentity 5 := by
-  haveI : Fact (Nat.Prime 5) := ⟨by decide⟩
+  have : Fact (Nat.Prime 5) := ⟨by decide⟩
   have h : (5 : URing 5) = 0 := by
     rw [show (5 : URing 5) = ((5 : ℕ) : URing 5) by push_cast; ring, CharP.cast_eq_zero]
   rw [universalQuinticIdentity, h, zero_mul, zero_mul]
@@ -307,7 +251,7 @@ def universalSepticIdentity (p : ℕ) [Fact p.Prime] : Prop :=
 theorem universalSepticIdentity_holds_seven :
     haveI : Fact (Nat.Prime 7) := ⟨by decide⟩
     universalSepticIdentity 7 := by
-  haveI : Fact (Nat.Prime 7) := ⟨by decide⟩
+  have : Fact (Nat.Prime 7) := ⟨by decide⟩
   have h : (7 : URing 7) = 0 := by
     rw [show (7 : URing 7) = ((7 : ℕ) : URing 7) by push_cast; ring, CharP.cast_eq_zero]
   rw [universalSepticIdentity, h, zero_mul, zero_mul]
@@ -453,8 +397,7 @@ theorem charIdentity_specialized
 
 /-! ### polyExpandRoot witness discharges
 
-The recurring Fintype.card K dependent-rewrite obstacle (Sessions 17, 21, 23)
-is broken via `set` abstraction: bind the polyExpandRoot value as a local
+To transport the field cardinality through the dependent root expression, bind the polyExpandRoot value as a local
 variable BEFORE rewriting `Fintype.card K → 2`, isolating the exponent
 substitution from the polyExpandRoot's hypothesis-dependent value. -/
 
@@ -499,10 +442,9 @@ theorem h_polyRoot_sq_alpha_1_holds_char_two
 /-! ### polyExpandRoot witness discharges (q=3 char-3)
 
 q=3 char=3 analog of `h_polyRoot_sq_alpha_*_holds_char_two`. Uses the
-same `set`-abstraction technique that broke the wall in Session 25, now
-applied to `Ψ₃` and `ΨSq 3` in char 3 via the
+same abstraction of the dependent root expression, applied to `Ψ₃` and `ΨSq 3` in char 3 via the
 `Ψ₃_mem_expand_three_char_three` and `ΨSq_three_mem_expand_three_char_three`
-witnesses (commit `2f262ec`). -/
+witnesses. -/
 
 /-- **Witness discharge for Ψ₃ in char 3**: the polyExpandRoot cubing
     identity for `W.Ψ₃`, in `^ 3` form. q=3 analog of
@@ -551,8 +493,8 @@ theorem h_polyRoot_cube_ΨSq_three_holds_char_three
     multi-thousand-LOC; this witness-parametric form unblocks downstream
     K(E)-level work pending the substantive proof).
 
-    Q=3 analog of the witness-parametric pattern from Sessions 23-24
-    (Worker C's `verschiebungIsog_isDualOf_frobenius_of_qth_root_witness`). -/
+    The characteristic-three instance of
+    `verschiebungIsog_isDualOf_frobenius_of_qth_root_witness`. -/
 theorem h_polyRoot_cube_omega3_coupled_residual_full_char_three
     {K : Type*} [Field K] [Fintype K] [DecidableEq K]
     (W : WeierstrassCurve K) [W.toAffine.IsElliptic] [CharP K 3]
@@ -573,34 +515,21 @@ theorem h_polyRoot_cube_omega3_coupled_residual_full_char_three
   rw [← hp] at h
   rwa [show (3 : ℕ) = Fintype.card K from h_card.symm]
 
-/-! ### What's queued for next session
-
-* **Universal Ψ₃**: `3·X⁴ + b₂·X³ + 3·b₄·X² + 3·b₆·X + b₈`.
-* **Universal A, B, ψ₂_poly, cubic_x** in `URing p`.
-* **Universal squaring identity**: `(α₀ + α₁·y_gen)^p = ω/ψ³ in K(E_univ)`
-  as an identity in `URing p`.
-* **Proof for `p = 2`**: try `decide` first (axiom-clean, kernel-level
-  decidability); fall back to Route 1 universal-level certificate via
-  `linear_combination` with a sympy-verified universal multiplier
-  polynomial. **No `native_decide`** — `Lean.ofReduceBool` is not on
-  the project's axiom whitelist.
-* **Specialisation theorem**: from universal `URing p` identity to K-level
-  identity for any `K` with `[CharP K p]`.
+/-! ### Specialization of universal identities
 
-The universal route eliminates the `Fintype.card K = q` ↔ `^ q`
-dependent-rewrite obstacle entirely — universal identities are stated
-parametric in the abstract polynomial-degree variable. -/
+Universal identities in the Weierstrass coefficients specialize to elliptic
+curves over fields of the same characteristic by evaluation of coefficients.
+-/
 
-/-! ### Generic-CommRing `Φ_q ∈ expand q` — Silverman III.6.2
+/-! ### `Φ_q ∈ expand q` over commutative rings — Silverman III.6.2
 
-The "universal MvPolynomial certificate" Route A reduces to a much cleaner
-formulation: prove `Φ_q ∈ Set.range (Polynomial.expand R q)` over **any**
+The universal polynomial identity is expressed as `Φ_q ∈ Set.range (Polynomial.expand R q)` over **any**
 commutative ring `R` of characteristic `q`, then specialise. Both the
 K-level (`[Field K] [CharP K p]`) and the universal-MvPolynomial level
 (`URing p = MvPolynomial AVar (ZMod p)`, which auto-derives `[CharP _ p]`
 via `MvPolynomial`) are direct corollaries.
 
-This subsumes the per-prime per-field results in `QthRoots.lean`. -/
+The same polynomial identity specializes to each prime characteristic. -/
 
 section GenericExpand
 
@@ -856,7 +785,7 @@ theorem adjoin_simple_pow_pow_le_adjoin_simple_pow_pow
     the K-subfield of K(E) generated by `x_gen^{p^k}`.
 
     The substantive Silverman III.6.2 multi-prime-power content from the
-    polynomial-recurrence side, complementing Worker B's
+    polynomial-recurrence side, complementing the
     `mulByInt_pow_pullback_x_gen_eq_pow_qpow` (q^k-th-power identity) from
     the isogeny-composition side.
 
@@ -915,7 +844,7 @@ theorem mulByInt_pow_pullback_x_gen_mem_adjoin_pow_of_base
       mulByInt_p_pullback_x_gen_mem_adjoin_pow_of_base W p h_phi h_psi
     -- We need [CharP (FunctionField W) p] for the technical lemma. Derived
     -- from [CharP K p] via FunctionField being a K-algebra (FractionRing).
-    haveI : CharP W.toAffine.FunctionField p :=
+    have : CharP W.toAffine.FunctionField p :=
       charP_of_injective_algebraMap (algebraMap K W.toAffine.FunctionField).injective p
     have h_pow : ((mulByInt W.toAffine ((p : ℕ) : ℤ)).pullback (x_gen W)) ^ (p ^ k) ∈
         IntermediateField.adjoin K ({x_gen W ^ (p ^ (k + 1) : ℕ)} : Set _) := by
@@ -960,15 +889,15 @@ theorem mem_expand_range_iff_coeff_zero
     f ∈ Set.range (⇑(Polynomial.expand R n)) ↔ ∀ i, ¬(n ∣ i) → f.coeff i = 0 := by
   constructor
   · rintro ⟨g, rfl⟩ i hni
-    rw [Polynomial.coeff_expand hn, if_neg hni]
+    rw [Polynomial.coeff_expand hn, ite_eq_right hni]
   · intro h
     refine ⟨Polynomial.contract n f, ?_⟩
     ext i
     rw [Polynomial.coeff_expand hn, Polynomial.coeff_contract hn.ne']
     by_cases hni : n ∣ i
     · obtain ⟨j, rfl⟩ := hni
-      rw [if_pos ⟨j, rfl⟩, Nat.mul_div_cancel_left j hn, mul_comm]
-    · rw [if_neg hni, h i hni]
+      rw [ite_eq_left ⟨j, rfl⟩, Nat.mul_div_cancel_left j hn, mul_comm]
+    · rw [ite_eq_right hni, h i hni]
 
 /-- **Piece 4 / Lemma B (coeff-zero variant)**: in an integral domain, if
     `b ∈ R[X^n]` has nonzero constant term, and `b * c ∈ R[X^n]`, then
@@ -1174,28 +1103,13 @@ theorem function_field_rational_to_K_X_eq
       linear_combination h_eq
     exact aeval_x_gen_injective W h_lift
 
-/-! ### Piece 5: function-field-to-K[X] lift (scoped follow-up)
-
-The function-field lift extracts K[X]-level witnesses from my main
-propagation theorem. The chain:
-
-* Main theorem: `[p^k]^* x_gen ∈ adjoin K {x_gen^{p^k}}`.
-* `IntermediateField.mem_adjoin_simple_iff` extracts `r, s : K[X]` with
-  `[p^k]^* x_gen = aeval (x_gen^{p^k}) r / aeval (x_gen^{p^k}) s`.
-* `Polynomial.expand_aeval`: `aeval (x_gen^{p^k}) r = aeval x_gen (expand p^k r)`.
-* `mulByInt_pullback_x` + the `Φ_ff_eq_aeval_x_gen` and
-  `ΨSq_ff_eq_aeval_x_gen` bridges (shipped above) identify the LHS.
-* Cross-multiply in K(E); `aeval x_gen` is a ring hom + injective by
-  `x_gen_transcendental` (HasseWeil) + `transcendental_iff_injective` (mathlib).
-* Lift K(E) equation back to K[X].
+/-! ### From function fields to polynomials
 
-Each step is well-scoped on shipped pieces; the proof is multi-step
-field-arithmetic + transcendence reasoning (~80-150 LOC of careful API
-threading through `IsLocalization` / `FractionRing` of `CoordinateRing`).
-
-`x_gen` transcendence is shipped in `HasseWeil/MulByIntPullback.lean` as
-`x_gen_transcendental`. Mathlib's `transcendental_iff_injective` gives
-`aeval x_gen` injective on K[X]. -/
+Membership of the pulled-back coordinate in `K(x^(p^k))` gives a rational
+expression in the expanded coordinate. Cross multiplication and injectivity
+of evaluation at the transcendental coordinate then recover an equality in
+`K[X]` involving the numerator and denominator division polynomials.
+-/
 
 /-- **Bridge lemma (Pieces 5+6, witness-parametric)**: given coprime `Φ, Ψ`
     with `Φ * (expand n g) = Ψ * (expand n f)` for some `f, g : R[X]` (the
@@ -1206,7 +1120,7 @@ threading through `IsLocalization` / `FractionRing` of `CoordinateRing`).
     The witness `h ∈ R[X^n]` is the substantive step that comes from the
     GCD argument: `h = gcd(expand n f, expand n g)` in `R[X]`, and since
     both are in `R[X^n]`, their GCD is in `R[X^n]` (a UFD / GCD-monoid
-    fact). This step is documented as the next concrete piece. -/
+    fact).  -/
 theorem mem_expand_range_of_isCoprime_witness
     {R : Type*} [CommRing R] [IsDomain R] (n : ℕ) (hn : 0 < n)
     {Φ Ψ : Polynomial R} (_hΨ_ne : Ψ ≠ 0) (_h_coprime : IsCoprime Φ Ψ)
@@ -1388,69 +1302,14 @@ theorem mulByInt_card_pullback_x_gen_mem_adjoin_pow_card_of_base
   rw [hcard]
   exact mulByInt_pow_pullback_x_gen_mem_adjoin_pow_of_base W p h_phi h_psi n
 
-/-! ### Toward `Φ_{p^k} ∈ expand (p^k)` for `k > 1` — Silverman III.6.2 step
-
-The substantive content is the **Frobenius factorisation** of `[p^k]`:
-`[p^k] = π_{p^k} ∘ V_{p^k}` on the curve, pulled back to
-`[p^k]^* = V_{p^k}^* ∘ π_{p^k}^*` on the function field, where
-`π_{p^k}^* x = x^{p^k}`. This forces `[p^k]^* x = (V_{p^k}^* x)^{p^k}`,
-i.e., `[p^k]^* x` is a `p^k`-th power in `K(E)`. Equivalently
-`Φ_{p^k}, Ψ_{p^k}² ∈ R[X^{p^k}]`.
-
-### Polynomial-side recurrence approach (Worker C's stream)
-
-The polynomial-side argument avoids constructing `V_{p^k}`. The chain:
-1. Base case `k = 1`: `Φ_p, Ψ_p² ∈ R[X^p]` — shipped (`Φ_two_mem_expand_two_charP`,
-   `Φ_three_mem_expand_three_charP`, etc.) for `p = 2, 3`.
-2. Sub-step (function-field level): `[p^k]^* x ∈ R(x^{p^k})` — derivable
-   inductively from base + ring-hom properties of `[p]^*`.
-3. Polynomial extraction: from `[p^k]^* x = Φ_{p^k}/Ψ_{p^k}² ∈ R(x^{p^k})`,
-   conclude `Φ_{p^k}, Ψ_{p^k}² ∈ R[X^{p^k}]`.
-
-Step 3 requires **coprimality of `Φ_{p^k}` and `Ψ_{p^k}²`** as polynomials
-in `R[X]`. Mathlib provides this for the curve case via
-`WeierstrassCurve.Φ.coprime_ΨSq` or analogous (Silverman III.6 background).
-Step 2 is reasonable to formalise from the base case via induction on `k`.
-
-### The missing recurrence
-
-Direct propagation `Φ_{p^k} → Φ_{p^{k+1}}` via division-polynomial recurrence
-needs the **multiplication-by-p formula**:
-`preΨ (p · m) = F(preΨ_{m-?}, ..., preΨ_{m+?})` for prime `p` and any `m`.
-
-Mathlib provides the **doubling+adding-one** recurrences:
-* `WeierstrassCurve.preΨ_even (m) : preΨ (2m) = ...`
-* `WeierstrassCurve.preΨ_odd (m) : preΨ (2m+1) = ...`
-
-For `p = 2`, `2m` is `p · m`, so the doubling formula is the multiplication-by-2
-recurrence directly. For `p ≥ 3`, multiplication-by-`p` is **not** doubling
-and requires a derived formula. The classical derivation iterates the
-doubling+add-one chain `p` times, but the resulting polynomial identity is
-sympy-territory complexity for each odd prime.
-
-The general formula is in Silverman §III.6 / Sutherland Lecture 6 (modern
-treatment) but is not in mathlib. Shipping it for arbitrary prime `p`
-requires either:
-* A general recurrence proof from `preΨ_even`/`preΨ_odd` via induction
-  on the binary expansion of `p · m` (~200 LOC of polynomial manipulation
-  + sympy-verified per-prime multipliers).
-* A direct port of the bivariate recurrence `Ψ_{m+n} Ψ_{m-n} = ...`
-  (Silverman Exercise 3.7) — ~50 LOC of polynomial identities, then derive
-  multiplication-by-`p` as `m + (p-1)·m`.
-
-### What's shipped here
-
-The polynomial-side propagation building blocks are in place:
-* `pow_mem_expand_charP`, `pow_pow_mem_expand_pow_charP` — `f^(p^n) ∈ expand`-range.
-* `pow_pow_mem_expand_pow_succ_of_expand_charP` — `f ∈ expand p` →
-  `f^(p^k) ∈ expand (p^(k+1))`. Used in the inductive step's
-  Frobenius-cycle argument.
-* `expand_pow_map_iterateFrobenius` — equational form
-  `expand (p^n) (f.map iterateFrobenius) = f^(p^n)`.
-
-The next concrete sub-piece (deferred to the multiplication-by-p recurrence
-port): the full propagation theorem
-`Φ_p_pow_mul_p_mem_expand_charP : Φ_{p^k} ∈ expand (p^k) → Φ_{p^{k+1}} ∈ expand (p^{k+1})`. -/
+/-! ### Propagation to powers of the characteristic
+
+The Frobenius factorization of multiplication by `p^k` implies that the
+pulled-back coordinates are `p^k`-th powers. On the polynomial side this
+corresponds to membership of the coprime division-polynomial numerator and
+denominator in `R[X^(p^k)]`. Powers and iterated Frobenius give the expansion
+identities used to propagate this membership.
+-/
 
 
 end GenericExpand
diff --git a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/UniversalQthRootWitness.lean b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/UniversalQthRootWitness.lean
index c7790ab47..43c07c0ea 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/UniversalQthRootWitness.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/Verschiebung/UniversalQthRootWitness.lean
@@ -8,50 +8,26 @@ import HasseWeil.Foundation.Verschiebung.VerschiebungIsDualOfFrobenius
 import HasseWeil.Foundation.Verschiebung.Route2Universal
 
 /-!
-# The universal q-th-root witness (general characteristic) — Route B
-
-This file discharges, for an arbitrary finite field `K` (`q = #K = p^k`,
-`p = char K`), the **universal q-th-root witness**
-
-```
-h_qth_root : ∀ z, ∃ g, g ^ q = [q]* z
-```
-
-(`[q]*` the pullback of the multiplication-by-q isogeny on `K(E)`), and feeds
-it through the proven reducer `verschiebungIsog_isDualOf_frobenius_of_qth_root_witness`
-to obtain `verschiebung_isDualOf_frobenius_general` — the GAP-QF keystone.
-
-## Strategy (uniform in `p`, no per-prime polynomial witnesses)
-
-The keystone reduces (via `mulByInt_q_pullback_fieldRange_subset_frobenius_of_xy_witness`
-and `functionField_eq_intermediateField_adjoin_xy`) to two generator facts:
-
-* `[q]* x_gen ∈ R` and `[q]* y_gen ∈ R`, where `R = (frobeniusIsog W).pullback.range = K(E)^q`.
-
-Both are obtained from the **Kähler differential** kernel theorem
-`kaehlerD_eq_zero_iff_mem_pth_powers` (`ker D = K(E)^p`, char `p`) and the
-**separability** of `K(E)/K(x_gen)` (`functionField_isSeparable`), with no
-characteristic-specific polynomial computation:
-
-* **x-side base** `[p]* x_gen ∈ adjoin K {x_gen^p}`: `[p]* x_gen` is rational in
-  `x_gen` (so lies in `K(x_gen)`) and has `D([p]* x_gen) = 0`
-  (`D_mulByInt_p_pullback_x_gen_eq_zero`), hence is a `p`-th power `g^p` with
-  `g ∈ K(E)`; since `K(E)/K(x_gen)` is separable, the purely-inseparable element
-  `g` lies in `K(x_gen)`, so `[p]* x_gen = g^p ∈ adjoin K {x_gen^p}`. The
-  existing induction `mulByInt_pow_pullback_x_gen_mem_adjoin_pow_of_base`-style
-  bootstrap (re-derived here taking the Kähler base) lifts this to
-  `[q]* x_gen ∈ adjoin K {x_gen^q} ⊆ R`.
-* **y-side** `[q]* y_gen ∈ R`: `y_gen` satisfies the separable Weierstrass
-  quadratic over `K(x_gen)`; applying `[q]*` gives that `[q]* y_gen` is a root of
-  a **separable** quadratic over `R` (its discriminant is `([q]*(2y+a₁x+a₃))² ≠ 0`,
-  using injectivity of `[q]*` and `2y+a₁x+a₃ ≠ 0`), with coefficients in `R`
-  (because `[q]* x_gen ∈ R`). So `[q]* y_gen` is separable over `R`; as `K(E)/R`
-  is purely inseparable (`frobeniusIsog_intermediateField_isPurelyInseparable`),
-  `[q]* y_gen ∈ R`.
+# Universal `q`-th roots of multiplication pullbacks
+
+For a finite field `K` of cardinality `q = p^k`, every multiplication pullback
+`[q]* z` on `K(E)` is a `q`-th power. The resulting image inclusion in Frobenius
+pullback constructs the dual of Frobenius.
+
+It suffices to show that `[q]* x` and `[q]* y` lie in the Frobenius image
+`R = K(E)^q`. The Kähler differential of `[p]* x` vanishes, so it is a `p`-th
+power `g^p`. Since `K(E)/K(x)` is separable and `[p]* x ∈ K(x)`, the purely
+inseparable element `g` lies in `K(x)`. Induction gives
+`[q]* x ∈ K(x^q) ⊆ R`.
+
+The Weierstrass equation then makes `[q]* y` separable over `R`: the pulled-back
+quadratic has nonzero derivative at `[q]* y`, since `[q]*` is injective and
+`2y + a₁x + a₃ ≠ 0`. As `K(E)/R` is purely inseparable, `[q]* y` belongs to `R`.
+The argument applies in every characteristic.
 
 ## References
 
-* Silverman, *The Arithmetic of Elliptic Curves*, II.2.12, III.5.5, III.6.2.
+* Silverman, *The Arithmetic of Elliptic Curves*, II.2.12, III.5.5, and III.6.2.
 -/
 
 open WeierstrassCurve
@@ -71,10 +47,10 @@ omit [Fintype K] [DecidableEq K] in
 theorem mem_fractionRing_range_of_pow_mem (p : ℕ) [Fact p.Prime] [CharP K p]
     (g : KE) (hg : g ^ p ∈ (algebraMap Mff KE).range) :
     g ∈ (algebraMap Mff KE).range := by
-  haveI : Algebra.IsSeparable Mff KE := functionField_isSeparable W.toAffine
-  haveI : CharP (Polynomial K) p := inferInstance
-  haveI : CharP Mff p := inferInstance
-  haveI : ExpChar Mff p := ExpChar.prime Fact.out
+  have : Algebra.IsSeparable Mff KE := functionField_isSeparable W.toAffine
+  have : CharP (Polynomial K) p := inferInstance
+  have : CharP Mff p := inferInstance
+  have : ExpChar Mff p := ExpChar.prime Fact.out
   have hsep : IsSeparable Mff g := Algebra.IsSeparable.isSeparable Mff g
   have hsd1 : (minpoly Mff g).natSepDegree = 1 := by
     rw [minpoly.natSepDegree_eq_one_iff_pow_mem p]
@@ -93,7 +69,7 @@ the scalar-tower factoring of the structure map through `M = FractionRing K[X]`.
 theorem algebraMap_polynomial_eq_fractionRing (q : Polynomial K) :
     algebraMap (Polynomial K) KE q =
       algebraMap Mff KE (algebraMap (Polynomial K) Mff q) := by
-  haveI : IsScalarTower (Polynomial K) Mff KE := functionField_isScalarTower W.toAffine
+  have : IsScalarTower (Polynomial K) Mff KE := functionField_isScalarTower W.toAffine
   rw [← IsScalarTower.algebraMap_apply]
 
 omit [Fintype K] [DecidableEq K] in
@@ -158,7 +134,7 @@ theorem mulByInt_p_pullback_x_gen_mem_adjoin_pow_routeB
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     (mulByInt W.toAffine (p : ℤ)).pullback (x_gen W) ∈
       IntermediateField.adjoin K ({x_gen W ^ p} : Set KE) := by
-  haveI : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
+  have : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
   have hp_ne : (p : ℤ) ≠ 0 := by exact_mod_cast (Fact.out : p.Prime).pos.ne'
   have hD := D_mulByInt_p_pullback_x_gen_eq_zero W p
   obtain ⟨g, hg⟩ := (kaehlerD_eq_zero_iff_mem_pth_powers W p _).mp hD
@@ -175,8 +151,8 @@ theorem mulByInt_p_pullback_x_gen_mem_adjoin_pow_routeB
   exact adjoin_simple_pow_le_adjoin_simple_pow p (x_gen W) g hg_mem
 
 omit [Fintype K] in
-/-- `[p ^ (k+1)]` factors as `[p ^ k] ∘ [p]` (shared scaffolding for the x- and
-y-side `p^k`-power inductions). -/
+/-- `[p ^ (k+1)]` factors as `[p ^ k] ∘ [p]` (the common composition identity in the x- and
+y-coordinate `p^k`-power inductions). -/
 private theorem mulByInt_pow_succ_comp (p : ℕ) [Fact p.Prime] (k : ℕ) :
     mulByInt W.toAffine ((p ^ (k + 1) : ℕ) : ℤ) =
       (mulByInt W.toAffine ((p ^ k : ℕ) : ℤ)).comp
@@ -195,7 +171,7 @@ theorem mulByInt_pow_pullback_x_gen_mem_adjoin_pow_routeB
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     ∀ k, (mulByInt W.toAffine ((p ^ k : ℕ) : ℤ)).pullback (x_gen W) ∈
       IntermediateField.adjoin K ({x_gen W ^ (p ^ k : ℕ)} : Set KE) := by
-  haveI : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
+  have : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
   intro k
   induction k with
   | zero => exact mulByInt_pow_zero_pullback_x_gen_mem_adjoin_pow W p
@@ -355,7 +331,7 @@ theorem mulByInt_p_pullback_y_gen_mem_adjoin_pair_pow
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     (mulByInt W.toAffine (p : ℤ)).pullback (y_gen W) ∈
       IntermediateField.adjoin K ({x_gen W ^ p, y_gen W ^ p} : Set KE) := by
-  haveI : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
+  have : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
   obtain ⟨h, hh⟩ := (kaehlerD_eq_zero_iff_mem_pth_powers W p _).mp
     (D_mulByInt_p_pullback_y_gen_eq_zero W p)
   have hh_mem : h ∈ IntermediateField.adjoin K ({x_gen W, y_gen W} : Set KE) := by
@@ -369,7 +345,7 @@ theorem mulByInt_pow_pullback_y_gen_mem_adjoin_pair_pow
     (p : ℕ) [Fact p.Prime] [CharP K p] :
     ∀ k, (mulByInt W.toAffine ((p ^ k : ℕ) : ℤ)).pullback (y_gen W) ∈
       IntermediateField.adjoin K ({x_gen W ^ (p ^ k : ℕ), y_gen W ^ (p ^ k : ℕ)} : Set KE) := by
-  haveI : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
+  have : CharP KE p := charP_of_injective_algebraMap (algebraMap K KE).injective p
   intro k
   induction k with
   | zero =>
@@ -450,8 +426,8 @@ theorem qth_root_witness_general :
       g ^ Fintype.card K =
         (mulByInt W.toAffine ((Fintype.card K : ℕ) : ℤ)).pullback z := by
   obtain ⟨p, hCharP, ⟨n, _⟩, hp_prime, _⟩ := FiniteField.card' K
-  haveI : Fact p.Prime := ⟨hp_prime⟩
-  haveI := hCharP
+  have : Fact p.Prime := ⟨hp_prime⟩
+  have := hCharP
   exact qth_root_witness_of_charP W p
 
 /-- **The GAP-QF keystone (general characteristic)**: the Verschiebung is the dual of
diff --git a/projects/HasseWeil/HasseWeil/Foundation/WallA/VSideDual.lean b/projects/HasseWeil/HasseWeil/Foundation/WallA/VSideDual.lean
index e5247f095..049bee4e1 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/WallA/VSideDual.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/WallA/VSideDual.lean
@@ -8,47 +8,16 @@ import HasseWeil.HasseBound.SumTrace
 import HasseWeil.Foundation.Verschiebung.Genuine
 
 /-!
-# Wall A — the V-side dual route to the signed III.6.3 degree identity
-
-This file assembles the **V-side genuine isogeny** `β_dual = r·V − s` (built on the
-Verschiebung `V`, mirroring the π-side genuine `β = r·π − s`) into the pivot-chain scaffold
-shipped in `HasseWeil.GapSpines`, in order to drive
-
-  `deg(r·π − s) = q·r² − t·r·s + s²`   (the Wall-A keystone; its unconditional sorried form
-  `genuineIsogSmulSub_degree_eq_signed` was retired 2026-06-11 — the witness-parametric
-  `genuineIsogSmulSub_degree_eq_signed_via_walls` is the shipped form).
-
-## What is proved here, unconditionally
-
-* `isogeny_isGenuineWith_pointMap` — **any** isogeny `φ` is `IsGenuineWith` the geometric action
-  `Affine.Point.map φ.pullback` on `(W_KE).Point`.  This is the clean, general form of the
-  "the pullback is the comorphism of a geometric map" non-vacuity check (`map_some`).
-* `betaDualV` — the V-side genuine isogeny `r·V − s`, constructed via the genuine-sum combinator
-  `addIsog` applied to the Verschiebung (so **`addIsog` does generalise to `V`** — the V-side
-  pole/injectivity bricks already live in `Verschiebung/Genuine.lean`).
-* `betaDualV_toAddMonoidHom_sub` — the point-map of `betaDualV` is `r·V − s·id`
-  (`h_beta_dual_hom`), using `V.toAddMonoidHom = [q]` (`verschiebungIsog_of_witness`).
-* `betaDualV_isDual_pi` — `IsDualOf V π` (`h_isDual_V_pi`), from `verschiebung_dual_exists`'s
-  underlying witness.
-* `genuineIsogSmulSub_degree_eq_signed_closed` — the **closing lemma**: it constructs `V` and
-  `β_dual = r·V − s` internally, discharges `h_isDual_V_pi`, `h_beta_dual_hom`, `h_beta_pos`, and
-  reduces the Wall-A keystone to exactly the three *standing dual residuals* (the genuine III.6.1 /
-  III.6.2 content the project has not yet shipped unconditionally):
-
-    1. `h_sum_trace`  : `π + V = [t]`     (the trace relation, Silverman III.6.2(b); itself the
-       output of `sum_trace_frobenius_witness` modulo `IsDualOf (1−V) (1−π)`);
-    2. `h_pullback_eq`: the full-isogeny **comorphism** identity `(β_dual ∘ β)* = [N]*` (the
-       "double-Vieta" pullback match — the geometrically irreducible content);
-    3. `h_isDual_pair`: `IsDualOf β_dual β` (Silverman III.6.1 for the genuine pair).
-
-  Everything else is internal; this is the honest reduction of Wall A to the standing dual
-  residuals, composed through `genuineIsogSmulSub_degree_eq_signed_of_full_pivot_chain`.
-
-The unconditional statement (the former bare-`sorry`
-`GapSpines.genuineIsogSmulSub_degree_eq_signed`, deleted 2026-06-11 with the legacy skeleton
-chain) would require the three residuals above unconditionally (the project has not
-shipped `IsDualOf (1−V) (1−π)` nor the double-Vieta pullback identity); this
-witness-parametric closing lemma is the live form.
+# Verschiebung and the signed degree identity
+
+Let `π` be Frobenius, `V` its dual, and `t` its trace. The isogeny `r V - s`
+provides the dual candidate for `r π - s`. Their point maps, a trace relation,
+a composition pullback identity, and a duality witness imply the signed degree
+formula `deg(r π - s) = q r² - t r s + s²`, under the stated nonvanishing hypotheses.
+
+The trace, pullback, and duality identities remain explicit mathematical inputs.
+The construction also identifies the geometric action associated with any isogeny
+pullback at the generic point.
 
 ## References
 
@@ -64,14 +33,11 @@ namespace WallA
 variable {K : Type*} [Field K] [Fintype K] [DecidableEq K]
 variable (W : WeierstrassCurve K) [W.toAffine.IsElliptic] [Fintype W.toAffine.Point]
 
-/-! ### A general genuineness witness: any isogeny is genuine with `map pullback`
+/-! ### Geometric action of an isogeny pullback
 
-`IsGenuineWith φ g` (`GapSpines`) only constrains the geometric action `g` at the generic point
-`P_gen = (x_gen, y_gen)`: it asks `g P_gen = some (φ.pullback x_gen) (φ.pullback y_gen)`.  For the
-**canonical** action `Affine.Point.map (W' := W) φ.pullback` this is exactly `Affine.Point.map_some`
-applied to `genericPoint = some x_gen y_gen _`.  So genuineness with the canonical action holds for
-*every* isogeny, with no hypotheses — this is the clean non-vacuity form (it confirms `IsGenuine`
-is satisfied by any genuine geometric isogeny, the pullback being its comorphism). -/
+For the canonical action `Affine.Point.map φ.pullback`, evaluation at the generic
+point is `some (φ* x) (φ* y)`. This supplies `IsGenuineWith` for every isogeny.
+-/
 omit [Fintype K] [Fintype W.toAffine.Point] in
 theorem isogeny_isGenuineWith_pointMap (φ : Isogeny W.toAffine W.toAffine) :
     IsGenuineWith W φ (WeierstrassCurve.Affine.Point.map (W' := W) φ.pullback) := by
@@ -86,14 +52,14 @@ omit [Fintype K] [Fintype W.toAffine.Point] in
 theorem isogeny_isGenuine (φ : Isogeny W.toAffine W.toAffine) : IsGenuine W φ :=
   ⟨_, isogeny_isGenuineWith_pointMap W φ⟩
 
-/-! ### The Verschiebung as a concrete `V` with `IsDualOf V π`
+/-! ### Verschiebung duality
 
-We pin a single concrete Verschiebung witness, `verschiebungV`, from the connected (axiom-clean
-modulo the upstream `[q] = V ∘ π` factorisation) inclusion `mulByInt_q_pullback_subset_frobenius`.
-Its `IsDualOf V π` is `verschiebung_dual_exists`'s underlying witness, and its point map is the
-`[q]`-point map (`verschiebungIsog_of_witness.toAddMonoidHom = (mulByInt q).toAddMonoidHom`). -/
+The image inclusion `Im([q]*) ⊆ Im(π*)` constructs a Verschiebung `V` dual to
+Frobenius. On rational points over the finite field, Frobenius is the identity,
+so the point map of `V` is multiplication by `q`.
+-/
 
-/-- The inclusion `Im([q]*) ⊆ Im(π*)` (Silverman II.2.11/III.6.2), the connected witness. -/
+/-- The image inclusion `Im([q]*) ⊆ Im(π*)`. See Silverman II.2.11 and III.6.2. -/
 noncomputable abbrev hSubset (hq : 2 ≤ Fintype.card K) :
     (mulByInt W.toAffine ((Fintype.card K : ℕ) : ℤ)).pullback.range ≤
       (frobeniusIsog W).pullback.range :=
@@ -104,11 +70,13 @@ noncomputable def verschiebungV (hq : 2 ≤ Fintype.card K) :
     Isogeny W.toAffine W.toAffine :=
   verschiebungIsog_of_witness W (hSubset W hq)
 
-/-- `IsDualOf V π` for the concrete Verschiebung (`h_isDual_V_pi`, Silverman III.6.1 Case 2). -/
+omit [Fintype W.toAffine.Point] in
+/-- The concrete Verschiebung is dual to Frobenius. See Silverman III.6.1. -/
 theorem verschiebungV_isDual (hq : 2 ≤ Fintype.card K) :
     IsDualOf W.toAffine (verschiebungV W hq) (frobeniusIsog W) :=
   verschiebungIsog_of_witness_isDualOf_frobenius W (hSubset W hq)
 
+omit [Fintype W.toAffine.Point] in
 /-- The point map of the concrete Verschiebung is the `[q]`-point map (`= q • ·`).  Frobenius on
 `F_q`-rational points is the identity, so the dual `V` (`V ∘ π = [q]`) must carry the `[q]`-point
 map. -/
@@ -117,12 +85,12 @@ map. -/
       (mulByInt W.toAffine ((Fintype.card K : ℕ) : ℤ)).toAddMonoidHom :=
   rfl
 
-/-! ### The V-side genuine isogeny `β_dual = r·V − s`
+/-! ### The isogeny `r V - s`
 
-This is `addIsog` of the genuine pair `(V.zsmul r, [−s])` — the **exact mirror** of the π-side
-`genuineIsogSmulSub = addIsog (π.zsmul r, [−s])`.  So `addIsog` *does* generalise to `V`: the V-side
-non-inverse / injectivity / pole bricks are the ones shipped in `Verschiebung/Genuine.lean`
-(`genuineIsogSmulSubV_universal_unconditional`).  Its point map is `r·V + (−s) = r·V − s`. -/
+The addition construction applied to `(V.zsmul r, [-s])` gives an isogeny with
+point map `r V - s id`, using the distinct-coordinate and pole identities for
+Verschiebung.
+-/
 
 /-- The V-side genuine isogeny `r·V − s` on the concrete Verschiebung `V`. -/
 noncomputable def betaDualV (hq : 2 ≤ Fintype.card K)
@@ -131,6 +99,7 @@ noncomputable def betaDualV (hq : 2 ≤ Fintype.card K)
   genuineIsogSmulSubV_universal_unconditional W (verschiebungV W hq)
     (verschiebungV_isDual W hq) r s hr hs hrK hsK
 
+omit [Fintype W.toAffine.Point] in
 /-- The point map of `betaDualV` is `r·V − s·id` (`h_beta_dual_hom`).
 
 `betaDualV.toAddMonoidHom = (V.zsmul r) + [−s]`, and `(V.zsmul r) = [r] ∘ V`, so pointwise this is
@@ -150,28 +119,19 @@ theorem betaDualV_toAddMonoidHom_sub (hq : 2 ≤ Fintype.card K)
     AddMonoidHom.id_apply, Isogeny.zsmul_apply, mulByInt_apply]
   rw [neg_smul, sub_eq_add_neg]
 
-/-! ### The Wall-A keystone, reduced to the standing dual residuals
-
-`genuineIsogSmulSub_degree_eq_signed_closed` is the live, witness-parametric form of the Wall-A
-keystone (whose unconditional sorried form was deleted 2026-06-11).  It constructs the concrete
-Verschiebung
-`V` and the V-side genuine isogeny `β_dual = r·V − s` internally, discharges the three *structural*
-pivot inputs (`h_isDual_V_pi`, `h_beta_dual_hom`, `h_beta_pos`), and reduces the keystone to exactly
-the three **standing dual residuals** plus the nonvanishing of `N`:
-
-* `h_sum_trace`  — the trace relation `π + V = [t]` (Silverman III.6.2(b); the output of
-  `sum_trace_frobenius_witness` once `IsDualOf (1−V) (1−π)` ships);
-* `h_pullback_eq` — the comorphism identity `(β_dual ∘ β)* = [N]*` (the double-Vieta pullback match,
-  the geometrically irreducible content of Wall A — equivalently, that `β_dual ∘ β` is genuine with
-  the `[N]` action; see `GapSpines.genuine_dual_comp_eq_mulByInt_of_isGenuineWith`);
-* `h_isDual_pair` — `IsDualOf β_dual β` (Silverman III.6.1 for the genuine pair);
-* `h_N_ne`        — `N = q·r² − t·r·s + s² ≠ 0` (automatic in the Hasse assembly: `β_dual ∘ β = [N]`
-  forces `deg β · deg β_dual = N²`, so `N ≠ 0`; carried here as a hypothesis).
-
-The composition is `genuineIsogSmulSub_degree_eq_signed_of_full_pivot_chain` (GapSpines), with the
-internally-built `V = verschiebungV`, `β_dual = betaDualV`.  This is the honest reduction of Wall A:
-the only inputs are the genuine III.6.1/III.6.2 facts the project has not yet shipped
-unconditionally. -/
+/-! ### The signed degree identity
+
+For `β = r π - s` and `β_dual = r V - s`, assume:
+
+* the trace relation `π + V = [t]` on points;
+* the pullback identity `(β_dual ∘ β)* = [N]*`, where `N = q r² - t r s + s²`;
+* the duality relation `IsDualOf β_dual β`;
+* `N ≠ 0`.
+
+These inputs, together with the constructed point map of `β_dual` and positivity
+of the degree of `β`, give `deg β = N`. See Silverman III.6.1–III.6.3.
+-/
+omit [Fintype W.toAffine.Point] in
 theorem genuineIsogSmulSub_degree_eq_signed_closed (hq : 2 ≤ Fintype.card K)
     (r s : ℤ) (hr : r ≠ 0) (hs : s ≠ 0) (hrK : (r : K) ≠ 0) (hsK : (s : K) ≠ 0)
     (h_sum_trace : (frobeniusIsog W).toAddMonoidHom + (verschiebungV W hq).toAddMonoidHom =
diff --git a/projects/HasseWeil/HasseWeil/Foundation/WronskianAux.lean b/projects/HasseWeil/HasseWeil/Foundation/WronskianAux.lean
index 2c92295af..6cd2db847 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/WronskianAux.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/WronskianAux.lean
@@ -1,120 +1,32 @@
+import HasseWeil.Foundation.WronskianAux.Prefix
 /-
 Copyright (c) 2026 Chris Birkbeck. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Birkbeck
 -/
-import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
-import Mathlib.Algebra.Polynomial.Derivation
-import HasseWeil.Foundation.WronskianAux.CNorm
-
-/-!
-# Wronskian identities for Weierstrass division polynomials
-
-Two auxiliary polynomial identities used in the proof of the division-polynomial
-Wronskian identity (Silverman III.3 Exercise 3.7):
-
-* `wronskian_aux_three` — the `m = 3` case.
-* `wronskian_aux_four` — the `m = 4` case.
-
-## Strategy
-
-Both identities are pure polynomial identities in `R[X]` that hold because of
-`b_relation : 4·b₈ = b₂·b₆ - b₄²`. Concretely, `LHS - RHS` factors as
-`M(X) · (4b₈ - b₂b₆ + b₄²)` in `ℤ[b₂,b₄,b₆,b₈,X]` for an explicit polynomial
-`M`. The multipliers `M` were computed by dividing the expanded difference by
-`(4b₈ - b₂b₆ + b₄²)` over `ℤ[b₂,b₄,b₆,b₈]` (with `b₈` as the leading variable);
-see `scripts/compute_multipliers.py`.
-
-The proof uses `linear_combination M · h_P` where `h_P` is `b_relation` lifted
-to `R[X]`. The `C`-normalization lemmas imported from `CNorm.lean` handle the
-`C (Nat.cast n : R)` vs `C (OfNat.ofNat n : R)` atomization issue that
-otherwise blocks `ring` from closing the residual.
-
-Resource usage (vs original):
-
-* `wronskian_aux_three`: default `maxHeartbeats 200000` (was 32M, 160× reduction).
-* `wronskian_aux_four`: `maxHeartbeats 350000` (was 64M, ~180× reduction; measured
-  minimum — 300000 fails). See TODO below for reducing to default.
-
-TODO (cleanup): reduce `wronskian_aux_four` `maxHeartbeats` to 200K (default).
-
-The 350K is required because `ring` must normalize a single degree-30 polynomial
-identity against the degree-26 multiplier `M`; the `evalMulProd` step in ring's
-normalization exceeds 300K heartbeats. Options to fit default:
-
-* Coefficient-wise approach via `Polynomial.ext_iff_natDegree_le`: works for
-  `coeff 0` at default (using `eval 0` trick, which is a ring hom), but
-  `Polynomial.coeff_mul` on nested products (`Ψ₃^3`, `preΨ₄ * Ψ₂Sq^2`, etc.)
-  makes simp hit `max_steps` for `coeff i > 0`. Over arbitrary `CommRing R`,
-  `(derivative^i p).eval 0 = i! · coeff i p` can't be inverted (no division).
-* New tactic: bounded-degree coefficient extraction avoiding simp's antidiagonal
-  explosion — doesn't exist in mathlib yet.
-* Formalize in the quotient ring `ℤ[b₂,b₄,b₆,b₈] / (4b₈ - b₂b₆ + b₄²)` where
-  `b_relation` becomes a ring identity — would work on 3 atoms instead of 4,
-  smaller ring work, likely fits default. Requires building the quotient.
-* Manual algebraic split of m=4 into smaller sub-identities — research-level
-  mathematical work.
-
-RAM usage: ~1-2 GB (was ~57 GB, ~30× reduction).
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, III.3 Exercise 3.7.
--/
-
 open WeierstrassCurve Polynomial
-
 namespace HasseWeil
-
 variable {R : Type*} [CommRing R] (W : WeierstrassCurve R)
 
--- `b_relation` lifted to an equality in `R[X]`. The `C`-distributed statement shape is
--- load-bearing: the `wronskian_aux_*` proofs below pass this to `linear_combination`,
--- whose `ring` normalization treats each `C _` application as an atom.
-private lemma b_relation_poly :
-    ((4 : R[X]) * Polynomial.C W.b₈ : R[X]) =
-      Polynomial.C W.b₂ * Polynomial.C W.b₆ - Polynomial.C W.b₄ ^ 2 := by
-  simpa [Polynomial.C_ofNat] using congrArg Polynomial.C W.b_relation
-
-/-- Wronskian auxiliary identity, `m = 3` case (Silverman III.3.7).
-`4·Ψ₃³ + 2·preΨ₄·Ψ₂Sq·Ψ₃' − (preΨ₄·Ψ₂Sq)'·Ψ₃
-= 3·preΨ₄·Ψ₂Sq² − 3·preΨ₄²`.
+lemma wronskian_derivative_three : Polynomial.derivative W.Ψ₃ = 3 * W.Ψ₂Sq := by
+  simp only [Ψ₃, Ψ₂Sq, Polynomial.derivative_add, Polynomial.derivative_mul,
+    Polynomial.derivative_pow, Polynomial.derivative_X, Polynomial.derivative_C,
+    Polynomial.derivative_ofNat, Polynomial.C_mul,
+    Polynomial.C_ofNat, Nat.cast_ofNat]
+  ring
 
-Multiplier: `M = b₈² + 4b₆b₈·X + 6b₄b₈·X² + 4b₂b₈·X³ + (b₂b₆ + 34b₈)·X⁴
-               + 36b₆·X⁵ + 18b₄·X⁶ + 4b₂·X⁷ + 9·X⁸`. -/
-lemma wronskian_aux_three :
-    4 * W.Ψ₃ ^ 3 + 2 * W.preΨ₄ * W.Ψ₂Sq * Polynomial.derivative W.Ψ₃ -
-      Polynomial.derivative (W.preΨ₄ * W.Ψ₂Sq) * W.Ψ₃ =
-    Polynomial.C 3 * W.preΨ₄ * W.Ψ₂Sq ^ 2 - Polynomial.C 3 * W.preΨ₄ ^ 2 := by
+lemma wronskian_three_derivative_two :
+    W.Ψ₃ * Polynomial.derivative W.Ψ₂Sq = 2 * W.preΨ₄ + 2 * W.Ψ₂Sq ^ 2 := by
   linear_combination (norm := (
-    simp only [Ψ₃, preΨ₄, Ψ₂Sq,
-      Polynomial.derivative_add, Polynomial.derivative_sub,
-      Polynomial.derivative_mul, Polynomial.derivative_pow, Polynomial.derivative_X,
-      Polynomial.derivative_C, Polynomial.derivative_ofNat,
-      Polynomial.C_add, Polynomial.C_sub, Polynomial.C_mul, Polynomial.C_pow,
-      Polynomial.C_ofNat, Nat.cast_ofNat]
-    ring))
-    (Polynomial.C (W.b₈ ^ 2)
-    + Polynomial.C (4 * W.b₆ * W.b₈) * Polynomial.X
-    + Polynomial.C (6 * W.b₄ * W.b₈) * Polynomial.X ^ 2
-    + Polynomial.C (4 * W.b₂ * W.b₈) * Polynomial.X ^ 3
-    + Polynomial.C (W.b₂ * W.b₆ + 34 * W.b₈) * Polynomial.X ^ 4
-    + Polynomial.C (36 * W.b₆) * Polynomial.X ^ 5
-    + Polynomial.C (18 * W.b₄) * Polynomial.X ^ 6
-    + Polynomial.C (4 * W.b₂) * Polynomial.X ^ 7
-    + Polynomial.C 9 * Polynomial.X ^ 8) * b_relation_poly W
-
--- Measured minimum (2026-07-16, #7205): 300000 fails at `synthesize pending MVars`
--- (ring's normalization of the degree-30 residual), 350000 passes. Removal attempts:
--- default budget and a type-ascribed multiplier both time out — the structural fix is the
--- quotient-ring route documented above (producer/decompose work).
-set_option maxHeartbeats 350000 in
-/-- Wronskian auxiliary identity, `m = 4` case (Silverman III.3.7).
-
-Multiplier `M(X)` is a degree-26 polynomial in `W.b₂, W.b₄, W.b₆, W.b₈` with
-integer coefficients, computed offline by polynomial division of `LHS - RHS`
-by `(4b₈ - b₂b₆ + b₄²)` over `ℤ[b₂, b₄, b₆, b₈]`. See
-`scripts/compute_multipliers.py` for the derivation. -/
+    simp only [Ψ₃, preΨ₄, Ψ₂Sq, Polynomial.derivative_add, Polynomial.derivative_mul,
+      Polynomial.derivative_pow, Polynomial.derivative_X, Polynomial.derivative_C,
+      Polynomial.derivative_ofNat, Polynomial.C_sub,
+      Polynomial.C_mul, Polynomial.C_pow, Polynomial.C_ofNat, Nat.cast_ofNat]
+    ring)) (-2 * Polynomial.X ^ 2) * wronskian_b_relation_poly W
+
+/-- The fourth Wronskian identity follows from the third identity and the two
+small derivative relations above. This factorization avoids expansion of the
+original degree-26 multiplier. -/
 lemma wronskian_aux_four :
     (W.preΨ₄ ^ 2 * W.Ψ₂Sq) ^ 2 -
     (Polynomial.derivative (W.Ψ₃ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3)) *
@@ -126,315 +38,12 @@ lemma wronskian_aux_four :
           (W.Ψ₃ * ((W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3) - W.preΨ₄ ^ 2)) -
         W.preΨ₄ * (W.preΨ₄ * W.Ψ₂Sq ^ 2 - W.Ψ₃ ^ 3) ^ 2) := by
   linear_combination (norm := (
-    simp only [Ψ₃, preΨ₄, Ψ₂Sq,
-      Polynomial.derivative_mul, Polynomial.derivative_pow,
-      Polynomial.derivative_add, Polynomial.derivative_sub,
-      Polynomial.derivative_X, Polynomial.derivative_C, Polynomial.derivative_ofNat,
-      Polynomial.C_mul, Polynomial.C_sub, Polynomial.C_add, Polynomial.C_pow,
-      Polynomial.C_neg, Polynomial.C_ofNat, Nat.cast_ofNat]
+    simp only [Polynomial.derivative_mul, Polynomial.derivative_sub,
+      Polynomial.derivative_pow, Polynomial.C_ofNat, Nat.cast_ofNat]
     ring))
-    (Polynomial.C (-W.b₆ ^ 6 * W.b₈ ^ 2 + (-2 * W.b₆ ^ 2 * W.b₈ ^ 5)
-                     + 2 * W.b₄ * W.b₈ ^ 6
-                     + (-W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 4)
-                     + 2 * W.b₄ * W.b₆ ^ 4 * W.b₈ ^ 3)
-    + Polynomial.C (-26 * W.b₆ ^ 3 * W.b₈ ^ 4 + (-4 * W.b₆ ^ 7 * W.b₈)
-                      + 2 * W.b₂ * W.b₈ ^ 6
-                      + (-4 * W.b₄ ^ 3 * W.b₆ * W.b₈ ^ 4)
-                      + 2 * W.b₂ * W.b₆ ^ 4 * W.b₈ ^ 3
-                      + 2 * W.b₄ * W.b₆ ^ 5 * W.b₈ ^ 2
-                      + 6 * W.b₄ ^ 2 * W.b₆ ^ 3 * W.b₈ ^ 3
-                      + 24 * W.b₄ * W.b₆ * W.b₈ ^ 5
-                      + (-2 * W.b₂ * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 4)) * Polynomial.X
-    + Polynomial.C (-4 * W.b₆ ^ 8 + 20 * W.b₈ ^ 6 + (-108 * W.b₆ ^ 4 * W.b₈ ^ 3)
-                      + (-4 * W.b₄ ^ 4 * W.b₈ ^ 4) + 28 * W.b₄ ^ 2 * W.b₈ ^ 5
-                      + (-W.b₂ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 4)
-                      + (-18 * W.b₄ * W.b₆ ^ 6 * W.b₈)
-                      + (-2 * W.b₄ ^ 3 * W.b₆ ^ 2 * W.b₈ ^ 3)
-                      + 6 * W.b₂ * W.b₆ ^ 5 * W.b₈ ^ 2
-                      + 26 * W.b₂ * W.b₆ * W.b₈ ^ 5
-                      + 27 * W.b₄ ^ 2 * W.b₆ ^ 4 * W.b₈ ^ 2
-                      + 54 * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 4
-                      + (-10 * W.b₂ * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 4)
-                      + 6 * W.b₂ * W.b₄ * W.b₆ ^ 3 * W.b₈ ^ 3) * Polynomial.X ^ 2
-    + Polynomial.C (-196 * W.b₆ ^ 5 * W.b₈ ^ 2 + (-28 * W.b₄ * W.b₆ ^ 7)
-                      + 280 * W.b₆ * W.b₈ ^ 5
-                      + (-96 * W.b₄ * W.b₆ ^ 3 * W.b₈ ^ 3)
-                      + (-16 * W.b₄ ^ 4 * W.b₆ * W.b₈ ^ 3)
-                      + (-16 * W.b₄ ^ 2 * W.b₆ ^ 5 * W.b₈)
-                      + (-12 * W.b₂ * W.b₄ ^ 3 * W.b₈ ^ 4)
-                      + 40 * W.b₂ * W.b₄ * W.b₈ ^ 5
-                      + 52 * W.b₄ ^ 3 * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + 96 * W.b₂ * W.b₆ ^ 2 * W.b₈ ^ 4
-                      + 156 * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 4
-                      + (-16 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 3)
-                      + (-8 * W.b₂ ^ 2 * W.b₄ * W.b₆ * W.b₈ ^ 4)
-                      + 44 * W.b₂ * W.b₄ * W.b₆ ^ 4 * W.b₈ ^ 2) * Polynomial.X ^ 3
-    + Polynomial.C (-136 * W.b₆ ^ 6 * W.b₈ + (-76 * W.b₄ ^ 2 * W.b₆ ^ 6)
-                      + (-9 * W.b₂ * W.b₆ ^ 7)
-                      + (-8 * W.b₄ ^ 5 * W.b₈ ^ 3) + 12 * W.b₂ ^ 2 * W.b₈ ^ 5
-                      + 48 * W.b₄ ^ 3 * W.b₈ ^ 4 + 356 * W.b₄ * W.b₈ ^ 5
-                      + 1374 * W.b₆ ^ 2 * W.b₈ ^ 4 + (-508 * W.b₄ * W.b₆ ^ 4 * W.b₈ ^ 2)
-                      + (-13 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₈ ^ 4)
-                      + (-2 * W.b₂ ^ 3 * W.b₆ * W.b₈ ^ 4)
-                      + 11 * W.b₂ ^ 2 * W.b₆ ^ 4 * W.b₈ ^ 2
-                      + 28 * W.b₄ ^ 4 * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + 30 * W.b₄ ^ 3 * W.b₆ ^ 4 * W.b₈
-                      + 128 * W.b₂ * W.b₆ ^ 3 * W.b₈ ^ 3
-                      + 216 * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 3
-                      + (-52 * W.b₂ * W.b₄ ^ 3 * W.b₆ * W.b₈ ^ 3)
-                      + (-20 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 3)
-                      + 30 * W.b₂ * W.b₄ * W.b₆ ^ 5 * W.b₈
-                      + 81 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + 240 * W.b₂ * W.b₄ * W.b₆ * W.b₈ ^ 4) * Polynomial.X ^ 4
-    + Polynomial.C (-2 * W.b₆ ^ 7 + (-100 * W.b₄ ^ 3 * W.b₆ ^ 5) + 188 * W.b₂ * W.b₈ ^ 5
-                      + 3056 * W.b₆ ^ 3 * W.b₈ ^ 3 + (-468 * W.b₄ * W.b₆ ^ 5 * W.b₈)
-                      + (-410 * W.b₄ ^ 2 * W.b₆ ^ 3 * W.b₈ ^ 2)
-                      + (-46 * W.b₂ * W.b₄ * W.b₆ ^ 6)
-                      + (-28 * W.b₂ * W.b₆ ^ 4 * W.b₈ ^ 2)
-                      + (-24 * W.b₂ * W.b₄ ^ 4 * W.b₈ ^ 3)
-                      + (-6 * W.b₂ ^ 3 * W.b₄ * W.b₈ ^ 4)
-                      + (-6 * W.b₂ ^ 3 * W.b₆ ^ 2 * W.b₈ ^ 3)
-                      + 14 * W.b₂ ^ 2 * W.b₆ ^ 5 * W.b₈
-                      + 56 * W.b₄ ^ 4 * W.b₆ ^ 3 * W.b₈
-                      + 64 * W.b₂ * W.b₄ ^ 2 * W.b₈ ^ 4
-                      + 68 * W.b₂ ^ 2 * W.b₆ * W.b₈ ^ 4
-                      + 248 * W.b₄ ^ 3 * W.b₆ * W.b₈ ^ 3 + 3168 * W.b₄ * W.b₆ * W.b₈ ^ 4
-                      + (-58 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 3)
-                      + 24 * W.b₂ * W.b₄ ^ 3 * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + 24 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + 122 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 4 * W.b₈
-                      + 528 * W.b₂ * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 3) * Polynomial.X ^ 5
-    + Polynomial.C (664 * W.b₈ ^ 5 + (-W.b₂ ^ 4 * W.b₈ ^ 4) + (-64 * W.b₄ ^ 4 * W.b₆ ^ 4)
-                      + (-6 * W.b₂ ^ 2 * W.b₆ ^ 6) + 116 * W.b₄ * W.b₆ ^ 6
-                      + 116 * W.b₄ ^ 4 * W.b₈ ^ 3 + 1500 * W.b₄ ^ 2 * W.b₈ ^ 4
-                      + 2572 * W.b₆ ^ 4 * W.b₈ ^ 2 + (-W.b₂ ^ 3 * W.b₆ ^ 3 * W.b₈ ^ 2)
-                      + (-758 * W.b₄ ^ 2 * W.b₆ ^ 4 * W.b₈)
-                      + (-110 * W.b₂ * W.b₆ ^ 5 * W.b₈)
-                      + (-106 * W.b₄ ^ 3 * W.b₆ ^ 2 * W.b₈ ^ 2)
-                      + (-85 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 5)
-                      + (-26 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₈ ^ 3)
-                      + 12 * W.b₂ ^ 2 * W.b₄ * W.b₈ ^ 4
-                      + 24 * W.b₄ ^ 5 * W.b₆ ^ 2 * W.b₈
-                      + 168 * W.b₂ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 3
-                      + 1610 * W.b₂ * W.b₆ * W.b₈ ^ 4
-                      + 10544 * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 3
-                      + (-27 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 2)
-                      + (-26 * W.b₂ ^ 3 * W.b₄ * W.b₆ * W.b₈ ^ 3)
-                      + (-20 * W.b₂ * W.b₄ ^ 4 * W.b₆ * W.b₈ ^ 2)
-                      + 76 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 4 * W.b₈
-                      + 78 * W.b₂ * W.b₄ * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + 156 * W.b₂ * W.b₄ ^ 3 * W.b₆ ^ 3 * W.b₈
-                      + 484 * W.b₂ * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 3) * Polynomial.X ^ 6
-    + Polynomial.C (-336 * W.b₆ ^ 5 * W.b₈ + (-16 * W.b₄ ^ 5 * W.b₆ ^ 3)
-                      + (-4 * W.b₂ ^ 3 * W.b₈ ^ 4) + 48 * W.b₂ * W.b₆ ^ 6
-                      + 516 * W.b₄ ^ 2 * W.b₆ ^ 5 + 6552 * W.b₆ * W.b₈ ^ 4
-                      + (-1032 * W.b₄ ^ 3 * W.b₆ ^ 3 * W.b₈)
-                      + (-68 * W.b₂ * W.b₄ ^ 3 * W.b₆ ^ 4)
-                      + (-20 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 5)
-                      + (-12 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₈ ^ 3)
-                      + (-4 * W.b₂ ^ 4 * W.b₆ * W.b₈ ^ 3)
-                      + 12 * W.b₂ ^ 3 * W.b₆ ^ 4 * W.b₈
-                      + 80 * W.b₂ ^ 2 * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + 212 * W.b₄ ^ 4 * W.b₆ * W.b₈ ^ 2
-                      + 304 * W.b₂ * W.b₄ ^ 3 * W.b₈ ^ 3 + 1272 * W.b₂ * W.b₄ * W.b₈ ^ 4
-                      + 5552 * W.b₂ * W.b₆ ^ 2 * W.b₈ ^ 3
-                      + 11312 * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 3
-                      + 12336 * W.b₄ * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + (-472 * W.b₂ * W.b₄ * W.b₆ ^ 4 * W.b₈)
-                      + (-44 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₆ * W.b₈ ^ 2)
-                      + (-28 * W.b₂ ^ 3 * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 2)
-                      + 64 * W.b₂ * W.b₄ ^ 4 * W.b₆ ^ 2 * W.b₈
-                      + 108 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + 116 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ ^ 3 * W.b₈
-                      + 240 * W.b₂ ^ 2 * W.b₄ * W.b₆ * W.b₈ ^ 3) * Polynomial.X ^ 7
-    + Polynomial.C (-883 * W.b₆ ^ 6 + (-W.b₂ ^ 3 * W.b₆ ^ 5) + 198 * W.b₄ ^ 5 * W.b₈ ^ 2
-                      + 216 * W.b₂ ^ 2 * W.b₈ ^ 4 + 790 * W.b₄ ^ 3 * W.b₆ ^ 4
-                      + 4720 * W.b₄ ^ 3 * W.b₈ ^ 3 + 5526 * W.b₄ * W.b₈ ^ 4
-                      + 26236 * W.b₆ ^ 2 * W.b₈ ^ 3 + (-918 * W.b₄ ^ 4 * W.b₆ ^ 2 * W.b₈)
-                      + (-270 * W.b₄ * W.b₆ ^ 4 * W.b₈)
-                      + (-70 * W.b₂ ^ 2 * W.b₆ ^ 4 * W.b₈)
-                      + (-22 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ ^ 4)
-                      + (-20 * W.b₂ * W.b₄ ^ 4 * W.b₆ ^ 3)
-                      + (-6 * W.b₂ ^ 4 * W.b₆ ^ 2 * W.b₈ ^ 2)
-                      + (-2 * W.b₂ ^ 4 * W.b₄ * W.b₈ ^ 3)
-                      + 32 * W.b₂ ^ 3 * W.b₆ * W.b₈ ^ 3
-                      + 300 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₈ ^ 3
-                      + 354 * W.b₂ * W.b₄ * W.b₆ ^ 5 + 7316 * W.b₂ * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + 21030 * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + (-1346 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 3 * W.b₈)
-                      + (-33 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 2)
-                      + 30 * W.b₂ ^ 3 * W.b₄ * W.b₆ ^ 3 * W.b₈
-                      + 54 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₆ ^ 2 * W.b₈
-                      + 72 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + 588 * W.b₂ * W.b₄ ^ 3 * W.b₆ * W.b₈ ^ 2
-                      + 10848 * W.b₂ * W.b₄ * W.b₆ * W.b₈ ^ 3) * Polynomial.X ^ 8
-    + Polynomial.C (-4494 * W.b₄ * W.b₆ ^ 5 + 62 * W.b₂ ^ 2 * W.b₆ ^ 5
-                      + 506 * W.b₄ ^ 4 * W.b₆ ^ 3
-                      + 1870 * W.b₂ * W.b₈ ^ 4 + 47004 * W.b₆ ^ 3 * W.b₈ ^ 2
-                      + (-312 * W.b₄ ^ 5 * W.b₆ * W.b₈)
-                      + (-8 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₆ ^ 3)
-                      + (-2 * W.b₂ ^ 3 * W.b₄ * W.b₆ ^ 4)
-                      + (-2 * W.b₂ ^ 3 * W.b₆ ^ 2 * W.b₈ ^ 2)
-                      + 2 * W.b₂ ^ 4 * W.b₆ ^ 3 * W.b₈
-                      + 136 * W.b₂ ^ 3 * W.b₄ * W.b₈ ^ 3
-                      + 550 * W.b₂ * W.b₄ ^ 4 * W.b₈ ^ 2
-                      + 706 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 4
-                      + 1610 * W.b₄ ^ 2 * W.b₆ ^ 3 * W.b₈
-                      + 2042 * W.b₂ * W.b₆ ^ 4 * W.b₈
-                      + 2344 * W.b₂ ^ 2 * W.b₆ * W.b₈ ^ 3
-                      + 7136 * W.b₂ * W.b₄ ^ 2 * W.b₈ ^ 3
-                      + 18064 * W.b₄ ^ 3 * W.b₆ * W.b₈ ^ 2
-                      + 49456 * W.b₄ * W.b₆ * W.b₈ ^ 3
-                      + (-1592 * W.b₂ * W.b₄ ^ 3 * W.b₆ ^ 2 * W.b₈)
-                      + (-600 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 3 * W.b₈)
-                      + (-10 * W.b₂ ^ 4 * W.b₄ * W.b₆ * W.b₈ ^ 2)
-                      + 18 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈
-                      + 546 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 2
-                      + 21708 * W.b₂ * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 2) * Polynomial.X ^ 9
-    + Polynomial.C (3532 * W.b₈ ^ 4 + (-9537 * W.b₄ ^ 2 * W.b₆ ^ 4)
-                      + (-936 * W.b₂ * W.b₆ ^ 5)
-                      + 24 * W.b₂ ^ 4 * W.b₈ ^ 3 + 114 * W.b₄ ^ 5 * W.b₆ ^ 2
-                      + 6304 * W.b₄ ^ 4 * W.b₈ ^ 2 + 27864 * W.b₄ ^ 2 * W.b₈ ^ 3
-                      + 35644 * W.b₆ ^ 4 * W.b₈ + (-W.b₂ ^ 5 * W.b₆ * W.b₈ ^ 2)
-                      + (-W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₆ ^ 3)
-                      + (-98 * W.b₂ ^ 3 * W.b₆ ^ 3 * W.b₈)
-                      + 192 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 4
-                      + 532 * W.b₂ * W.b₄ ^ 3 * W.b₆ ^ 3
-                      + 578 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₈ ^ 2
-                      + 3608 * W.b₂ ^ 2 * W.b₄ * W.b₈ ^ 3
-                      + 4818 * W.b₂ ^ 2 * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + 5778 * W.b₄ ^ 3 * W.b₆ ^ 2 * W.b₈
-                      + 19028 * W.b₂ * W.b₆ * W.b₈ ^ 3
-                      + 131228 * W.b₄ * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + (-966 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈)
-                      + (-598 * W.b₂ * W.b₄ ^ 4 * W.b₆ * W.b₈)
-                      + 2 * W.b₂ ^ 4 * W.b₄ * W.b₆ ^ 2 * W.b₈
-                      + 222 * W.b₂ ^ 3 * W.b₄ * W.b₆ * W.b₈ ^ 2
-                      + 6986 * W.b₂ * W.b₄ * W.b₆ ^ 3 * W.b₈
-                      + 26712 * W.b₂ * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 2) * Polynomial.X ^ 10
-    + Polynomial.C (8884 * W.b₆ ^ 5 + (-9916 * W.b₄ ^ 3 * W.b₆ ^ 3)
-                      + 12 * W.b₂ ^ 3 * W.b₆ ^ 4
-                      + 640 * W.b₂ ^ 3 * W.b₈ ^ 3 + 36848 * W.b₆ * W.b₈ ^ 3
-                      + (-4596 * W.b₂ * W.b₄ * W.b₆ ^ 4)
-                      + 36 * W.b₂ ^ 4 * W.b₆ * W.b₈ ^ 2
-                      + 132 * W.b₂ * W.b₄ ^ 4 * W.b₆ ^ 2
-                      + 180 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ ^ 3
-                      + 288 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₈ ^ 2
-                      + 1968 * W.b₂ ^ 2 * W.b₆ ^ 3 * W.b₈
-                      + 6096 * W.b₄ ^ 4 * W.b₆ * W.b₈
-                      + 11592 * W.b₂ * W.b₄ ^ 3 * W.b₈ ^ 2
-                      + 24400 * W.b₂ * W.b₄ * W.b₈ ^ 3
-                      + 49728 * W.b₂ * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + 124416 * W.b₄ * W.b₆ ^ 3 * W.b₈
-                      + 134568 * W.b₄ ^ 2 * W.b₆ * W.b₈ ^ 2
-                      + (-408 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₆ * W.b₈)
-                      + (-240 * W.b₂ ^ 3 * W.b₄ * W.b₆ ^ 2 * W.b₈)
-                      + 12480 * W.b₂ ^ 2 * W.b₄ * W.b₆ * W.b₈ ^ 2
-                      + 13560 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈) * Polynomial.X ^ 11
-    + Polynomial.C (-4902 * W.b₄ ^ 4 * W.b₆ ^ 2 + (-545 * W.b₂ ^ 2 * W.b₆ ^ 4)
-                      + 1932 * W.b₄ ^ 5 * W.b₈ + 5736 * W.b₂ ^ 2 * W.b₈ ^ 3
-                      + 35344 * W.b₄ * W.b₆ ^ 4 + 45144 * W.b₄ * W.b₈ ^ 3
-                      + 46752 * W.b₄ ^ 3 * W.b₈ ^ 2 + 105356 * W.b₆ ^ 2 * W.b₈ ^ 2
-                      + (-7315 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 3)
-                      + (-18 * W.b₂ ^ 4 * W.b₆ ^ 2 * W.b₈)
-                      + 18 * W.b₂ ^ 3 * W.b₄ * W.b₆ ^ 3
-                      + 50 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₆ ^ 2
-                      + 68 * W.b₂ ^ 4 * W.b₄ * W.b₈ ^ 2
-                      + 1960 * W.b₂ ^ 3 * W.b₆ * W.b₈ ^ 2
-                      + 7602 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₈ ^ 2
-                      + 45016 * W.b₂ * W.b₆ ^ 3 * W.b₈
-                      + 173648 * W.b₄ ^ 2 * W.b₆ ^ 2 * W.b₈
-                      + (-118 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₆ * W.b₈)
-                      + 7284 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 2 * W.b₈
-                      + 12548 * W.b₂ * W.b₄ ^ 3 * W.b₆ * W.b₈
-                      + 107360 * W.b₂ * W.b₄ * W.b₆ * W.b₈ ^ 2) * Polynomial.X ^ 12
-    + Polynomial.C (-912 * W.b₄ ^ 5 * W.b₆ + 6 * W.b₂ ^ 5 * W.b₈ ^ 2
-                      + 12024 * W.b₂ * W.b₆ ^ 4
-                      + 20232 * W.b₂ * W.b₈ ^ 3 + 59770 * W.b₄ ^ 2 * W.b₆ ^ 3
-                      + 112576 * W.b₆ ^ 3 * W.b₈ + (-4656 * W.b₂ * W.b₄ ^ 3 * W.b₆ ^ 2)
-                      + (-1600 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 3)
-                      + 6 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₆ ^ 2
-                      + 1206 * W.b₂ ^ 3 * W.b₆ ^ 2 * W.b₈
-                      + 2124 * W.b₂ ^ 3 * W.b₄ * W.b₈ ^ 2
-                      + 3892 * W.b₂ * W.b₄ ^ 4 * W.b₈
-                      + 22208 * W.b₂ ^ 2 * W.b₆ * W.b₈ ^ 2
-                      + 54192 * W.b₂ * W.b₄ ^ 2 * W.b₈ ^ 2
-                      + 108840 * W.b₄ ^ 3 * W.b₆ * W.b₈
-                      + 217088 * W.b₄ * W.b₆ * W.b₈ ^ 2
-                      + (-12 * W.b₂ ^ 4 * W.b₄ * W.b₆ * W.b₈)
-                      + 8514 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ * W.b₈
-                      + 128256 * W.b₂ * W.b₄ * W.b₆ ^ 2 * W.b₈) * Polynomial.X ^ 13
-    + Polynomial.C (24720 * W.b₈ ^ 3 + 39812 * W.b₆ ^ 4 + (-69 * W.b₂ ^ 3 * W.b₆ ^ 3)
-                      + 210 * W.b₂ ^ 4 * W.b₈ ^ 2 + 24836 * W.b₄ ^ 4 * W.b₈
-                      + 51722 * W.b₄ ^ 3 * W.b₆ ^ 2 + 101464 * W.b₄ ^ 2 * W.b₈ ^ 2
-                      + (-1395 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆ ^ 2)
-                      + (-1026 * W.b₂ * W.b₄ ^ 4 * W.b₆)
-                      + 2946 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₈
-                      + 20544 * W.b₂ ^ 2 * W.b₄ * W.b₈ ^ 2
-                      + 24296 * W.b₂ ^ 2 * W.b₆ ^ 2 * W.b₈
-                      + 40930 * W.b₂ * W.b₄ * W.b₆ ^ 3
-                      + 85700 * W.b₂ * W.b₆ * W.b₈ ^ 2 + 308736 * W.b₄ * W.b₆ ^ 2 * W.b₈
-                      + 2394 * W.b₂ ^ 3 * W.b₄ * W.b₆ * W.b₈
-                      + 116300 * W.b₂ * W.b₄ ^ 2 * W.b₆ * W.b₈) * Polynomial.X ^ 14
-    + Polynomial.C (2472 * W.b₂ ^ 3 * W.b₈ ^ 2 + 7232 * W.b₂ ^ 2 * W.b₆ ^ 3
-                      + 22172 * W.b₄ ^ 4 * W.b₆ + 108016 * W.b₆ * W.b₈ ^ 2
-                      + 132592 * W.b₄ * W.b₆ ^ 3 + (-380 * W.b₂ ^ 2 * W.b₄ ^ 3 * W.b₆)
-                      + (-100 * W.b₂ ^ 3 * W.b₄ * W.b₆ ^ 2)
-                      + 236 * W.b₂ ^ 4 * W.b₆ * W.b₈
-                      + 1060 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₈
-                      + 33232 * W.b₂ * W.b₄ ^ 3 * W.b₈
-                      + 50972 * W.b₂ * W.b₄ ^ 2 * W.b₆ ^ 2
-                      + 72816 * W.b₂ * W.b₄ * W.b₈ ^ 2
-                      + 110896 * W.b₂ * W.b₆ ^ 2 * W.b₈
-                      + 263536 * W.b₄ ^ 2 * W.b₆ * W.b₈
-                      + 40912 * W.b₂ ^ 2 * W.b₄ * W.b₆ * W.b₈) * Polynomial.X ^ 15
-    + Polynomial.C (3710 * W.b₄ ^ 5 + 8 * W.b₂ ^ 4 * W.b₆ ^ 2
-                      + 12320 * W.b₂ ^ 2 * W.b₈ ^ 2
-                      + 43908 * W.b₂ * W.b₆ ^ 3 + 70896 * W.b₄ ^ 3 * W.b₈
-                      + 85422 * W.b₄ * W.b₈ ^ 2
-                      + 149190 * W.b₆ ^ 2 * W.b₈ + 157539 * W.b₄ ^ 2 * W.b₆ ^ 2
-                      + (-37 * W.b₂ ^ 3 * W.b₄ ^ 2 * W.b₆)
-                      + 182 * W.b₂ ^ 4 * W.b₄ * W.b₈
-                      + 4632 * W.b₂ ^ 3 * W.b₆ * W.b₈
-                      + 16268 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₈
-                      + 16744 * W.b₂ ^ 2 * W.b₄ * W.b₆ ^ 2
-                      + 27268 * W.b₂ * W.b₄ ^ 3 * W.b₆
-                      + 175520 * W.b₂ * W.b₄ * W.b₆ * W.b₈) * Polynomial.X ^ 16
-    + Polynomial.C (65230 * W.b₆ ^ 3 + 12 * W.b₂ ^ 5 * W.b₈ + 1794 * W.b₂ ^ 3 * W.b₆ ^ 2
-                      + 5302 * W.b₂ * W.b₄ ^ 4 + 27230 * W.b₂ * W.b₈ ^ 2
-                      + 79828 * W.b₄ ^ 3 * W.b₆
-                      + 6 * W.b₂ ^ 4 * W.b₄ * W.b₆ + 3432 * W.b₂ ^ 3 * W.b₄ * W.b₈
-                      + 12422 * W.b₂ ^ 2 * W.b₄ ^ 2 * W.b₆
-                      + 27992 * W.b₂ ^ 2 * W.b₆ * W.b₈
-                      + 66496 * W.b₂ * W.b₄ ^ 2 * W.b₈ + 97734 * W.b₂ * W.b₄ * W.b₆ ^ 2
-                      + 223992 * W.b₄ * W.b₆ * W.b₈) * Polynomial.X ^ 17
-    + Polynomial.C (14684 * W.b₄ ^ 4 + 21868 * W.b₈ ^ 2 + W.b₂ ^ 5 * W.b₆
-                      + 264 * W.b₂ ^ 4 * W.b₈
-                      + 2982 * W.b₂ ^ 2 * W.b₄ ^ 3 + 14687 * W.b₂ ^ 2 * W.b₆ ^ 2
-                      + 81548 * W.b₄ ^ 2 * W.b₈ + 139902 * W.b₄ * W.b₆ ^ 2
-                      + 2466 * W.b₂ ^ 3 * W.b₄ * W.b₆ + 20136 * W.b₂ ^ 2 * W.b₄ * W.b₈
-                      + 67922 * W.b₂ * W.b₆ * W.b₈
-                      + 70114 * W.b₂ * W.b₄ ^ 2 * W.b₆) * Polynomial.X ^ 18
-    + Polynomial.C (180 * W.b₂ ^ 4 * W.b₆ + 824 * W.b₂ ^ 3 * W.b₄ ^ 2
-                      + 1984 * W.b₂ ^ 3 * W.b₈
-                      + 16324 * W.b₂ * W.b₄ ^ 3 + 40320 * W.b₂ * W.b₆ ^ 2
-                      + 57464 * W.b₆ * W.b₈
-                      + 97500 * W.b₄ ^ 2 * W.b₆ + 20040 * W.b₂ ^ 2 * W.b₄ * W.b₆
-                      + 47624 * W.b₂ * W.b₄ * W.b₈) * Polynomial.X ^ 19
-    + Polynomial.C (22160 * W.b₄ ^ 3 + 36374 * W.b₆ ^ 2 + 112 * W.b₂ ^ 4 * W.b₄
-                      + 1874 * W.b₂ ^ 3 * W.b₆ + 6675 * W.b₂ ^ 2 * W.b₄ ^ 2
-                      + 6828 * W.b₂ ^ 2 * W.b₈
-                      + 39492 * W.b₄ * W.b₈
-                      + 54192 * W.b₂ * W.b₄ * W.b₆) * Polynomial.X ^ 20
-    + Polynomial.C (6 * W.b₂ ^ 5 + 1194 * W.b₂ ^ 3 * W.b₄ + 7420 * W.b₂ ^ 2 * W.b₆
-                      + 11068 * W.b₂ * W.b₈ + 17840 * W.b₂ * W.b₄ ^ 2
-                      + 48224 * W.b₄ * W.b₆) * Polynomial.X ^ 21
-    + Polynomial.C (79 * W.b₂ ^ 4 + 6872 * W.b₈ + 15692 * W.b₄ ^ 2
-                      + 4724 * W.b₂ ^ 2 * W.b₄
-                      + 12978 * W.b₂ * W.b₆) * Polynomial.X ^ 22
-    + Polynomial.C (412 * W.b₂ ^ 3 + 8440 * W.b₆ + 8216 * W.b₂ * W.b₄) * Polynomial.X ^ 23
-    + Polynomial.C (1064 * W.b₂ ^ 2 + 5306 * W.b₄) * Polynomial.X ^ 24
-    + Polynomial.C (1362 * W.b₂) * Polynomial.X ^ 25
-    + Polynomial.C 692 * Polynomial.X ^ 26) * b_relation_poly W
+    (2 * W.Ψ₃ ^ 3 * W.preΨ₄ - W.preΨ₄ ^ 2 * W.Ψ₂Sq ^ 2) * wronskian_aux_three W +
+    (W.preΨ₄ ^ 3 * W.Ψ₂Sq ^ 3) * wronskian_derivative_three W +
+    (W.Ψ₃ ^ 3 * W.preΨ₄ ^ 2 - 2 * W.preΨ₄ ^ 3 * W.Ψ₂Sq ^ 2) *
+      wronskian_three_derivative_two W
 
 end HasseWeil
diff --git a/projects/HasseWeil/HasseWeil/Foundation/WronskianAux/CNorm.lean b/projects/HasseWeil/HasseWeil/Foundation/WronskianAux/CNorm.lean
index 4dee067bc..528b604a5 100644
--- a/projects/HasseWeil/HasseWeil/Foundation/WronskianAux/CNorm.lean
+++ b/projects/HasseWeil/HasseWeil/Foundation/WronskianAux/CNorm.lean
@@ -1,25 +1,11 @@
 import Mathlib.Algebra.Polynomial.Basic
 
 /-!
-# C-normalization simp set for polynomial ring tactics
+# Normalization of polynomial coefficients
 
-`Polynomial.derivative_pow` produces `C ((n : ℕ) : R) * p^(n-1) * derivative p`,
-where `((n : ℕ) : R)` is `Nat.cast n`. On the other side of an identity,
-`Polynomial.C` is often applied to literals `(n : R)` which use `OfNat.ofNat`.
-`ring` sees `C (Nat.cast n : R)` and `C (OfNat.ofNat n : R)` as distinct atoms,
-blocking cancellation.
-
-The pair `Nat.cast_ofNat` (already `@[simp]` in mathlib) plus
-`Polynomial.C_ofNat` (not `@[simp]`) normalizes every `C k` for numeric `k` to
-the polynomial literal `(k : Polynomial R)`. `Polynomial.C_ofNat` must be
-explicitly included in `simp only` at call sites; this file just documents the
-intended usage.
-
-Typical use:
-```
-simp only [..., Polynomial.C_ofNat, Polynomial.C_mul, Polynomial.C_sub,
-           Polynomial.C_pow, Polynomial.C_add, ...]
-```
-(`Nat.cast_ofNat` fires automatically since it is `@[simp]` — `simp only`
-actually does not pick it up, so include it too for safety.)
+`Polynomial.derivative_pow` introduces coefficients of the form
+`Polynomial.C (Nat.cast n : R)`. Numerical coefficients may instead be written
+with `OfNat.ofNat`, which polynomial ring normalization can treat as a distinct
+atom. Explicit simplification by `Nat.cast_ofNat` and `Polynomial.C_ofNat`
+identifies both forms with the corresponding polynomial numeral.
 -/
diff --git a/projects/HasseWeil/HasseWeil/Foundation/WronskianAux/Prefix.lean b/projects/HasseWeil/HasseWeil/Foundation/WronskianAux/Prefix.lean
new file mode 100644
index 000000000..b1df73ddd
--- /dev/null
+++ b/projects/HasseWeil/HasseWeil/Foundation/WronskianAux/Prefix.lean
@@ -0,0 +1,62 @@
+import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
+import Mathlib.Algebra.Polynomial.Derivation
+import Mathlib.Tactic.LinearCombination
+import HasseWeil.Foundation.WronskianAux.CNorm
+
+
+/-
+Copyright (c) 2026 Chris Birkbeck. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Birkbeck
+-/
+
+/-!
+# Third Wronskian identity
+
+The third division-polynomial Wronskian identity follows from the Weierstrass
+coefficient relation. Its polynomial multiplier has degree eight.
+-/
+
+open WeierstrassCurve Polynomial
+
+namespace HasseWeil
+
+variable {R : Type*} [CommRing R] (W : WeierstrassCurve R)
+
+-- The Weierstrass coefficient relation lifted to the polynomial ring, with
+-- coefficient maps distributed for polynomial normalization.
+lemma wronskian_b_relation_poly :
+    ((4 : R[X]) * Polynomial.C W.b₈ : R[X]) =
+      Polynomial.C W.b₂ * Polynomial.C W.b₆ - Polynomial.C W.b₄ ^ 2 := by
+  simpa [Polynomial.C_ofNat] using congrArg Polynomial.C W.b_relation
+
+/-- Wronskian auxiliary identity, `m = 3` case (Silverman III.3.7).
+`4·Ψ₃³ + 2·preΨ₄·Ψ₂Sq·Ψ₃' − (preΨ₄·Ψ₂Sq)'·Ψ₃
+= 3·preΨ₄·Ψ₂Sq² − 3·preΨ₄²`.
+
+Multiplier: `M = b₈² + 4b₆b₈·X + 6b₄b₈·X² + 4b₂b₈·X³ + (b₂b₆ + 34b₈)·X⁴
+               + 36b₆·X⁵ + 18b₄·X⁶ + 4b₂·X⁷ + 9·X⁸`. -/
+lemma wronskian_aux_three :
+    4 * W.Ψ₃ ^ 3 + 2 * W.preΨ₄ * W.Ψ₂Sq * Polynomial.derivative W.Ψ₃ -
+      Polynomial.derivative (W.preΨ₄ * W.Ψ₂Sq) * W.Ψ₃ =
+    Polynomial.C 3 * W.preΨ₄ * W.Ψ₂Sq ^ 2 - Polynomial.C 3 * W.preΨ₄ ^ 2 := by
+  linear_combination (norm := (
+    simp only [Ψ₃, preΨ₄, Ψ₂Sq,
+      Polynomial.derivative_add, Polynomial.derivative_sub,
+      Polynomial.derivative_mul, Polynomial.derivative_pow, Polynomial.derivative_X,
+      Polynomial.derivative_C, Polynomial.derivative_ofNat,
+      Polynomial.C_add, Polynomial.C_sub, Polynomial.C_mul, Polynomial.C_pow,
+      Polynomial.C_ofNat, Nat.cast_ofNat]
+    ring))
+    (Polynomial.C (W.b₈ ^ 2)
+    + Polynomial.C (4 * W.b₆ * W.b₈) * Polynomial.X
+    + Polynomial.C (6 * W.b₄ * W.b₈) * Polynomial.X ^ 2
+    + Polynomial.C (4 * W.b₂ * W.b₈) * Polynomial.X ^ 3
+    + Polynomial.C (W.b₂ * W.b₆ + 34 * W.b₈) * Polynomial.X ^ 4
+    + Polynomial.C (36 * W.b₆) * Polynomial.X ^ 5
+    + Polynomial.C (18 * W.b₄) * Polynomial.X ^ 6
+    + Polynomial.C (4 * W.b₂) * Polynomial.X ^ 7
+    + Polynomial.C 9 * Polynomial.X ^ 8) * wronskian_b_relation_poly W
+
+
+end HasseWeil
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/Infrastructure.lean b/projects/HasseWeil/HasseWeil/HasseBound/Infrastructure.lean
index 9facfb997176c5ac380d29e70848729672edc3fd..8c198b73959a7180a7e5398bfbe82ea2777afe6b 100644
GIT binary patch
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F{{V?d-7EkA

diff --git a/projects/HasseWeil/HasseWeil/HasseBound/PointCount.lean b/projects/HasseWeil/HasseWeil/HasseBound/PointCount.lean
index e004de056..c8fa5b0d3 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/PointCount.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/PointCount.lean
@@ -149,7 +149,7 @@ theorem card_kernel_eq_pointCount_of_kernel_eq_top
 
 /-- **Sub-helper III-4-015-S3** (fiber witness via degree = pointCount):
 under separability + `β.degree = pointCount`, the fiber witness for the
-Hasse bound consumer ships axiom-clean.
+Hasse bound consumer exists.
 
 Proof chain:
 1. `β.degree = pointCount` (input).
@@ -215,7 +215,7 @@ omit [Fintype K] in
 given `Aut(K(E₁)/α*K(E₂)) ≃ α.kernel` (the substantive Galois-correspondence
 content for elliptic isogenies) and a witness for `Nat.card Aut = α.degree`
 (the IsGalois card_aut_eq_finrank applied to our specific algebra), derive
-`Nat.card α.kernel = α.degree` axiom-clean.
+`Nat.card α.kernel = α.degree` .
 
 Reduces the III.4.10(a) wall to two named witnesses:
 1. `Nat.card Aut = α.degree` (IsGalois + finrank).
@@ -263,9 +263,9 @@ theorem card_aut_eq_degree_of_isGalois
       W.toAffine.FunctionField _ _ β.toAlgebra.toModule) :
     Nat.card (@AlgEquiv W.toAffine.FunctionField W.toAffine.FunctionField
         W.toAffine.FunctionField _ _ _ β.toAlgebra β.toAlgebra) = β.degree := by
-  letI := β.toAlgebra
-  haveI := hgal
-  haveI := hfin
+  let := β.toAlgebra
+  have := hgal
+  have := hfin
   exact IsGalois.card_aut_eq_finrank W.toAffine.FunctionField W.toAffine.FunctionField
 
 /-- **Sub-helper III-4-015-S9** (full chain: IsGalois + bijection → fiber witness):
@@ -275,7 +275,7 @@ final closing-arc consumer. Takes:
 - Aut ≃ β.kernel bijection (the substantive Galois-correspondence content).
 
 Produces the fiber witness used by the historical witness-parametric
-Hasse-bound route, axiom-clean.
+Hasse-bound route, .
 
 The cascade reduces T-III-4-015 to two named witnesses via this consumer:
 1. `IsGalois (α*K(E₂)) K(E₁)` (Mathlib has the structure; needs Normal).
@@ -346,7 +346,7 @@ theorem fiber_witness_via_inverse_witnesses
     (aut_kernel_equiv_of_inverse_witnesses W β _ forward inverse
       h_left_inv h_right_inv)
 
-/-- **Forward map at k = 0** (axiom-clean): the translation by 0 is the identity
+/-- **Forward map at k = 0** (): the translation by 0 is the identity
 algebra automorphism. -/
 noncomputable def aut_of_kernel_zero
     {F : Type*} [Field F] [DecidableEq F]
@@ -354,7 +354,7 @@ noncomputable def aut_of_kernel_zero
     W.toAffine.FunctionField ≃ₐ[F] W.toAffine.FunctionField :=
   AlgEquiv.refl
 
-/-- **Forward map at k = 0 is the identity** (axiom-clean). -/
+/-- **Forward map at k = 0 is the identity** (). -/
 @[simp] theorem aut_of_kernel_zero_apply
     {F : Type*} [Field F] [DecidableEq F]
     (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
@@ -362,21 +362,9 @@ noncomputable def aut_of_kernel_zero
     aut_of_kernel_zero W f = f :=
   rfl
 
-/-- **Substantive partial forward-map construction** (Sub-helper III-4-015-S12):
-the forward map for the substantive cases requires the curve-translation
-algebra automorphism `τ_k : K(E) ≃ₐ[F] K(E)` defined by
-`τ_k(x_gen) = x-coord(P_gen + k)` and `τ_k(y_gen) = y-coord(P_gen + k)`.
-
-Construction parallels Worker A's `addPullbackAlgHom_negFrobenius`:
-1. Define `τ_k` via the addition formula (`addX`, `addY`).
-2. Show it's a K-algebra hom (well-defined modulo Weierstrass).
-3. Show it has inverse `τ_(-k)` (since τ is a group action).
-4. For k ∈ β.kernel, show `τ_k` fixes `β.pullback`'s image.
-
-This sub-helper takes the substantive `τ_k` AS A WITNESS — a precise
-statement of what's needed beyond what's currently in mathlib. The
-construction itself is ~200-300 LOC of `addCoordAlgHom`-style infrastructure
-applied to the translation map. -/
+/-- A kernel-indexed family of translation algebra automorphisms.
+For a kernel point `k`, translation by `k` fixes the image of the pullback;
+the family is supplied as an explicit witness. -/
 noncomputable def aut_of_kernel_construction_witness
     (β : Isogeny W.toAffine W.toAffine)
     (translation_at : β.kernel →
@@ -400,7 +388,7 @@ noncomputable def kernelTranslateForward
     kernelTranslateForward W β ⟨0, h_zero_mem⟩ = AlgEquiv.refl := rfl
 
 /-- **Layer 2 forward map under `β.toAlgebra`** (witness-parametric).
-Takes Worker A's `translateAlgEquivOfPoint W k.val` (an F-AlgEquiv) and
+Takes `translateAlgEquivOfPoint W k.val` (an F-AlgEquiv) and
 the covariance identity `τ_k ∘ β.pullback = β.pullback` (provided as
 hypothesis), and produces the `K(E)`-AlgEquiv (under `β.toAlgebra`)
 consumed by the S10 bijection. -/
@@ -425,7 +413,7 @@ noncomputable def kernelTranslateForwardAsAut
       W.toAffine.FunctionField _ _ _ β.toAlgebra β.toAlgebra) :=
   fun k ↦ kernelTranslateAsAut W β k (h_invariance_family k)
 
-/-- **`kernelTranslateAsAut` at `k = 0`** is the identity AlgEquiv (axiom-clean).
+/-- **`kernelTranslateAsAut` at `k = 0`** is the identity AlgEquiv ().
 The `h_invariance` hypothesis at k = 0 is automatic since `τ_0 = refl`. -/
 @[simp] theorem kernelTranslateAsAut_zero
     (β : Isogeny W.toAffine W.toAffine)
@@ -436,7 +424,7 @@ The `h_invariance` hypothesis at k = 0 is automatic since `τ_0 = refl`. -/
     kernelTranslateAsAut W β ⟨0, h_zero_mem⟩ h_invariance =
       @AlgEquiv.refl W.toAffine.FunctionField W.toAffine.FunctionField
         _ _ β.toAlgebra := by
-  letI := β.toAlgebra
+  let := β.toAlgebra
   apply AlgEquiv.ext
   intro f
   change translateAlgEquivOfPoint W (0 : W.toAffine.Point) f = f
@@ -474,6 +462,7 @@ omit [DecidableEq F] [W.toAffine.IsElliptic] in
     (x : W.toAffine.FunctionField) :
     x ∈ algHomFieldEqualizer W f g ↔ f x = g x := Iff.rfl
 
+omit [DecidableEq F] in
 /-- **Two `F`-AlgHoms `K(E) → K(E)` agreeing on `x_gen`, `y_gen` are equal**:
 `K(E)` is generated over `F` by `x_gen` and `y_gen`. -/
 theorem algHom_ext_of_eq_on_xy [Fintype F]
@@ -553,7 +542,7 @@ theorem kernel_pullback_invariance_id
   rw [translateAlgEquivOfPoint_zero]
   rfl
 
-/-- **`SMulCommClass` for Worker A's master action**: the translation
+/-- **Commuting scalar multiplication for the translation action**: the translation
 action commutes with `F`-scalar multiplication because every
 `translateAlgEquivOfPoint W k` is an F-AlgEquiv (hence F-linear). -/
 instance translateMulSemiringAction_smulCommClass :
@@ -564,7 +553,7 @@ instance translateMulSemiringAction_smulCommClass :
     rw [Algebra.smul_def, Algebra.smul_def, map_mul, AlgEquiv.commutes]
 
 /-- **Restricted action**: `Multiplicative β.kernel` acts on `K(E)` via
-the inclusion `β.kernel → E.Point` composed with Worker A's master action
+the inclusion `β.kernel → E.Point` composed with the translation action
 `translateMulSemiringAction`. -/
 noncomputable instance kernelMulSemiringAction
     (β : Isogeny W.toAffine W.toAffine) :
@@ -632,12 +621,12 @@ theorem xy_family_zero
         (β.pullback (y_gen W)) = β.pullback (y_gen W)) :=
   ⟨rfl, rfl⟩
 
-/-- **Layer 2 closure (witness-parametric on finrank match)**: under the
+/-- **Fixed-field equality from a rank equality**: under the
 xy-covariance family + finrank equality, `β.pullback.fieldRange =
 FixedPoints.intermediateField (Multiplicative β.kernel)`.
 
 Direct application of `IntermediateField.eq_of_le_of_finrank_eq'` to the
-forward inclusion (just shipped) plus the finrank-match witness. -/
+forward inclusion  plus the finrank-match witness. -/
 theorem pullback_fieldRange_eq_fixedField_of_finrank_match [Fintype F]
     (β : Isogeny W.toAffine W.toAffine)
     [hfindim : FiniteDimensional ↥β.pullback.fieldRange W.toAffine.FunctionField]
@@ -659,11 +648,11 @@ theorem pullback_fieldRange_eq_fixedField_of_finrank_match [Fintype F]
     (pullback_fieldRange_le_fixedField_of_xy_family W β h_xy_family)
     h_finrank_match
 
-/-- **Layer 2 closure (Artin-machinery sourced)**: the finrank match
+/-- **Fixed-field equality from Artin's theorem**: the finrank match
 witness can be sourced from Mathlib's `FixedPoints.finrank_eq_card` (which
-gives `[K(E) : FixedPoints] = |G|` axiom-clean from Faithful + Fintype) and
+gives `[K(E) : FixedPoints] = |G|`  from Faithful + Fintype) and
 the cardinality match `|β.kernel| = β.degree` (Silverman III.4.10(b),
-T-V-1-003 / V.1.3) plus the intrinsic relation
+V.1.3) plus the intrinsic relation
 `[K(E) : β.pullback.fieldRange] = β.degree` (witness-parametric below).
 
 This is the **packaged Artin-route closure**: takes the Hasse-content card
@@ -707,11 +696,10 @@ theorem faithfulSMul_kernel_of_translate_inj
 
 /-- **`FaithfulSMul` (UNCONDITIONAL)**: discharges the witness in
 `faithfulSMul_kernel_of_translate_inj` using
-`translateAlgEquivOfPoint_injective` (just shipped in `EC/TranslationOrd`).
+`translateAlgEquivOfPoint_injective` .
 
 Direct corollary: for any isogeny β over a finite field, the restricted
-action `kernelMulSemiringAction β` is faithful axiom-clean. This collapses
-the FaithfulSMul witness in the Layer-2 cascade — combined with finiteness
+action `kernelMulSemiringAction β` is faithful . Combined with finiteness
 of β.kernel (over finite F), Mathlib's Artin machinery
 (`FixedPoints.finrank_eq_card`) applies unconditionally. -/
 instance faithfulSMul_kernel
@@ -730,10 +718,10 @@ closure given the cardinality match `Fintype.card β.kernel = β.degree`
 theorem finrank_pullback_fieldRange_eq_degree
     (β : Isogeny W.toAffine W.toAffine) :
     Module.finrank ↥β.pullback.fieldRange W.toAffine.FunctionField = β.degree := by
-  letI inst1 : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := β.toAlgebra
-  letI inst_im : Algebra ↥β.pullback.fieldRange W.toAffine.FunctionField :=
+  let inst1 : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := β.toAlgebra
+  let inst_im : Algebra ↥β.pullback.fieldRange W.toAffine.FunctionField :=
     IntermediateField.toAlgebra _
-  letI mod_im : Module ↥β.pullback.fieldRange W.toAffine.FunctionField :=
+  let mod_im : Module ↥β.pullback.fieldRange W.toAffine.FunctionField :=
     inst_im.toModule
   change @Module.finrank ↥β.pullback.fieldRange W.toAffine.FunctionField _ _ mod_im =
     β.degree
@@ -758,7 +746,7 @@ theorem finrank_pullback_fieldRange_eq_degree
   exact (Algebra.finrank_eq_of_equiv_equiv i j h_compat).symm
 
 /-- **Layer 2 closure (UNCONDITIONAL on the intrinsic part)**: combines
-`pullback_fieldRange_eq_fixedField_of_finrank_match` with the just-shipped
+`pullback_fieldRange_eq_fixedField_of_finrank_match` with the
 intrinsic finrank relation, leaving only the Hasse-content cardinality
 match `Nat.card β.kernel = β.degree` as a witness.
 
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/PoleDivisorFallback.lean b/projects/HasseWeil/HasseWeil/HasseBound/PoleDivisorFallback.lean
index 6c96651f4795880dc78a3532568e069e75c9f7d2..cc60cff7787880be5b3c3d19acf7c586481df8fa 100644
GIT binary patch
delta 1398
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zi-bBi$s$M=ZkD@<i5gwz9}63~AvfwsHYH$!gf1XXH{1mO5)w(&_=lKzUt1(`UHtFf
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Hfdc3sx_cCT

diff --git a/projects/HasseWeil/HasseWeil/HasseBound/PoleDivisorTwoTorsion.lean b/projects/HasseWeil/HasseWeil/HasseBound/PoleDivisorTwoTorsion.lean
index 25f0c6948..31fbb612c 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/PoleDivisorTwoTorsion.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/PoleDivisorTwoTorsion.lean
@@ -6,49 +6,20 @@ Authors: Chris Birkbeck
 import HasseWeil.HasseBound.PoleDivisorFallback
 
 /-!
-# Bridge at `addPullbackNumerator_negFrobenius` for 2-torsion (T-T21-2TORSION-BRIDGE)
-
-The non-2-torsion bridge `bridge_at_addPullbackNumerator_negFrobenius_of_non_2_tor`
-(`PoleDivisorFallback.lean:2353`) goes through `bridge_at_x_gen_of_non_2_tor`
-(`PoleDivisorFallback.lean:606`), which uses
-`ord_P_translateX_xy_eq_neg_two_of_non_2_tor` (`EC/TranslationOrd.lean:1379`)
-as the substantive `= -2` identity at `−T` (with `T` non-2-torsion).
-
-This file ships the **2-torsion analogue**:
-`ord_P (translateX_xy W xT yT) = -2` at the smooth point `−T = T` (where
-`−T = T` modulo `Nonsingular`-prop-irrelevance for 2-torsion), and
-propagates it to the base bridges and the full
-`bridge_at_addPullbackNumerator_negFrobenius` at 2-torsion.
-
-The chain is:
-
-* `ord_P_x_gen_sub_const_ge_two_at_2tor` — `ord_P (x_gen − xk) ≥ 2` from the
-  curve identity `(y − yk)·A = (x − xk)·(B − a₁·yk)` at 2-torsion T with
-  `ord_P (y − yk) = 1`, `ord_P A ≥ 1`, `ord_P (B − a₁·yk) = 0`.
-* `ord_P_A_eq_one_at_2tor` — `ord_P A = 1` exactly via strict comparison
-  `ord_P (y − yk) = 1 < 2 ≤ ord_P (a₁·(x − xk))` in `A = (y − yk) + a₁·(x − xk)`.
-* `ord_P_translateSlope_xy_eq_neg_one_at_2tor` — `ord_P slope = −1` exactly
-  via `slope = (B − a₁·yk) / A` with `ord = 0 − 1 = −1`.
-* `ord_P_translateX_xy_eq_neg_two_at_2tor` — `ord_P translateX_xy = −2`
-  exactly via `translateX_xy = slope² + rest` with strict comparison.
-* `bridge_at_x_gen_of_2_tor` — bridge `ord_P (τ_{−T} x_gen) = ord_∞ x_gen`.
-* `bridge_at_x_gen_pow_card_of_2_tor` — bridge for `x_gen^q`.
-* `bridge_at_y_gen_of_2_tor` and `bridge_at_y_gen_pow_card_of_2_tor` —
-  y-side analogues, derived via `translateY_xy` analysis.
-* `bridge_at_addPullbackNumerator_negFrobenius_of_2_tor` — the full Num
-  bridge at 2-torsion via the dominant-T7 strict comparison.
-
-After this lands, the chain `T21 → T22 → T23 → T24` (L6 closer) is
-unblocked: `support_card_eq_pointCount_of_two_torsion_ord_witness`
-(`L6Witnesses.lean:497`) consumes the 2-torsion ord witness, and the
-T22 composer becomes unconditional.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, II.1, III.2.
-* `tickets/hasse/T-T21-2TORSION-BRIDGE.md` — the spawned sub-ticket.
-* `EC/TranslationOrd.lean:2566+` — the 2-torsion `A`-factorisation and
-  per-piece order bounds.
+# Orders of translated coordinates at smooth two-torsion points
+
+At a smooth two-torsion point `T`, the curve identity
+`(y − y_T)A = (x − x_T)(B − a₁y_T)` gives `ord_T(x − x_T) ≥ 2`.
+The factorization `A = (y − y_T) + a₁(x − x_T)` then gives `ord_T A = 1`,
+so the translation slope has order `-1`. Strict comparison in the addition
+formulas yields orders `-2` and `-3` for the translated x- and y-coordinates.
+
+These coordinate orders identify local orders after translation with orders at
+infinity, including powers, products, and the reduced addition numerator for
+negative Frobenius. In that numerator the term `x · (π*x)²`, of order `-2-4q`,
+strictly dominates the other terms.
+
+References: Silverman, *The Arithmetic of Elliptic Curves*, II.1 and III.2.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
@@ -76,7 +47,7 @@ private lemma pi_y_gen_ne_zero_aux (hq : 2 ≤ Fintype.card K) :
   rw [ordAtInfty_negFrobeniusIsog_pullback_y_gen W hq] at h_top
   exact WithTop.coe_ne_top h_top
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **`ord_P (x_gen − xk) ≥ 2` at smooth 2-torsion `T`** via the curve
 identity `(y − yk)·A = (x − xk)·(B − a₁·yk)` and the ord facts at
 2-torsion. -/
@@ -117,7 +88,9 @@ theorem ord_P_x_gen_sub_const_ge_two_at_2tor (xk yk : K)
         (y_gen W + algebraMap K KE yk +
           algebraMap K KE W.a₁ * x_gen W + algebraMap K KE W.a₃)) := by
     rw [h_LHS_mul, h_yd_ord]
-    exact add_le_add (le_refl _) h_A_ge
+    calc
+      ((2 : ℤ) : WithTop ℤ) = ((1 : ℤ) : WithTop ℤ) + ((1 : ℤ) : WithTop ℤ) := by norm_num
+      _ ≤ _ := add_le_add le_rfl h_A_ge
   have h_RHS_ord : ((2 : ℤ) : WithTop ℤ) ≤ (W_smooth W).ord_P P
       ((x_gen W - algebraMap K KE xk) *
         (x_gen W ^ 2 + x_gen W * algebraMap K KE xk +
@@ -143,7 +116,7 @@ theorem ord_P_x_gen_sub_const_ge_two_at_2tor (xk yk : K)
   rw [h_mul_eq, h_Bma_ord, add_zero] at h_RHS_ord
   exact h_RHS_ord
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **`ord_P A = 1` exactly at smooth 2-torsion `T`** via strict comparison
 `ord_P (y − yk) = 1 < 2 ≤ ord_P (a₁·(x − xk))` in the factorisation
 `A = (y − yk) + a₁·(x − xk)`. -/
@@ -185,7 +158,7 @@ theorem ord_P_A_eq_one_at_2tor (xk yk : K)
   have h_add := SmoothPlaneCurve.ord_P_add_eq_of_lt h_strict
   exact h_add.trans h_yd_ord
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **`ord_P slope = -1` exactly at smooth 2-torsion `T`** via
 `slope = (B − a₁·yk) / A` with `ord = 0 − 1 = −1` (using the
 sharpened `ord_P A = 1`). -/
@@ -233,7 +206,7 @@ theorem ord_P_translateSlope_xy_eq_neg_one_at_2tor (xk yk : K)
   rw [h_mul, h_inv, h_Bma_ord, h_A_ord, zero_add]
   rfl
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **`ord_P (translateX_xy) = -2` exactly at smooth 2-torsion `T`**:
 `translateX_xy = slope² + (a₁·slope − a₂ − x_gen − xk)`. With
 `ord_P slope = −1` exactly, `ord_P (slope²) = −2`, and the rest has
@@ -383,7 +356,7 @@ theorem ord_T_translateAlgEquivOfPoint_neg_x_gen_eq_neg_two_at_2tor (xT yT : K)
 omit [Fintype K] in
 /-- **Bridge at `f = x_gen` for 2-torsion `T`** (clean version): the analog of
 `bridge_at_x_gen_of_non_2_tor` for 2-torsion. Composes the 2-torsion `ord_T`
-value with the shipped `ordAtInfty_x_gen = -2`. -/
+value with the `ordAtInfty_x_gen = -2`. -/
 theorem bridge_at_x_gen_of_2_tor (xT yT : K)
     (h_ns : W.toAffine.Nonsingular xT yT) (h_2_tor : yT = W.toAffine.negY xT yT) :
     (W_smooth W).ord_P (⟨xT, yT, h_ns⟩ : (W_smooth W).SmoothPoint)
@@ -474,7 +447,7 @@ theorem bridge_at_x_gen_sub_negFrobeniusIsog_pullback_x_gen_of_2_tor (xT yT : K)
   exact bridge_at_x_gen_sub_x_gen_pow_card_of_2_tor W xT yT h_ns h_2_tor hq
 
 /-- **Bridge at `(x_gen - (negFrob).pullback x_gen)^2` (slope-denominator squared)**
-for 2-torsion `T`. Pow on the just-shipped bridge for `x_gen - negFrob.pullback x_gen`.
+for 2-torsion `T`. Pow on the bridge for `x_gen - negFrob.pullback x_gen`.
 Mirrors `bridge_at_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq_of_non_2_tor`. -/
 theorem bridge_at_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq_of_2_tor (xT yT : K)
     (h_ns : W.toAffine.Nonsingular xT yT) (h_2_tor : yT = W.toAffine.negY xT yT)
@@ -492,7 +465,7 @@ theorem bridge_at_x_gen_sub_negFrobeniusIsog_pullback_x_gen_sq_of_2_tor (xT yT :
       W xT yT h_ns h_2_tor hq) 2
 
 /-- **Bridge at `((negFrob).pullback x_gen)^2` for 2-torsion `T`**: pow on
-the just-shipped bridge for `(negFrob).pullback x_gen`. Mirrors the non-2-tor
+the bridge for `(negFrob).pullback x_gen`. Mirrors the non-2-tor
 version (`bridge_at_negFrobeniusIsog_pullback_x_gen_sq_of_non_2_tor`). -/
 theorem bridge_at_negFrobeniusIsog_pullback_x_gen_sq_of_2_tor (xT yT : K)
     (h_ns : W.toAffine.Nonsingular xT yT) (h_2_tor : yT = W.toAffine.negY xT yT) :
@@ -525,7 +498,7 @@ theorem bridge_at_x_gen_mul_negFrobeniusIsog_pullback_x_gen_sq_of_2_tor (xT yT :
     (bridge_at_negFrobeniusIsog_pullback_x_gen_sq_of_2_tor W xT yT h_ns h_2_tor)
 
 /-- **Bridge at T6 = `x_gen² · (negFrob).pullback x_gen`** for 2-torsion `T`:
-mul on bridges for `x_gen²` (pow) and `(negFrob).pullback x_gen` (shipped). -/
+mul on bridges for `x_gen²` (pow) and `(negFrob).pullback x_gen` . -/
 theorem bridge_at_x_gen_sq_mul_negFrobeniusIsog_pullback_x_gen_of_2_tor (xT yT : K)
     (h_ns : W.toAffine.Nonsingular xT yT) (h_2_tor : yT = W.toAffine.negY xT yT) :
     (W_smooth W).ord_P (⟨xT, yT, h_ns⟩ : (W_smooth W).SmoothPoint)
@@ -563,11 +536,10 @@ theorem bridge_at_x_gen_mul_negFrobeniusIsog_pullback_x_gen_of_2_tor (xT yT : K)
     (bridge_at_x_gen_of_2_tor W xT yT h_ns h_2_tor)
     (bridge_at_negFrobeniusIsog_pullback_x_gen_of_2_tor W xT yT h_ns h_2_tor)
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **`ord_P (x_gen − xk) = 2` EXACTLY at smooth 2-torsion `T`** via the curve
 identity `(y − yk)·A = (x − xk)·(B − a₁·yk)` with the EXACT `ord_P A = 1`
-(shipped at `ord_P_A_eq_one_at_2tor`) and `ord_P (B − a₁·yk) = 0` (shipped at
-`ord_P_B_minus_a1_yk_eq_zero_at_2tor`). Tightens the shipped `≥ 2` to equality. -/
+(given by `ord_P_A_eq_one_at_2tor`) and `ord_P (B − a₁·yk) = 0` (given by `ord_P_B_minus_a1_yk_eq_zero_at_2tor`). Tightens the `≥ 2` to equality. -/
 theorem ord_P_x_gen_sub_const_eq_two_at_2tor (xk yk : K)
     (h_ns : W.toAffine.Nonsingular xk yk) (h_2_tor : yk = W.toAffine.negY xk yk) :
     (W_smooth W).ord_P (negSmoothPoint W xk yk h_ns)
@@ -631,10 +603,9 @@ theorem ord_P_x_gen_sub_const_eq_two_at_2tor (xk yk : K)
   rw [h_mul_eq, h_Bma_ord, add_zero] at h_RHS_ord
   exact h_RHS_ord
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Nonnegativity of `ord_P` of the `T3` coefficient `yd·(a₂ + 2x + xk') − a₁²·yd`, given the basic
-order data.  Extracted from `ord_P_translateY_xy_eq_neg_three_at_2tor` to keep its per-term `ord_P`
-bound bookkeeping under the default heartbeat budget. -/
+order data. -/
 theorem ord_P_translateY_T3_coef_nonneg
     {P : (W_smooth W).SmoothPoint} {yd a1 a2 xk' : KE}
     (h_yd_ord : (W_smooth W).ord_P P yd = ((1 : ℤ) : WithTop ℤ))
@@ -717,8 +688,7 @@ theorem ord_P_translateY_T3_coef_nonneg
 
 omit [Fintype K] [DecidableEq K] [W.toAffine.IsElliptic] in
 /-- Nonnegativity of `ord_P` of the `T4` coefficient `−y + a₁·(a₂ + x + xk') − a₃`, given the basic
-order data.  Extracted from `ord_P_translateY_xy_eq_neg_three_at_2tor` to keep its per-term `ord_P`
-bound bookkeeping under the default heartbeat budget. -/
+order data. -/
 theorem ord_P_translateY_T4_coef_nonneg
     {P : (W_smooth W).SmoothPoint} {a1 a2 a3 xk' : KE}
     (h_xg_nn : (0 : WithTop ℤ) ≤ (W_smooth W).ord_P P (x_gen W))
@@ -779,9 +749,9 @@ theorem ord_P_translateY_T4_coef_nonneg
     SmoothPlaneCurve.ord_P_add_le (P := P) _ _
   exact (le_min h12' h_neg_a3_nn).trans h123
 
-omit [Fintype K] in
+omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Nonnegativity of `ord_P (y_gen)`: the generic `y`-coordinate has no pole at any smooth point,
-since its `pointValuation` is `≤ 1`.  Extracted from `ord_P_translateY_xy_eq_neg_three_at_2tor`. -/
+since its `pointValuation` is `≤ 1`. -/
 private theorem ord_P_y_gen_nonneg {P : (W_smooth W).SmoothPoint} :
     (0 : WithTop ℤ) ≤ (W_smooth W).ord_P P (y_gen W) := by
   by_cases hf : y_gen W = 0
@@ -794,7 +764,7 @@ private theorem ord_P_y_gen_nonneg {P : (W_smooth W).SmoothPoint} :
     have hv : (W_smooth W).pointValuation P (y_gen W) ≠ 0 :=
       ((W_smooth W).pointValuation P).ne_zero_iff.mpr hf
     simp only [SmoothPlaneCurve.ord_P]
-    rw [dif_neg hv]
+    rw [dite_eq_right hv]
     rw [show (0 : WithTop ℤ) = ((0 : ℤ) : WithTop ℤ) from rfl,
         WithTop.coe_le_coe]
     have h_unz_le : WithZero.unzero hv ≤ 1 := by
@@ -807,9 +777,9 @@ private theorem ord_P_y_gen_nonneg {P : (W_smooth W).SmoothPoint} :
       rw [h1] at h2; exact h2
     omega
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The `T2` term `-(2·a₁·yd²·xd)` has `ord_P ≥ 4` at 2-torsion, given `ord_P yd = 1`,
-`ord_P xd = 2` and `ord_P a₁ ≥ 0`.  Extracted from `ord_P_translateY_xy_eq_neg_three_at_2tor`. -/
+`ord_P xd = 2` and `ord_P a₁ ≥ 0`. -/
 private theorem ord_P_translateY_T2_term_ge_four
     {P : (W_smooth W).SmoothPoint} {a1 yd xd : KE}
     (h_yd_sq : (W_smooth W).ord_P P (yd ^ 2) = ((2 : ℤ) : WithTop ℤ))
@@ -857,10 +827,9 @@ private theorem ord_P_translateY_T2_term_ge_four
           (((2 : ℤ) : WithTop ℤ) + ((2 : ℤ) : WithTop ℤ)) :=
         add_le_add h_2a1_nn (le_refl _)
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The `T3` term `(yd·(a₂ + 2x + xk') - a₁²·yd)·xd²` has `ord_P ≥ 4` at 2-torsion: the
-coefficient is nonnegative (`ord_P_translateY_T3_coef_nonneg`) and `ord_P xd² = 4`.  Extracted from
-`ord_P_translateY_xy_eq_neg_three_at_2tor`. -/
+coefficient is nonnegative (`ord_P_translateY_T3_coef_nonneg`) and `ord_P xd² = 4`. -/
 private theorem ord_P_translateY_T3_term_ge_four
     {P : (W_smooth W).SmoothPoint} {a1 a2 yd xd xk' : KE}
     (h_yd_ord : (W_smooth W).ord_P P yd = ((1 : ℤ) : WithTop ℤ))
@@ -889,8 +858,7 @@ private theorem ord_P_translateY_T3_term_ge_four
 
 omit [Fintype K] [DecidableEq K] [W.toAffine.IsElliptic] in
 /-- The `T4` term `(-y + a₁·(a₂ + x + xk') - a₃)·xd³` has `ord_P ≥ 6` at 2-torsion: the
-coefficient is nonnegative (`ord_P_translateY_T4_coef_nonneg`) and `ord_P xd³ = 6`.  Extracted from
-`ord_P_translateY_xy_eq_neg_three_at_2tor`. -/
+coefficient is nonnegative (`ord_P_translateY_T4_coef_nonneg`) and `ord_P xd³ = 6`. -/
 private theorem ord_P_translateY_T4_term_ge_six
     {P : (W_smooth W).SmoothPoint} {a1 a2 a3 xd xk' : KE}
     (h_xd_cube_eq : (W_smooth W).ord_P P (xd ^ 3) = ((6 : ℤ) : WithTop ℤ))
@@ -920,8 +888,7 @@ private theorem ord_P_translateY_T4_term_ge_six
 
 omit [Fintype K] [DecidableEq K] [W.toAffine.IsElliptic] in
 /-- Extract `ord_P f = -3` from `ord_P (f · g) = 3` and `ord_P g = 6`: since `f · g` has finite
-order, `f ≠ 0`, so `ord_P f` is a finite integer `k` with `k + 6 = 3`.  Extracted from
-`ord_P_translateY_xy_eq_neg_three_at_2tor` (with `g = xd³`). -/
+order, `f ≠ 0`, so `ord_P f` is a finite integer `k` with `k + 6 = 3`. -/
 private theorem ord_P_eq_neg_three_of_mul_eq_three
     {P : (W_smooth W).SmoothPoint} {f g : KE}
     (h_LHS_ord : (W_smooth W).ord_P P (f * g) = ((3 : ℤ) : WithTop ℤ))
@@ -949,8 +916,7 @@ private theorem ord_P_eq_neg_three_of_mul_eq_three
 
 omit [Fintype K] [DecidableEq K] [W.toAffine.IsElliptic] in
 /-- A four-term sum `d + (A + (B + C))` has `ord_P` equal to that of its dominant term `d` when
-`ord_P d = 3` and each of `A`, `B`, `C` has `ord_P ≥ 4`.  Extracted from
-`ord_P_translateY_xy_eq_neg_three_at_2tor`. -/
+`ord_P d = 3` and each of `A`, `B`, `C` has `ord_P ≥ 4`. -/
 private theorem ord_P_dominant_sum_eq_three
     {P : (W_smooth W).SmoothPoint} {d A B C : KE}
     (h_d : (W_smooth W).ord_P P d = ((3 : ℤ) : WithTop ℤ))
@@ -969,15 +935,14 @@ private theorem ord_P_dominant_sum_eq_three
     exact lt_of_lt_of_le (by exact_mod_cast (show (3 : ℤ) < 4 by norm_num)) h_ABC
   exact (SmoothPlaneCurve.ord_P_add_eq_of_lt h_strict).trans h_d
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **`ord_P (translateY_xy) = -3` at smooth 2-torsion `T`** via the algebraic
 identity `translateY_xy_mul_cube_eq` and the 2-tor ord values
-`ord_P (yd) = 1` (shipped at `ord_P_y_gen_sub_negY_const_eq_one_of_2_tor`)
-+ `ord_P (xd) = 2` exactly (shipped at `ord_P_x_gen_sub_const_eq_two_at_2tor`).
+`ord_P (yd) = 1` (given by `ord_P_y_gen_sub_negY_const_eq_one_of_2_tor`)
++ `ord_P (xd) = 2` exactly (given by `ord_P_x_gen_sub_const_eq_two_at_2tor`).
 At 2-tor the RHS dominant is `-yd³` with ord = 3, all other RHS terms have
 ord ≥ 4 (strict); LHS = `translateY · xd³` has ord = `ord(translateY) + 6`,
-forcing `ord(translateY) = -3`. Mirrors the non-2-tor proof at
-`EC/TranslationOrd.lean:1536`. -/
+forcing `ord(translateY) = -3`. Mirrors the non-2-tor proof at -/
 theorem ord_P_translateY_xy_eq_neg_three_at_2tor (xk yk : K)
     (h_ns : W.toAffine.Nonsingular xk yk) (h_2_tor : yk = W.toAffine.negY xk yk) :
     (W_smooth W).ord_P (negSmoothPoint W xk yk h_ns)
@@ -1072,11 +1037,7 @@ theorem ord_P_translateY_xy_eq_neg_three_at_2tor (xk yk : K)
   -- ord_P translateY + 6 = 3 ⟹ ord_P translateY = -3.
   exact ord_P_eq_neg_three_of_mul_eq_three W h_LHS_ord h_xd_cube_eq
 
-/-- **Type abbreviation for the y-side substantive value at 2-torsion T**:
-`ord_P (translateY_xy) = -3` at the negated point. This is the single
-substantive witness the y-side chain reduces to.
-
-⟶ Now DISCHARGEABLE via `ord_P_translateY_xy_eq_neg_three_at_2tor` (just shipped). -/
+/-- The y-coordinate after translation has order `-3` at the negated point. -/
 abbrev TwoTorYValueWitness (xT yT : K) (h_ns : W.toAffine.Nonsingular xT yT) : Prop :=
   (W_smooth W).ord_P
       (negSmoothPoint W xT (W.toAffine.negY xT yT)
@@ -1084,7 +1045,7 @@ abbrev TwoTorYValueWitness (xT yT : K) (h_ns : W.toAffine.Nonsingular xT yT) : P
       (translateY_xy W xT (W.toAffine.negY xT yT)) =
     ((-3 : ℤ) : WithTop ℤ)
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- **Discharge `TwoTorYValueWitness` from the substantive y-side lemma**: at smooth
 2-torsion `T`, the y-side ord = -3 value follows directly from
 `ord_P_translateY_xy_eq_neg_three_at_2tor` applied at the negated coordinates. -/
@@ -1222,7 +1183,7 @@ theorem bridge_at_y_gen_of_2_tor (xT yT : K)
     (twoTorYValueWitness_discharge W xT yT h_ns h_2_tor)
 
 /-- **Bridge at `f = y_gen^q` for 2-torsion `T` (UNCONDITIONAL)**: pow on the
-just-shipped unconditional y-side base bridge. -/
+the y-coordinate order identity. -/
 theorem bridge_at_y_gen_pow_card_of_2_tor (xT yT : K)
     (h_ns : W.toAffine.Nonsingular xT yT) (h_2_tor : yT = W.toAffine.negY xT yT) :
     (W_smooth W).ord_P (⟨xT, yT, h_ns⟩ : (W_smooth W).SmoothPoint)
@@ -1246,7 +1207,7 @@ theorem bridge_at_y_gen_sub_y_gen_pow_card_of_2_tor (xT yT : K)
     (twoTorYValueWitness_discharge W xT yT h_ns h_2_tor) hq
 
 /-- **Bridge at `(negFrob).pullback y_gen` for 2-torsion `T` (UNCONDITIONAL)**:
-mirrors `bridge_at_negFrobeniusIsog_pullback_y_gen_of_non_2_tor` (`PoleDivisorFallback.lean:1034`)
+mirrors `bridge_at_negFrobeniusIsog_pullback_y_gen_of_non_2_tor`
 using the now-unconditional 2-tor y-side bridges. The pullback expands as
 `-y_gen^q - a₁·x_gen^q - a₃`; strict-comparison isolates `-y_gen^q` with ord `-3q`
 as the dominant term. -/
@@ -1393,7 +1354,7 @@ theorem bridge_at_negFrobeniusIsog_pullback_y_gen_of_2_tor (xT yT : K)
     (Conditional.ordAtInfty_neg_y_gen_pow_card_lt_rest W hq)
 
 /-- **Bridge at T1 = `a₄ · (x_gen + (negFrob).pullback x_gen)` for 2-torsion `T` (UNCONDITIONAL)**:
-const_mul on the strict-add bridge (already shipped). -/
+const_mul on the strict-add bridge (proved). -/
 theorem bridge_at_T1_a4_x_add_pi_x_of_2_tor (xT yT : K)
     (h_ns : W.toAffine.Nonsingular xT yT) (h_2_tor : yT = W.toAffine.negY xT yT)
     (hq : 2 ≤ Fintype.card K) :
@@ -1719,7 +1680,7 @@ under `hq`, **UNCONDITIONAL**.
 Mirror of `bridge_at_addPullbackNumerator_negFrobenius_of_non_2_tor`: apply
 `ord_P_translateAlgEquivOfPoint_sum_dominant` with dom = T7 (x · π·x²,
 ord = -2-4q) and the same seven other terms; the per-term bridges are the
-2-torsion versions just shipped, and the strict-comparison framework is
+2-torsion versions proved, and the strict-comparison framework is
 identical because the `ordAtInfty_T*_ge` bounds are universal in T. -/
 theorem bridge_at_addPullbackNumerator_negFrobenius_of_2_tor (xT yT : K)
     (h_ns : W.toAffine.Nonsingular xT yT) (h_2_tor : yT = W.toAffine.negY xT yT)
@@ -1811,7 +1772,7 @@ theorem bridge_at_addPullbackNumerator_negFrobenius_of_2_tor (xT yT : K)
 /-- **Bridge at `addPullback_x W (negFrobeniusIsog W)`** for 2-torsion T
 under `hq`, **UNCONDITIONAL**.  Discharges the
 `bridge_at_addPullback_x_negFrobenius_of_bridge_at_Num` Conditional consumer
-with the just-shipped 2-tor Num bridge.
+with the 2-tor Num bridge.
 
 Note: `bridge_at_addPullback_x_negFrobenius_of_bridge_at_Num` requires a
 non-2-torsion hypothesis only for its appeal to
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/Primitives.lean b/projects/HasseWeil/HasseWeil/HasseBound/Primitives.lean
index 7b7c9f9bdd2de81fd56ed1658ddd95a0088e6cef..bc3351f74e8f72d1c306fa97c7af59533db0a65e 100644
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diff --git a/projects/HasseWeil/HasseWeil/HasseBound/Separability.lean b/projects/HasseWeil/HasseWeil/HasseBound/Separability.lean
index b4d1cde93..c55ab26f7 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/Separability.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/Separability.lean
@@ -12,38 +12,19 @@ import HasseWeil.Foundation.PullbackCoeff
 import HasseWeil.Foundation.OmegaCoeffMulByIntGeneral
 
 /-!
-# Witness-parametric Silverman III.5.5 and V.1.2
-
-Silverman III.5.5: for an elliptic curve `E/F_q` with `q`-power Frobenius `π` and
-integers `m, n`, the isogeny `m + n·π` is separable iff `p ∤ m`.
-
-The (unconditional) Silverman proof chains:
-1. III.5.2 additivity  `(φ+ψ)*ω = φ*ω + ψ*ω`
-2. III.5.3 scalar      `[m]*ω = m·ω`
-3. II.2.14 + II.4.4    `π* ω = 0`  (Frobenius is purely inseparable)
-4. II.4.4              separability ⇔ pullback of `ω` nonzero
-
-Steps 1–3 give `(m + n·π)* ω = m · ω`, i.e. the ω-pullback coefficient is
-`(m : F_q)`. Step 4 then gives separability ⇔ `m ≠ 0` in `F_q` ⇔ `¬ p ∣ m`.
-
-Those steps are currently blocked (T-III-5-002 OPEN, T-III-5-003 PARTIAL,
-T-II-4-004 OPEN), so this file provides *witness-parametric* closures:
-
-* `isSeparable_iff_of_coeff_witness` — given the ω-pullback coefficient of `β`
-  as `algebraMap F _ c` and the T-II-4-004 criterion for `β`, concludes
-  `β.IsSeparable ↔ c ≠ 0`.
-* `m_plus_n_frob_isSeparable_iff_of_witness` — specialization with coefficient
-  `(m : K)` (mirrors Silverman III.5.5 for `m + n·π`).
-* `oneSubFrobeniusIsog_isSeparable_of_witness` — specialization with `m = 1,
-  n = -1` (T-V-1-002, `1 − π` separable).
-
-Each uses the same design as existing `_of_witness` theorems
-(`pointCount_eq_of_witness`, `degree_quadratic_nonneg_of_witness`): the caller
-supplies the key external fact; this file chains it to the separability
-conclusion by pure rewriting.
-
-## References
-* [Silverman, *The Arithmetic of Elliptic Curves*], III.5.5, V.1.2.
+# Separability and invariant differentials
+
+For an elliptic curve over a finite field of characteristic `p`, the
+pullback of the invariant differential by `m + n·π` has coefficient `m`.
+Thus separability is equivalent to `m ≠ 0` in the base field, or to
+`p ∤ m`. The coefficient calculations use additivity, the scalar formula,
+and vanishing of the Frobenius pullback.
+
+The witness-parametric forms retain the coefficient identity and the
+nonzero-pullback criterion as explicit hypotheses. In particular the
+coefficient `1` gives separability of `1 − π`.
+
+References: Silverman, *The Arithmetic of Elliptic Curves*, III.5.5, V.1.2.
 -/
 
 open WeierstrassCurve
@@ -57,7 +38,7 @@ variable {F : Type*} [Field F] [DecidableEq F]
 Given `β : Isogeny E E` and `c : F`, suppose:
 * the ω-pullback coefficient of `β` is `algebraMap F _ c`;
 * separability of `β` is equivalent to the ω-pullback coefficient being nonzero
-  (this is T-II-4-004 specialized to `β`).
+  (the nonzero-pullback criterion specialized to `β`).
 
 Then `β.IsSeparable ↔ c ≠ 0` in `F`.
 
@@ -75,7 +56,7 @@ theorem isSeparable_iff_of_coeff_witness
 /-- **Witness-parametric Silverman III.5.5 for `m + n·π`**.
 
 Given an endomorphism isogeny `β` whose ω-pullback coefficient is `(m : K)` (in
-the algebra-map sense), and given the T-II-4-004 criterion for `β`,
+the algebra-map sense), and given the nonzero-pullback criterion for `β`,
 `β.IsSeparable ↔ (m : K) ≠ 0`. When `K` has characteristic `p`, the right-hand
 side is equivalent to `¬ (p : ℤ) ∣ m` via `CharP.intCast_eq_zero_iff`.
 
@@ -167,7 +148,7 @@ For `m : ℤ` nonzero in `K`, the multiplication-by-`m` isogeny `[m]` is separab
 given its ω-pullback coefficient `= algebraMap K _ m` (T-III-5-003 for the
 `[m]` case) and the T-II-4-004 separability criterion.
 
-The coefficient hypothesis is discharged by the axiom-clean field-general
+The coefficient hypothesis is discharged by the field-general
 Route-B chain `omegaCoeff_mulByInt` (`RouteBGeneral.lean`, for `m ≠ 0 : ℤ`);
 passing it as a hypothesis here keeps the present theorem parametric.
 
@@ -184,7 +165,7 @@ theorem mulByInt_isSeparable_of_witness
   (m_plus_n_frob_isSeparable_iff_of_witness W m _ h_coeff h_sep_iff).mpr hm
 
 /-- **T-FROB-OMEGA-ZERO**: in characteristic `p`, the ω-pullback coefficient of
-    `[p]` vanishes. One-line corollary of the axiom-clean `omegaCoeff_mulByInt`
+    `[p]` vanishes. One-line corollary of the `omegaCoeff_mulByInt`
     (`RouteBGeneral.lean`) and `CharP.cast_eq_zero`. -/
 theorem omegaPullbackCoeff_mulByNat_p_eq_zero
     {k : Type*} [Field k] [DecidableEq k] (p : ℕ) [CharP k p] [Fact (Nat.Prime p)]
@@ -220,7 +201,7 @@ theorem inseparableDegree_eq_one_iff_isSeparable (φ : Isogeny W₁ W₂)
     (hfin : @FiniteDimensional W₂.FunctionField W₁.FunctionField _ _
       φ.toAlgebra.toModule) :
     φ.inseparableDegree = 1 ↔ φ.IsSeparable := by
-  letI : Algebra W₂.FunctionField W₁.FunctionField := φ.toAlgebra
+  let : Algebra W₂.FunctionField W₁.FunctionField := φ.toAlgebra
   rw [show φ.IsSeparable ↔ φ.sepDegree = φ.degree from
     isSeparable_iff_sepDegree_eq_degree φ hfin]
   have h_mul : φ.sepDegree * Field.finInsepDegree W₂.FunctionField W₁.FunctionField =
@@ -255,12 +236,12 @@ theorem inseparableDegree_isPow_of_charP
     (α : Isogeny W.toAffine W.toAffine)
     (h_deg_pos : 0 < α.degree) :
     ∃ e : ℕ, α.inseparableDegree = p ^ e := by
-  letI alg : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
-  haveI : CharP W.toAffine.FunctionField p :=
+  let alg : Algebra W.toAffine.FunctionField W.toAffine.FunctionField := α.toAlgebra
+  have : CharP W.toAffine.FunctionField p :=
     charP_of_injective_algebraMap
       (algebraMap K W.toAffine.FunctionField).injective p
-  haveI : ExpChar W.toAffine.FunctionField p := ExpChar.prime Fact.out
-  haveI hfin : @FiniteDimensional W.toAffine.FunctionField
+  have : ExpChar W.toAffine.FunctionField p := ExpChar.prime Fact.out
+  have hfin : @FiniteDimensional W.toAffine.FunctionField
       W.toAffine.FunctionField _ _ alg.toModule :=
     @FiniteDimensional.of_finrank_pos _ _ _ _ alg.toModule h_deg_pos
   obtain ⟨e, he⟩ : ∃ n, @Field.finInsepDegree W.toAffine.FunctionField
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Constancy.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Constancy.lean
index bedd183a7..b22f0976f 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Constancy.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Constancy.lean
@@ -32,6 +32,7 @@ open Curves
 
 variable {F : Type*} [Field F] [DecidableEq F] {W : WeierstrassCurve.Affine F}
 
+omit [DecidableEq F] in
 /-- A nonzero function with trivial projective divisor is induced by a base-field scalar. -/
 theorem const_of_projectiveDivisorOf_eq_zero [IsAlgClosed F] [W.IsElliptic]
     [IsDedekindDomain (⟨W⟩ : SmoothPlaneCurve F).CoordinateRing]
@@ -64,6 +65,7 @@ theorem const_of_projectiveDivisorOf_eq_zero [IsAlgClosed F] [W.IsElliptic]
   rw [← hP, ← pointValuation_eq_heightOneValuation W P f]
   exact pointValuation_le_one_of_ord_nonneg W hf P (hord P)
 
+omit [DecidableEq F] in
 /-- A nonzero function with trivial projective divisor is a nonzero base-field scalar. -/
 theorem const_unit_of_projectiveDivisorOf_eq_zero [IsAlgClosed F] [W.IsElliptic]
     [IsDedekindDomain (⟨W⟩ : SmoothPlaneCurve F).CoordinateRing]
@@ -74,6 +76,7 @@ theorem const_unit_of_projectiveDivisorOf_eq_zero [IsAlgClosed F] [W.IsElliptic]
   refine ⟨c, fun h => hf ?_, hc⟩
   simpa [h] using hc
 
+omit [DecidableEq F] in
 /-- If `τ g / g` has trivial projective divisor, then `τ g` is a nonzero scalar multiple
 of `g`. -/
 theorem pairing_const_of_transport [IsAlgClosed F] [W.IsElliptic]
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorPullback.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorPullback.lean
index aa75199b4..8789a8842 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorPullback.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorPullback.lean
@@ -10,55 +10,23 @@ import HasseWeil.HasseBound.WeilPairing.SigmaBridge
 import HasseWeil.HasseBound.WeilPairing.TorsionGeometric
 
 /-!
-# Divisor-pullback functoriality for the separable multiplication-by-`ℓ` isogeny
+# Divisor pullback for multiplication by `ℓ`
 
-For a separable isogeny `φ = [ℓ]` (`(ℓ : F) ≠ 0`, `[IsAlgClosed F]`) and a nonzero rational
-function `h ∈ K(E)`, the divisor of the pulled-back function `φ.pullback h` equals the geometric
-pullback of `div h`:
-```
-divisorOf (φ.pullback h) = pullback-of-divisor (divisorOf h).
-```
-Because `[ℓ]` is **separable** (unramified — every fibre is étale, multiplicity `1`), the pullback
-of a single place `(Q)` is the multiplicity-free fibre sum `Σ_{φP=Q} (P)` (the repo's
-`pullbackDiv`, `WeilPairing/Pullback.lean`). The geometric heart is the per-place
-**unramified order-transport**
-```
-ord_P (φ.pullback h) = ord_{φ(P)} (h)        (no ramification factor `e_P > 1`),
-```
-which, summed over all places, yields the divisor identity. This is Silverman III.4.10c /
-III.8.1–2 in divisor language: the geometric content of the separability of `[ℓ]`.
-
-## Architecture
-
-* **Item 1 — the core (`OrdTransport` / `ordTransport_mulByInt`).** The per-place transport
-  `ord_P P (φ.pullback h) = (ord of h at φ(P))` is the deep geometric unramifiedness statement.
-  It is the exact isogeny analogue of the translation order-transport proven in
-  `EC/TranslateValuation.lean` + `EC/TranslateOrdInfty.lean` (≈5000 lines): a valuation on `K(E)`
-  pinned by its values on `x_gen`, `y_gen`. For an isogeny the base values
-  `ord_P P (φ.pullback x_gen)`, `ord_P P (φ.pullback y_gen)` are the `ℓ`-division-polynomial orders,
-  whose computation is the genuine ramification content. We **isolate** this as the named predicate
-  `OrdTransport` (a per-`(P, φ)` hypothesis bundling the projective place transport) and prove the
-  full divisor assembly on top of it.
-
-* **Item 2 — the assembly (`projectiveDivisorOf_pullback_eq_pullbackDivisor`).** Given the core
-  for every place, the projective divisor of `φ.pullback h` equals the fibre-pullback of the
-  projective divisor of `h`. This is a pure `Finsupp` computation: every coefficient matches by
-  the core, and the supports are related by the (finite) fibre structure. Fully proven modulo the
-  core.
-
-* **Item 3 — the pairing-facing corollaries.** `pullback_divisorOf_eq_of_divisorOf_eq` (the
-  shape bilinearity-in-`T` and nondegeneracy consume) and the III.8.1 relation
-  `weilFunction_pow_mem_pullback_range`-style statement, parametric on the core.
-
-Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.4.10c, III.8.1, III.8.2.
+For a separable multiplication-by-`ℓ` isogeny over an algebraically closed field,
+the geometric pullback of a place is the multiplicity-free fiber sum.
+The per-place hypothesis `OrdTransport` expresses the unramified order identity
+`ord_P (φ*h) = ord_{φ(P)} h`, including the projective place at infinity.
+Summing these identities over all places identifies the projective divisor of
+`φ*h` with the fiber pullback of the divisor of `h`.
+
+References: Silverman, *The Arithmetic of Elliptic Curves*, III.4.10c,
+III.8.1, and III.8.2.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
 
 namespace HasseWeil.WeilPairing.DivisorPullback
 
-set_option linter.unusedSectionVars false
-
 variable {F : Type*} [Field F] [DecidableEq F] {W : WeierstrassCurve.Affine F} [W.IsElliptic]
 
 /-- Local abbreviation for the function field `K(E)` (`= (⟨W⟩ : SmoothPlaneCurve F).FunctionField`),
@@ -70,6 +38,7 @@ local notation3 "KE" => W.toAffine.FunctionField
 noncomputable def projOrdAt (f : (⟨W⟩ : SmoothPlaneCurve F).FunctionField) (Q : W.Point) : ℤ :=
   (⟨W⟩ : SmoothPlaneCurve F).projectiveDivisorOf f Q.toProjectiveSmoothPoint
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W] in
 theorem projOrdAt_some (f : (⟨W⟩ : SmoothPlaneCurve F).FunctionField) {x y : F}
     (h : W.Nonsingular x y) :
     projOrdAt f (Affine.Point.some x y h) =
@@ -77,6 +46,7 @@ theorem projOrdAt_some (f : (⟨W⟩ : SmoothPlaneCurve F).FunctionField) {x y :
   rw [projOrdAt, Affine.Point.toProjectiveSmoothPoint_some,
     (⟨W⟩ : SmoothPlaneCurve F).projectiveDivisorOf_apply_affine f ⟨x, y, h⟩]
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W] in
 theorem projOrdAt_zero (f : (⟨W⟩ : SmoothPlaneCurve F).FunctionField) :
     projOrdAt f (0 : W.Point) = ((⟨W⟩ : SmoothPlaneCurve F).ordAtInfty f).untopD 0 := by
   rw [projOrdAt, Affine.Point.toProjectiveSmoothPoint_zero,
@@ -107,7 +77,7 @@ theorem coeff_affine_pullback_eq {φ : Isogeny W W}
 
 /-- **The fibre-pullback of a projective divisor** under a point-map endomorphism with finite
 kernel: `φ*(D) = Σ_v D(v) · pullbackDiv(v)`, the `ℤ`-linear extension of the multiplicity-free
-fibre pullback `pullbackDiv` (`WeilPairing/Pullback.lean`) over the (projective) places of `D`.
+fibre pullback `pullbackDiv` over the (projective) places of `D`.
 The place at infinity pulls back to `pullbackDiv f h 0 = Σ_{φP=O}(P)` (the kernel). -/
 noncomputable def pullbackDivisor (f : W.Point →+ W.Point) (hf : Finite f.ker)
     (D : ProjectiveDivisor (⟨W⟩ : SmoothPlaneCurve F)) :
@@ -154,10 +124,10 @@ theorem pullbackDiv_apply (f : W.Point →+ W.Point) (hf : Finite f.ker)
     (Q : W.Point) (w : ProjectiveSmoothPoint (⟨W⟩ : SmoothPlaneCurve F)) :
     pullbackDiv f hf Q w = if f w.toAffinePoint = Q then 1 else 0 := by
   classical
-  letI : Fintype {R : W.Point // f R = Q} := @Fintype.ofFinite _ (fiber_finite f hf Q)
+  let : Fintype {R : W.Point // f R = Q} := @Fintype.ofFinite _ (fiber_finite f hf Q)
   rw [pullbackDiv, Finsupp.finsetSum_apply]
   by_cases hwQ : f w.toAffinePoint = Q
-  · rw [if_pos hwQ]
+  · rw [ite_eq_left hwQ]
     have hwproj : (⟨w.toAffinePoint, hwQ⟩ : {R : W.Point // f R = Q}).val.toProjectiveSmoothPoint =
         w := Affine.Point.toAffinePoint_toProjectiveSmoothPoint w
     rw [Finset.sum_eq_single (⟨w.toAffinePoint, hwQ⟩ : {R : W.Point // f R = Q})]
@@ -173,7 +143,7 @@ theorem pullbackDiv_apply (f : W.Point →+ W.Point) (hf : Finite f.ker)
       exact Subtype.ext hRval
     · intro hni
       exact absurd (Finset.mem_univ _) hni
-  · rw [if_neg hwQ]
+  · rw [ite_eq_right hwQ]
     refine Finset.sum_eq_zero (fun R _ ↦ ?_)
     rw [Finsupp.single_eq_of_ne]
     intro hcontra
@@ -209,9 +179,9 @@ theorem pullbackDivisor_apply (f : W.Point →+ W.Point) (hf : Finite f.ker)
     intro v _
     rw [Finsupp.coe_smul, Pi.smul_apply, pullbackDiv_apply, smul_eq_mul]
     by_cases hv : f w.toAffinePoint = v.toAffinePoint
-    · rw [if_pos hv, mul_one, if_pos]
+    · rw [ite_eq_left hv, mul_one, ite_eq_left]
       rw [hv, Affine.Point.toAffinePoint_toProjectiveSmoothPoint]
-    · rw [if_neg hv, mul_zero, if_neg]
+    · rw [ite_eq_right hv, mul_zero, ite_eq_right]
       intro hcontra
       apply hv
       have := congrArg ProjectiveSmoothPoint.toAffinePoint hcontra
@@ -219,8 +189,8 @@ theorem pullbackDivisor_apply (f : W.Point →+ W.Point) (hf : Finite f.ker)
   rw [Finset.sum_congr rfl hterm,
     Finset.sum_ite_eq D.support (f w.toAffinePoint).toProjectiveSmoothPoint (fun v ↦ D v)]
   by_cases hmem : (f w.toAffinePoint).toProjectiveSmoothPoint ∈ D.support
-  · rw [if_pos hmem]
-  · rw [if_neg hmem, Finsupp.notMem_support_iff.mp hmem]
+  · rw [ite_eq_left hmem]
+  · rw [ite_eq_right hmem, Finsupp.notMem_support_iff.mp hmem]
 
 /-- **The per-place core, projective form.** For the isogeny `φ`, the coefficient of
 `projectiveDivisorOf (φ.pullback h)` at the projective place `w` equals the coefficient of
@@ -336,7 +306,7 @@ theorem inftyOrdTransport_mulByInt (ℓ : ℤ) (hℓ : ℓ ≠ 0) (hℓF : (ℓ
     rw [hm, hn, WithTop.untopD_coe, WithTop.untopD_coe]
     omega
 
-/-- **Comap-valuation identity, affine-image case** (proven, axiom-clean). The comap of
+/-- **Comap-valuation identity, affine-image case** . The comap of
 `pointValuation P` along `[ℓ].pullback` equals `pointValuation` at the affine image smooth point
 `⟨x, y, h_ns⟩`. This is the unramifiedness of `[ℓ]` at an affine image (Silverman III.4.10c). -/
 theorem comap_pointValuation_mulByInt_eq_affine [IsAlgClosed F] (ℓ : ℤ) (hℓ : (ℓ : F) ≠ 0)
@@ -529,8 +499,7 @@ Hence `φ.pullback f_T` and `g_T^ℓ` have the *same* divisor, so they differ by
 (`Constancy`): the relation `f_T ∘ [ℓ] = c · g_T^ℓ` that nondegeneracy consumes.
 
 Parametric on the per-place core `ProjOrdTransport [ℓ]` (`= projOrdTransport_mulByInt`), and on the
-finiteness of `ker[ℓ]` (the caller supplies `mulByInt_ker_finite`, keeping this file decoupled from
-`Pairing.lean`/`TorsionCardEll`). The conclusion's right side is exactly `ℓ • (div g_T)`. -/
+finiteness of `ker[ℓ]`. The conclusion's right side is exactly `ℓ • (div g_T)`. -/
 theorem projectiveDivisorOf_pullback_weilFunction (ℓ : ℤ)
     [hker : Finite (mulByInt W ℓ).toAddMonoidHom.ker] (hcore : ProjOrdTransport (mulByInt W ℓ))
     (T : W.Point) {fT : (⟨W⟩ : SmoothPlaneCurve F).FunctionField}
@@ -573,7 +542,7 @@ divisor the Abel–Jacobi divisor `D = (T₁+T₂) − (T₁) − (T₂) + (O)`,
 they differ by a constant.
 
 Parametric on the per-place core `ProjOrdTransport [ℓ]` (`= projOrdTransport_mulByInt`) and the
-finiteness of `ker[ℓ]` (caller-supplied, decoupling this file from `Pairing.lean`). -/
+finiteness of `ker[ℓ]`. -/
 theorem projectiveDivisorOf_pullback_bilinFunction (ℓ : ℤ)
     [hker : Finite (mulByInt W ℓ).toAddMonoidHom.ker] (hcore : ProjOrdTransport (mulByInt W ℓ))
     (T₁ T₂ : W.Point) {k : (⟨W⟩ : SmoothPlaneCurve F).FunctionField}
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorTranslate.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorTranslate.lean
index 5aa369762..ba8eced54 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorTranslate.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/DivisorTranslate.lean
@@ -11,7 +11,7 @@ import HasseWeil.HasseBound.WeilPairing.WeilFunction
 /-!
 # Divisor transport under translation
 
-This file proves the projective divisor transport facts for translation by a
+We prove the projective divisor transport facts for translation by a
 point on a Weierstrass curve, as used in the Weil pairing construction.
 
 ## Main definitions
@@ -38,8 +38,6 @@ namespace HasseWeil
 
 open HasseWeil.WeilPairing
 
-set_option linter.unusedSectionVars false
-
 variable {F : Type*} [Field F] [DecidableEq F]
 variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
 
@@ -53,6 +51,7 @@ noncomputable def placeTranslate (S : W.toAffine.Point) :
     ((Equiv.addRight S).trans
       (WeierstrassCurve.Affine.Point.equivProjectiveSmoothPoint (W := W.toAffine)))
 
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 theorem placeTranslate_apply (S : W.toAffine.Point)
     (v : ProjectiveSmoothPoint (W_smooth W)) :
     placeTranslate W S v =
@@ -60,6 +59,7 @@ theorem placeTranslate_apply (S : W.toAffine.Point)
         (v.toAffinePoint + S) := by
   rfl
 
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 @[simp] theorem placeTranslate_affine (S : W.toAffine.Point)
     (P : (W_smooth W).SmoothPoint) :
     placeTranslate W S (ProjectiveSmoothPoint.affine P) =
@@ -68,6 +68,7 @@ theorem placeTranslate_apply (S : W.toAffine.Point)
   rw [placeTranslate_apply]
   rfl
 
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 @[simp] theorem placeTranslate_infinity (S : W.toAffine.Point) :
     placeTranslate W S ProjectiveSmoothPoint.infinity =
       S.toProjectiveSmoothPoint := by
@@ -75,6 +76,7 @@ theorem placeTranslate_apply (S : W.toAffine.Point)
   change (((0 : W.toAffine.Point) + S)).toProjectiveSmoothPoint = _
   rw [zero_add]
 
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Translation by `O` is the identity place permutation. -/
 @[simp] theorem placeTranslate_zero :
     placeTranslate W (0 : W.toAffine.Point) = Equiv.refl _ := by
@@ -89,13 +91,16 @@ noncomputable def ordProj (v : ProjectiveSmoothPoint (W_smooth W))
   | ProjectiveSmoothPoint.affine P => (W_smooth W).ord_P P f
   | ProjectiveSmoothPoint.infinity => (W_smooth W).ordAtInfty f
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 @[simp] theorem ordProj_affine (P : (W_smooth W).SmoothPoint) (f : KE) :
     ordProj W (ProjectiveSmoothPoint.affine P) f = (W_smooth W).ord_P P f := rfl
 
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 @[simp] theorem ordProj_infinity (f : KE) :
     ordProj W ProjectiveSmoothPoint.infinity f = (W_smooth W).ordAtInfty f := rfl
 
 set_option backward.isDefEq.respectTransparency.types false in
+omit [DecidableEq F] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The coefficient of `projectiveDivisorOf f` at `v` is `(ordProj v f).untopD 0`. -/
 theorem projectiveDivisorOf_apply_ordProj (f : KE)
     (v : ProjectiveSmoothPoint (W_smooth W)) :
@@ -119,6 +124,7 @@ theorem ord_P_translate (P : (W_smooth W).SmoothPoint)
   · exact translate_ord_eq_all_nonzero W P xk yk h_ns h f hf
 
 set_option backward.isDefEq.respectTransparency.types false in
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- If `P + S` is finite, `placeTranslate W S (affine P)` is the translated affine place. -/
 theorem placeTranslate_affine_of_isSome (S : (W_smooth W).toAffine.Point)
     (P : (W_smooth W).SmoothPoint)
@@ -132,6 +138,7 @@ theorem placeTranslate_affine_of_isSome (S : (W_smooth W).toAffine.Point)
     P S h).symm
 
 set_option backward.isDefEq.respectTransparency.types false in
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- When `P + S = O` (the translate hits infinity), the place
 `placeTranslate W S (affine P)` is the place at infinity. -/
 theorem placeTranslate_affine_eq_infinity (S : (W_smooth W).toAffine.Point)
@@ -283,6 +290,7 @@ theorem projectiveDivisorOf_translate_div_eq_zero_of_invariant
     _ = 0 := by rw [hself, sub_self]
 
 set_option backward.isDefEq.respectTransparency.types false in
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The `toAffinePoint` of a translated place is `w.toAffinePoint + S`. -/
 theorem placeTranslate_toAffinePoint (S : (W_smooth W).toAffine.Point)
     (w : ProjectiveSmoothPoint (W_smooth W)) :
@@ -299,7 +307,7 @@ theorem pullbackDiv_apply (f : W.toAffine.Point →+ W.toAffine.Point)
     (w : ProjectiveSmoothPoint (W_smooth W)) :
     pullbackDiv (W := W.toAffine) f hker Q w =
       if f w.toAffinePoint = Q then (1 : ℤ) else 0 := by
-  letI : Fintype {P : W.toAffine.Point // f P = Q} :=
+  let : Fintype {P : W.toAffine.Point // f P = Q} :=
     @Fintype.ofFinite _ (fiber_finite f hker Q)
   rw [pullbackDiv, Finset.sum_apply']
   simp only [Finsupp.single_apply]
@@ -314,19 +322,19 @@ theorem pullbackDiv_apply (f : W.toAffine.Point →+ W.toAffine.Point)
       rw [hPeq,
         WeierstrassCurve.Affine.Point.toAffinePoint_toProjectiveSmoothPoint]
   by_cases hQ : f w.toAffinePoint = Q
-  · rw [if_pos hQ, Finset.sum_eq_single (⟨w.toAffinePoint, hQ⟩ :
+  · rw [ite_eq_left hQ, Finset.sum_eq_single (⟨w.toAffinePoint, hQ⟩ :
       {P : W.toAffine.Point // f P = Q})]
-    · rw [if_pos ((hkey w.toAffinePoint).mpr rfl)]
+    · rw [ite_eq_left ((hkey w.toAffinePoint).mpr rfl)]
     · rintro ⟨P, hP⟩ _ hne
-      rw [if_neg]
+      rw [ite_eq_right]
       intro hPw
       exact hne (Subtype.ext ((hkey P).mp hPw))
     · intro h
       exact absurd (Finset.mem_univ _) h
-  · rw [if_neg hQ]
+  · rw [ite_eq_right hQ]
     apply Finset.sum_eq_zero
     rintro ⟨P, hP⟩ _
-    rw [if_neg]
+    rw [ite_eq_right]
     intro hPw
     exact hQ (by
       rw [← (hkey P).mp hPw]
@@ -366,6 +374,7 @@ theorem equivMapDomain_placeTranslate_pullbackDiv (S : (W_smooth W).toAffine.Poi
   rw [Finsupp.equivMapDomain_symm_apply]
   exact pullbackDiv_placeTranslate_apply_general W S f hker Q w
 
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- `equivMapDomain` fixes a divisor invariant under `placeTranslate W S`. -/
 theorem equivMapDomain_placeTranslate_symm_eq_self
     (S : (W_smooth W).toAffine.Point)
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Fiber.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Fiber.lean
index 2d0e9d56e..5a54b5281 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Fiber.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Fiber.lean
@@ -6,20 +6,6 @@ Authors: Chris Birkbeck
 import Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point
 import Mathlib.Tactic
 
-/-!
-# Route 2A — fibres of point-map endomorphisms as kernel cosets (keystone foundation)
-
-The multiplicity-free geometric divisor pullback `β*((Q)) = Σ_{βP=Q} (P)` (separable case, used by
-both the Weil-pairing construction and the separable adjoint, Silverman III.8) is summed over the
-**fibre** `{P : βP = Q}`. This file ships the group-theoretic foundation: a fibre of a point-map
-endomorphism is a coset of the kernel, hence finite when the kernel is finite.
-
-This is the point-map (`AddMonoidHom`) form of the project's `EC.Isogeny` fibre machinery — the
-level at which the Weil pairing and the pullback divisor operate.
-
-Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.4 (fibres of an isogeny are cosets of
-the kernel).
--/
 
 namespace HasseWeil.WeilPairing
 
@@ -29,8 +15,7 @@ open WeierstrassCurve
 
 variable {F : Type*} [Field F] [DecidableEq F] {W : WeierstrassCurve.Affine F} [W.IsElliptic]
 
-/-- **Fibre as a kernel coset.** Given `f P₀ = Q`, the fibre `{P : f P = Q}` is in canonical
-bijection with `ker f` via `P ↦ P − P₀`. -/
+
 def fiberEquivKer (f : W.Point →+ W.Point) {P₀ Q : W.Point} (hP₀ : f P₀ = Q) :
     {P : W.Point // f P = Q} ≃ f.ker where
   toFun := fun ⟨P, hP⟩ => ⟨P - P₀, by rw [AddMonoidHom.mem_ker, map_sub, hP, hP₀, sub_self]⟩
@@ -38,16 +23,15 @@ def fiberEquivKer (f : W.Point →+ W.Point) {P₀ Q : W.Point} (hP₀ : f P₀
   left_inv := fun ⟨P, hP⟩ => by simp
   right_inv := fun ⟨T, hT⟩ => by simp
 
-/-- **Fibres are finite when the kernel is.** -/
+
 theorem fiber_finite (f : W.Point →+ W.Point) (h : Finite f.ker) (Q : W.Point) :
     Finite {P : W.Point // f P = Q} := by
   rcases Classical.em (∃ P₀, f P₀ = Q) with ⟨P₀, hP₀⟩ | hempty
   · exact Finite.of_equiv _ (fiberEquivKer f hP₀).symm
-  · haveI : IsEmpty {P : W.Point // f P = Q} := ⟨fun ⟨P, hP⟩ => hempty ⟨P, hP⟩⟩
+  · have : IsEmpty {P : W.Point // f P = Q} := ⟨fun ⟨P, hP⟩ => hempty ⟨P, hP⟩⟩
     infer_instance
 
-/-- **Fibre cardinality = kernel cardinality** (for a nonempty fibre). The constant fibre size that
-makes `deg = #ker` for a separable isogeny (Silverman III.4.10c). -/
+
 theorem fiber_card_eq_ker_card (f : W.Point →+ W.Point) {P₀ Q : W.Point} (hP₀ : f P₀ = Q) :
     Nat.card {P : W.Point // f P = Q} = Nat.card f.ker :=
   Nat.card_eq_of_bijective _ (fiberEquivKer f hP₀).bijective
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobMatrixData.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobMatrixData.lean
index 30cf18d00..7da5769f2 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobMatrixData.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobMatrixData.lean
@@ -9,7 +9,7 @@ import HasseWeil.Isogeny.BaseChange.Basic
 /-!
 # Frobenius matrix data over the algebraic closure
 
-This file supplies the per-`ℓ` Frobenius-matrix determinant data used by
+The per-`ℓ` Frobenius-matrix determinant data give the hypotheses of
 `hasse_bound_via_weil_pairing`, after base-changing `E/K` to `AlgebraicClosure K`.
 
 ## Main definitions
@@ -29,10 +29,6 @@ This file supplies the per-`ℓ` Frobenius-matrix determinant data used by
 * Silverman, *The Arithmetic of Elliptic Curves*, III.8.1 (the pairing + Galois equivariance),
   III.8.6 (`det φ_ℓ = deg φ`), V.1.1 / V.2.3.1 (the Hasse bound assembly).
 
-## Name-clash note
-
-`pullbackDivisor_kappaDivisor` is declared in both `HfactLemma.lean` and `PairingNondeg.lean`.
-This file imports neither directly, so no clash arises and no rename is needed.
 -/
 
 open WeierstrassCurve Real Matrix
@@ -155,7 +151,7 @@ theorem frob_det_residual_baseChange
       (1 - M).det = ((Fintype.card K + 1 -
           isogTrace (frobeniusIsog W) (isogOneSub_negFrobenius W hq) : ℤ) : ZMod ℓ) ∧
       ((r' : ZMod ℓ) • M - (s' : ZMod ℓ) • 1).det = (deg r' s' : ZMod ℓ) := by
-  letI : Fact ℓ.Prime := ⟨hℓp⟩
+  let : Fact ℓ.Prime := ⟨hℓp⟩
   obtain ⟨hπ, h1, hrs⟩ := hsc
   have hDd : ((deg r' s').toNat : ℤ) = deg r' s' := Int.toNat_of_nonneg (hdeg_nonneg r' s')
   exact frob_det_residual_of_weil_scaling (W.baseChange L) ℓ hℓF
@@ -193,7 +189,7 @@ theorem hres_of_baseChange_scalings
         (1 - M).det = ((Fintype.card K + 1 -
             isogTrace (frobeniusIsog W) (isogOneSub_negFrobenius W hq) : ℤ) : ZMod ℓ) ∧
         ((r' : ZMod ℓ) • M - (s' : ZMod ℓ) • 1).det = (deg r' s' : ZMod ℓ) := by
-  haveI : CharP L p := charP_of_injective_algebraMap (FaithfulSMul.algebraMap_injective K L) p
+  have : CharP L p := charP_of_injective_algebraMap (FaithfulSMul.algebraMap_injective K L) p
   obtain ⟨hFrob, hOneSub, hPencil⟩ := hscale
   intro r' s' hps ℓ hℓp hℓne
   have hℓF : (ℓ : L) ≠ 0 :=
@@ -218,7 +214,7 @@ theorem hres_of_baseChange_scalings_coprime
         (1 - M).det = ((Fintype.card K + 1 -
             isogTrace (frobeniusIsog W) (isogOneSub_negFrobenius W hq) : ℤ) : ZMod ℓ) ∧
         ((r' : ZMod ℓ) • M - (s' : ZMod ℓ) • 1).det = (deg r' s' : ZMod ℓ) := by
-  haveI : CharP L p := charP_of_injective_algebraMap (FaithfulSMul.algebraMap_injective K L) p
+  have : CharP L p := charP_of_injective_algebraMap (FaithfulSMul.algebraMap_injective K L) p
   obtain ⟨hFrob, hOneSub, hPencil⟩ := hscale
   intro r' s' hpr hps ℓ hℓp hℓne
   have hℓF : (ℓ : L) ≠ 0 :=
@@ -238,16 +234,16 @@ theorem hasse_bound_unconditional_of_baseChange_scalings
     |(↑(pointCount W.toAffine) - ↑(Fintype.card K) - 1 : ℝ)| ≤
       2 * Real.sqrt (Fintype.card K : ℝ) := by
   obtain ⟨p, hCharP, ⟨n, _hn⟩, hp_prime, hcard⟩ := FiniteField.card' K
-  haveI : Fact p.Prime := ⟨hp_prime⟩
-  haveI : CharP K p := hCharP
-  haveI : Fact (Fintype.card K = p ^ (n : ℕ)) := ⟨hcard⟩
+  have : Fact p.Prime := ⟨hp_prime⟩
+  have : CharP K p := hCharP
+  have : Fact (Fintype.card K = p ^ (n : ℕ)) := ⟨hcard⟩
   have hpchar : ringChar K = p := by rw [ringChar.eq_iff]; exact hCharP
-  haveI : ExpChar (AlgebraicClosure K) p :=
+  have : ExpChar (AlgebraicClosure K) p :=
     haveI : CharP (AlgebraicClosure K) p :=
       charP_of_injective_algebraMap
         (FaithfulSMul.algebraMap_injective K (AlgebraicClosure K)) p
     ExpChar.prime hp_prime
-  haveI : (W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic := inferInstance
+  have : (W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic := inferInstance
   exact hasse_bound_via_weil_pairing W hq deg hdeg_nonneg
     (hres_of_baseChange_scalings W p (n : ℕ) (AlgebraicClosure K) hq deg hdeg_nonneg hpchar
       (hscale p (n : ℕ) ⟨hp_prime⟩ hCharP ⟨hcard⟩))
@@ -261,16 +257,16 @@ theorem hasse_bound_unconditional_of_baseChange_scalings_coprime
     |(↑(pointCount W.toAffine) - ↑(Fintype.card K) - 1 : ℝ)| ≤
       2 * Real.sqrt (Fintype.card K : ℝ) := by
   obtain ⟨p, hCharP, ⟨n, _hn⟩, hp_prime, hcard⟩ := FiniteField.card' K
-  haveI : Fact p.Prime := ⟨hp_prime⟩
-  haveI : CharP K p := hCharP
-  haveI : Fact (Fintype.card K = p ^ (n : ℕ)) := ⟨hcard⟩
+  have : Fact p.Prime := ⟨hp_prime⟩
+  have : CharP K p := hCharP
+  have : Fact (Fintype.card K = p ^ (n : ℕ)) := ⟨hcard⟩
   have hpchar : ringChar K = p := by rw [ringChar.eq_iff]; exact hCharP
-  haveI : ExpChar (AlgebraicClosure K) p :=
+  have : ExpChar (AlgebraicClosure K) p :=
     haveI : CharP (AlgebraicClosure K) p :=
       charP_of_injective_algebraMap
         (FaithfulSMul.algebraMap_injective K (AlgebraicClosure K)) p
     ExpChar.prime hp_prime
-  haveI : (W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic := inferInstance
+  have : (W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic := inferInstance
   exact hasse_bound_via_weil_pairing_both W hq deg hdeg_nonneg
     (hres_of_baseChange_scalings_coprime W p (n : ℕ) (AlgebraicClosure K) hq deg
       hdeg_nonneg hpchar
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobeniusDivisorGalois.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobeniusDivisorGalois.lean
index a5cecf0c5..2cc3630be 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobeniusDivisorGalois.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobeniusDivisorGalois.lean
@@ -14,7 +14,7 @@ import HasseWeil.Foundation.Curves.Valuation.NoFinitePolesBridge
 /-!
 # Divisor Galois descent for the arithmetic Frobenius `σ` of `K̄(E)`
 
-This file applies the abstract divisor-Galois-descent engine (`DivisorGalois.lean`) to the concrete
+Divisor Galois descent for the concrete
 arithmetic Frobenius `σ = frobeniusFunctionFieldEquiv W` of the function field `K̄(E)`, proving the
 affine order transport
 
@@ -112,19 +112,19 @@ theorem pointValuation_frobeniusFunctionFieldEquiv
         (frobeniusFunctionFieldEquiv W g) =
       (⟨(W.baseChange (AlgebraicClosure K)).toAffine⟩ :
         SmoothPlaneCurve (AlgebraicClosure K)).pointValuation Q g := by
-  haveI hMell : ((W.baseChange (AlgebraicClosure K)).map
+  have hMell : ((W.baseChange (AlgebraicClosure K)).map
       (coeffFrobEquiv (K := K) :
         AlgebraicClosure K →+* AlgebraicClosure K)).toAffine.IsElliptic := by
     rw [map_coeffFrobEquiv_eq W]; infer_instance
-  haveI hMic : IsIntegrallyClosed (⟨((W.baseChange (AlgebraicClosure K)).map
+  have hMic : IsIntegrallyClosed (⟨((W.baseChange (AlgebraicClosure K)).map
       (coeffFrobEquiv (K := K) : AlgebraicClosure K →+* AlgebraicClosure K)).toAffine⟩ :
       SmoothPlaneCurve (AlgebraicClosure K)).CoordinateRing := by
     rw [map_coeffFrobEquiv_eq W]; infer_instance
-  haveI hMdd : IsDedekindDomain (⟨((W.baseChange (AlgebraicClosure K)).map
+  have hMdd : IsDedekindDomain (⟨((W.baseChange (AlgebraicClosure K)).map
       (coeffFrobEquiv (K := K) : AlgebraicClosure K →+* AlgebraicClosure K)).toAffine⟩ :
       SmoothPlaneCurve (AlgebraicClosure K)).CoordinateRing :=
     SmoothPlaneCurve.isDedekindDomain_coordinateRing _
-  haveI hEdd : IsDedekindDomain (⟨(W.baseChange (AlgebraicClosure K)).toAffine⟩ :
+  have hEdd : IsDedekindDomain (⟨(W.baseChange (AlgebraicClosure K)).toAffine⟩ :
       SmoothPlaneCurve (AlgebraicClosure K)).CoordinateRing :=
     SmoothPlaneCurve.isDedekindDomain_coordinateRing _
   rw [frobeniusFunctionFieldEquiv, RingEquiv.trans_apply, ffFrobCast,
@@ -144,9 +144,8 @@ theorem pointValuation_frobeniusFunctionFieldEquiv
         (by rw [pointOnMapped_x, hPx]) (by rw [pointOnMapped_y, hPy])).symm) g
 
 /- The order-at-infinity transport `ordAtInfty (σ g) = ordAtInfty g`: the arithmetic Frobenius `σ`
-fixes the place at infinity. Mirrors `OrdAtInftyBaseChange.lean` (the base-change `ord_∞`
-transport), replacing the injective `algebraMap K L` by the coefficient Frobenius `coeffFrobEquiv`
-(also injective). -/
+fixes the place at infinity. Injectivity of the coefficient Frobenius identifies the
+orders of the corresponding norm polynomials. -/
 
 /-- `crFrobEquiv` (= `CoordinateRing.map e`) on the `K̄[X]`-basis decomposition `p • 1 + q • y`:
 it sends the coefficients `p, q` through `Polynomial.map e` and fixes the basis `{1, y}`.
@@ -305,7 +304,7 @@ omit [DecidableEq K] [W.toAffine.IsElliptic]
     ((FiniteField.frobeniusAlgEquivOfAlgebraic K (AlgebraicClosure K)).symm P.y) = P.y
   rw [AlgEquiv.coe_ringEquiv, AlgEquiv.apply_symm_apply]
 
-omit [W.toAffine.IsElliptic] [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic] in
+omit [W.toAffine.IsElliptic] [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic] [DecidableEq K] in
 /-- **`π̄` recovers `P` from its inverse-Frobenius point** (at the affine-point level):
 `geomFrobeniusPointFun (geomFrobSmoothPointInv P).toAffinePoint = P.toAffinePoint`. -/
 theorem geomFrobeniusPointFun_geomFrobSmoothPointInv
@@ -341,6 +340,7 @@ open HasseWeil
 
 omit [W.toAffine.IsElliptic] in
 set_option backward.isDefEq.respectTransparency false in
+omit [DecidableEq K] in
 /-- **The fibre-divisor place comparison** (the combinatorial heart of σ-naturality). The
 coefficient of the fibre divisor `[ℓ]^*(π̄T)` at an affine place `P` equals the coefficient of
 `[ℓ]^*(T)` at the inverse-Frobenius place `geomFrobSmoothPointInv P`, because
@@ -382,6 +382,7 @@ theorem pullbackDiv_geomFrobInv_eq (ℓ : ℤ) (hℓ : (ℓ : AlgebraicClosure K
 
 omit [W.toAffine.IsElliptic] in
 set_option backward.isDefEq.respectTransparency false in
+omit [DecidableEq K] in
 /-- **The fibre-divisor place comparison at infinity**:
 `pullbackDiv [ℓ] (π̄T) ∞ = pullbackDiv [ℓ] T ∞` (both equal `if 0 = Q`, and `0 = π̄T ⟺ 0 = T` since
 π̄ is injective with `π̄ 0 = 0`). -/
@@ -500,7 +501,7 @@ theorem frobeniusFunctionFieldEquiv_weilFunction_eq_smul
             (W.baseChange (AlgebraicClosure K)).toAffine.FunctionField c *
           weilFunction (W.baseChange (AlgebraicClosure K)) ℓ hℓ
             (HasseWeil.geomFrobeniusPoint W T) hπT := by
-  haveI hEdd : IsDedekindDomain (⟨(W.baseChange (AlgebraicClosure K)).toAffine⟩ :
+  have hEdd : IsDedekindDomain (⟨(W.baseChange (AlgebraicClosure K)).toAffine⟩ :
       SmoothPlaneCurve (AlgebraicClosure K)).CoordinateRing :=
     SmoothPlaneCurve.isDedekindDomain_coordinateRing _
   set gT := weilFunction (W.baseChange (AlgebraicClosure K)) ℓ hℓ T hT with hgT
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobeniusGenericCovariance.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/FrobeniusGenericCovariance.lean
index 2e94ad5e465c6dada76ce68439b6c5d25480b4c3..86c212da5910eadae69fd7f1828430aaa2ee4c24 100644
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P9^d50>SBl89v}V#6%nlU

diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/HfactLemma.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/HfactLemma.lean
index 66737344a..5d4f80a0a 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/HfactLemma.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/HfactLemma.lean
@@ -9,57 +9,21 @@ import HasseWeil.HasseBound.WeilPairing.Constancy
 import HasseWeil.Pic0.PicDual
 
 /-!
-# The separable divisor factorisation `hfact` (Silverman III.8.2 / III.6.1b)
+# Divisor factorization for the separable Weil-pairing adjoint
 
-This file discharges the **divisor factorisation hypothesis** `hfact` of
-`HasseWeil.WeilPairing.weilPairing_adjoint_core` (the separable adjoint, Silverman III.8.2): for a
-separable isogeny `φ : E → E`, the Weil functions `g_T = weilFunction W ℓ T`, `g_U = weilFunction
-W ℓ U` (with `U = picDual φ T = φ̂ T`), and an appropriate `k ∈ K(E)`,
-```
-φ^* g_T = c · (g_U · ([ℓ]^* k))      for some c ∈ Fˣ.
-```
+Let `g_T` and `g_U` be Weil functions, and suppose the projective-divisor class
+identity `φ*((T) − (O)) ∼ (U) − (O)` holds. Divisor pullback functoriality and
+commutation with multiplication by `ℓ` then give
+`div(φ*g_T) = div(g_U · ([ℓ]*k))` for a suitable function `k`.
+Functions with the same projective divisor differ by a nonzero base-field
+constant, so `φ*g_T = c · g_U · ([ℓ]*k)`.
 
-## The mathematics (Silverman III.8.2 proof, divisor language)
-
-Write `div g_T = [ℓ]^*((T) − (O))` (`weilFunction_divisor`, the fibre-difference divisor). Then
-
-* `div(φ^* g_T) = φ^*(div g_T) = φ^*([ℓ]^*((T) − (O)))` — the divisor-pullback functoriality
-  `div(φ^* h) = φ^*(div h)` for the separable `φ` (`DivisorPullback.ProjOrdTransport φ`, the
-  analogue of `projOrdTransport_mulByInt` for `[ℓ]`);
-* `φ^* ∘ [ℓ]^* = [ℓ]^* ∘ φ^*` on divisors (the function-field shadow of `[ℓ] ∘ φ = φ ∘ [ℓ]`, which
-  holds for *any* additive point maps: `[ℓ]` commutes with every homomorphism) — proven here as
-  `pullbackDivisor_comm`;
-* the **`picDual` divisor-class identity** (Silverman III.6.1b, the genuine separability content)
-  `φ^*((T) − (O)) ∼ (U) − (O)` with `U = φ̂ T`, i.e. `φ^*(kappaDivisor T) − kappaDivisor U` is
-  principal `= div k₀` — isolated below as the single minimal hypothesis `PicDualDivisorClass φ`
-  (a projective-divisor avatar of `PicDual.Naturality`, see the residual note).
-
-Combining: `div(φ^* g_T) = [ℓ]^*(φ^*(kappaDivisor T)) = [ℓ]^*(kappaDivisor U + div k₀) = div(g_U) +
-div([ℓ]^* k₀) = div(g_U · [ℓ]^* k₀)`. Two functions with the same projective divisor differ by a
-nonzero constant `c ∈ Fˣ` (`Constancy.const_unit_of_projectiveDivisorOf_eq_zero`), which is exactly
-`hfact`.
-
-## The isolated minimal residual `PicDualDivisorClass`
-
-The decisive ingredient is the divisor-class identity `φ^*((T) − (O)) ∼ (φ̂ T) − (O)` *expressed in
-the projective-divisor model* `ProjectiveDivisor` (where `weilFunction_divisor`/`hfact` speak),
-namely `(⟨W⟩).ProjIsPrincipal (pullbackDivisor φ (kappaDivisor T) − kappaDivisor (φ̂ T))`.
-
-This is the **honest frontier**: `picDual = κ⁻¹ ∘ classMap ∘ κ` is defined in the *affine ideal
-class group* model `ClassGroup R` (`PicDual.lean`), where `classMap = ClassGroup.map` is the ideal
-extension `𝔪 ↦ 𝔪·𝒪` — Silverman's divisor pullback `φ^*`. But `hfact` lives in the *projective
-divisor* model `PicProj₀`/`projectiveDivisorOf`. The two `Pic⁰(E)` models are connected by `κ`
-(the canonical `E ≅ Pic⁰(E)` of III.3.4), but the repo carries no bridge `ClassGroup.map ↔
-pullbackDivisor`; building it is exactly the `comap`-vs-`relNorm` (residue-degree) bookkeeping that
-`PicDual.Naturality` documents as carried per-isogeny data. We therefore isolate the divisor-class
-identity as the one minimal hypothesis `PicDualDivisorClass φ ch hinj hfin`, dischargeable per
-isogeny just as `ProjOrdTransport`/`Naturality` are throughout the project, and prove the *entire*
-`hfact` factorisation on top of it.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, III.6.1 (the dual isogeny, `φ̂ = κ⁻¹ ∘ φ^* ∘ κ`),
-  III.8.2 (Prop 8.2, the separable adjoint via the multiplicity-free pullback).
+The projective-divisor class identity is an explicit hypothesis. The ideal-class
+definition of `picDual` uses extension of ideals in the affine coordinate ring;
+identifying it with projective divisor pullback requires compatibility of those
+two models of `Pic⁰`. The factorization proofs retain this compatibility input.
+
+References: Silverman, *The Arithmetic of Elliptic Curves*, III.6.1 and III.8.2.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
@@ -68,8 +32,6 @@ namespace HasseWeil.WeilPairing
 
 open HasseWeil HasseWeil.WeilPairing.DivisorPullback
 
-set_option linter.unusedSectionVars false
-
 variable {F : Type*} [Field F] [DecidableEq F]
 variable (W : WeierstrassCurve F) [W.toAffine.IsElliptic]
   [IsIntegrallyClosed (⟨W.toAffine⟩ : SmoothPlaneCurve F).CoordinateRing]
@@ -78,13 +40,14 @@ local notation "KE" => W.toAffine.FunctionField
 
 /-! ### Commutation of fibre-pullback divisors `φ^* ∘ ψ^* = (ψ ∘ φ)^*`
 
-The fibre-pullback `pullbackDivisor f` (`DivisorPullback.lean`) transports a coefficient along the
+The fibre-pullback `pullbackDivisor f` transports a coefficient along the
 point map `f`: `(pullbackDivisor f D) w = D (f w)` (`pullbackDivisor_apply`). Hence two such
 pullbacks compose by composing the point maps in the *opposite* order:
 `pullbackDivisor f (pullbackDivisor g D) w = D (g (f w))`. We only need the special case where the
 two maps **commute** as point endomorphisms (`g ∘ f = f ∘ g`), which is automatic when one of them
 is `[ℓ]` (multiplication-by-`ℓ` commutes with every additive hom). -/
 
+omit [IsIntegrallyClosed (⟨W.toAffine⟩ : SmoothPlaneCurve F).CoordinateRing] in
 /-- **Fibre-pullback divisors of commuting point maps commute.** If `f g : E.Point →+ E.Point`
 commute (`g.comp f = f.comp g` — automatic when one is `[ℓ]`), then their fibre-pullback divisor
 operators commute: `pullbackDivisor f (pullbackDivisor g D) = pullbackDivisor g (pullbackDivisor f
@@ -110,11 +73,10 @@ theorem pullbackDivisor_comm {f g : W.toAffine.Point →+ W.toAffine.Point}
 `pullbackDivisor [ℓ] (kappaDivisor T)` of the Abel–Jacobi divisor `(T) − (O) = kappaDivisor T`
 (`pullbackDivisor` distributes over the difference of the two `single`s; `∞.toAffinePoint = O`). -/
 
+omit [IsIntegrallyClosed (⟨W.toAffine⟩ : SmoothPlaneCurve F).CoordinateRing] in
 /-- **The fibre-pullback of `(T) − (O)` is `[ℓ]^*(T) − [ℓ]^*(O)`** (= `div g_T`). The fibre-pullback
 `pullbackDivisor [ℓ]` of `kappaDivisor T = (T) − (O)` equals `pullbackDiv [ℓ] T − pullbackDiv [ℓ]
-O`. (Local copy of `PairingNondeg.pullbackDivisor_kappaDivisor`, kept here to avoid the import;
-named `…_local` to avoid the cross-file name clash with the `PairingNondeg` copy, so that downstream
-files importing both `HfactLemma` and `DetDeg`/`PairingNondeg` — e.g. `SeparableScaling` — compile.)
+O`.
 -/
 theorem pullbackDivisor_kappaDivisor_local (ℓ : ℤ)
     [hker : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker] (T : W.toAffine.Point) :
@@ -135,7 +97,7 @@ theorem weilFunction_divisor_eq_pullbackDivisor_kappaDivisor [IsAlgClosed F]
     (⟨W.toAffine⟩ : SmoothPlaneCurve F).projectiveDivisorOf (weilFunction W ℓ hℓ T hT) =
       pullbackDivisor (W := W.toAffine) (mulByInt W.toAffine ℓ).toAddMonoidHom
           (mulByInt_ker_finite W ℓ hℓ) (Curves.kappaDivisor W.toAffine T) := by
-  haveI : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
+  have : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
   rw [show (⟨W.toAffine⟩ : SmoothPlaneCurve F).projectiveDivisorOf (weilFunction W ℓ hℓ T hT) =
       (W_smooth W).projectiveDivisorOf (weilFunction W ℓ hℓ T hT) from rfl,
     weilFunction_divisor W ℓ hℓ T hT, pullbackDivisor_kappaDivisor_local W ℓ T]
@@ -146,7 +108,7 @@ theorem weilFunction_divisor_eq_pullbackDivisor_kappaDivisor [IsAlgClosed F]
 fibre-pullback `φ^*((T) − (O))` is *linearly equivalent* to `(φ̂ T) − (O)` (with `φ̂ = picDual φ`),
 i.e. their difference `pullbackDivisor φ (kappaDivisor T) − kappaDivisor (φ̂ T)` is principal. This
 is the one ingredient that cannot be assembled from the project's *projective*-model API: `picDual`
-is built from the *affine* ideal-class-group divisor pullback `classMap` (`PicDual.lean`), and the
+is built from the *affine* ideal-class-group divisor pullback `classMap`, and the
 ClassGroup↔ProjectiveDivisor bridge is the carried `PicDual.Naturality` data (the `comap`-vs-
 `relNorm`/residue-degree bookkeeping). Everything else in `hfact` is proven on top of this. -/
 
@@ -209,7 +171,7 @@ theorem hfact_projectiveDivisorOf_eq (ℓ : ℤ) (hℓ : (ℓ : F) ≠ 0)
         (φ.pullback (weilFunction W ℓ hℓ T hT)) =
       (⟨W.toAffine⟩ : SmoothPlaneCurve F).projectiveDivisorOf
         (weilFunction W ℓ hℓ U hU * (mulByInt W.toAffine ℓ).pullback k₀) := by
-  haveI hker : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
+  have hker : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
   have hcoreℓ : ProjOrdTransport (mulByInt W.toAffine ℓ) := projOrdTransport_mulByInt ℓ hℓ
   set κT := Curves.kappaDivisor W.toAffine T
   set κU := Curves.kappaDivisor W.toAffine U with hκU
@@ -274,7 +236,7 @@ theorem hfact_of_picDualDivisorClass (ℓ : ℤ) (hℓ : (ℓ : F) ≠ 0)
           (weilFunction W ℓ hℓ ((φ.picDual ch hinj hfin) T)
               (by rw [← map_zsmul, hT, map_zero]) *
             (mulByInt W.toAffine ℓ).pullback k) := by
-  haveI : IsDedekindDomain (⟨W.toAffine⟩ : SmoothPlaneCurve F).CoordinateRing := inferInstance
+  have : IsDedekindDomain (⟨W.toAffine⟩ : SmoothPlaneCurve F).CoordinateRing := inferInstance
   have hU : ℓ • (φ.picDual ch hinj hfin) T = 0 := by rw [← map_zsmul, hT, map_zero]
   -- The Abel–Jacobi function `k₀` of the `picDual` divisor-class identity, and its divisor.
   obtain ⟨k₀, hk₀_ne, hk₀_div⟩ := hpd T
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/OneSubDualDivisor.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/OneSubDualDivisor.lean
index d3000b32d30e10c2b46fc50f6a717384d41259f1..4a6db4566facbd1bd32e68b7dfa22044d93fa66b 100644
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diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/OneSubFrobeniusBaseChangeWitnesses.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/OneSubFrobeniusBaseChangeWitnesses.lean
index 63ca18b80..6a943b56a 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/OneSubFrobeniusBaseChangeWitnesses.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/OneSubFrobeniusBaseChangeWitnesses.lean
@@ -7,73 +7,22 @@ import HasseWeil.Foundation.Curves.Frobenius.FrobeniusFixedPoint
 import HasseWeil.Isogeny.BaseChange.Concrete
 
 /-!
-# Discharging the point/divisor witnesses of `OneSubScalingData` over `K̄` (CoordHom-free)
-
-This file connects the abstract base-changed isogeny `(1 − π)_{K̄}` of
-`OneSubScaling.lean`/`IsogenyBaseChangeConcrete.lean` to the concrete **geometric Frobenius**
-machinery of `Curves/FrobeniusFixedPoint.lean`, and uses that connection to **prove** several of
-the genuinely-deep K̄-level geometric witnesses that `mkOneSubScalingDataConcrete` consumes,
-isolating precisely the ones that remain open.
-
-## The linchpin: `frobeniusHomBaseChange = geomFrobeniusPoint`
-
-The point map carried by `oneSubFrobeniusIsogBaseChange` is `id − π̄` with
-`π̄ = frobeniusHomBaseChange W p r L`
-(`= (frobeniusIsog_baseChange_charP_pow p r W L).toAddMonoidHom`,
-the iterated relative `p`-Frobenius point map).  Over `L = AlgebraicClosure K` this is the same map
-as the **geometric Frobenius** `geomFrobeniusPoint W` of `FrobeniusFixedPoint.lean` (mathlib's
-`Affine.Point.map (frobeniusAlgHom K K̄)`, the literal `q`-power on coordinates), but the two are
-*not* definitionally equal — `frobeniusHomBaseChange` is built through a `cast` over `r` iterations
-of the relative Frobenius.  `frobeniusHomBaseChange_eq_geomFrobeniusPoint` proves the equality by:
-
-* `iterate_apply_some_heq` — an induction on `r` showing the iterated relative Frobenius point map
-  sends `(x, y) ↦ (x^{p^r}, y^{p^r})` (transporting the codomain-curve `cast` at each step via the
-  generic `isogeny_cast_apply_heq`/`some_heq_of_curve_eq` `HEq` lemmas), then
-* peeling the outer `charP_pow` `cast` and matching with `geomFrobeniusPointFun_some`
-  (`x^{p^r} = x^{#K}` since `#K = p^r`).
-
-This is **axiom-clean** (`[propext, Classical.choice, Quot.sound]`).
-
-## What is proved vs. isolated
-
-With the connection, `(1 − π)_{K̄}`'s kernel becomes the geometric-Frobenius fixed locus
-`ker(id − π̄) = ker(oneSubGeomFrobHom) = Fix(π̄) = range(includePointBC) ≅ E(𝔽_q)`:
-
-* **`finiteKer` — PROVED** (axiom-clean): the kernel is the image of the finite `E(𝔽_q)` under the
-  base-change inclusion (`fixedLocus_geomFrobenius_eq_range_includePointBC` + `Set.finite_range`).
-* **`hkerdeg` — REDUCED to V.1.3**: `#ker(id − π̄) = #E(𝔽_q) = pointCount` is *proved* clean
-  (`ncard_ker_oneSubGeomFrobHom_eq_pointCount`), so `hkerdeg : #ker = φ_L.degree` follows once
-  `φ_L.degree = pointCount` — which is exactly Silverman V.1.3
-  (`isogOneSub_negFrobenius_degree_eq_pointCount`, the known sharp residual carrying `sorryAx`).
-  `hkerdeg_of_degree_eq_pointCount` packages this reduction, taking the degree-identity as a
-  hypothesis so the V.1.3 dependence is explicit and the lemma itself is axiom-clean.
-
-* **`hsurj` — REDUCED to the dual** (axiom-clean reduction): surjectivity of `id − π̄` is *not*
-  reachable from the fixed-locus kernel facts, but it **is** reachable from the divisor-pushforward
-  dual via its *other* composition `φ ∘ δ = [#ker]`: given `Q`, pick `R` with `[#ker] R = Q` (the
-  *shipped* `mulByInt_point_surjective`, Silverman III.4.10b over `K̄`); then `φ (δ R) = Q`.
-  `hsurj_of_self_comp_dual` packages this, so once the dual is bundled as a genuine `IsDualOf`
-  (carrying *both* `δ ∘ φ = [#ker]` and `φ ∘ δ = [#ker]`), `hsurj` comes for free alongside `hdc`.
-
-The genuinely-deep residuals that remain open (each a named hypothesis), not reachable from the
-fixed-locus / `[ℓ]`-surjectivity machinery:
-
-* **`δ`/`hdc`/`hself`** — the divisor-pushforward dual `1 − V̄` as an `IsDualOf`, i.e. *both*
-  composition identities `δ ∘ φ = [#ker φ]` and `φ ∘ δ = [#ker φ]` (Silverman III.6.2(a)); needs
-  the base-change of the `K`-level Verschiebung dual relation;
-* **`hproj`** — `ProjOrdTransport` (the multiplicity-free divisor-pullback functoriality);
-* **`hcomm'`** — the translation covariance (Silverman III.8.2).
-
-`mkOneSubScalingDataConcrete_of_witnesses` assembles the full `OneSubScalingData` over
-`L = AlgebraicClosure K` consuming **only** the still-open residuals (`δ`/`hdc`/`hself`, `hproj`,
-`hcomm'`) plus the V.1.3 degree identity, with `finiteKer`, `hkerdeg`, `hsurj`, and `hdeg_bc`
-(the curve-free base-change-of-finrank, now proved as `finrankBaseChange`) discharged internally.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, III.4.10a/c, III.6.2(a), III.8.2, III.8.6.1,
-  V.1.1.
-* `HasseWeil/Curves/FrobeniusFixedPoint.lean` (the geometric-Frobenius fixed-locus theory).
+# Frobenius base change and geometric witnesses over an algebraic closure
+
+For an elliptic curve over a finite field of cardinality `q = p ^ r`, the iterated relative
+`p`-Frobenius point map after base change agrees with geometric Frobenius: both send
+`(x, y)` to `(x ^ q, y ^ q)`. Heterogeneous equalities transport points across the curve
+equalities in the iterated construction.
+
+Consequently, the kernel of `1 − π` over the algebraic closure is the finite image of the
+rational points, and its cardinality is the rational point count. A supplied degree identity
+identifies this cardinality with the isogeny degree. A dual composition `φ ∘ δ = [N]`, with
+`N` nonzero in the algebraic closure, gives surjectivity through surjectivity of multiplication
+by `N`. These facts construct the scaling data from the two dual composition identities,
+divisor-pullback functoriality, translation covariance, and the degree identity.
+
+The geometric ingredients correspond to Silverman, *The Arithmetic of Elliptic Curves*,
+III.4.10, III.6.2(a), III.8.2, III.8.6.1, and V.1.1–3.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
@@ -193,18 +142,14 @@ variable (p r : ℕ) [Fact p.Prime] [CharP K p] [Fact (Fintype.card K = p ^ r)]
 
 noncomputable local instance instDecEqAC : DecidableEq (AlgebraicClosure K) := Classical.decEq _
 
-/-- **The linchpin** (axiom-clean): over the algebraic closure, the base-changed Frobenius point map
-`frobeniusHomBaseChange` (the iterated relative `p`-Frobenius, carried through a `cast`) equals the
-**geometric Frobenius** `geomFrobeniusPoint W` (mathlib's `Affine.Point.map (frobeniusAlgHom K K̄)`,
-the literal `q`-power on coordinates).
-
-Both send `0 ↦ 0` and `(x, y) ↦ (x^{#K}, y^{#K})`; the proof peels the outer `charP_pow` `cast`
-(`isogeny_cast_apply_heq`), applies `iterate_apply_some_heq` (`(x, y) ↦ (x^{p^r}, y^{p^r})`), and
-matches `geomFrobeniusPointFun_some` using `#K = p^r`. -/
+omit [DecidableEq K] in
+/-- Over the algebraic closure, iterated relative Frobenius agrees with geometric Frobenius.
+Both maps send `(x, y)` to `(x ^ q, y ^ q)`, where `q = #K = p ^ r`; heterogeneous equalities
+transport the iterated point map across the equalities of its codomain curve. -/
 theorem frobeniusHomBaseChange_eq_geomFrobeniusPoint
     [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic] :
     frobeniusHomBaseChange W p r (AlgebraicClosure K) = geomFrobeniusPoint W := by
-  haveI : ExpChar (AlgebraicClosure K) p := ExpChar.prime Fact.out
+  have : ExpChar (AlgebraicClosure K) p := ExpChar.prime Fact.out
   have hcz : (W.baseChange (AlgebraicClosure K)).map (iterateFrobenius (AlgebraicClosure K) p r) =
       W.baseChange (AlgebraicClosure K) :=
     Isogeny.frobeniusTwistIterate_baseChange_eq_self_of_charP_pow (k := K) p r W
@@ -238,9 +183,9 @@ theorem frobeniusHomBaseChange_eq_geomFrobeniusPoint
     · refine heq_of_eq ?_
       congr 1 <;> simp only [hax, hay]
 
-/-- **The base-changed `(1 − π)_{K̄}` point map is `id − geomFrobenius`** (= `oneSubGeomFrobHom`),
-over `L = AlgebraicClosure K`.  Combines the constructional `toAddMonoidHom = id − π̄`
-(`oneSubFrobeniusIsogBaseChange_toAddMonoidHom`) with the linchpin `π̄ = geomFrobeniusPoint`. -/
+omit [DecidableEq K] in
+/-- The base-changed `1 − π` point map is the difference of the identity and geometric
+Frobenius over the algebraic closure. -/
 theorem oneSubFrobeniusIsogBaseChange_toAddMonoidHom_eq_oneSubGeomFrobHom
     [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic]
     (pullback_L : (W.baseChange (AlgebraicClosure K)).toAffine.FunctionField →ₐ[AlgebraicClosure K]
@@ -260,13 +205,9 @@ variable (p r : ℕ) [Fact p.Prime] [CharP K p] [Fact (Fintype.card K = p ^ r)]
 
 noncomputable local instance instDecEqAC' : DecidableEq (AlgebraicClosure K) := Classical.decEq _
 
-/-- **Witness 1 — `finiteKer`** (PROVED, axiom-clean): the kernel of `(1 − π)_{K̄}` is finite.
-
-Via the linchpin, `ker(id − π̄) = ker(oneSubGeomFrobHom)`, whose underlying set is the
-geometric-Frobenius fixed locus `= range(includePointBC)` (Step S2/S3 of
-`FrobeniusFixedPoint.lean`),
-i.e. the image of the **finite** point set `E(𝔽_q) = W.toAffine.Point` under the base-change
-inclusion — hence finite. -/
+omit [DecidableEq K] in
+/-- The kernel of base-changed `1 − π` is finite: it is the geometric-Frobenius fixed locus,
+which is the image of the finite rational point set under base change. -/
 theorem oneSubFrobeniusIsogBaseChange_finiteKer
     [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic]
     (pullback_L : (W.baseChange (AlgebraicClosure K)).toAffine.FunctionField →ₐ[AlgebraicClosure K]
@@ -281,9 +222,9 @@ theorem oneSubFrobeniusIsogBaseChange_finiteKer
   rw [← hset] at hfin
   exact hfin.to_subtype
 
-/-- **`#ker(1 − π)_{K̄} = pointCount W`** (PROVED, axiom-clean).  Via the linchpin the kernel is the
-geometric-Frobenius fixed locus, whose cardinality is the `𝔽_q`-rational point count
-(`ncard_ker_oneSubGeomFrobHom_eq_pointCount`).  This is the *clean half* of `hkerdeg`. -/
+omit [DecidableEq K] in
+/-- The cardinality of the kernel of base-changed `1 − π` equals the rational point count,
+by its identification with the geometric-Frobenius fixed locus. -/
 theorem oneSubFrobeniusIsogBaseChange_nat_card_ker_eq_pointCount
     [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic]
     (pullback_L : (W.baseChange (AlgebraicClosure K)).toAffine.FunctionField →ₐ[AlgebraicClosure K]
@@ -299,16 +240,9 @@ theorem oneSubFrobeniusIsogBaseChange_nat_card_ker_eq_pointCount
   rw [hcard, ncard_ker_oneSubGeomFrobHom_eq_pointCount]
   rfl
 
-/-- **Witness 3 — `hkerdeg`, reduced to V.1.3** (axiom-clean reduction).  Given the degree identity
-`φ_L.degree = pointCount W` (Silverman V.1.3, `isogOneSub_negFrobenius_degree_eq_pointCount`
-base-changed through `hdeg_bc` — the project's known sharp residual), the separable degree match
-`#ker(1 − π)_{K̄} = φ_L.degree` follows from the *proved* clean count
-`#ker = pointCount`.
-
-The V.1.3 degree identity is taken as a hypothesis so this reduction is itself axiom-clean and the
-V.1.3 dependence is explicit (the caller supplies `hdeg_eq` from
-`(oneSubFrobeniusIsogBaseChange_degree_eq_of_finrankBaseChange W p r L hq).trans
-  (isogOneSub_negFrobenius_degree_eq_pointCount W hq)` once `hq` is available). -/
+omit [DecidableEq K] in
+/-- The degree identity of Silverman V.1.3, supplied as a hypothesis, identifies the
+kernel cardinality with the degree of the base-changed `1 − π` isogeny. -/
 theorem oneSubFrobeniusIsogBaseChange_hkerdeg_of_degree_eq_pointCount
     [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic]
     (pullback_L : (W.baseChange (AlgebraicClosure K)).toAffine.FunctionField →ₐ[AlgebraicClosure K]
@@ -321,11 +255,9 @@ theorem oneSubFrobeniusIsogBaseChange_hkerdeg_of_degree_eq_pointCount
       (oneSubFrobeniusIsogBaseChange W p r (AlgebraicClosure K) pullback_L).degree := by
   rw [oneSubFrobeniusIsogBaseChange_nat_card_ker_eq_pointCount, hdeg_eq]
 
-omit [Fintype W.toAffine.Point] in
-/-- **Witness 2 — `hsurj` from the dual composition `φ ∘ δ = [N]`** (axiom-clean reduction).  For
-any `N` with `(N : K̄) ≠ 0`, surjectivity of `[N]` on `E(K̄)`-points (`mulByInt_point_surjective`)
-turns the dual composition `φ ∘ δ = [N]` into surjectivity of `φ`.  This is the second consequence
-(alongside `hdc`) of bundling the divisor-pushforward dual as an `IsDualOf`. -/
+omit [Fintype W.toAffine.Point] [DecidableEq K] in
+/-- A dual composition `φ ∘ δ = [N]`, with `N` nonzero in the algebraic closure, implies
+surjectivity of `φ` by surjectivity of multiplication by `N` (Silverman III.4.10). -/
 theorem oneSubFrobeniusIsogBaseChange_hsurj_of_self_comp_dual
     [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic]
     (pullback_L : (W.baseChange (AlgebraicClosure K)).toAffine.FunctionField →ₐ[AlgebraicClosure K]
@@ -363,14 +295,10 @@ variable [(W.baseChange (AlgebraicClosure K)).toAffine.IsElliptic]
     (⟨(W.baseChange (AlgebraicClosure K)).toAffine⟩ : SmoothPlaneCurve
       (AlgebraicClosure K)).CoordinateRing]
 
-/-- **Assemble `OneSubScalingData` over `K̄` from only the still-open witnesses.**
-
-Discharges `pullback_L` (concrete `oneSubFrobeniusPullback_L`), `hdeg_bc` (proved, no hypothesis),
-`finiteKer` (proved), `hkerdeg` (from the V.1.3 identity `hdeg_eq`), and `hsurj` (from the
-dual composition
-`hself` + `mulByInt_point_surjective`).  The caller supplies only the genuinely-deep divisor-level
-residuals: the dual `δ` with *both* composition identities `hdc`/`hself` (an `IsDualOf`), the
-divisor-pullback functoriality `hproj`, and the translation covariance `hcomm'`. -/
+/-- Scaling data over the algebraic closure constructed from the degree identity, both dual
+composition identities, divisor-pullback functoriality, and translation covariance. The
+geometric-Frobenius fixed locus supplies the finite kernel and its cardinality; the second
+dual composition supplies surjectivity. -/
 noncomputable def mkOneSubScalingDataConcrete_of_witnesses (hq : 2 ≤ Fintype.card K)
     (hdeg_eq :
       (oneSubFrobeniusIsogBaseChange W p r (AlgebraicClosure K)
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingNondeg.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingNondeg.lean
index 18c4b2749..327e9d223 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingNondeg.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingNondeg.lean
@@ -13,7 +13,7 @@ import HasseWeil.HasseBound.WeilPairing.TorsionCardEll
 /-!
 # Nondegeneracy of the Weil pairing
 
-This file proves that the finite-level Weil pairing over an algebraically closed field is
+We prove that the finite-level Weil pairing over an algebraically closed field is
 nondegenerate in its second argument. The proof uses surjectivity of multiplication on curve
 points, injectivity of divisor pullback, the Galois fixed-field description, and Abel--Jacobi.
 
@@ -154,7 +154,7 @@ theorem aut_eq_translate (ℓ : ℤ) (hℓ0 : ℓ ≠ 0)
       (mulByInt W.toAffine ℓ).toAlgebra (mulByInt W.toAffine ℓ).toAlgebra) :
     ∃ k : W.toAffine.Point, ℓ • k = 0 ∧
       ∀ z : KE, σ z = translateAlgEquivOfPoint W k z := by
-  letI := (mulByInt W.toAffine ℓ).toAlgebra
+  let := (mulByInt W.toAffine ℓ).toAlgebra
   have hcov := hcov_mulByInt_of_xy W ℓ hℓ0 (hxy_mulByInt W ℓ hℓ0)
   set forward := kernelTranslateForwardAut W (mulByInt W.toAffine ℓ) hcov with hfwd_def
   obtain ⟨k, hk_mem, hk_lift⟩ := hdesc_mulByInt W ℓ hℓ0 σ
@@ -183,10 +183,10 @@ theorem mem_pullback_range_of_translate_fixed (ℓ : ℤ) (hℓ : (ℓ : F) ≠
       translateAlgEquivOfPoint W S g = g) :
     ∃ h : KE, (mulByInt W.toAffine ℓ).pullback h = g := by
   have hℓ0 : ℓ ≠ 0 := by rintro rfl; norm_num at hℓ
-  letI := (mulByInt W.toAffine ℓ).toAlgebra
-  haveI hfin : @FiniteDimensional KE KE _ _ (mulByInt W.toAffine ℓ).toAlgebra.toModule :=
+  let := (mulByInt W.toAffine ℓ).toAlgebra
+  have hfin : @FiniteDimensional KE KE _ _ (mulByInt W.toAffine ℓ).toAlgebra.toModule :=
     isogeny_finiteDimensional W (mulByInt W.toAffine ℓ)
-  haveI hgal : @IsGalois KE _ KE _ (mulByInt W.toAffine ℓ).toAlgebra :=
+  have hgal : @IsGalois KE _ KE _ (mulByInt W.toAffine ℓ).toAlgebra :=
     Isogeny.isGalois_of_isSeparable_and_normal (mulByInt W.toAffine ℓ)
       (mulByInt_isSeparable W ℓ hℓ) (h_normal_mulByInt W ℓ hℓ0)
   have hfix : ∀ σ : @AlgEquiv KE KE KE _ _ _
@@ -217,7 +217,7 @@ theorem weilPairing_nondegenerate (ℓ : ℤ) (hℓ : (ℓ : F) ≠ 0)
     (h_deg : ∀ S : W.toAffine.Point, (hS : ℓ • S = 0) →
       weilPairing W ℓ hℓ S T hS hT = 1) :
     T = 0 := by
-  haveI hker : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
+  have hker : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
   obtain ⟨h, hh⟩ := mem_pullback_range_of_translate_fixed W ℓ hℓ (fun S hS ↦ by
     rw [weilPairing_translate W ℓ hℓ S T hS hT, h_deg S hS, map_one, one_mul])
   apply eq_zero_of_kappaDivisor_principal W
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingProps.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingProps.lean
index 403c142bc..70880d716 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingProps.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PairingProps.lean
@@ -9,9 +9,9 @@ import HasseWeil.HasseBound.WeilPairing.Pairing
 /-!
 # Weil-pairing properties: bilinearity in the second slot and alternating (Silverman III.8.1)
 
-This file proves the two remaining structural properties of the finite-level Weil pairing
-`e_ℓ : E[ℓ] × E[ℓ] → F` (ticket `T-R2-PAIRING-PROPS`), over an algebraically closed field `F`,
-extending the slot-1 bilinearity `weilPairing_mul_left` from `Pairing.lean`:
+We prove bilinearity and alternation of the finite-level Weil pairing
+`e_ℓ : E[ℓ] × E[ℓ] → F`, over an algebraically closed field `F`,
+extending the slot-1 bilinearity `weilPairing_mul_left`:
 
 * `weilPairing_mul_right` — **bilinearity in the second slot** (Silverman III.8.1b):
   `e_ℓ(S, T₁ + T₂) = e_ℓ(S, T₁) · e_ℓ(S, T₂)`.
@@ -112,7 +112,7 @@ theorem weilPairing_mul_right (ℓ : ℤ) (hℓ : (ℓ : F) ≠ 0)
     weilPairing W ℓ hℓ S (T₁ + T₂) hS h₁₂ =
       weilPairing W ℓ hℓ S T₁ hS hT₁ * weilPairing W ℓ hℓ S T₂ hS hT₂ := by
   have hℓ0 : ℓ ≠ 0 := by rintro rfl; simp at hℓ
-  haveI hker : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
+  have hker : Finite (mulByInt W.toAffine ℓ).toAddMonoidHom.ker := mulByInt_ker_finite W ℓ hℓ
   obtain ⟨k, hk_ne, hk_div⟩ := bilinDivisor_isPrincipal W T₁ T₂
   set u : KE := (mulByInt W.toAffine ℓ).pullback k
   have hu_ne : u ≠ 0 :=
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PencilDualDivisor.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/PencilDualDivisor.lean
index 5f516ab8e619e3cdfcf79eb728f53efdfb81a2c5..4401ef64fd20246d7f73d595ce01960cce1b673b 100644
GIT binary patch
delta 2332
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diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Pullback.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Pullback.lean
index 6a6379829..9fe8fdd6a 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Pullback.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Pullback.lean
@@ -7,22 +7,16 @@ import HasseWeil.HasseBound.WeilPairing.Fiber
 import HasseWeil.Foundation.Curves.Divisor.PicZero
 
 /-!
-# Route 2A — the multiplicity-free geometric divisor pullback (keystone)
+# Multiplicity-free geometric divisor pullback
 
-For a **separable** point-map endomorphism `f` (étale fibres, `e = 1`), the pullback divisor is
-the mult-1 fibre sum `f*((Q)) = Σ_{fP=Q} (P)`. This is the central new object of the Weil-pairing
-route — the same divisor pullback Route 1 lacked, now in the multiplicity-free regime — used by
-both the pairing construction (`div g = [ℓ]*((T)) − [ℓ]*((O))`) and the separable adjoint
-(Silverman III.8.2's function `h`).
-
-This file ships the definition and its **degree**: `deg(f*((Q))) = #fibre = #ker f` (Silverman
-III.4.10c, the separable case). The `σ`-bridge and the addition-formula linkage are downstream.
+For a separable point-map endomorphism `f`, the pullback of `(Q)` is the sum
+of the points in its finite fibre, each with multiplicity one. Its degree
+is the cardinality of the kernel, as in Silverman III.4.10(c). This divisor
+also gives the geometric group sum used in the dual isogeny formula.
 -/
 
 namespace HasseWeil.WeilPairing
 
-set_option linter.unusedSectionVars false
-
 open WeierstrassCurve
 
 variable {F : Type*} [Field F] [DecidableEq F] {W : WeierstrassCurve.Affine F} [W.IsElliptic]
@@ -44,7 +38,7 @@ noncomputable def pullbackDiv (f : W.Point →+ W.Point) (h : Finite f.ker) (Q :
 /-- **Degree of the mult-1 pullback** (Silverman III.4.10c, separable): `deg(f*((Q))) = #ker f`. -/
 theorem degree_pullbackDiv (f : W.Point →+ W.Point) (h : Finite f.ker) {P₀ Q : W.Point}
     (hP₀ : f P₀ = Q) : (pullbackDiv f h Q).degree = Nat.card f.ker := by
-  letI : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
+  let : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
   rw [pullbackDiv, ← Curves.ProjectiveDivisor.degreeHom_apply, map_sum]
   simp only [Curves.ProjectiveDivisor.degreeHom_apply, degree_single, Finset.sum_const,
     Finset.card_univ, nsmul_eq_mul, mul_one]
@@ -55,7 +49,7 @@ fibre). The start of the III.6.1(b) `σ`-bridge `σ(f*((Q))−f*((O))) = f̂(Q)`
 theorem projectiveDivisorSum_pullbackDiv (f : W.Point →+ W.Point) (h : Finite f.ker) (Q : W.Point) :
     letI : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
     Curves.projectiveDivisorSum W (pullbackDiv f h Q) = ∑ P : {P : W.Point // f P = Q}, P.val := by
-  letI : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
+  let : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
   rw [pullbackDiv, ← Curves.projectiveDivisorSumHom_apply, map_sum]
   refine Finset.sum_congr rfl (fun P _ => ?_)
   rw [Curves.projectiveDivisorSumHom_apply, Curves.projectiveDivisorSum_single, one_zsmul,
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Scaling/FrobeniusGalois.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Scaling/FrobeniusGalois.lean
index ef66a86e649eed9ded1a96df6f25baa29b62948a..8e90832e3d316d0e2eae1846034a1632be1da6fb 100644
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diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Scaling/OneSub.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Scaling/OneSub.lean
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diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Scaling/Separable.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/Scaling/Separable.lean
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diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/SigmaBridge.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/SigmaBridge.lean
index fd8f80407..1a65ebc5b 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/SigmaBridge.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/SigmaBridge.lean
@@ -6,18 +6,16 @@ Authors: Chris Birkbeck
 import HasseWeil.HasseBound.WeilPairing.Pullback
 
 /-!
-# Route 2A — the σ-bridge computation (Silverman III.6.1(b))
+# The σ-bridge computation (Silverman III.6.1(b))
 
 `σ(f*((Q)) − f*((O))) = (#ker f) · P₀` for any `P₀` with `f P₀ = Q`. This is the geometric heart of
 the dual: `σ` of the pullback divisor of `(Q) − (O)` is `[deg_s f] P₀`, which for a separable `f`
-(`deg_s = deg`) is `f̂(Q)`. Pure group theory over the kernel coset — the bridge the separable
-adjoint (Silverman III.8.2) consumes once linked to the genuine dual.
+(`deg_s = deg`) is `f̂(Q)`. This follows by reindexing the fibre as a kernel coset and forms the group-theoretic
+part of the separable adjoint formula in Silverman III.8.2.
 -/
 
 namespace HasseWeil.WeilPairing
 
-set_option linter.unusedSectionVars false
-
 open WeierstrassCurve
 
 variable {F : Type*} [Field F] [DecidableEq F] {W : WeierstrassCurve.Affine F} [W.IsElliptic]
@@ -28,8 +26,8 @@ theorem fiber_sum_eq_ker_sum (f : W.Point →+ W.Point) (h : Finite f.ker) {P₀
     letI : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
     letI : Fintype f.ker := @Fintype.ofFinite _ h
     (∑ P : {P : W.Point // f P = Q}, P.val) = ∑ T : f.ker, (P₀ + (T : W.Point)) := by
-  letI : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
-  letI : Fintype f.ker := @Fintype.ofFinite _ h
+  let : Fintype {P : W.Point // f P = Q} := @Fintype.ofFinite _ (fiber_finite f h Q)
+  let : Fintype f.ker := @Fintype.ofFinite _ h
   refine Fintype.sum_equiv (fiberEquivKer f hP₀) (fun P => P.val)
     (fun T => P₀ + (T : W.Point)) (fun P => ?_)
   simp only [fiberEquivKer, Equiv.coe_fn_mk]
@@ -41,7 +39,7 @@ theorem sigma_pullbackDiv_sub (f : W.Point →+ W.Point) (h : Finite f.ker) {P
     (hP₀ : f P₀ = Q) :
     Curves.projectiveDivisorSum W (pullbackDiv f h Q) -
         Curves.projectiveDivisorSum W (pullbackDiv f h 0) = Nat.card f.ker • P₀ := by
-  letI : Fintype f.ker := @Fintype.ofFinite _ h
+  let : Fintype f.ker := @Fintype.ofFinite _ h
   have hQ := projectiveDivisorSum_pullbackDiv f h Q
   have hO := projectiveDivisorSum_pullbackDiv f h 0
   rw [hQ, hO, fiber_sum_eq_ker_sum f h hP₀,
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionKernelRational.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionKernelRational.lean
index 378e2e72b..f1d4ef6ca 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionKernelRational.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionKernelRational.lean
@@ -11,26 +11,19 @@ import HasseWeil.Foundation.EC.GenericPointZsmul
 import HasseWeil.Foundation.EC.SeparableKernelTorsor
 
 /-!
-# Kernel-rationality of `[ℓ]` over `K̄` (Silverman III.4.10c)
+# Rationality of torsion kernels over an algebraically closed field
 
-For the multiplication-by-`ℓ` isogeny `[ℓ] = mulByInt W.toAffine ℓ` over an algebraically closed
-field `F = K̄`, we discharge the two genuine-isogeny coherence inputs of the capstone
-`HasseWeil.card_kernel_eq_degree_of_separable_concrete` (`SeparableKernelTorsor.lean`):
+For a nonzero integer `ℓ`, every point in the kernel of multiplication
+by `ℓ` over an extension of an algebraically closed base field already
+comes from a torsion point over that base field. The division-polynomial
+identity makes its x-coordinate algebraic, and the Weierstrass equation
+then makes its y-coordinate algebraic.
 
-* `hdesc_mulByInt` — the inverse-witness descent: every fibre `σ(P_gen) − P_gen` (`σ` an
-  automorphism of `K(E)/[ℓ]*K(E)`) is an `F`-rational kernel point of `[ℓ]`.
-* `h_normal_mulByInt` — the function-field extension `K(E)/[ℓ]*K(E)` is normal.
+Consequently, automorphisms of `K(E)/[ℓ]*K(E)` differ from the generic
+point by rational kernel points. The minimal polynomials of the generic
+coordinates split in `K(E)`, proving normality of this extension.
 
-Both rest on **kernel-rationality** (`kernelDescends_general`, specialised to
-`kernelOverKE_descends` at `L = K(E)`): over `K̄`, every point `Q` of `E` with coordinates in a
-field extension `L` that is killed by `[ℓ]` already descends to a `K̄`-rational `ℓ`-torsion point.
-The reason is the division-polynomial connection `[ℓ]Q = O ⟹ ψ_ℓ(x(Q), y(Q)) = 0 ⟹ Ψ²_ℓ(x(Q)) = 0`:
-the `x`-coordinate of `Q` is a root of the `ℓ`-division polynomial `Ψ²_ℓ`, whose coefficients lie
-in `K̄` (not just `L`); since `K̄` is algebraically closed, `Ψ²_ℓ` splits over `K̄`, so `x(Q)` is
-the image of a `K̄`-element. The `y`-coordinate then descends via the Weierstrass
-equation over `K̄`.
-
-Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.4.10c.
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.4.10(c).
 -/
 
 open WeierstrassCurve Polynomial
@@ -228,7 +221,7 @@ of `[ℓ]`.
 
 The geometric action `g` of `[ℓ]` on `(W_KE).Point` is multiplication-by-`ℓ` (`zsmulAddGroupHom ℓ`),
 whose value at the generic point is `(mulByInt_x ℓ, mulByInt_y ℓ) = ([ℓ]*x_gen, [ℓ]*y_gen)`
-(`zsmul_genericPoint_eq` + `mulByInt_pullback_x/y`); this is the genuine data. The shipped
+(`zsmul_genericPoint_eq` + `mulByInt_pullback_x/y`); the generic-point identity. The lemma
 `genericPointAct_mem_ker_g` then yields `g(σ P_gen) = g(P_gen)` — using `σ.commutes` (it fixes the
 pullback range) and the equivariance `g ∘ (Point.map σ) = (Point.map σ) ∘ g`, which here is just
 `map_zsmul`. Thus `ℓ • (σ P_gen − P_gen) = 0`, and `kernelOverKE_descends` produces the descended
@@ -241,7 +234,7 @@ theorem hdesc_mulByInt [IsAlgClosed F] (ℓ : ℤ) (hℓ : ℓ ≠ 0) :
         liftPointToKE W k =
           genericPointAct W (mulByInt W.toAffine ℓ) σ - genericPoint W := by
   intro σ
-  letI := (mulByInt W.toAffine ℓ).toAlgebra
+  let := (mulByInt W.toAffine ℓ).toAlgebra
   set g : (W_KE W).toAffine.Point →+ (W_KE W).toAffine.Point := zsmulAddGroupHom ℓ
   obtain ⟨hns, hsmul⟩ := zsmul_genericPoint_eq W ℓ hℓ
   have hsmul' : ℓ • genericPoint W =
@@ -449,16 +442,15 @@ theorem minpoly_gen_splits_of_mem_range [IsAlgClosed F] (ℓ : ℤ) (_hℓ : ℓ
     letI := (mulByInt W.toAffine ℓ).toAlgebra
     Polynomial.Splits ((minpoly W.toAffine.FunctionField a).map
       (algebraMap W.toAffine.FunctionField W.toAffine.FunctionField)) := by
-  letI algB := (mulByInt W.toAffine ℓ).toAlgebra
-  -- Re-pin the canonical `Algebra K(E) Ω` to use `Algebra.id` for the `[Algebra K(E) K(E)]` slot
-  -- of `AlgebraicClosure.instAlgebra` (otherwise the ambient `letI := pullback` poisons it, making
-  -- the proof's `algebraMap K(E) Ω` disagree with the one baked into `hmem`).
-  letI stdΩ : Algebra W.toAffine.FunctionField (AlgebraicClosure W.toAffine.FunctionField) :=
+  let algB := (mulByInt W.toAffine ℓ).toAlgebra
+  -- Use the standard embedding into the algebraic closure for the coordinate field,
+  -- while the base-field action on that coordinate field is the isogeny pullback.
+  let stdΩ : Algebra W.toAffine.FunctionField (AlgebraicClosure W.toAffine.FunctionField) :=
     @AlgebraicClosure.instAlgebra W.toAffine.FunctionField _ W.toAffine.FunctionField _
       (Algebra.id W.toAffine.FunctionField)
-  haveI hfin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
+  have hfin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ algB.toModule := isogeny_finiteDimensional W (mulByInt W.toAffine ℓ)
-  haveI halg : Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField :=
+  have halg : Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField :=
     Algebra.IsAlgebraic.of_finite _ _
   -- `Ω = AlgebraicClosure K(E)` is made a *base*-algebra via the ring hom `algebraMap ∘ pullback`,
   -- supplied *inline* to the abstract criterion (never a scope instance, so the standard
@@ -486,7 +478,7 @@ theorem minpoly_gen_splits_of_mem_range [IsAlgClosed F] (ℓ : ℤ) (_hℓ : ℓ
     rfl
   -- Build the `F`-algHom `g` by restricting scalars along `F → B`, using the inline scalar tower
   -- `F → B → Ω` (`of_algHom` for `(algebraMap) ∘ pullback`); supplied `@`-explicitly.
-  haveI htower : @IsScalarTower F W.toAffine.FunctionField
+  have htower : @IsScalarTower F W.toAffine.FunctionField
       (AlgebraicClosure W.toAffine.FunctionField) _
       (((algebraMap W.toAffine.FunctionField (AlgebraicClosure W.toAffine.FunctionField)).comp
         (mulByInt W.toAffine ℓ).pullback.toRingHom).toAlgebra).toSMul _ :=
@@ -537,11 +529,11 @@ once the two generator minimal polynomials do (`mulByInt_genCoords_minpoly_split
 theorem h_normal_mulByInt [IsAlgClosed F] (ℓ : ℤ) (hℓ : ℓ ≠ 0) :
     letI := (mulByInt W.toAffine ℓ).toAlgebra
     Normal W.toAffine.FunctionField W.toAffine.FunctionField := by
-  letI := (mulByInt W.toAffine ℓ).toAlgebra
-  haveI hfin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
+  let := (mulByInt W.toAffine ℓ).toAlgebra
+  have hfin : @FiniteDimensional W.toAffine.FunctionField W.toAffine.FunctionField
       _ _ (mulByInt W.toAffine ℓ).toAlgebra.toModule :=
     isogeny_finiteDimensional W (mulByInt W.toAffine ℓ)
-  haveI halg : Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField :=
+  have halg : Algebra.IsAlgebraic W.toAffine.FunctionField W.toAffine.FunctionField :=
     Algebra.IsAlgebraic.of_finite _ _
   obtain ⟨hsx, hsy⟩ := mulByInt_genCoords_minpoly_splits W ℓ hℓ
   refine normal_iff.mpr fun z ↦ ⟨(halg.isAlgebraic z).isIntegral, ?_⟩
diff --git a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionModule.lean b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionModule.lean
index e011591e1..dcc33eba4 100644
--- a/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionModule.lean
+++ b/projects/HasseWeil/HasseWeil/HasseBound/WeilPairing/TorsionModule.lean
@@ -10,27 +10,16 @@ import Mathlib.FieldTheory.Finiteness
 import Mathlib.LinearAlgebra.Dimension.Free
 
 /-!
-# `E[ℓ] ≅ (ZMod ℓ)²` — the geometric `ℓ`-torsion as a 2-dimensional `ZMod ℓ`-vector space
+# The two-dimensional module of prime torsion points
 
-For `ℓ` prime and `F` algebraically closed with `(ℓ : F) ≠ 0` (i.e. `ℓ ≠ char F`), the geometric
-`ℓ`-torsion `E[ℓ] = W.toAffine[(ℓ : ℤ)]` is a 2-dimensional vector space over the field `ZMod ℓ`.
+For a prime `ℓ` different from the characteristic of an algebraically
+closed field, the geometric torsion subgroup `E[ℓ]` has cardinality
+`ℓ²`. Since every element is killed by `ℓ`, it is a vector space over
+`ZMod ℓ`. The finite-vector-space cardinality formula gives dimension
+two, hence a basis indexed by `Fin 2` and an identification with
+`Fin 2 → ZMod ℓ`.
 
-This builds on the axiom-clean cardinality theorem
-`HasseWeil.WeilPairing.TorsionGeometric.card_torsion_ell` (`#E[ℓ] = ℓ²`, Silverman III.6.4(a))
-and packages the structure-theoretic consequences:
-
-* `card_torsion_ell_nat` — `Nat.card E[ℓ] = ℓ²` (no `ℤ`-coercion);
-* `torsion_ell_finite` — `E[ℓ]` is finite;
-* the `Module (ZMod ℓ)` structure on `E[ℓ]` (every element is killed by `ℓ`), via
-  `AddCommGroup.zmodModule`;
-* `finrank_torsion_ell` — `finrank (ZMod ℓ) E[ℓ] = 2`, via `Module.natCard_eq_pow_finrank`;
-* `torsion_ell_basis` — a `Basis (Fin 2) (ZMod ℓ) E[ℓ]`, via `Module.finBasisOfFinrankEq`;
-* `torsion_ell_linearEquiv` — `E[ℓ] ≃ₗ[ZMod ℓ] (Fin 2 → ZMod ℓ)`, via `Basis.equivFun`.
-
-These are the structures the downstream mod-`ℓ` Galois representation
-`ρ_ℓ : End(E) → GL₂(ZMod ℓ)` consumes.
-
-Reference: Silverman III.6.4(a), III.7.
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.6.4(a), III.7.
 -/
 
 open WeierstrassCurve
@@ -97,13 +86,13 @@ theorem torsion_ell_finite : Finite W.toAffine[(ℓ : ℤ)] := by
 forces `ℓ^finrank = ℓ²`, hence `finrank = 2` since `ℓ ≥ 2`. -/
 theorem finrank_torsion_ell :
     Module.finrank (ZMod ℓ) W.toAffine[(ℓ : ℤ)] = 2 := by
-  haveI := torsion_ell_finite W ℓ hℓF
+  have := torsion_ell_finite W ℓ hℓF
   have hcard : Nat.card W.toAffine[(ℓ : ℤ)]
       = Nat.card (ZMod ℓ) ^ Module.finrank (ZMod ℓ) W.toAffine[(ℓ : ℤ)] :=
     Module.natCard_eq_pow_finrank
   rw [card_torsion_ell_nat W ℓ hℓF] at hcard
   have hZcard : Nat.card (ZMod ℓ) = ℓ := by
-    haveI : NeZero ℓ := ⟨hℓ.out.pos.ne'⟩
+    have : NeZero ℓ := ⟨hℓ.out.pos.ne'⟩
     rw [Nat.card_eq_fintype_card, ZMod.card]
   rw [hZcard] at hcard
   -- `ℓ ^ 2 = ℓ ^ finrank`, and `ℓ ≥ 2`, so the exponents agree.
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/BaseChange/Basic.lean b/projects/HasseWeil/HasseWeil/Isogeny/BaseChange/Basic.lean
index 5e7b88094453689a80fefbc79c4a6dcb9032a838..e2365c39fcec4a07f8b7e2aac47dfbd1ec0b90ad 100644
GIT binary patch
delta 481
zcmX^2mhsOf#tjvWlTF!QZ@$9$o-sB*H?u?`+9fqPGbJ%8C)Kr3!7WxHIy^NqwWzqH
zD6zQMxwNP(RnN27H76&tpd>R{A^cHOBVi>?u?m@alN)&DHb3BANMma@H}GC(+$=39
z!8Ey1v}W^jNllK;b}DHCo1d8f;1b95VK&HZn{PWkXT&La&?Sa>bGz4C7Mv=K{jam(
zl;jLkV!|n@6)}ZRo(qSwxb$^(6|}S_CuGZP*2>v~+mxqy0!+*bRtl2^!=)#S6mHu5
Iw)i0v0F`R9Hvj+t

diff --git a/projects/HasseWeil/HasseWeil/Isogeny/BaseChange/Concrete.lean b/projects/HasseWeil/HasseWeil/Isogeny/BaseChange/Concrete.lean
index a56daff33..930b5c5e6 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/BaseChange/Concrete.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/BaseChange/Concrete.lean
@@ -6,72 +6,23 @@ Authors: Chris Birkbeck
 import HasseWeil.HasseBound.WeilPairing.Scaling.OneSub
 
 /-!
-# Concrete base-change of a function-field pullback (CoordHom-free)
+# Base change of function-field pullbacks
 
-For a smooth plane curve `C / F` and an `F`-algebra extension `L` with `L/F` algebraic, the
-function-field scalar-extension iso `K(C_L) ≅ L ⊗_F K(C)` is **already** available, axiom-clean, in
-`HasseWeil/Curves/CurveMapBaseChange.lean`:
+For an algebraic field extension `L/F`, the canonical scalar-extension
+isomorphism `Φ : L ⊗[F] K(C) ≃ₐ[L] K(C_L)` transports a function-field
+endomorphism `f` to `Φ ∘ (id_L ⊗ f) ∘ Φ⁻¹`.
 
-* `functionField_tensor_locBaseChange L : (L ⊗ K(C)) ≃ₐ[L] FractionRing (L ⊗ C.CR)`,
-* `functionField_baseChange_fracEquiv L : FractionRing (L ⊗ C.CR) ≃ₐ[L] K(C_L)`.
+The type synonym `Twist f` carries the scalar action through `f`. An
+explicit tensor-product linear equivalence identifies its base change
+with `Twist (id_L ⊗ f)`, proving preservation of the twisted module rank
+and hence of the degree of the pullback.
 
-Composing them gives the iso `Φ : (L ⊗ K(C)) ≃ₐ[L] K(C_L)` (`tensorFunctionFieldEquiv`).
+For `1 - π`, this constructs the base-changed pullback and its degree
+identity. The scaling data retain explicit hypotheses for the geometric
+kernel, divisor transport, duality, surjectivity and translation covariance.
 
-This file uses `Φ` to **construct** the base-change of an arbitrary function-field `F`-algebra hom
-`f : K(C) →ₐ[F] K(C)` — *no `CoordHom` required* — as the conjugate
-
-  `baseChangePullback f := Φ ∘ (id_L ⊗ f) ∘ Φ⁻¹ : K(C_L) →ₐ[L] K(C_L)`,
-
-an honest `L`-algebra hom (`Algebra.TensorProduct.map (AlgHom.id L) f` is the `L`-linear scalar
-extension of the `F`-linear `f`, conjugated by the `L`-algebra equiv `Φ`).
-
-The payoff (`oneSubFrobeniusPullback_L`) is the concrete `pullback_L` field of `OneSubScalingData`
-for the separable genuine isogeny `1 − π`, whose pullback has poles at the affine kernel so admits
-**no** `CoordHom` — hence the whole CoordHom-free route.
-
-## Degree preservation
-
-`finrank_baseChangePullback_eq_finrank_lTensorMap` reduces the degree of the conjugate
-`baseChangePullback f` (as an `Isogeny`, via `Isogeny.degree_eq_of_finrank_eq`) to the `finrank` of
-the *scalar extension* `id_L ⊗ f` over `L ⊗ K(C)`, by transporting the finrank along the `L`-algebra
-equiv `Φ` (`Algebra.finrank_eq_of_equiv_equiv`).  This **conjugation step is proved here in full**.
-
-The remaining equality `finrank_{L ⊗ K(C), via id_L ⊗ f}(L ⊗ K(C)) = finrank_{K(C), via f} K(C)`
-is the curve-free **base-change-of-finrank** content; it is **proved here in full** as
-`finrankBaseChange`, by realizing the `f`-twisted self-module `K(C)` as the type synonym `Twist f`
-and exhibiting `(L ⊗ K(C)) ⊗_{K(C)} Twist f ≃ₗ[L ⊗ K(C)] Twist (id_L ⊗ f)` (so
-`Module.finrank_baseChange` applies).  Hence `baseChangePullback_finrank_eq` (and the degree
-preservation it feeds) carries **no** finrank hypothesis.
-
-## What this file discharges for `OneSubScalingData`
-
-`mkOneSubScalingDataConcrete` assembles a full `OneSubScalingData` for the genuine `1 − π` base
-change in which:
-
-* `pullback_L` is **constructed** as the concrete `oneSubFrobeniusPullback_L` (the conjugate
-  `Φ ∘ (id_L ⊗ (1−π).pullback) ∘ Φ⁻¹`) — *no longer a raw input* (axiom-clean);
-* `hdeg_bc` is **fully discharged** by the proved conjugation step
-  `finrank_baseChangePullback_eq_finrank_lTensorMap` chained with the proved curve-free
-  `finrankBaseChange` — no carried hypothesis.
-
-The remaining inputs are exactly the genuinely-deep **K̄-level geometric** residuals, each stated
-against the concrete pullback:
-
-* `finiteKer` — finiteness of `ker(1 − π)_{K̄}` (a separable isogeny has finite kernel over `K̄`);
-* `hproj` — `ProjOrdTransport` (multiplicity-free divisor pullback);
-* `δ`/`hdc` — the divisor-pushforward dual `1 − V̄` and Silverman III.6.2(a) `δ ∘ φ = [#ker φ]`;
-* `hsurj` — surjectivity over `K̄` (Silverman III.4.10a);
-* `hkerdeg` — the separable degree match `#ker = deg` (Silverman III.4.10c);
-* `hcomm'` — the translation covariance (Silverman III.8.2).
-
-These are the genuine geometric content of the base-changed separable isogeny; the project ships no
-concrete base-change of a non-Frobenius isogeny's *point map* (only the witness-parametric
-`mkBaseChange`), so they remain carried (cf. `ProjOrdTransport`/`Naturality` elsewhere).
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, II.2.11 (degree under base change),
-  III.4.10a/c, III.6.2(a), III.8.2, III.8.6.1.
+References: Silverman, *The Arithmetic of Elliptic Curves*, II.2.11,
+III.4.10, III.6.2(a), and III.8.2–6.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
@@ -149,8 +100,7 @@ local notation "lTM" => Algebra.TensorProduct.map (AlgHom.id L L) f
 attribute [local instance] Algebra.TensorProduct.rightAlgebra
 
 /-- `eFwd` is surjective, from its action on pure tensors (`heFwd`) and the section `eInv`
-(`heInv`).  Stated with the forward/inverse maps as opaque parameters so it elaborates within the
-default heartbeat budget. -/
+(`heInv`).  The forward and inverse maps are explicit parameters. -/
 private theorem twistLTensor_surjective [Module A (Twist (lTM))]
     [IsScalarTower A (L ⊗[F] A) (Twist (lTM))]
     (eFwd : ((L ⊗[F] A) ⊗[A] (Twist f)) →ₗ[L ⊗[F] A] (Twist (lTM)))
@@ -164,7 +114,6 @@ private theorem twistLTensor_surjective [Module A (Twist (lTM))]
   refine ⟨eInv (Twist.toB (lTM) z), ?_⟩
   apply Twist.toB_injective (lTM)
   induction (Twist.toB (lTM) z) with
-  | zero => simp only [map_zero]; rfl
   | add x y hx hy =>
       rw [map_add, map_add, Twist.toB_add]
       rw [show Twist.toB (lTM) (eFwd (eInv x)) = x from hx,
@@ -176,7 +125,7 @@ private theorem twistLTensor_surjective [Module A (Twist (lTM))]
         Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul]
 
 /-- `eInv ∘ toB ∘ eFwd = id`, from the action on pure tensors.  The left inverse, giving
-injectivity of `eFwd`.  Forward/inverse maps are opaque parameters for heartbeat budget. -/
+injectivity of `eFwd`.  The forward and inverse maps are explicit parameters. -/
 private theorem twistLTensor_leftInverse [Module A (Twist (lTM))]
     [IsScalarTower A (L ⊗[F] A) (Twist (lTM))]
     (eFwd : ((L ⊗[F] A) ⊗[A] (Twist f)) →ₗ[L ⊗[F] A] (Twist (lTM)))
@@ -188,14 +137,9 @@ private theorem twistLTensor_leftInverse [Module A (Twist (lTM))]
     ∀ t : (L ⊗[F] A) ⊗[A] (Twist f), eInv (Twist.toB (lTM) (eFwd t)) = t := by
   intro t
   induction t with
-  | zero => rw [map_zero (f := eFwd), show Twist.toB (lTM) 0 = 0 from rfl,
-      map_zero (f := eInv)]
   | add x y hx hy => rw [map_add (f := eFwd), Twist.toB_add, map_add (f := eInv), hx, hy]
   | tmul w m =>
       induction w with
-      | zero =>
-          rw [TensorProduct.zero_tmul, map_zero (f := eFwd),
-            show Twist.toB (lTM) 0 = 0 from rfl, map_zero (f := eInv)]
       | add x y hx hy =>
           rw [TensorProduct.add_tmul, map_add (f := eFwd), Twist.toB_add, map_add (f := eInv),
             hx, hy]
@@ -222,9 +166,9 @@ equals the `A`-module finrank of `A` via `f`.  Proved by exhibiting the `L ⊗ A
 private theorem finrank_lTensorMap_eq_finrank :
     @Module.finrank (L ⊗[F] A) (L ⊗[F] A) _ _ (lTM).toRingHom.toAlgebra.toModule =
       @Module.finrank A A _ _ f.toRingHom.toAlgebra.toModule := by
-  letI iA_TwL : Module A (Twist (lTM)) :=
+  let iA_TwL : Module A (Twist (lTM)) :=
     Module.compHom (Twist (lTM)) (algebraMap A (L ⊗[F] A))
-  haveI iA_TwL_tower : IsScalarTower A (L ⊗[F] A) (Twist (lTM)) :=
+  have iA_TwL_tower : IsScalarTower A (L ⊗[F] A) (Twist (lTM)) :=
     SMul.comp.isScalarTower (algebraMap A (L ⊗[F] A))
   let l : (Twist f) →ₗ[A] (Twist (lTM)) :=
     { toFun := fun m ↦ Twist.ofB (lTM) (1 ⊗ₜ[F] Twist.toB f m)
@@ -292,8 +236,7 @@ variable (L : Type*) [Field L] [Algebra F L] [Algebra.IsAlgebraic F L]
 
 /-- **The function-field scalar-extension iso** `Φ : (L ⊗_F K(C)) ≃ₐ[L] K(C_L)`, the composite of
 `functionField_tensor_locBaseChange` (`L ⊗ K(C) ≅ FractionRing (L ⊗ C.CR)`) and
-`functionField_baseChange_fracEquiv` (`FractionRing (L ⊗ C.CR) ≅ K(C_L)`).  Both are shipped
-axiom-clean in `CurveMapBaseChange.lean`. -/
+`functionField_baseChange_fracEquiv` (`FractionRing (L ⊗ C.CR) ≅ K(C_L)`).  These are the canonical tensor and fraction-field identifications. -/
 noncomputable def tensorFunctionFieldEquiv :
     letI := C.isDomain_tensorCoordRing L
     (L ⊗[F] C.toAffine.FunctionField) ≃ₐ[L] (C.baseChange L).toAffine.FunctionField :=
@@ -307,7 +250,7 @@ noncomputable def lTensorMap (f : C.toAffine.FunctionField →ₐ[F] C.toAffine.
     (L ⊗[F] C.toAffine.FunctionField) →ₐ[L] (L ⊗[F] C.toAffine.FunctionField) :=
   Algebra.TensorProduct.map (AlgHom.id L L) f
 
-set_option linter.unusedSectionVars false in
+omit [Algebra.IsAlgebraic F L] in
 @[simp] theorem lTensorMap_tmul
     (f : C.toAffine.FunctionField →ₐ[F] C.toAffine.FunctionField) (l : L)
     (u : C.toAffine.FunctionField) :
@@ -355,10 +298,10 @@ theorem finrank_baseChangePullback_eq_finrank_lTensorMap
         (baseChangePullback C L f).toRingHom.toAlgebra.toModule =
       @Module.finrank (L ⊗[F] C.toAffine.FunctionField) (L ⊗[F] C.toAffine.FunctionField) _ _
         (lTensorMap C L f).toRingHom.toAlgebra.toModule := by
-  letI := C.isDomain_tensorCoordRing L
-  letI algBcp : Algebra (C.baseChange L).toAffine.FunctionField
+  let := C.isDomain_tensorCoordRing L
+  let algBcp : Algebra (C.baseChange L).toAffine.FunctionField
       (C.baseChange L).toAffine.FunctionField := (baseChangePullback C L f).toRingHom.toAlgebra
-  letI algLt : Algebra (L ⊗[F] C.toAffine.FunctionField) (L ⊗[F] C.toAffine.FunctionField) :=
+  let algLt : Algebra (L ⊗[F] C.toAffine.FunctionField) (L ⊗[F] C.toAffine.FunctionField) :=
     (lTensorMap C L f).toRingHom.toAlgebra
   refine Algebra.finrank_eq_of_equiv_equiv (tensorFunctionFieldEquiv C L).symm.toRingEquiv
     (tensorFunctionFieldEquiv C L).symm.toRingEquiv ?_
@@ -380,7 +323,7 @@ def FinrankBaseChange (f : C.toAffine.FunctionField →ₐ[F] C.toAffine.Functio
     @Module.finrank C.toAffine.FunctionField C.toAffine.FunctionField _ _
       f.toRingHom.toAlgebra.toModule
 
-set_option linter.unusedSectionVars false in
+omit [Algebra.IsAlgebraic F L] in
 /-- **The base-change-of-finrank fact, proved (curve-free).**  For an `F`-algebra endomorphism `f`
 of the field `K(C)`, base change along `F → L` preserves the degree of the (injective) field
 endomorphism: `[L ⊗ K(C) : (id_L ⊗ f)(L ⊗ K(C))] = [K(C) : f(K(C))]`. -/
@@ -422,7 +365,7 @@ noncomputable def oneSubFrobeniusPullback_L (hq : 2 ≤ Fintype.card K) :
 
 /-- **Degree preservation for the concrete `1 − π` base change** (curve-free, fully discharged).
 Chains the proved conjugation step (inside `baseChangePullback_finrank_eq`, which now invokes the
-proved `finrankBaseChange`) into the shipped witness-parametric
+proved `finrankBaseChange`) into the corresponding
 `oneSubFrobeniusIsogBaseChange_degree_eq_of_finrank`.  This **discharges the `hdeg_bc` field** of
 `OneSubScalingData` with no carried `FinrankBaseChange` hypothesis. -/
 theorem oneSubFrobeniusIsogBaseChange_degree_eq_of_finrankBaseChange (hq : 2 ≤ Fintype.card K) :
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/Basic.lean b/projects/HasseWeil/HasseWeil/Isogeny/Basic.lean
index 01ae90e29..fc74956a0 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/Basic.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/Basic.lean
@@ -45,7 +45,7 @@ image of `x` has poles at the `n`-torsion).
 The current `pullback_ordAtInfty_nonneg` field captures "morphism is defined at
 the basepoint" (regular functions pull back to regular functions). It does not
 strictly capture `φ(O) = O` — that would require the strict-positive form
-`0 < ord_∞ f → 0 < ord_∞ pullback f`. A future refinement may strengthen this.
+`0 < ord_∞ f → 0 < ord_∞ pullback f`.
 
 ## References
 
@@ -296,6 +296,7 @@ noncomputable def Isogeny.frobenius : Isogeny W W where
     · rw [(⟨W⟩ : Curves.SmoothPlaneCurve K).ordAtInfty_pow hf]
       exact nsmul_nonneg h _
 
+omit [DecidableEq K] in
 @[simp] theorem Isogeny.frobenius_pullback (f : W.FunctionField) :
     (Isogeny.frobenius W).toCurveMap.pullback f = f ^ Fintype.card K :=
   congr_fun (FiniteField.coe_frobeniusAlgHom (K := K) (R := W.FunctionField)) f
@@ -313,12 +314,14 @@ noncomputable def Isogeny.frobeniusCoordHom :
         (FiniteField.frobeniusAlgHom K W.CoordinateRing u)
     simp only [FiniteField.coe_frobeniusAlgHom, map_pow]
 
+omit [DecidableEq K] in
 /-- The Frobenius isogeny has degree `q := #K`. Inherits from
 `HasseWeil.frobenius_finrank_functionField`. Reference: Silverman III.4.6. -/
 @[simp] theorem Isogeny.frobenius_degree :
     (Isogeny.frobenius W).degree = Fintype.card K :=
   HasseWeil.frobenius_finrank_functionField K W
 
+omit [DecidableEq K] in
 /-- **Frobenius coordinate-evaluation step.** Evaluating a pulled-back
 coordinate-ring element `r` through the Frobenius coordinate hom at a smooth
 point `(x, y)` equals the `q`-th power (`q := #K`) of the plain evaluation:
@@ -336,6 +339,7 @@ private theorem Isogeny.frobenius_evalAtPullback {x y : K}
       ⟨x, y, h⟩ (FiniteField.frobeniusAlgHom K W.CoordinateRing r) = _
   simp only [FiniteField.coe_frobeniusAlgHom, map_pow]
 
+omit [DecidableEq K] in
 /-- The Frobenius isogeny acts as the **identity** on `K`-rational points: for
 any `(x, y) ∈ W(K)`, Frobenius sends `(x, y) ↦ (x^q, y^q) = (x, y)` since
 `x^q = x` for all `x ∈ K` by `FiniteField.pow_card`. The basepoint is
@@ -383,23 +387,17 @@ end Frobenius
 
 /-! ### Multiplication-by-`n` isogeny
 
-The endomorphism `[n] : E → E` is constructed via the function-field pullback
-`mulByInt_pullbackAlgHom` from `HasseWeil/MulByIntPullback.lean` (built from
-division polynomials).
+The endomorphism `[n] : E → E` is constructed from the division-polynomial
+function-field pullback `mulByInt_pullbackAlgHom`.
 
-The basepoint condition `pullback_ordAtInfty_nonneg` for `[n]` is genuinely
-non-trivial — it requires the ramification-theoretic fact that
-`ord_∞([n]*f) = e · ord_∞(f)` for some positive integer `e` (the local
-ramification index of `[n]` at `O`). For now we expose this as an explicit
-hypothesis (`mulByIntOfBasepoint`); a follow-up ticket will discharge it via
-the existing `mulByInt_finrank` infrastructure or a dedicated valuation
-extension argument.
+The construction `mulByIntOfBasepoint` takes an explicit hypothesis that the
+pullback preserves functions regular at infinity. The corresponding local
+ramification formula is `ord_∞([n]*f) = e · ord_∞(f)` with positive ramification
+index `e`.
 
-Note that `[n]` does *not* admit a `coordHom` witness in the affine model —
-its pullback of `x` is `Φ_n / Ψ_n²`, a fraction with poles at the
-`n`-torsion. Hence `Isogeny.toPointMap` cannot be invoked for `mulByInt`
-directly via the affine route; this is the design reason for separating
-`coordHom` from the structure. -/
+For `n ≥ 2`, the pullback of `x` is `Φ_n / Ψ_n²`, with poles at the `n`-torsion.
+Thus this projective morphism does not generally induce a map of affine
+coordinate rings. -/
 
 section MulByInt
 
@@ -428,6 +426,7 @@ noncomputable def Isogeny.mulByIntOfBasepoint {n : ℤ} (hn : n ≠ 0)
   toCurveMap := { pullback := HasseWeil.mulByInt_pullbackAlgHom W n hn }
   pullback_ordAtInfty_nonneg := h_basepoint
 
+omit [DecidableEq F] in
 @[simp] theorem Isogeny.mulByIntOfBasepoint_pullback {n : ℤ} (hn : n ≠ 0)
     (h_basepoint : MulByIntBasepoint W hn) (f : W.FunctionField) :
     (Isogeny.mulByIntOfBasepoint W hn h_basepoint).toCurveMap.pullback f =
@@ -435,48 +434,14 @@ noncomputable def Isogeny.mulByIntOfBasepoint {n : ℤ} (hn : n ≠ 0)
 
 end MulByInt
 
-/-! ### Universal group-homomorphism property: bundled `WithHom` isogenies
+/-! ### Bundled group-homomorphism property
 
-This is the working framework for **Silverman III.4.8** (every isogeny is a
-group homomorphism). Silverman's argument uses `Pic⁰(E) ≅ E` and the
-contravariant functoriality of the Picard variety on morphisms; alternative
-routes go via the formal group of `E` or direct addition-formula computation.
+The `WithHom` structure bundles an isogeny with an affine coordinate-ring map
+and a proof of `AddHomProperty`. Identity, composition and Frobenius preserve
+this data, yielding additive homomorphisms on rational points.
 
-In our setup, the unconditional theorem is genuinely hard — it reduces to one
-of those upstream infrastructures, none of which are fully in the project yet.
-Instead, we package the **closure under standard constructions**: any isogeny
-built from `id`, `compose`, and `frobenius` automatically carries an
-`AddHomProperty`, hence a bundled `AddMonoidHom` on points.
-
-The `WithHom` bundle below is the canonical recipient of these closure
-operations. Future extensions (mulByInt, neg, dual, factor-through-Frobenius,
-…) should provide their own `WithHom` instances by discharging the
-`AddHomProperty` witness.
-
-## Proof routes for the universal version
-
-Silverman's three known routes for the universal Silverman III.4.8:
-
-1. **Pic⁰ route** (Silverman III.4.8 proper). Uses
-   `σ : Pic⁰(E) ≅ E` (Silverman III.3.4) — gated on `T-III-3-004` in this
-   project. The induced pullback `φ_* : Pic⁰(E₁) → Pic⁰(E₂)` is a hom by
-   functoriality, and the diagram with σ commutes when `φ(O) = O`.
-
-2. **Formal-group route** (deformation argument). An isogeny pulls back to a
-   formal-group hom on `Ê`. The hom property holds on the formal neighborhood
-   of `O`, then extends to all of `E` by translation invariance and density.
-   Requires the bridge `HasseWeil.Isogeny → HasseWeil.FormalGroupHom`, not yet
-   built.
-
-3. **Addition-formula route** (direct computation). Use the explicit
-   `Affine.addX/addY` formulas; show that any `F`-algebra hom on `K(E)` that
-   restricts to coord rings preserves these (an algebra-homs-respect-rational-
-   functions argument). Requires `addPullbackAlgHom` from
-   `HasseWeil/AdditionPullback.lean` (currently has 3 transcendence sorries).
-
-All three reduce the universal claim to upstream infrastructure that is
-modular: once any one route's gates are filled, `silvermanIII48_via_<route>`
-can be added as a one-liner taking the upstream output as a witness. -/
+Silverman III.4.8 proves the group-homomorphism property for isogenies using
+`Pic⁰(E) ≅ E` and the functoriality of the Picard variety. -/
 
 namespace Isogeny
 
@@ -627,23 +592,6 @@ noncomputable def Isogeny.WithHom.frobenius
   show (Isogeny.frobenius W).toPointMap (Isogeny.frobeniusCoordHom W) P = P
   exact Isogeny.frobenius_toPointMap W P
 
-/-! ### Future extensions (mulByInt, neg, dual)
-
-For each of the following, providing a `WithHom` instance requires both a
-coord-ring lift (often non-trivial) and the `AddHomProperty` proof:
-
-* `mulByInt n` for `n ≥ 2`: the affine coord-ring lift doesn't naturally
-  exist (image of `x` is `Φₙ/Ψₙ²`, with poles at the `n`-torsion). Resolution
-  requires a localization-based `CoordHom` or a projective coord model.
-
-* `neg` (the `[-1]` isogeny): `(x, y) ↦ (x, -y - a₁x - a₃)`. Has a clean
-  affine coord-ring lift via `Affine.negY`. The `AddHomProperty` is
-  `-(P+Q) = -P + -Q`, which holds by `neg_add` on the abelian group `W.Point`.
-  Construction is straightforward in principle but requires explicit
-  `AlgHom` building from the negY formula on the function field.
 
-* `dual α`: requires the existence-uniqueness of the dual isogeny
-  (Silverman III.6.1), gated on either Pic⁰ or kernel/factorization
-  infrastructure (`T-III-6-001` in the project ticket board). -/
 
 end HasseWeil.EC
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/ClassGroup.lean b/projects/HasseWeil/HasseWeil/Isogeny/ClassGroup.lean
index 9b2003a2e..5d5eaceb9 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/ClassGroup.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/ClassGroup.lean
@@ -8,59 +8,12 @@ import HasseWeil.Foundation.Ramification
 import HasseWeil.Pic0.ClassGroupNorm
 
 /-!
-# The isogeny ↔ class-group bridge (reusable Pic⁰ infrastructure)
+An injective finite coordinate-ring pullback associated to an elliptic-curve
+endomorphism induces extension and relative-norm maps on the ideal class group.
+Their composite raises each class to the coordinate-ring extension degree.
+A compatible fraction-field tower identifies this degree with the isogeny degree.
 
-This file packages, for an endomorphism `α : Isogeny E E` of an elliptic curve `E`, the data and
-lemmas needed to push `α` down to the ideal class group `ClassGroup E.CoordinateRing` and recover
-the dual relation `α̂ ∘ α = [deg α]` at the level of `ClassGroup`.
-
-The function-field pullback `α.pullback : E.FunctionField →ₐ[F] E.FunctionField` does **not** in
-general restrict to the coordinate ring `R := E.CoordinateRing` (this is the
-integrality-preservation content of Silverman III.3.4, deliberately *not* discharged here).  We
-therefore carry the restriction as **data**, an `Isogeny.CoordHom`, mirroring
-`Curves.CurveMap.CoordHom`.
-
-The deep obligations are carried as **hypotheses** (witness-parametric), not discharged universally:
-
-* injectivity of the coordinate-ring restriction (`hinj`), giving `FaithfulSMul`/`IsTorsionFree`;
-* finiteness of `R` over itself through the restriction (`Module.Finite`), giving the `ClassGroup`
-  norm/extension maps and matching `α.degree` to the coordinate-ring `finrank`.
-
-These are instantiable per isogeny (e.g. Frobenius, multiplication-by-`n`); the universal versions
-are separate, flagged-deep tickets.
-
-## Main definitions
-
-* `HasseWeil.Isogeny.CoordHom`: the coordinate-ring restriction witness for `α : Isogeny E E`.
-* `HasseWeil.Isogeny.CoordHom.toAlgebra`: the induced `R`-algebra structure on `R`.
-* `HasseWeil.Isogeny.classNorm` / `Isogeny.classMap`: the relative norm / extension maps on
-  `ClassGroup R` induced by a `CoordHom`.
-
-## Main results
-
-* `HasseWeil.Isogeny.degree_eq_finrank_coordinateRing_of_tower_eq`: `α.degree` equals the
-  coordinate-ring `finrank` of `ch.toAlgebra`, given a fraction-field tower witness for the
-  `ch`-twisted coordinate ring (witness-parametric, to sidestep a same-type instance diamond — see
-  the lemma's docstring).
-* `HasseWeil.Isogeny.classNorm_comp_classMap`: `classNorm (classMap c) = c ^ (finrank R R)` — the
-  class-group shadow of `α̂ ∘ α = [deg α]`, exponent as the coordinate-ring degree.
-* `HasseWeil.Isogeny.classNorm_comp_classMap_degree`: the same with exponent `α.degree` (combines
-  the previous two; carries the tower witness as hypotheses).
-
-## Design note
-
-The function-field pullback restricted to coordinate rings and lifted back to the fraction field
-gives, on the *same* carriers `R` and `FF`, a *non-identity* algebra structure.  Several would-be
-"auto-derived" scalar towers (`R → R → FF`, `R → FF → FF`) are therefore **false** as stated with
-the canonical base action, and `finrank_of_isFractionRing` cannot be applied with `S = R`,
-`S' = FF` literally (the `IsFractionRing` typeclass cannot distinguish the canonical inclusion from
-the twisted map).  The deep obligations (injectivity `hinj`, finiteness `Module.Finite`, and the
-fraction-field tower witness for the degree bridge) are consequently carried as **hypotheses**;
-they are instantiable per isogeny (Frobenius, multiplication-by-`n`).
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], III.3.4, III.4, III.6.
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, III.3.4.
 -/
 
 open WeierstrassCurve Polynomial
@@ -146,7 +99,7 @@ noncomputable def classNorm (ch : α.CoordHom) (hinj : Function.Injective ch.toA
   haveI hfinI : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
   haveI htfI : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
     ch.isTorsionFree hinj
-  @ClassGroup.relNorm E.CoordinateRing E.CoordinateRing _ _ _ _ _ _ _ _ ch.toAlgebra hfinI htfI
+  @ClassGroup.relNorm E.CoordinateRing E.CoordinateRing _ _ _ _ ch.toAlgebra hfinI htfI
 
 /-- The **extension map on the class group** induced by `α : Isogeny E E` and a coordinate-ring
 restriction witness `ch`, with the finiteness obligation carried as a hypothesis.
@@ -160,7 +113,7 @@ noncomputable def classMap (ch : α.CoordHom) (hinj : Function.Injective ch.toAl
   haveI : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
   haveI htfI : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
     ch.isTorsionFree hinj
-  @ClassGroup.map E.CoordinateRing E.CoordinateRing _ _ _ _ _ _ ch.toAlgebra htfI
+  @ClassGroup.map E.CoordinateRing E.CoordinateRing _ _ _ _ ch.toAlgebra htfI
 
 /-- **Function-field ↔ coordinate-ring degree bridge (witness-parametric `_of_tower_eq` form).**
 
@@ -168,23 +121,11 @@ The function-field degree `α.degree` equals the coordinate-ring degree
 `Module.finrank R R` (with `R` carrying `ch.toAlgebra`), *given* a fraction-field tower witness for
 the `ch`-twisted coordinate ring.
 
-The deep obstruction (see the file-level note and the round report) is a genuine **same-type
-instance diamond**: `Algebra.IsAlgebraic.finrank_of_isFractionRing` needs the source/target rings
-`R, S` and their fraction fields `R', S'` to be *distinguishable type pairs* so that
-`IsFractionRing R R'` and `IsFractionRing S S'` can coexist with the canonical inclusion (for `R'`)
-and the `ch`-twisted map (for the `S → S'` tower) respectively.  With `S = R` and `S' = R' = FF`
-*literally*, Lean's instance resolution cannot keep the two `Algebra R FF` structures apart, even
-though the required equation `α.pullback (algebraMap R FF r) = algebraMap R FF (ch.toAlgHom r)` is
-exactly `ch.compat`.
-
-We therefore expose the bridge witness-parametrically: the caller supplies a *distinct nominal copy*
-`S` of the `ch`-twisted coordinate ring together with its fraction field `S'` (a copy of
-`E.FunctionField` carrying `α.toAlgebra`), the standard fraction-field tower instances, and the two
-`Module.finrank`-transfer equalities `hSR`, `hS'FF` (each a `LinearEquiv.finrank_eq` away once the
-copies are chosen).  These are dischargeable per isogeny (Frobenius, multiplication-by-`n`).
-
-The conclusion is then a one-line application of `finrank_of_isFractionRing` between the four
-distinct type slots `R, FF, S, S'`. -/
+The coordinate-ring algebra induced by the endomorphism differs from the
+canonical algebra on the same carrier. Distinct nominal copies `S` and `S'`
+represent the twisted coordinate ring and its fraction field. Compatible
+scalar towers and the rank-transfer equalities `hSR` and `hS'FF` allow the
+fraction-field rank theorem to compare their degrees. -/
 theorem degree_eq_finrank_coordinateRing_of_tower_eq (ch : α.CoordHom)
     (S : Type*) [CommRing S] [Algebra E.CoordinateRing S]
     [FaithfulSMul E.CoordinateRing S] [Algebra.IsAlgebraic E.CoordinateRing S] [NoZeroDivisors S]
@@ -197,7 +138,7 @@ theorem degree_eq_finrank_coordinateRing_of_tower_eq (ch : α.CoordHom)
     (hS'FF : @Module.finrank E.FunctionField S' _ _ _ = α.degree) :
     α.degree = @Module.finrank E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := by
   rw [← hSR, ← hS'FF]
-  exact Algebra.IsAlgebraic.finrank_of_isFractionRing E.CoordinateRing E.FunctionField S S'
+  exact IsFractionRing.finrank_eq E.CoordinateRing E.FunctionField S S'
 
 /-- **The class-group dual relation (coordinate-ring `finrank` form).**
 
@@ -209,17 +150,17 @@ the `ch.toAlgebra` structure):
 This is the class-group shadow of `α̂ ∘ α = [deg α]`, with the exponent expressed as the
 coordinate-ring degree.  To express the exponent as `α.degree` (the function-field degree), compose
 with `degree_eq_finrank_coordinateRing_of_tower_eq` (the function-field ↔ coordinate-ring degree
-bridge, separately flagged). -/
+bridge). -/
 theorem classNorm_comp_classMap (ch : α.CoordHom) (hinj : Function.Injective ch.toAlgHom)
     (hfin : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule)
     (c : ClassGroup E.CoordinateRing) :
     α.classNorm ch hinj hfin (α.classMap ch hinj hfin c) =
       c ^ @Module.finrank E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := by
-  letI := ch.toAlgebra
-  haveI hfinI : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
-  haveI htfI : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
+  let := ch.toAlgebra
+  have hfinI : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
+  have htfI : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
     ch.isTorsionFree hinj
-  exact @ClassGroup.relNorm_comp_map E.CoordinateRing E.CoordinateRing _ _ _ _ _ _ _ _
+  exact @ClassGroup.relNorm_comp_map E.CoordinateRing E.CoordinateRing _ _ _ _
     ch.toAlgebra hfinI htfI c
 
 /-- **The class-group dual relation (`α.degree` form).**
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/FormalSeries.lean b/projects/HasseWeil/HasseWeil/Isogeny/FormalSeries.lean
index 0cabc394abc4ca7608ba92b737cb85363078b59b..dcd6e5526f0503f078f119bafd1bb4a80e1148f6 100644
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diff --git a/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/FunctionField.lean b/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/FunctionField.lean
index 6c27d435c..f83025581 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/FunctionField.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/FunctionField.lean
@@ -15,43 +15,6 @@ import Mathlib.RingTheory.IsTensorProduct
 import HasseWeil.Foundation.Auxiliary.Universal
 import HasseWeil.Isogeny.FunctionField
 
-/-!
-# The Frobenius Isogeny via Function Fields
-
-We construct the q-th power Frobenius as a concrete isogeny `π : E → E` using the
-function field pullback `π* : K(E) →ₐ[K] K(E)` given by `x ↦ x^q`.
-
-## Main Definitions
-
-* `HasseWeil.frobeniusIsogeny`: The Frobenius as a `PullbackIsogeny K W W`, with pullback
-  given by `frobeniusAlgHom K K(E)`.
-
-## Main Results
-
-* `HasseWeil.frobeniusIsogeny_degree`: The Frobenius isogeny has degree `q = #K`.
-* `HasseWeil.frobeniusIsogeny_pow_mem_fieldRange`: pure inseparability of `K(E) / K(E)^q`
-  (Silverman II.2.11(b)), stated as a power-membership fact.
-* `HasseWeil.frobeniusIsogeny_pullback_range`: the pullback image is exactly the `q`-th
-  powers in `K(E)` (Silverman II.2.11(a)).
-
-## Implementation notes
-
-The degree `[K(E) : K(E)^q] = q` is obtained from the tower law. Writing `x` for the image
-of `X` in the coordinate ring, so `K(E) = K(x, y)`:
-
-* `[K(E) : K(x)] = 2` and `[K(E)^q : K(x^q)] = 2` (the Weierstrass equation is the minimal
-  polynomial of `y` over `K(x)`, and of `y^q` over `K(x^q)`);
-* `[K(x) : K(x^q)] = q` (computed via `RatFunc.finrank_eq_max_natDegree`, after identifying
-  the Frobenius range in `RatFunc K` with `K⟮X^q⟯`);
-* the tower law gives `[K(E) : K(E)^q] · 2 = 2 · q`, hence `[K(E) : K(E)^q] = q`.
-
-The Frobenius-twisted `Module.finrank` is reduced to the plain field extension `[K(E) : K(E)^q]`
-via `Algebra.finrank_eq_of_equiv_equiv`.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*][silverman2009], III.4.6, II.2.11
--/
 
 open WeierstrassCurve FiniteField
 open scoped Polynomial
@@ -98,7 +61,7 @@ private theorem frobenius_finrank_eq_fieldRange_finrank :
     (frobeniusAlgHom K W.toAffine.FunctionField).toRingHom x
   simp [AlgEquiv.ofInjective_apply]
 
-/-- Explicit module instance: `K[X]` acts on `CoordinateRing` via the algebra structure. -/
+
 noncomputable instance coordinateRing_module :
     Module K[X] W.toAffine.CoordinateRing :=
   @Algebra.toModule K[X] W.toAffine.CoordinateRing _ _ inferInstance
@@ -107,7 +70,7 @@ instance coordinateRing_finite : Module.Finite K[X] W.toAffine.CoordinateRing :=
   Module.Finite.of_basis (Affine.CoordinateRing.basis W.toAffine)
 
 omit [DecidableEq K] [Fintype K] [W.toAffine.IsElliptic] in
-/-- The coordinate ring `K[x, y]/(W)` is a free `K[X]`-module of rank `2` (Silverman II.2). -/
+
 theorem finrank_coordinateRing_eq_two :
     Module.finrank K[X] W.toAffine.CoordinateRing = 2 :=
   (Module.finrank_eq_card_basis (Affine.CoordinateRing.basis W.toAffine)).trans
@@ -206,7 +169,7 @@ private theorem frobenius_fieldRange_ratFunc :
     obtain ⟨p, r, _hr, rfl⟩ : ∃ p r : K[X], algebraMap K[X] (RatFunc K) r ≠ 0 ∧
       g = algebraMap K[X] (RatFunc K) p / algebraMap K[X] (RatFunc K) r :=
       ⟨g.num, g.denom, RatFunc.algebraMap_ne_zero g.denom_ne_zero, g.num_div_denom.symm⟩
-    simp only [div_pow, ← map_pow, ← FiniteField.expand_card (K := K)]
+    simp only [div_pow, ← map_pow, ← FiniteField.Polynomial.expand_card (K := K)]
     have hmem : ∀ f : K[X], algebraMap K[X] (RatFunc K) (Polynomial.expand K q f) ∈ A := by
       intro f
       have h_amap : ∀ g : K[X],
@@ -340,8 +303,8 @@ private theorem finrank_over_frobenius_image :
   lia
 
 omit [DecidableEq K] [W.toAffine.IsElliptic] in
-/-- **Core algebraic fact**: `[K(E) : K(E)^q] = q`, where the module structure on `K(E)` over
-itself is via the `q`-th power Frobenius (Silverman II.2.11(a)). -/
+
+
 theorem frobenius_finrank_functionField :
     @Module.finrank W.toAffine.FunctionField W.toAffine.FunctionField _ _
       (frobeniusAlgHom K W.toAffine.FunctionField).toRingHom.toAlgebra.toModule =
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/OrdAtInfty.lean b/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/OrdAtInfty.lean
index 166040c82..a3a20a730 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/OrdAtInfty.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/Frobenius/OrdAtInfty.lean
@@ -10,23 +10,19 @@ import HasseWeil.Foundation.EC.GenericPointZsmul
 import HasseWeil.Foundation.EC.MulByIntBaseCase
 
 /-!
-# Frobenius-specialized addition-pullback ord-at-infinity computations
+# Orders at infinity for Frobenius addition
 
-For `α = frobeniusIsog W` (the Frobenius endomorphism over a finite field
-`K` with `q = #K ≥ 2`), we compute the order at infinity of the addition-
-formula slope and related quantities. These are the building blocks for the
-pole bound `ord_∞(addPullback_x W (frobeniusIsog W)) ≤ -2`, which closes
-Sorry 1 of `HasseWeil/AdditionPullback.lean` for the case actually used by
-HOLE D in the unconditional Hasse-Weil bound.
+For Frobenius `π` over a finite field of size `q`, the coordinate differences
+have orders `−2q` and `−3q`, so their slope has order `−q`. Clearing the
+addition-formula denominator and reducing by the Weierstrass equation gives
+the precise coordinate pole orders.
 
-## Main lemmas
+The same arguments apply to negative Frobenius and the pencil `r·π − s·id`.
+Curve-negation invariance places the summed x-coordinate in `K(x)`; its pole
+then proves transcendence and the injectivity required for the function-field
+pullback.
 
-* `ordAtInfty_addSlope_frobenius`: `ord_∞(L) = -q` where `L = addSlope W π`.
-* `ordAtInfty_addSlope_sq_frobenius`: `ord_∞(L²) = -2q`.
-
-## References
-
-* Silverman, *The Arithmetic of Elliptic Curves*, IV.1 (orders at infinity).
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, IV.1.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
@@ -64,7 +60,7 @@ theorem addSlope_frobenius_eq :
   simp only [addSlope]
   exact Affine.slope_of_X_ne (x_gen_ne_frobeniusIsog_pullback_x_gen W)
 
-/-- `ord_∞(x_gen − π·x) = -2q` (sign-flipped form of the previously-shipped
+/-- `ord_∞(x_gen − π·x) = -2q` (sign-flipped form of the corresponding
 `ordAtInfty_frobeniusIsog_pullback_x_gen_sub_x_gen`). -/
 theorem ordAtInfty_x_gen_sub_frobeniusIsog_pullback_x_gen :
     (W_smooth W).ordAtInfty (x_gen W - (frobeniusIsog W).pullback (x_gen W)) =
@@ -98,7 +94,7 @@ theorem x_gen_sub_frobeniusIsog_pullback_x_gen_ne_zero :
   exact WithTop.coe_ne_top h_top
 
 /-- **`ord_∞(addSlope W π) = -q`** where `q = #K`. Direct from
-`ordAtInfty_div_of_ord_eq` with the shipped sub-ord lemmas. -/
+`ordAtInfty_div_of_ord_eq` with the sub-ord lemmas. -/
 theorem ordAtInfty_addSlope_frobenius :
     (W_smooth W).ordAtInfty (addSlope W (frobeniusIsog W)) =
       ((-(Fintype.card K : ℤ)) : WithTop ℤ) := by
@@ -131,9 +127,8 @@ theorem ordAtInfty_addSlope_sq_frobenius :
   push_cast
   ring
 
-/-- **Witness-parametric Sorry 1 closure for the Frobenius case** (Path Y):
-given a pole hypothesis on `addPullback_x W (frobeniusIsog W)`, derive False
-from `addPullback_x = algebraMap K _ c`. -/
+/-- A pole of the Frobenius addition x-coordinate rules out every constant
+    value in the base field. -/
 theorem addPullback_x_ne_const_frobenius_of_pole
     (hxy : AddNonInverse W (frobeniusIsog W))
     (h_pole :
@@ -142,7 +137,7 @@ theorem addPullback_x_ne_const_frobenius_of_pole
     False :=
   addPullback_x_ne_const_of_pole hxy c h_pole hc
 
-/-! ### Path X-prime: pole bound for q ≥ 3 with cancellation witness
+/-! ### pole bound for q ≥ 3 with cancellation witness
 
 `addPullback_x = L² + a₁L - a₂ - x_gen - π·x`. Regroup as
 `(L² - π·x) + (a₁L - a₂ - x_gen)`. The right group has ord exactly `-q`
@@ -231,26 +226,14 @@ private theorem ordAtInfty_add_le_neg_two {A B : KE} (hq : 3 ≤ Fintype.card K)
     have : (3 : ℤ) ≤ (Fintype.card K : ℤ) := by exact_mod_cast hq
     linarith
 
-/-- **Pole bound for q ≥ 3 with cancellation witness** (Path X-prime,
-**SUPERSEDED**): `ord_∞(addPullback_x W π) ≤ -2`, given the structural
-witness `ord(L² - π·x) ≠ -q`.
-
-**Status: SUPERSEDED.** The hypothesised witness is unprovable when
-`a₁ ≠ 0`. Direct term-by-term ord arithmetic on the `(y - π·y)²
-- π·x · (x - π·x)²` numerator shows that the term `-a₁·π·x·π·y` at
-ord `-5q` is the unique smallest (when `a₁ ≠ 0`), forcing
-`ord(L² - π·x) = -q` exactly. So `lt_or_gt_of_ne h_witness` cannot
-be applied: the witness is false in the case it is supposed to discharge.
-
-The replacement is the direct numerator-of-`addPullback_x` analysis
-shipped below as `addPullbackNumerator_frobenius`, which sidesteps the
-`L² - π·x` regrouping entirely. The lemmas immediately below are kept
-for now to preserve a stable API in case downstream callers reference
-them, but should not be used to close Sorry 1.
-
-The witness-parametric form remains formally true (vacuously, since
-the witness premise is unsatisfiable in the relevant case) but yields
-no usable closure. -/
+/-- A pole bound under the additional cancellation witness
+    `ord(L² − π·x) ≠ −q`.
+
+    When `a₁ ≠ 0`, direct numerator comparison forces
+    `ord(L² − π·x) = −q`: the term `−a₁·π·x·π·y` has uniquely minimal
+    order `−5q`. Thus the extra witness excludes that situation. The
+    reduced-numerator argument gives the coordinate pole without this
+    witness. -/
 theorem addPullback_x_pole_frobenius_of_lc_witness
     (hq : 3 ≤ Fintype.card K)
     (ha1 : W.toAffine.a₁ ≠ 0) (ha2 : W.toAffine.a₂ ≠ 0)
@@ -286,7 +269,7 @@ theorem addPullback_x_pole_frobenius_of_lc_witness
     ordAtInfty_a1L_minus_a2_minus_xgen_frobenius W hq ha1 ha2
   exact ordAtInfty_add_le_neg_two W hq h_ord_B h_witness
 
-/-- **Sorry 1 closure for Frobenius case (q ≥ 3)** with leading-coefficient
+/-- **Nonconstancy from the coordinate pole for Frobenius case (q ≥ 3)** with leading-coefficient
 witness. Composes the pole bound with `addPullback_x_ne_const_frobenius_of_pole`. -/
 theorem addPullback_x_ne_const_frobenius_q_ge_3_of_witness
     (hq : 3 ≤ Fintype.card K)
@@ -305,36 +288,19 @@ theorem addPullback_x_ne_const_frobenius_q_ge_3_of_witness
     exact_mod_cast (by norm_num : (-2 : ℤ) < 0)
   exact addPullback_x_ne_const_frobenius_of_pole W hxy h_pole c hc
 
-/-! ### Direct numerator approach (replaces the broken `_lc_witness` path)
-
-The witness `ord(L² - π·x) ≠ -q` is unprovable because `ord(L² - π·x) = -q`
-exactly when `a₁ ≠ 0` (the case of interest). Instead we work directly with
-the numerator of `addPullback_x · (x - π·x)²`.
-
-Multiplying the addition formula `addPullback_x = L² + a₁·L - a₂ - x_gen - π·x`
-through by `(x - π·x)²` (using `L = (y - π·y)/(x - π·x)`) yields the polynomial-
-shaped element
-
-```
-T₁ + T₂ - T₃
-  with T₁ = (y_gen - π·y)²,
-       T₂ = a₁ · (x_gen - π·x) · (y_gen - π·y),
-       T₃ = (x_gen - π·x)² · (a₂ + x_gen + π·x).
-```
+/-! ### Clearing the addition denominator
 
-The `ord_∞` values, for `q ≥ 2`:
+Multiplying `L² + a₁L − a₂ − x − π·x` by `(x − π·x)²` gives
+`T₁ + T₂ − T₃`, with
+`T₁ = (y − π·y)²`, `T₂ = a₁(x − π·x)(y − π·y)` and
+`T₃ = (x − π·x)²(a₂ + x + π·x)`.
+Their orders are `−6q`, `−5q` (when `a₁ ≠ 0`) and `−6q`.
 
-| term | ord  |
-|------|------|
-| `T₁` | `-6q` |
-| `T₂` | `-5q` (when `a₁ ≠ 0`) |
-| `T₃` | `-6q` |
-
-The leading-coefficient analysis at `-6q` (deferred to a follow-up session)
-gives `lc(T₁) + lc(-T₃) = 2·c_x^{3q} ≠ 0` in characteristic `≠ 2`, hence
-`ord(T₁ + T₂ - T₃) = -6q` and `ord(addPullback_x) = -6q + 4q = -2q < 0`.
-
-These lemmas ship the term-ord half of the analysis. -/
+The Weierstrass equation reduces this numerator so that `x(π·x)²`, of
+order `−2 − 4q`, strictly dominates the remaining terms. Since the
+squared denominator has order `−4q`, the summed x-coordinate has order
+`−2`, in every characteristic.
+-/
 
 /-- The denominator `y_gen − π·y` is nonzero (its ord is finite). -/
 theorem y_gen_sub_frobeniusIsog_pullback_y_gen_ne_zero :
@@ -396,7 +362,7 @@ theorem addPullbackNumerator_frobenius_eq :
   ring
 
 /-- **`ord_∞(T₁) = -6q`** where `T₁ = (y_gen − π·y)²` and `q = #K`.
-Direct from `ordAtInfty_pow_of_ord_eq` applied to the shipped
+Direct from `ordAtInfty_pow_of_ord_eq` applied to
 `ordAtInfty_y_gen_sub_frobeniusIsog_pullback_y_gen`. -/
 theorem ordAtInfty_T1_frobenius :
     (W_smooth W).ordAtInfty
@@ -411,7 +377,7 @@ theorem ordAtInfty_T1_frobenius :
   ring
 
 /-- **`ord_∞(T₂) = -5q`** where `T₂ = a₁·(x_gen − π·x)·(y_gen − π·y)`
-and `a₁ ≠ 0`. Multiplicativity of `ord_∞` plus the shipped sub-ord lemmas. -/
+and `a₁ ≠ 0`. Multiplicativity of `ord_∞` plus the sub-ord lemmas. -/
 theorem ordAtInfty_T2_frobenius (ha₁ : W.toAffine.a₁ ≠ 0) :
     (W_smooth W).ordAtInfty
       (algebraMap K KE W.toAffine.a₁ *
@@ -551,7 +517,7 @@ a polynomial-shaped expression whose unique smallest-order term is
 `x_gen · (π·x)²` at order `-2 - 4q`.
 
 This reduction yields `ord_∞(addPullback_x) = -2` (independent of q!),
-which immediately closes Sorry 1 for `q ≥ 3` in **any characteristic**
+which immediately closes the nonconstancy condition for `q ≥ 3` in **any characteristic**
 (the `x_gen · (π·x)²` term has coefficient `1`, never vanishing). -/
 
 /-- The Weierstrass-reduced form of `addPullbackNumerator_frobenius`.
@@ -651,7 +617,7 @@ than `x_gen · (π·x)²` has `ord_∞ ≥ -3 - 3q`. The dominant term has
 `ord_∞ = -2 - 4q`. Since `-3 - 3q > -2 - 4q` for `q ≥ 2`, strict
 non-archimedean additivity gives `ord(reduced) = -2 - 4q`. -/
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Helper: `ord(algebraMap c · f) ≥ ord(f)`. The `algebraMap K KE c`
 factor never makes the order more negative — when `c = 0` the product
 is zero (ord `⊤`), and when `c ≠ 0` the algebraMap value has ord 0. -/
@@ -860,6 +826,7 @@ private lemma ord_a4_mul_x_add_pi_x_ge (hq : 2 ≤ Fintype.card K) :
   rw [ord_x_gen_add_pi_x_eq W hq]
   exact WithTop.coe_le_coe.mpr h_int
 
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Helper: `ord_∞(2 · a₆) ≥ -3 - 3q` for `q ≥ 2`. -/
 private lemma ord_two_mul_a6_ge (hq : 2 ≤ Fintype.card K) :
     (((-3 - 3 * (Fintype.card K : ℤ)) : ℤ) : WithTop ℤ) ≤
@@ -1020,7 +987,7 @@ theorem ordAtInfty_addPullbackNumerator_reduced_frobenius_eq
     ((W_smooth W).ordAtInfty_add_eq_of_lt (ord_dom_lt_rest W hq)).trans
       (ord_x_gen_mul_pi_x_sq_eq W)
 
-/-! ### Sorry 1 closure for Frobenius case (q ≥ 2, any characteristic) -/
+/-! ### Nonconstancy from the coordinate pole for Frobenius case (q ≥ 2, any characteristic) -/
 
 /-- **`ord_∞(addPullback_x W π) = -2`** for `q ≥ 2` (any characteristic).
 
@@ -1061,7 +1028,7 @@ theorem ord_addPullback_x_frobenius (hq : 2 ≤ Fintype.card K) :
   congr 1
   ring
 
-/-- **Sorry 1 closure for the Frobenius case** (`q ≥ 2`, any characteristic):
+/-- **Nonconstancy from the coordinate pole for the Frobenius case** (`q ≥ 2`, any characteristic):
 `addPullback_x W (frobeniusIsog W)` is not the image of any constant `c ∈ K`.
 
 Direct from `ord_addPullback_x_frobenius` (which gives `ord = -2 < 0`,
@@ -1074,44 +1041,21 @@ theorem addPullback_x_ne_const_frobenius
   rw [ord_addPullback_x_frobenius W hq]
   exact_mod_cast (by norm_num : (-2 : ℤ) < 0)
 
-/-- **Inequality wrapper** for `ord_addPullback_x_frobenius`.
-
-Some downstream consumers (CLOSE-A-1 ticket spec) want the `≤ -2` form
-rather than the equality. Trivial corollary. -/
+/-- The Frobenius addition x-coordinate has order at most minus two. -/
 theorem ordAtInfty_addPullback_x_frobenius_le_neg_two
     (hq : 2 ≤ Fintype.card K) :
     (W_smooth W).ordAtInfty (addPullback_x W (frobeniusIsog W)) ≤
       ((-2 : ℤ) : WithTop ℤ) :=
   (ord_addPullback_x_frobenius W hq).le
 
-/-! ### `negFrobeniusIsog`: the `−π` isogeny (HOLE-D-bound work, Day 1)
-
-For HOLE D, we ultimately need the addition-pullback for `1 − π`, which
-is `id + α` with `α = −π`. The `−π` isogeny is constructed as the
-composition `[-1] ∘ π` via the existing `Isogeny.comp` and `mulByInt _ (-1)`.
-
-By `Isogeny.comp_algebraMap_eq`, the resulting pullback is
-`f ↦ frobeniusIsog.pullback (mulByInt _ (-1)).pullback f`. For the two
-generators:
-* `(negFrobeniusIsog W).pullback (x_gen W)` reduces to
-  `(frobeniusIsog W).pullback (x_gen W) = π·x_gen` (since `[-1]` fixes
-  `x_gen`).
-* `(negFrobeniusIsog W).pullback (y_gen W)` reduces to
-  `(frobeniusIsog W).pullback (negY x_gen y_gen) =
-   (frobeniusIsog W).pullback (-y_gen - a₁·x_gen - a₃)`. By Frobenius
-  being a ring hom and `a_i^q = a_i` in `K = F_q`, this equals
-  `-π·y_gen - a₁·π·x_gen - a₃` in any characteristic (in char 2 the
-  signs are no-ops).
-
-Both pullbacks have the same `ord_∞` as the Frobenius case (`-2q` and
-`-3q` respectively), so the term-ord chain shipped earlier in this file
-templates over to `negFrobeniusIsog` with name-prefix swaps. The reduced-
-numerator analysis (Weierstrass identity certificate) also carries
-forward, since `(π·x_gen, negY π·x π·y)` is on the curve by `equation_neg`.
-
-This file ships only the structural definition tonight. The Day 2 work
-(ord lemmas + pullback formulas + Sorry 1 closure for `α = -π`) is
-factored separately and reuses the existing helpers verbatim. -/
+/-! ### Negative Frobenius
+
+The isogeny `−π` is the composition `[-1] ∘ π`. Its pullback sends
+`x` to `π·x` and `y` to `−π·y − a₁·π·x − a₃`, in every characteristic.
+These coordinates have orders `−2q` and `−3q`. The Weierstrass equation
+for the negated point gives the corresponding reduced-numerator pole
+calculation for `1 − π`.
+-/
 
 /-- The `-π` isogeny as `[-1] ∘ π`, via `Isogeny.comp` of the existing
 `mulByInt _ (-1)` and `frobeniusIsog W` isogenies. -/
@@ -1131,6 +1075,7 @@ theorem negFrobeniusIsog_toAddMonoidHom_apply (P : W.toAffine.Point) :
   rw [mulByInt_apply]
   exact (neg_one_zsmul _).trans rfl
 
+omit [Fintype K] in
 /-- The `[-1]`-pullback sends `y_gen` to `negY x_gen y_gen = -y_gen − a₁·x_gen
 − a₃` (the curve-negation formula).
 
@@ -1188,7 +1133,7 @@ theorem negFrobeniusIsog_pullback_y_gen :
   simp only [map_sub, map_neg, map_mul,
     AlgHom.commutes (frobeniusIsog W).pullback]
 
-/-! ### `ord_∞` lemmas for the negFrobeniusIsog pullbacks (Day 2 first half)
+/-! ### `ord_∞` lemmas for the negFrobeniusIsog pullbacks
 
 With the pullback formulas in place, the ord values match the Frobenius
 case:
@@ -1265,7 +1210,7 @@ theorem ordAtInfty_negFrobeniusIsog_pullback_y_gen
   exact ordAtInfty_sub_eq_of_coe_lt_le W h_lt_q h_neg_πy
     (a1_pi_x_plus_a3_ge_neg_two_q W hq)
 
-/-! ### addPullbackNumerator for negFrobenius (Day 2 second half opener)
+/-! ### addPullbackNumerator for negFrobenius
 
 Mirrors the Frobenius numerator + reduction identity, with `α = negFrobeniusIsog W`
 instead of `α = frobeniusIsog W`. The (u, v) point used in the reduction is
@@ -1640,7 +1585,7 @@ theorem ordAtInfty_addPullbackNumerator_reduced_negFrobenius_eq
     ((W_smooth W).ordAtInfty_add_eq_of_lt (ord_dom_lt_negFrob_rest W hq)).trans
       (ord_x_gen_mul_negFrob_pi_x_sq_eq W)
 
-/-! ### Sorry 1 closure for the negFrobenius case (Day 2 final)
+/-! ### Nonconstancy from the coordinate pole for the negFrobenius case
 
 `ord(addPullback_x W (-π)) = -2` follows by dividing the reduced numerator
 (ord = -2 - 4q) through `(x_gen − negFrob.pb x_gen)²` (ord = -4q). Then
@@ -1740,12 +1685,11 @@ theorem ord_addPullback_x_negFrobenius (hq : 2 ≤ Fintype.card K) :
   congr 1
   ring
 
-/-- **Sorry 1 closure for the negFrobenius case** (`q ≥ 2`, any characteristic):
+/-- **Nonconstancy from the coordinate pole for the negFrobenius case** (`q ≥ 2`, any characteristic):
 `addPullback_x W (negFrobeniusIsog W)` is not the image of any constant `c ∈ K`.
 
 Direct from `ord_addPullback_x_negFrobenius` (which gives `ord = -2 < 0`,
-a pole) plus `addPullback_x_ne_const_of_pole`. This is the closure needed
-for HOLE D's `1 − π` use case (since `1 − π = id + (-π)`). -/
+a pole) plus `addPullback_x_ne_const_of_pole`. For `1 − π`, the addition formula is `id + (−π)`. -/
 theorem addPullback_x_ne_const_negFrobenius
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) (c : K)
@@ -1754,11 +1698,7 @@ theorem addPullback_x_ne_const_negFrobenius
   rw [ord_addPullback_x_negFrobenius W hq]
   exact_mod_cast (by norm_num : (-2 : ℤ) < 0)
 
-/-- **Inequality wrapper** for `ord_addPullback_x_negFrobenius`.
-
-Companion to `ordAtInfty_addPullback_x_frobenius_le_neg_two` for the
-`−π` (negFrobenius) case — the load-bearing case for HOLE D's `1 − π`
-analysis. Trivial corollary of the equality form. -/
+/-- The Frobenius addition x-coordinate has order at most minus two. -/
 theorem ordAtInfty_addPullback_x_negFrobenius_le_neg_two
     (hq : 2 ≤ Fintype.card K) :
     (W_smooth W).ordAtInfty (addPullback_x W (negFrobeniusIsog W)) ≤
@@ -1838,7 +1778,7 @@ private theorem ord_RHS_lower_ge_neg_four_negFrobenius
   exact ord_add_ge_of_both_ge W
     (ord_add_ge_of_both_ge W h_a₂_term h_a₄_term) h_a₆_term
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Helper: `ord(a₆) ≥ 0` (where `a₆` is a constant in K, viewed in KE). -/
 private theorem ord_a₆_ge_zero :
     (0 : WithTop ℤ) ≤ (W_smooth W).ordAtInfty
@@ -2181,66 +2121,25 @@ theorem ord_addPullback_y_negFrobenius (hq : 2 ≤ Fintype.card K) :
   have h_m_eq : m = -3 := by omega
   exact_mod_cast h_m_eq
 
-/-! ### Day 3a: addPullbackAlgHom for the negFrobenius case
-
-The existing `addPullbackAlgHom W α hxy hinj` construction
-(`HasseWeil.AdditionPullback`) is generic over `α`. For HOLE D we want
-this for `α = negFrobeniusIsog W`. The injectivity input
-`hinj : Function.Injective (addCoordAlgHom hxy)` was generically
-provided by a sorry-bearing `addCoordAlgHom_injective` chain (general-`α`
-transcendence of `addPullback_x` resting on an open III.3.6 pole witness) —
-that chain has since been **removed**; only the witness-parametric and
-negFrobenius forms remain.
-
-We ship `addPullbackAlgHom_negFrobenius_of_inj` as a **witness-parametric
-form** taking the (axiom-clean) injectivity proof as a hypothesis.
-Discharging this hypothesis axiom-clean for `α = negFrobeniusIsog W`
-requires one of:
-
-1. Closing Sorry 2 + Sorry 3 generically (~350 LOC, mathlib `minpoly`
-   machinery).
-
-2. A negFrobenius-specific σ-invariance argument: prove
-   `(mulByInt W (-1)).pullback (addPullback_x W (negFrobeniusIsog W))
-   = addPullback_x W (negFrobeniusIsog W)`. This says addPullback_x is
-   fixed by the curve-negation involution `σ : K(E) → K(E)`. By Galois
-   theory (K(E) over K(x) is degree-2, σ generates the Galois group),
-   σ-fixed elements lie in `K(x) = F(x_gen)`, which discharges the
-   `px ∉ F(x_gen)` case of the transcendence argument for the
-   negFrobenius case. Then `addPullback_x_ne_const_negFrobenius`
-   closes Case 1, completing the chain.
-
-Path 2 is more focused (~100-200 LOC) and closes only what HOLE D
-needs. The σ-invariance lemma proof structure: apply
-`(mulByInt W (-1)).pullback` to the addX formula
-`addPullback_x = L² + a₁L − a₂ − x_gen − π·x`, distribute the AlgHom
-over operations, substitute known values:
-* `(mulByInt W (-1)).pullback x_gen = x_gen` (`mulByInt_pullback_x_neg_one`).
-* `(mulByInt W (-1)).pullback y_gen = −y_gen − a₁·x_gen − a₃`
-  (`mulByInt_pullback_y_neg_one`).
-* `(mulByInt W (-1)).pullback (π·x_gen) = π·x_gen` (since `π·x_gen
-  = x_gen^q` and AlgHom preserves powers and fixes `x_gen`).
-* `(mulByInt W (-1)).pullback (π·y_gen) = -π·y - a₁·π·x - a₃` (via
-  Frobenius distributing over `−y − a₁·x − a₃` in char p, with
-  `a^q = a` in `K = F_q`).
-
-Substituting and using `(σ(L_neg)+L_neg) = -a₁` (a curve-arithmetic
-identity, computable via `field_simp; ring` after the algebra-hom
-rewrites), we get `σ(L_neg² + a₁·L_neg) = L_neg² + a₁·L_neg`. The
-remaining terms in addPullback_x (`a₂`, `x_gen`, `π·x_gen`) are all
-σ-fixed. So addPullback_x is σ-invariant. ∎ -/
-
-/-- Witness-parametric `addPullbackAlgHom` for `α = negFrobeniusIsog W`,
-parametric on the `addCoordAlgHom hxy`-injectivity hypothesis. Once
-that hypothesis is dischargeable axiom-clean (via Path 2 above or
-Sorry 2 + Sorry 3 closure), the resulting algebra-hom feeds the
-`isogOneSub` placeholder replacement and closes HOLE D. -/
+/-! ### Injectivity of the negative-Frobenius addition pullback
+
+Curve negation fixes `x` and sends `y` to `−y − a₁x − a₃`. Applying it to
+the negative-Frobenius slope gives `σ(L) + L = −a₁`, hence fixes
+`L² + a₁L` and the summed x-coordinate. The fixed field is `K(x)`.
+An algebraic element of this rational function field is constant, whereas
+the summed x-coordinate has a pole. It is therefore transcendental over
+`K`, which proves injectivity of its polynomial evaluation homomorphism
+and then of the coordinate-ring homomorphism.
+-/
+
+/-- The function-field addition pullback for negative Frobenius, given
+    injectivity of the coordinate-ring homomorphism. -/
 noncomputable def addPullbackAlgHom_negFrobenius_of_inj
     (hxy : AddNonInverse W (negFrobeniusIsog W))
     (hinj : Function.Injective (addCoordAlgHom hxy)) : KE →ₐ[K] KE :=
   addPullbackAlgHom hxy hinj
 
-/-! ### σ-invariance lemmas (Path 2 to discharge `hinj` axiom-clean) -/
+/-! ### Curve-negation invariance -/
 
 /-- **σ-invariance Step 1**:
 `σ(π·y_gen) = (negFrobeniusIsog W).pullback (y_gen W)` where
@@ -2360,7 +2259,7 @@ theorem addPullback_x_negFrobenius_sigma_invariant :
 
 /-! ### Galois group identification: {1, σ} on K(E)/K(x)
 
-Day 3a-final opener. To deduce that σ-fixed elements lie in K(x), we
+To deduce that σ-fixed elements lie in K(x), we
 need to know that {1, σ} is the full Galois group of K(E)/K(x).
 
 The two structural facts:
@@ -2370,6 +2269,7 @@ The two structural facts:
 
 Combined with `[K(E) : K(x)] = 2`, this is the full Galois group. -/
 
+omit [Fintype K] in
 /-- **σ has order 2**: `(mulByInt W (-1)).pullback.comp (mulByInt W (-1)).pullback
 = AlgHom.id`. The Galois-group order-2 statement, derived from the isogeny
 identity `[-1] ∘ [-1] = [1]` (`mulByInt_comp_eq_mul`) plus
@@ -2395,6 +2295,7 @@ theorem mulByInt_neg_one_pullback_comp_self :
         from rfl] at h_pb
   rw [h_pb, mulByInt_one_pullback_eq_id]
 
+omit [Fintype K] in
 /-- **σ ∘ σ = id pointwise**: for any `z ∈ KE`,
 `(mulByInt W (-1)).pullback ((mulByInt W (-1)).pullback z) = z`. -/
 theorem mulByInt_neg_one_pullback_pow_two_apply (z : KE) :
@@ -2413,6 +2314,7 @@ forces non-triviality). In characteristic 2 we need the discriminant: a curve
 with `a₁ = a₃ = 0` would have `Δ = 0` (by `Δ_of_char_two`), contradicting
 `[IsElliptic]`. -/
 
+omit [Fintype K] in
 /-- **σ ≠ id in characteristic 2**: the curve-negation involution
 `σ = (mulByInt W (-1)).pullback` does not fix `y_gen`.
 
@@ -2481,6 +2383,7 @@ private lemma two_K_eq_zero_of_two_fractionRing
   rw [map_zero, map_ofNat]
   exact h
 
+omit [Fintype K] in
 /-- **σ ≠ id in characteristic ≠ 2**: in char ≠ 2, the curve-negation involution
 σ does not fix `y_gen`.
 
@@ -2515,6 +2418,7 @@ theorem mulByInt_neg_one_pullback_y_gen_ne_y_gen_char_ne_two
   obtain ⟨_, hq_zero⟩ := (W_smooth W).decomp_zero_iff h_zero_smul
   exact h2 (two_K_eq_zero_of_two_fractionRing hq_zero)
 
+omit [Fintype K] in
 /-- **σ ≠ id (unified)**: in any characteristic, the curve-negation involution
 `σ = (mulByInt W (-1)).pullback` does not fix `y_gen`.
 
@@ -2525,7 +2429,7 @@ theorem mulByInt_neg_one_pullback_y_gen_ne_y_gen :
     (mulByInt W.toAffine (-1)).pullback (y_gen W) ≠ y_gen W := by
   by_cases h2 : (2 : K) = 0
   · -- Char 2 case via `[CharP K 2]` derived from `h2`.
-    haveI : CharP K 2 := CharTwo.of_one_ne_zero_of_two_eq_zero one_ne_zero h2
+    have : CharP K 2 := CharTwo.of_one_ne_zero_of_two_eq_zero one_ne_zero h2
     exact mulByInt_neg_one_pullback_y_gen_ne_y_gen_char_two W
   · -- Char ≠ 2 case directly.
     exact mulByInt_neg_one_pullback_y_gen_ne_y_gen_char_ne_two W h2
@@ -2535,6 +2439,7 @@ theorem mulByInt_neg_one_pullback_y_gen_ne_y_gen :
 A small first piece toward σ-fixed → K(x): σ commutes with the constant
 embedding `K → K(E)`. -/
 
+omit [Fintype K] in
 /-- σ fixes the image of `K → K(E)`: for any `c : K`,
 `σ(algebraMap c) = algebraMap c`. Immediate from σ being a `K`-algebra hom. -/
 @[simp] theorem mulByInt_neg_one_pullback_algebraMap_K (c : K) :
@@ -2542,6 +2447,7 @@ embedding `K → K(E)`. -/
       algebraMap K KE c :=
   AlgHom.commutes (mulByInt W.toAffine (-1)).pullback c
 
+omit [Fintype K] in
 /-- σ fixes `algebraMap (Polynomial K) KE p` for any polynomial `p`.
 Using `algebraMap p = aeval x_gen p` (via algebra-hom uniqueness on the
 generator `X ↦ x_gen`) plus σ commutes with `aeval` and σ fixes `x_gen`. -/
@@ -2566,6 +2472,7 @@ theorem mulByInt_neg_one_pullback_algebraMap_polyK (p : Polynomial K) :
     simpa using this
   rw [h_eval, ← Polynomial.aeval_algHom_apply, mulByInt_pullback_x_neg_one]
 
+omit [Fintype K] in
 /-- σ fixes the image of `Frac(K[X]) → K(E)`: for any `r ∈ Frac(K[X])`,
 `σ(algebraMap r) = algebraMap r`. Lifts `mulByInt_neg_one_pullback_algebraMap_polyK`
 via `IsLocalization.surj` denominator clearing. -/
@@ -2598,6 +2505,7 @@ theorem mulByInt_neg_one_pullback_algebraMap_kx
 
 /-! ### Path 2 Step 3 Piece 3b: σ acts on coordY -/
 
+omit [Fintype K] in
 /-- σ acts on `y_gen` as `-y_gen - a₁·x - a₃`. (Re-statement of
 `mulByInt_pullback_y_neg_one` for use with the basis decomposition machinery.) -/
 theorem mulByInt_neg_one_pullback_y_gen_eq :
@@ -2608,6 +2516,7 @@ theorem mulByInt_neg_one_pullback_y_gen_eq :
 
 /-! ### Path 2 Step 3 Piece 3c helpers -/
 
+omit [Fintype K] [DecidableEq K] in
 /-- The polynomial `a₁ X + a₃` is nonzero in characteristic 2 for an elliptic
 curve. If it were zero, then `a₁ = a₃ = 0`, and `Δ_of_char_two` would give
 `Δ = 0`, contradicting `[IsElliptic]`. -/
@@ -2625,6 +2534,7 @@ theorem a₁X_plus_a₃_ne_zero_char_two [CharP K 2] :
     rw [WeierstrassCurve.Δ_of_char_two, h_a1, h_a3]; ring
   exact W.toAffine.isUnit_Δ.ne_zero h_delta
 
+omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Image of the polynomial `a₁ X + a₃` under `K[X] → KE`: equals
 `a₁ · x_gen + a₃` in `KE`. -/
 theorem algebraMap_a₁X_plus_a₃ :
@@ -2642,6 +2552,7 @@ theorem algebraMap_a₁X_plus_a₃ :
     intro c; rw [Polynomial.C_eq_algebraMap, ← IsScalarTower.algebraMap_apply]
   rw [map_add, map_mul, h_C, h_C, h_x_alg]
 
+omit [Fintype K] [DecidableEq K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The image of `a₁ X + a₃` under the composite `K[X] → Frac(K[X]) → KE` equals
 `a₁ · x_gen + a₃` in `KE`. Lifts `algebraMap_a₁X_plus_a₃` along the scalar tower
 `K[X] → Frac(K[X]) → KE`. -/
@@ -2665,6 +2576,7 @@ private lemma smul_one_add_smul_y_gen_eq_algebraMap
       algebraMap (FractionRing (Polynomial K)) KE b * y_gen W := by
   simp only [Algebra.smul_def, mul_one]
 
+omit [Fintype K] in
 /-- σ acting on an `algebraMap`-decomposition: applying `σ = (mulByInt W (-1)).pullback`
 to `algebraMap a + (algebraMap b) · y_gen` gives
 `algebraMap a + (algebraMap b) · (-y_gen - a₁·x - a₃)`, since σ fixes the
@@ -2684,6 +2596,7 @@ private lemma mulByInt_neg_one_pullback_algebraMap_add_algebraMap_mul_y_gen
     mulByInt_neg_one_pullback_algebraMap_kx,
     mulByInt_neg_one_pullback_y_gen_eq]
 
+omit [Fintype K] in
 /-- Helper for Piece 3c: the σ-equation as a vanishing `{1, Y}`-decomposition.
 
 If `f = a • 1 + b • Y` is fixed by `σ = (mulByInt W (-1)).pullback`, then applying
@@ -2719,6 +2632,7 @@ private lemma sigma_fixed_decomp_coeffs_vanish {a b : FractionRing (Polynomial K
     map_mul, map_mul, algebraMap_fractionRing_a₁X_plus_a₃ W, map_ofNat]
   linear_combination -h_combine
 
+omit [Fintype K] in
 /-- Helper for Piece 3c: char-split forcing `b = 0`.
 
 From the two vanishing coefficients of `sigma_fixed_decomp_coeffs_vanish`,
@@ -2733,7 +2647,7 @@ private lemma eq_zero_of_mul_a₁X_plus_a₃_and_two_mul_eq_zero
     (hqb : 2 * b = 0) : b = 0 := by
   by_cases h2 : (2 : K) = 0
   · -- Char 2: `a₁ X + a₃ ≠ 0`, so `b · (a₁X+a₃) = 0` forces `b = 0`.
-    haveI : CharP K 2 := CharTwo.of_one_ne_zero_of_two_eq_zero one_ne_zero h2
+    have : CharP K 2 := CharTwo.of_one_ne_zero_of_two_eq_zero one_ne_zero h2
     have h_bXf_ne : algebraMap (Polynomial K) (FractionRing (Polynomial K))
         (Polynomial.C W.toAffine.a₁ * Polynomial.X + Polynomial.C W.toAffine.a₃) ≠ 0 :=
       fun h_eq ↦ a₁X_plus_a₃_ne_zero_char_two W
@@ -2756,6 +2670,7 @@ the image of `Frac(K[X]) → K(E)`. Combines the σ-action on the basis
 decomposition `{1, Y}` (Pieces 0/1/2/3b) with `decomp_zero_iff` and a
 char-split. -/
 
+omit [Fintype K] in
 /-- **Path 2 Step 3 Piece 3c**: If `f ∈ KE` is fixed by `σ`, then
 `f` is in the image of `algebraMap (Frac K[X]) KE`.
 
@@ -2796,18 +2711,18 @@ theorem addPullback_x_negFrobenius_in_KX_image :
   sigma_fixed_implies_in_KX_image W _
     (addPullback_x_negFrobenius_sigma_invariant W)
 
-/-! ### Path 2 Steps 4+5: addPullback_x_negFrobenius is transcendental
+/-! ### addPullback_x_negFrobenius is transcendental
 
-The consumer of `addPullback_x_negFrobenius_in_KX_image`. Combines that
+Apply `addPullback_x_negFrobenius_in_KX_image`. Combines that
 witness with `algebraic_in_fracRing_eq_const` (algebraic-closure-of-K-in-K(x))
 and `addPullback_x_ne_const_negFrobenius` (the non-constancy lemma) to
-discharge transcendence axiom-clean for the `α = -π` case. -/
+prove transcendence for the `α = -π` case. -/
 
-/-- **Path 2 Step 4+5 (axiom-clean closure of Sorry 2 for α = -π)**:
+/-- **Transcendence for negative Frobenius**:
 `addPullback_x W (negFrobeniusIsog W)` is transcendental over `K`.
 
 Proof outline:
-* By `addPullback_x_negFrobenius_in_KX_image` (Piece 3d): there exists
+* By `addPullback_x_negFrobenius_in_KX_image`: there exists
   `r : Frac(K[X])` with `addPullback_x = algebraMap _ KE r`.
 * If `addPullback_x` were algebraic over `K`, so would `r` be (since
   `algebraMap _ KE` is injective and pulls back transcendence).
@@ -2836,18 +2751,15 @@ theorem addPullback_x_transcendental_negFrobenius
     rw [hr, hc, ← IsScalarTower.algebraMap_apply K (FractionRing (Polynomial K)) KE]
   exact addPullback_x_ne_const_negFrobenius W hq hxy c hc'
 
-/-! ### Day 3b: addCoordAlgHom injectivity for the negFrobenius case
+/-! ### Coordinate-ring injectivity from transcendence
 
-Consumer of `addPullback_x_transcendental_negFrobenius` (Steps 4+5). Uses the
-witness-parametric `addCoordAlgHom_injective_of_baseHom_inj` with the
-unconditional base-hom injectivity below (the sorry-bearing generic
-`addBaseHom_injective` has been removed). -/
+Transcendence of the summed x-coordinate gives injectivity of polynomial
+evaluation, and the integral coordinate-ring extension gives injectivity
+of the coordinate homomorphism.
+-/
 
-/-- **Day 3b helper**: `addBaseHom W (negFrobeniusIsog W)` is injective.
-Axiom-clean negFrobenius base-hom injectivity: rewrites the base-hom as
-`Polynomial.aeval (addPullback_x W (negFrob W))` (`addBaseHom_eq_aeval`)
-and applies `transcendental_iff_injective` to the unconditional
-`addPullback_x_transcendental_negFrobenius`. -/
+/-- Transcendence of the negative-Frobenius addition x-coordinate makes its
+    polynomial evaluation homomorphism injective. -/
 theorem addBaseHom_injective_negFrobenius
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) :
@@ -2856,15 +2768,8 @@ theorem addBaseHom_injective_negFrobenius
   exact transcendental_iff_injective.mp
     (addPullback_x_transcendental_negFrobenius W hq hxy)
 
-/-- **Day 3b**: `addCoordAlgHom` for `α = negFrobeniusIsog W` is injective.
-One-line consumer of `addCoordAlgHom_injective_of_baseHom_inj` and
-`addBaseHom_injective_negFrobenius`. Closes one of the three sorries in HOLE D's
-downstream chain axiom-clean.
-
-Together with `addPullback_x_transcendental_negFrobenius` and Path 2 Step 3,
-the negFrobenius `addCoordAlgHom`-injectivity is now fully
-unconditional — the input needed by `addPullbackAlgHom_negFrobenius_of_inj`
-to build a non-witness-parametric `addPullbackAlgHom_negFrobenius`. -/
+/-- The negative-Frobenius addition coordinate-ring homomorphism is
+    injective, by injectivity of its polynomial restriction. -/
 theorem addCoordAlgHom_injective_negFrobenius
     (hq : 2 ≤ Fintype.card K)
     (hxy : AddNonInverse W (negFrobeniusIsog W)) :
@@ -2872,15 +2777,14 @@ theorem addCoordAlgHom_injective_negFrobenius
   addCoordAlgHom_injective_of_baseHom_inj hxy
     (addBaseHom_injective_negFrobenius W hq hxy)
 
-/-! ### Day 3c Piece A: AddNonInverse witness + specialized addPullbackAlgHom
+/-! ### The function-field pullback of `1 − π`
 
-The negFrobenius variant of `addPullbackAlgHom` requires both
-`AddNonInverse W (negFrobeniusIsog W)` and the corresponding injectivity
-of `addCoordAlgHom`. Both are dischargeable axiom-clean: the non-inverse
-condition reduces to `x_gen ≠ π·x_gen` (different orders at infinity),
-and injectivity is `addCoordAlgHom_injective_negFrobenius` (c8b4b66). -/
+The identity and negative-Frobenius x-coordinates differ because their
+orders differ. Together with coordinate-ring injectivity, this constructs
+the addition pullback on the function field.
+-/
 
-/-- **Day 3c Piece A.1**: `AddNonInverse W (negFrobeniusIsog W)` holds
+/-- **Non-inverse coordinates**: `AddNonInverse W (negFrobeniusIsog W)` holds
 unconditionally. The negation hypothesis would require `x_gen W = π·x_gen`
 (the first conjunct), which contradicts
 `x_gen_ne_frobeniusIsog_pullback_x_gen` (their orders at infinity differ). -/
@@ -2890,11 +2794,8 @@ theorem negFrobeniusIsog_addNonInverse :
   rw [negFrobeniusIsog_pullback_x_gen] at h_x
   exact x_gen_ne_frobeniusIsog_pullback_x_gen W h_x
 
-/-- **Day 3c Piece A.2**: Specialized `addPullbackAlgHom` for `α = negFrobenius`,
-discharging both `AddNonInverse` (via `negFrobeniusIsog_addNonInverse`) and
-`addCoordAlgHom`-injectivity (via `addCoordAlgHom_injective_negFrobenius`)
-axiom-clean. The unconditional algebra hom corresponding to the rational
-map `P ↦ P + (-π)(P) = P - π(P)` (i.e., `1 - π` on rational points). -/
+/-- The function-field pullback of `P ↦ P − π(P)`, constructed from the
+    non-inverse coordinate witness and the injective coordinate map. -/
 noncomputable def addPullbackAlgHom_negFrobenius
     (hq : 2 ≤ Fintype.card K) : KE →ₐ[K] KE :=
   addPullbackAlgHom_negFrobenius_of_inj W
@@ -2902,22 +2803,14 @@ noncomputable def addPullbackAlgHom_negFrobenius
     (addCoordAlgHom_injective_negFrobenius W hq
       (negFrobeniusIsog_addNonInverse W))
 
-/-! ### Day 3c Piece B: isogOneSub_negFrobenius (replacement for the placeholder)
+/-! ### The isogeny `1 − π`
 
-The unconditional `1 − π` isogeny: pullback from `addPullbackAlgHom_negFrobenius`
-(real addition pullback for `id + (-π)`), rational-point map from
-`(AddMonoidHom.id _) - (frobeniusIsog W).toAddMonoidHom`. Replaces the
-`isogOneSub (frobeniusIsog W)` placeholder (whose pullback was `AlgHom.id`)
-with the mathematically correct pullback. -/
-
-/-- **Day 3c Piece B**: The isogeny `1 − π` with the unconditional addition-formula
-pullback. Built from `addPullbackAlgHom_negFrobenius` (the pullback for
-`id + (−π) = 1 − π`) and the standard rational-point map
-`(AddMonoidHom.id _) - (frobeniusIsog W).toAddMonoidHom`.
+Its function-field pullback is the addition-formula pullback for
+`id + (−π)`, and its point map is `id − π`.
+-/
 
-Mathematically this is `1 − π` with the correct pullback that the placeholder
-`isogOneSub (frobeniusIsog W)` lacks (it uses `AlgHom.id` as a stub). HOLE D
-wire-up consumes this directly via `Isogeny.toAlgebra`/`degree`. -/
+/-- The isogeny `1 − π` with its addition-formula function-field pullback
+    and the point map `id − π`. -/
 noncomputable def isogOneSub_negFrobenius
     (hq : 2 ≤ Fintype.card K) : Isogeny W.toAffine W.toAffine where
   pullback := addPullbackAlgHom_negFrobenius W hq
@@ -2932,20 +2825,12 @@ noncomputable def isogOneSub_negFrobenius
     (isogOneSub_negFrobenius W hq).toAddMonoidHom =
       (AddMonoidHom.id _) - (frobeniusIsog W).toAddMonoidHom := rfl
 
-/-! ### Day 3c Piece C — REMOVED (2026-05-28 placeholder grind)
 
-The bridge lemma `isogOneSub_negFrobenius_toAddMonoidHom_eq_oneSubFrobeniusIsog`
-(genuine `1−π` and the placeholder agree on `toAddMonoidHom`) existed only to
-transfer the placeholder's rational-point API to the genuine isogeny. All
-consumers now prove their kernel/point-map facts directly on
-`isogOneSub_negFrobenius` (its `toAddMonoidHom = id − frob`, frob's point map
-is the identity), so the bridge is no longer needed and has been deleted as
-part of removing the placeholder `oneSubFrobeniusIsog`. -/
 
-/-! ### D3b base: `AddNonInversePair` for `(zsmul 1 π, mulByInt (-1))`
+/-! ### `AddNonInversePair` for `(zsmul 1 π, mulByInt (-1))`
 
 The `(r, s) = (1, 1)` specialisation of the genuine `(zsmul r π, mulByInt (-s))`
-family that D3b/D4 will assemble into `genuineIsogSmulSub r s`. Both pullbacks
+family defining `genuineIsogSmulSub r s`. Both pullbacks
 reduce on x-coord to `x_gen W` and `x_gen W ^ q` respectively via
 `mulByInt_x_one` / `mulByInt_x_neg` / `frobeniusIsog_pullback_apply`; the
 mismatch is `x_gen_ne_frobeniusIsog_pullback_x_gen`. -/
@@ -2977,12 +2862,12 @@ theorem AddNonInversePair_zsmul_one_frobenius_mulByInt_neg_one :
   rw [frobeniusIsog_pullback_apply] at hne
   exact hne h.symm
 
-/-! ### D3b general consumer: AddNonInversePair for `(zsmul r π, mulByInt -s)`
+/-! ### Non-inverse coordinate pairs: AddNonInversePair for `(zsmul r π, mulByInt -s)`
 
 Generalises the (1, 1) base case to arbitrary `(r, s) ≠ 0` (with `(r : K) ≠ 0`
 and `(s : K) ≠ 0`). The x-coord pullbacks are `(mulByInt_x r)^q` (LHS) and
 `mulByInt_x (-s)` (RHS); their orders at infinity are `-2q` and `-2` respectively
-(via `ordAtInfty_mulByInt_x` from `OrdAtInftyBridge.lean:207` and
+(via `ordAtInfty_mulByInt_x` for the coordinate pole and
 `ordAtInfty_pow`). For `q ≥ 2`, these are unequal, so the pullbacks are
 unequal — Silverman III.6 reflection of Frobenius pole-multiplication. -/
 
@@ -3043,7 +2928,7 @@ theorem AddNonInversePair_zsmul_frobenius_mulByInt_neg
     exact_mod_cast Fintype.one_lt_card_iff_nontrivial.mpr inferInstance
   linarith
 
-/-! ### D3c step 1: σ-action on `(mulByInt n)` pullbacks (generic in `n ≠ 0`)
+/-! ### σ-action on `(mulByInt n)` pullbacks (generic in `n ≠ 0`)
 
 For any nonzero `n`, σ commutes with `(mulByInt n).pullback`: their composition
 on `x_gen` is `(mulByInt (-n)).pullback x_gen = mulByInt_x W (-n) = mulByInt_x W n`,
@@ -3064,6 +2949,7 @@ private theorem mulByInt_comp_mulByInt_neg_one
     (by simpa using neg_ne_zero.mpr hn)
   rw [h]; congr 1; ring
 
+omit [Fintype K] in
 /-- **σ.pb fixes `(mulByInt n).pb x_gen`** for `n ≠ 0`. By `mulByInt`
 commutativity, `σ.pb ([n].pb x_gen) = ([n].comp σ).pb x_gen
 = (mulByInt (-n)).pb x_gen = mulByInt_x W (-n) = mulByInt_x W n` (the last by
@@ -3080,6 +2966,7 @@ theorem sigma_mulByInt_pullback_x_eq (n : ℤ) (hn : n ≠ 0) :
   rw [mulByInt_pullback_x W (-n) (neg_ne_zero.mpr hn), mulByInt_x_neg]
   exact (mulByInt_pullback_x W n hn).symm
 
+omit [Fintype K] in
 /-- **σ.pb on `(mulByInt n).pb y_gen`** for `n ≠ 0`. Same composition path
 as the x-version but using `mulByInt_y_neg`: the σ-image is
 `negY (mulByInt_x n) (mulByInt_y n) = -mulByInt_y n - a₁·mulByInt_x n - a₃`. -/
@@ -3117,7 +3004,7 @@ theorem sigma_mulByInt_pullback_y_eq (n : ℤ) (hn : n ≠ 0) :
   simp only [WeierstrassCurve.Affine.negY]
   rfl
 
-/-! ### D3c step 2: σ-action on `(zsmul r π).pb`
+/-! ### σ-action on `(zsmul r π).pb`
 
 For `α₁ = (frobeniusIsog W).zsmul r = (mulByInt r).comp π`, the σ-action
 follows from `sigma_mulByInt_pullback_x_eq` / `_y_eq` plus σ commuting
@@ -3176,13 +3063,14 @@ theorem sigma_zsmul_frobenius_pullback_y_eq (r : ℤ) (hr : r ≠ 0) :
   simp only [map_sub, map_neg, map_mul,
     AlgHom.commutes (frobeniusIsog W).pullback]
 
-/-! ### D3c step 3: x-ne mismatch witness for `(zsmul r π, mulByInt -s)`
+/-! ### x-ne mismatch witness for `(zsmul r π, mulByInt -s)`
 
 Extracted from `AddNonInversePair_zsmul_frobenius_mulByInt_neg`'s internal
 proof: the x-coord pullbacks differ at `ord_∞` (`-2q` vs `-2`). Used
 directly as the `h_x_ne` argument to the generic
 `addPullback_x_pair_sigma_invariant`. -/
 
+omit [DecidableEq K] in
 /-- The `ord_∞` contradiction behind the `x`-coord mismatch: `(mulByInt_x W r)^|K|`
 (order `-2q`) cannot equal `mulByInt_x W (-s)` (order `-2`), since `|K|·(-2) = -2`
 would force `|K| ≤ 1`. -/
@@ -3224,16 +3112,16 @@ theorem zsmul_frobenius_pullback_x_ne_mulByInt_neg_pullback_x
   rw [zsmul_frobenius_pullback_x_gen W r hr, mulByInt_neg_pullback_x_gen W s hs]
   exact mulByInt_x_pow_card_ne_mulByInt_x_neg W r s hr hrK hs hsK
 
-/-! ### D3c step 4: σ-invariance specialised + K(x) image
+/-! ### σ-invariance specialised + K(x) image
 
 Specialise `addPullback_x_pair_sigma_invariant` to
 `(α₁, α₂) = (zsmul r π, mulByInt -s)` using the four σ-symmetry helpers
-shipped above, then chain with `sigma_fixed_implies_in_KX_image` to extract
+above, then chain with `sigma_fixed_implies_in_KX_image` to extract
 the K(x) image. -/
 
 /-- **σ-invariance of `addPullback_x_pair (zsmul r π, mulByInt -s)`**.
 Specialisation of the generic `addPullback_x_pair_sigma_invariant` for
-the family used by D3c/D4. -/
+this pencil family. -/
 theorem addPullback_x_pair_zsmul_frobenius_mulByInt_neg_sigma_invariant
     (r s : ℤ) (hr : r ≠ 0) (hs : s ≠ 0)
     (hrK : (r : K) ≠ 0) (hsK : (s : K) ≠ 0) :
@@ -3262,7 +3150,7 @@ theorem addPullback_x_pair_zsmul_frobenius_mulByInt_neg_in_KX_image
     (addPullback_x_pair_zsmul_frobenius_mulByInt_neg_sigma_invariant
       W r s hr hs hrK hsK)
 
-/-! ### D3c step 5: ord helpers for `(zsmul r π).pb x_gen` and
+/-! ### ord helpers for `(zsmul r π).pb x_gen` and
 `(mulByInt -s).pb x_gen`
 
 Concrete ord values needed by the pole-bound argument: `(zsmul r π).pb x_gen
@@ -3302,6 +3190,7 @@ theorem ordAtInfty_zsmul_frobenius_pullback_x_gen
     push_cast
     rfl
 
+omit [Fintype K] in
 /-- `ord_∞((mulByInt (-s)).pb x_gen) = -2` for `s ≠ 0` with `(s : K) ≠ 0`.
 Direct from `(mulByInt -s).pb x_gen = mulByInt_x W (-s)` plus
 `ordAtInfty_mulByInt_x`. -/
@@ -3361,7 +3250,7 @@ private lemma ordAtInfty_mulByInt_x_mul_y (r : ℤ) {M m : ℤ}
   refine ((W_smooth W).ordAtInfty_mul hX_ne hY_ne).trans ?_
   rw [hX_ord, hm]; push_cast; rfl
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- For `M < 0` the right-hand side of the Weierstrass equation is dominated by `X³`, so
 `ord_∞(X³ + a₂·X² + a₄·X + a₆) = 3M` where `M = ord_∞(mulByInt_x W r)`.  Repeated strict
 non-archimedean additivity: `3M < 2M < M < 0`, so each successive `aᵢ·Xⁱ` term has strictly larger
@@ -3434,7 +3323,7 @@ private lemma mulByInt_y_ne_zero_of_weierstrass_lhs_ord (r : ℤ) {M : ℤ}
     (congrArg (W_smooth W).ordAtInfty h_zero).trans (W_smooth W).ordAtInfty_zero
   exact WithTop.top_ne_coe (h_ord_eq.symm.trans h_lhs_ord)
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- The halving lower bound `2m ≤ 3M`, where `m = ord_∞(mulByInt_y W r)` and `M = ord_∞(mulByInt_x W
 r)`.  If instead `2m ≥ 3M + 1`, every Weierstrass LHS term (`Y²`, `a₁·X·Y`, `a₃·Y`) would have order
 `≥ 3M + 1`, so the LHS order would be `≥ 3M + 1`, contradicting `ord(LHS) = 3M`. -/
@@ -3477,7 +3366,7 @@ private lemma two_mul_ordAtInfty_mulByInt_y_le (r : ℤ) {M m : ℤ} (hM_neg : M
   have : (3 * M + 1 : ℤ) ≤ 3 * M := by exact_mod_cast h_lhs_ge
   omega
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- When `m < M` and `m < 0` (which hold once `2m ≤ 3M` with `M < 0`), the term `Y²` strictly
 dominates the other two Weierstrass LHS terms, so `ord_∞(Y² + a₁·X·Y + a₃·Y) = 2m`, where `m =
 ord_∞(mulByInt_y W r)`. -/
@@ -3520,6 +3409,7 @@ private lemma ordAtInfty_mulByInt_weierstrass_lhs_eq (r : ℤ) {M m : ℤ}
   refine (((W_smooth W).ordAtInfty_add_eq_of_lt h_a3y_gt').trans h_inner_eq).trans ?_
   exact hY_sq_ord
 
+omit [Fintype K] in
 /-- **`ord_∞(mulByInt_y W r) = 3M/2`** for `r ≠ 0`, where `M = ord_∞(mulByInt_x W r)` is the
 (even, `≤ -2`) `x`-order — **no** `(r : K) ≠ 0` hypothesis.  The curve point
 `(mulByInt_x W r, mulByInt_y W r)` satisfies the Weierstrass equation
@@ -3563,13 +3453,14 @@ theorem ordAtInfty_mulByInt_y_eq_of_x
   rw [hm, h_target]
   exact_mod_cast (by omega : m = 3 * M₂)
 
-/-! ### D3c step 6: ord helpers for `mulByInt_y` and the y-pullbacks
+/-! ### ord helpers for `mulByInt_y` and the y-pullbacks
 
 To compute `ord_∞` of the pair `addPullback_x`, we need ord values for
 the y-coordinate pullbacks: `ord((zsmul r π).pb y_gen) = -3q` and
 `ord((mulByInt -s).pb y_gen) = -3`. These chain through
 `ord(mulByInt_y W r) = -3` (curve-equation argument with x-ord = -2). -/
 
+omit [Fintype K] in
 /-- **`ord_∞(mulByInt_y W r) = -3`** for `r ≠ 0` with `(r : K) ≠ 0`.
 The `(mulByInt_x W r, mulByInt_y W r)` curve point has `ord(mulByInt_x) = -2`,
 so the curve equation `Y² + a₁·X·Y + a₃·Y = X³ + ...` forces
@@ -3624,6 +3515,7 @@ theorem ordAtInfty_zsmul_frobenius_pullback_y_gen
     push_cast
     rfl
 
+omit [Fintype K] in
 /-- `ord_∞((mulByInt -s).pb y_gen) = -3` for `s ≠ 0` with `(s : K) ≠ 0`.
 Direct from `(mulByInt -s).pb y_gen = mulByInt_y W (-s)` plus
 `ordAtInfty_mulByInt_y_eq_neg_three`. -/
@@ -3701,7 +3593,7 @@ theorem ordAtInfty_zsmul_frobenius_pullback_y_gen_of_x
   rw [h_eq]
   exact ((W_smooth W).ord_pow_concrete hY_ne ((3 * M) / 2) (Fintype.card K) hY_ord)
 
-/-! ### D3c step 7: ord helpers for the pair differences
+/-! ### ord helpers for the pair differences
 
 The slope `L = (Y₁ - Y₂) / (X₁ - X₂)` requires ord of the differences:
 `ord(α₁(x) - α₂(x)) = -2q` (α₁(x) dominates, ord = -2q < -2 = ord(α₂(x))),
@@ -3754,7 +3646,7 @@ theorem zsmul_frobenius_sub_mulByInt_neg_x_ne_zero
   rw [ordAtInfty_zsmul_frobenius_sub_mulByInt_neg_x W r s hr hs hrK hsK] at h_top
   exact WithTop.coe_ne_top h_top
 
-/-! ### D3c step 8: numerator + reduced form for the pair (zsmul r π, mulByInt -s)
+/-! ### numerator + reduced form for the pair (zsmul r π, mulByInt -s)
 
 Mirror of `addPullbackNumerator_negFrobenius` and `_reduced` for the pair
 version. The numerator is `(X₁-X₂)² · addPullback_x_pair` (cleared of the
@@ -3830,7 +3722,7 @@ private theorem weierstrass_relation_pullback (α : Isogeny W.toAffine W.toAffin
 `addPullbackNumerator_negFrobenius_eq_reduced`, with both Weierstrass
 equations applied via `linear_combination h_α₁ + h_α₂`. -/
 theorem addPullbackNumerator_pair_zsmul_frobenius_mulByInt_neg_eq_reduced
-    (r s : ℤ) (hr : r ≠ 0) (hs : s ≠ 0) :
+    (r s : ℤ) (_hr : r ≠ 0) (_hs : s ≠ 0) :
     addPullbackNumerator_pair_zsmul_frobenius_mulByInt_neg W r s =
       addPullbackNumerator_reduced_pair_zsmul_frobenius_mulByInt_neg W r s := by
   have h_α₁ := weierstrass_relation_pullback W ((frobeniusIsog W).zsmul r)
@@ -3885,7 +3777,7 @@ theorem addPullbackNumerator_pair_zsmul_frobenius_mulByInt_neg_eq
   field_simp
   ring
 
-/-! ### D3c step 9: `ord(reduced_pair) = -4q - 2`
+/-! ### `ord(reduced_pair) = -4q - 2`
 
 Mirror of `ordAtInfty_addPullbackNumerator_reduced_negFrobenius_eq` for
 the pair version. Dominant term: `α₁(x)² · α₂(x)` (ord = -4q-2). All
@@ -3914,6 +3806,7 @@ private lemma zsmul_frobenius_pullback_x_gen_ne_zero
   rw [ordAtInfty_zsmul_frobenius_pullback_x_gen W r hr hrK] at h_top
   exact WithTop.coe_ne_top h_top
 
+omit [Fintype K] in
 /-- `(mulByInt -s).pb y_gen ≠ 0` for `s ≠ 0` with `(s : K) ≠ 0`. -/
 private lemma mulByInt_neg_pullback_y_gen_ne_zero
     (s : ℤ) (hs : s ≠ 0) (hsK : (s : K) ≠ 0) :
@@ -3925,6 +3818,7 @@ private lemma mulByInt_neg_pullback_y_gen_ne_zero
   rw [ordAtInfty_mulByInt_neg_pullback_y_gen W s hs hsK] at h_top
   exact WithTop.coe_ne_top h_top
 
+omit [Fintype K] in
 /-- `(mulByInt -s).pb x_gen ≠ 0` for `s ≠ 0` with `(s : K) ≠ 0`. -/
 private lemma mulByInt_neg_pullback_x_gen_ne_zero
     (s : ℤ) (hs : s ≠ 0) (hsK : (s : K) ≠ 0) :
@@ -4120,6 +4014,7 @@ private lemma ord_zsmul_frobenius_mulByInt_neg_rest_term_a₄_ge
   exact_mod_cast (by linarith :
     (-3 * (Fintype.card K : ℤ) - 3 : ℤ) ≤ -2 * (Fintype.card K : ℤ))
 
+omit [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Rest term 2: `ord(2 · a₆) ≥ -3q - 3`.  The constant `a₆` has order `≥ 0` (or `⊤` if
 zero), which clears `-3q - 3`; `ord_two_mul_ge` handles the `2·` factor in any
 characteristic. -/
@@ -4385,7 +4280,7 @@ theorem ord_addPullback_x_pair_zsmul_frobenius_mulByInt_neg
     (-4 * (Fintype.card K : ℤ)) h_num_ord h_den_ord).trans ?_
   congr 1; ring
 
-/-! ### D3c: the `y`-coordinate `∞`-order for the pair `(zsmul r π, mulByInt -s)`
+/-! ### the `y`-coordinate `∞`-order for the pair `(zsmul r π, mulByInt -s)`
 
 The pair analogue of `ord_addPullback_y_negFrobenius = -3`.  The proof is the *same*
 Weierstrass-equation argument: from `ord_∞(addPullback_x_pair) = -2` and the curve equation
@@ -4400,7 +4295,7 @@ private theorem ord_addPullback_x_sq_pair_zsmul_frobenius_mulByInt_neg
     (W_smooth W).ordAtInfty
         (addPullback_x_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) ^ 2) =
       ((-4 : ℤ) : WithTop ℤ) := by
-  haveI : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
+  have : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
   have hX_ne : addPullback_x_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) ≠ 0 :=
     fun h ↦ WithTop.coe_ne_top
       (((W_smooth W).ordAtInfty_eq_top_iff _).mpr h ▸
@@ -4415,7 +4310,7 @@ private theorem ord_addPullback_x_cube_pair_zsmul_frobenius_mulByInt_neg
     (W_smooth W).ordAtInfty
         (addPullback_x_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) ^ 3) =
       ((-6 : ℤ) : WithTop ℤ) := by
-  haveI : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
+  have : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
   have hX_ne : addPullback_x_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) ≠ 0 :=
     fun h ↦ WithTop.coe_ne_top
       (((W_smooth W).ordAtInfty_eq_top_iff _).mpr h ▸
@@ -4435,7 +4330,7 @@ private theorem ord_RHS_pair_zsmul_frobenius_mulByInt_neg
            addPullback_x_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) +
          algebraMap K KE W.toAffine.a₆) =
       ((-6 : ℤ) : WithTop ℤ) := by
-  haveI : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
+  have : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
   set X := addPullback_x_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) with hX
   have h_X3 : (W_smooth W).ordAtInfty (X ^ 3) = ((-6 : ℤ) : WithTop ℤ) :=
     ord_addPullback_x_cube_pair_zsmul_frobenius_mulByInt_neg W r s hr hs hrK hsK
@@ -4525,7 +4420,7 @@ private theorem addPullback_y_pair_zsmul_frobenius_mulByInt_neg_ne_zero
     (h_ord_eq.symm.trans
       (ord_weierstrass_lhs_pair_zsmul_frobenius_mulByInt_neg W r s hr hs hrK hsK))
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Order-arithmetic helper (step (a)): if `ord(LHS) = -6`, the y-order `m` of the Weierstrass LHS
 satisfies `m ≤ -3`. Otherwise all three summands would have order `≥ -4`, forcing `ord(LHS) ≥ -4`,
 contradicting `-6`. Purely generic in `X`, `Y`, `m` and the per-term order data. -/
@@ -4558,7 +4453,7 @@ private theorem ord_y_le_neg_three_of_weierstrass_lhs_ord {X Y : KE} {m : ℤ}
   have h46 : (-4 : ℤ) ≤ -6 := by exact_mod_cast h_lhs_ge
   omega
 
-omit [Fintype K] in
+omit [Fintype K] [WeierstrassCurve.IsElliptic W.toAffine] in
 /-- Order-arithmetic helper (step (b)): when `m ≤ -3`, the `Y²` term strictly dominates both the
 `a₁·X·Y` and `a₃·Y` terms (`2m < -2 + m` and `2m < m`), so the order of the Weierstrass LHS equals
 `ord(Y²)`. Purely generic in `X`, `Y`, `m` and the per-term order data. -/
@@ -4603,7 +4498,7 @@ theorem ord_addPullback_y_pair_zsmul_frobenius_mulByInt_neg
     (W_smooth W).ordAtInfty
         ((addPullback_y_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s))) : KE) =
       ((-3 : ℤ) : WithTop ℤ) := by
-  haveI : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
+  have : (W_smooth W).toAffine.IsElliptic := inferInstanceAs W.toAffine.IsElliptic
   set X := addPullback_x_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) with hX
   set Y := addPullback_y_pair ((frobeniusIsog W).zsmul r) (mulByInt W.toAffine (-s)) with hY
   have hX_ne : X ≠ 0 :=
@@ -4633,21 +4528,13 @@ theorem ord_addPullback_y_pair_zsmul_frobenius_mulByInt_neg
   have h_2m : (2 * m : ℤ) = -6 := by exact_mod_cast h_lhs_ord
   rw [hm]; exact_mod_cast (by omega : m = -3)
 
-/-! ### D4: pole-bound-parametric `genuineIsogSmulSub`
+/-! ### The pencil `r·π − s·id`
 
-Worker D consumes `genuineIsogSmulSub r s` for the genuine `r·π - s·id`
-isogeny. The constructor needs:
-1. The `AddNonInversePair` witness — shipped axiom-clean as
-   `AddNonInversePair_zsmul_frobenius_mulByInt_neg`.
-2. `addCoordAlgHomPair`-injectivity — reducible (via base-hom + transcendence
-   chain) to a pole bound `ord_∞ < 0` on the pair pullback x-coord.
-
-We ship the **pole-bound-parametric** form. The pole bound discharges
-axiom-clean via the same numerator/denominator/Weierstrass-reduction
-chain as `ord_addPullback_x_negFrobenius` (the (1,1) base case here),
-mirrored to the pair version with α₁(x)²·α₂(x) as the dominant term
-(ord = -4q-2). Worker D supplies the pole bound (or waits for the
-follow-up axiom-clean discharge). -/
+Different coordinate orders provide the non-inverse-pair witness.
+A pole of the summed x-coordinate gives polynomial and coordinate-ring
+injectivity through curve-negation invariance and its `K(x)` image.
+These facts construct the pencil isogeny's function-field pullback.
+-/
 
 /-- **Witness-parametric base-hom injectivity** for the
 `(zsmul r π, mulByInt -s)` family: takes the pole bound hypothesis
@@ -4713,17 +4600,9 @@ theorem addCoordAlgHomPair_injective_zsmul_frobenius_mulByInt_neg_of_pole
     (addBaseHomPair_injective_zsmul_frobenius_mulByInt_neg_of_pole
       W r s hr hs hrK hsK h_pole)
 
-/-- **D4: genuine `r·π - s·id` isogeny constructor (pole-bound-parametric)**.
-
-The genuine `r·π - s·id` isogeny with the *real* function-field pullback
-(replacing the `AlgHom.id` placeholder of `Endomorphism.lean`'s
-`isogSmulSub`). On rational points equals
-`(zsmul r π).toAddMonoidHom + (mulByInt -s).toAddMonoidHom`.
-
-Worker D in `DegreeQuadraticForm.lean` consumes this as the `r·π - s·id`
-isogeny family. The pole bound discharges axiom-clean via the same
-numerator chain as `ord_addPullback_x_negFrobenius` (the (1,1) base
-case), mirrored to the pair version. -/
+/-- The isogeny `r·π − s·id`, given the pole of its addition x-coordinate.
+    Its function-field pullback comes from the addition formulas, and its
+    point map is `(r·π) + (−s·id)`. -/
 noncomputable def genuineIsogSmulSub_of_pole
     (r s : ℤ) (hr : r ≠ 0) (hs : s ≠ 0)
     (hrK : (r : K) ≠ 0) (hsK : (s : K) ≠ 0)
@@ -4747,12 +4626,11 @@ noncomputable def genuineIsogSmulSub_of_pole
         (mulByInt W.toAffine (-s)).toAddMonoidHom :=
   rfl
 
-/-! ### D4 unconditional: `genuineIsogSmulSub`
+/-! ### Constructing the pencil from its coordinate pole
 
-With `ord_addPullback_x_pair_zsmul_frobenius_mulByInt_neg = -2 < 0`
-discharged axiom-clean, the witness-parametric `genuineIsogSmulSub_of_pole`
-becomes unconditional. The unconditional form is what Worker D consumes
-in `DegreeQuadraticForm.lean` for the polarisation. -/
+The addition x-coordinate has order `−2`, hence a pole. This supplies
+the injectivity witness for the function-field pullback of `r·π − s·id`.
+-/
 
 /-- The pole bound `ord_∞ < 0` discharges from
 `ord_addPullback_x_pair_zsmul_frobenius_mulByInt_neg = -2`. -/
@@ -4765,11 +4643,9 @@ private theorem h_pole_discharge
   rw [ord_addPullback_x_pair_zsmul_frobenius_mulByInt_neg W r s hr hs hrK hsK]
   exact_mod_cast (by norm_num : (-2 : ℤ) < 0)
 
-/-- **D4 (unconditional)**: the genuine `r·π - s·id` isogeny for
-`r, s ≠ 0` with `(r : K), (s : K) ≠ 0`. Pullback is the *real*
-function-field pullback; on rational points equals
-`r · π + (-s) · id = r · π - s · id`. Worker D in
-`DegreeQuadraticForm.lean` consumes this directly. -/
+/-- The isogeny `r·π − s·id` when the two integers and their base-field
+    images are nonzero. Its pullback is the addition-formula pullback,
+    and its point map is `r·π − s·id`. -/
 noncomputable def genuineIsogSmulSub
     (r s : ℤ) (hr : r ≠ 0) (hs : s ≠ 0)
     (hrK : (r : K) ≠ 0) (hsK : (s : K) ≠ 0) :
@@ -4785,7 +4661,7 @@ noncomputable def genuineIsogSmulSub
         (mulByInt W.toAffine (-s)).toAddMonoidHom :=
   rfl
 
-/-! ### D4 inseparable: the `p ∣ r` pole bound `ord_∞(addPullback_x_pair) = -2`
+/-! ### Inseparable pencil: the `p ∣ r` pole bound `ord_∞(addPullback_x_pair) = -2`
 
 For the **inseparable** pencil summand `r·π` (when `p ∣ r`, so `(r : K) = 0`), the `x`-pole of
 `α₁ = (frobeniusIsog).zsmul r` is `ord_∞(α₁^* x_gen) = q·M` with
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/FunctionField.lean b/projects/HasseWeil/HasseWeil/Isogeny/FunctionField.lean
index 390492635..fba307a06 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/FunctionField.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/FunctionField.lean
@@ -7,30 +7,6 @@ import Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point
 import Mathlib.LinearAlgebra.Dimension.Finrank
 import Mathlib.LinearAlgebra.Basis.VectorSpace
 
-/-!
-# Isogenies via Function Field Extensions
-
-We define isogenies between elliptic curves as injective `K`-algebra homomorphisms
-on function fields, following Silverman III.4. The degree of an isogeny is the degree
-of the corresponding field extension.
-
-## Main Definitions
-
-* `HasseWeil.Isogeny`: An isogeny from `E₁` to `E₂`, represented as an injective
-  `K`-algebra homomorphism `K(E₂) →ₐ[K] K(E₁)` on function fields.
-* `HasseWeil.Isogeny.degree`: The degree `[K(E₁) : φ*K(E₂)]`.
-
-## Design Note
-
-The degree is defined as `Module.finrank K(E₂) K(E₁)` where `K(E₁)` is given a
-`K(E₂)`-algebra structure via the pullback `φ* : K(E₂) →ₐ[F] K(E₁)`. This avoids
-working with subalgebra ranges and makes the tower law (degree multiplicativity)
-a direct application of `Module.finrank_mul_finrank`.
-
-## References
-
-* [Silverman, *The Arithmetic of Elliptic Curves*], III.4
--/
 
 open WeierstrassCurve Polynomial
 
@@ -49,8 +25,7 @@ namespace PullbackIsogeny
 variable {F : Type*} [Field F]
 variable {W₁ W₂ W₃ : Affine F} [W₁.IsElliptic] [W₂.IsElliptic] [W₃.IsElliptic]
 
-/-- The pullback of an isogeny is injective: any algebra homomorphism from a field
-    is injective because the kernel of a ring homomorphism from a field is trivial. -/
+
 theorem pullback_injective (φ : PullbackIsogeny F W₁ W₂) :
     Function.Injective φ.pullback :=
   φ.pullback.toRingHom.injective
@@ -61,8 +36,7 @@ noncomputable def toAlgebra (φ : PullbackIsogeny F W₁ W₂) :
     Algebra W₂.FunctionField W₁.FunctionField :=
   φ.pullback.toRingHom.toAlgebra
 
-/-- The degree of an isogeny, defined as `[K(E₁) : K(E₂)]` where `K(E₁)` is
-    a `K(E₂)`-module via the pullback. -/
+
 noncomputable def degree (φ : PullbackIsogeny F W₁ W₂) : ℕ :=
   @Module.finrank W₂.FunctionField W₁.FunctionField _ _ φ.toAlgebra.toModule
 
@@ -81,12 +55,12 @@ theorem comp_algebraMap_eq (ψ : PullbackIsogeny F W₂ W₃) (φ : PullbackIsog
 theorem comp_degree (ψ : PullbackIsogeny F W₂ W₃) (φ : PullbackIsogeny F W₁ W₂) :
     (ψ.comp φ).degree = φ.degree * ψ.degree := by
   simp only [degree]
-  letI inst₁ : Algebra W₂.FunctionField W₁.FunctionField := φ.toAlgebra
-  letI inst₂ : Algebra W₃.FunctionField W₂.FunctionField := ψ.toAlgebra
-  letI inst₃ : Algebra W₃.FunctionField W₁.FunctionField := (ψ.comp φ).toAlgebra
-  haveI : IsScalarTower W₃.FunctionField W₂.FunctionField W₁.FunctionField :=
+  let inst₁ : Algebra W₂.FunctionField W₁.FunctionField := φ.toAlgebra
+  let inst₂ : Algebra W₃.FunctionField W₂.FunctionField := ψ.toAlgebra
+  let inst₃ : Algebra W₃.FunctionField W₁.FunctionField := (ψ.comp φ).toAlgebra
+  have : IsScalarTower W₃.FunctionField W₂.FunctionField W₁.FunctionField :=
     IsScalarTower.of_algebraMap_eq fun x => rfl
-  haveI : Module.Free W₂.FunctionField W₁.FunctionField :=
+  have : Module.Free W₂.FunctionField W₁.FunctionField :=
     Module.Free.of_divisionRing _ _
   rw [mul_comm]
   exact (Module.finrank_mul_finrank
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/GroupHom/PicZero.lean b/projects/HasseWeil/HasseWeil/Isogeny/GroupHom/PicZero.lean
index 1b98e70eec533a4bac61cd0a445abb7372a63eea..603d41e1f0ecdaf549903d24babfdd67af8baf36 100644
GIT binary patch
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diff --git a/projects/HasseWeil/HasseWeil/Isogeny/Kernel.lean b/projects/HasseWeil/HasseWeil/Isogeny/Kernel.lean
index 2b2b0f7f5..f46fb56a6 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/Kernel.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/Kernel.lean
@@ -7,36 +7,6 @@ import HasseWeil.Foundation.Basic
 import Mathlib.FieldTheory.Galois.Basic
 import Mathlib.FieldTheory.SeparableDegree
 
-/-!
-# Kernel of an Isogeny
-
-For an isogeny `φ : E₁ → E₂`, we collect the structural facts about `ker φ`
-as an `AddSubgroup` of `E₁.Point`. These are foundational for the dual
-isogeny construction (Silverman III.6) and the Hasse bound (V.1).
-
-## Main definitions
-
-* `Isogeny.kernel φ` — the kernel of `φ.toAddMonoidHom`, viewed as an
-  `AddSubgroup` of `W₁.Point`. A thin wrapper around `AddMonoidHom.ker`.
-* `Isogeny.IsSeparable φ` — separability of the induced field extension.
-
-## Deep results (not yet proved)
-
-The following correspond to tickets `T-III-4-011/012/015`; they require
-substantial infrastructure (finiteness of fibers for finite morphisms of
-smooth curves, and the Galois theory of `K(E₁)/φ*K(E₂)`) and are deferred.
-
-* **Silverman III.4.10(a)** — nonzero isogeny has finite kernel.
-* **Silverman III.4.10(b)** — `#ker φ ≤ deg_s φ ≤ deg φ`.
-* **Silverman III.4.10(c)** — equality when `φ` is separable.
-
-Once available, these unblock the dual isogeny chain (T-III-6-001 etc.).
-
-## References
-
-* [J Silverman, *The Arithmetic of Elliptic Curves*][silverman2009], III.4.10.
-
--/
 
 open WeierstrassCurve
 
@@ -47,7 +17,7 @@ variable {W₁ W₂ : Affine F} [W₁.IsElliptic] [W₂.IsElliptic]
 
 namespace Isogeny
 
-/-- The kernel of an isogeny `φ`, as an `AddSubgroup` of `W₁.Point`. -/
+
 noncomputable def kernel (φ : Isogeny W₁ W₂) : AddSubgroup W₁.Point :=
   φ.toAddMonoidHom.ker
 
@@ -58,25 +28,20 @@ noncomputable def kernel (φ : Isogeny W₁ W₂) : AddSubgroup W₁.Point :=
 @[simp] theorem zero_mem_kernel (φ : Isogeny W₁ W₂) : (0 : W₁.Point) ∈ φ.kernel :=
   (φ.kernel).zero_mem
 
-/-- The kernel is preserved by composition (fiberwise): `ker φ ⊆ ker (ψ ∘ φ)`. -/
+
 theorem kernel_comp_le {W₃ : Affine F} [W₃.IsElliptic] (ψ : Isogeny W₂ W₃)
     (φ : Isogeny W₁ W₂) : φ.kernel ≤ (ψ.comp φ).kernel := by
   intro P hP
   simp only [mem_kernel_iff, comp_apply] at hP ⊢
   rw [hP, map_zero]
 
-/-- **T-III-4-011 witness form**: If the fiber `φ⁻¹({0})` (as a subtype
-    of `W₁.Point`) is finite, then `φ.kernel` is a finite additive subgroup.
 
-    The hypothesis `h_fiber` is a specific case of T-II-2-002 (finite fibers
-    for nonconstant morphisms of smooth curves) applied to the fiber over `0`.
-    Full unconditional version awaits that foundational curve-theory result. -/
 theorem kernel_finite_of_fiber_finite (φ : Isogeny W₁ W₂)
     (h_fiber : Finite {P : W₁.Point // φ.toAddMonoidHom P = 0}) :
     Finite φ.kernel :=
   Finite.of_equiv _ (Equiv.subtypeEquivRight (mem_kernel_iff φ)).symm
 
-/-- `ker [1] = ⊥` (trivial kernel, since [1] is the identity on points). -/
+
 @[simp] theorem kernel_mulByInt_one {W : WeierstrassCurve F} [W.toAffine.IsElliptic] :
     (mulByInt W.toAffine 1).kernel = ⊥ := by
   ext P
@@ -108,8 +73,7 @@ theorem mem_kernel_comp_of_mem_kernel {W₃ : Affine F} [W₃.IsElliptic]
     (hP : P ∈ φ.kernel) : P ∈ (ψ.comp φ).kernel :=
   kernel_comp_le ψ φ hP
 
-/-- If `ψ` has trivial kernel, `ker (ψ ∘ φ) = ker φ`: applying a "monomorphic"
-    post-composition doesn't enlarge the kernel. -/
+
 theorem kernel_comp_of_kernel_eq_bot {W₃ : Affine F} [W₃.IsElliptic]
     {ψ : Isogeny W₂ W₃} (hψ : ψ.kernel = ⊥) (φ : Isogeny W₁ W₂) :
     (ψ.comp φ).kernel = φ.kernel := by
@@ -123,9 +87,7 @@ theorem kernel_comp_of_kernel_eq_bot {W₃ : Affine F} [W₃.IsElliptic]
   · intro h
     rw [h, map_zero]
 
-/-- **Kernel finiteness over finite fields (unconditional)**: when `W₁.Point`
-    is finite (e.g., for an elliptic curve over a finite field), every kernel
-    is automatically finite. -/
+
 instance kernel_finite_of_point_finite [Finite W₁.Point] (φ : Isogeny W₁ W₂) :
     Finite φ.kernel :=
   inferInstance
@@ -135,15 +97,12 @@ theorem kernel_card_le_point_card [Finite W₁.Point] (φ : Isogeny W₁ W₂) :
     Nat.card φ.kernel ≤ Nat.card W₁.Point :=
   Nat.card_le_card_of_injective _ (Subtype.val_injective)
 
-/-- **Lagrange for kernel**: over a finite field, `#ker φ ∣ #W₁.Point`
-    (subgroup cardinality divides group cardinality). -/
+
 theorem kernel_card_dvd_point_card [Finite W₁.Point] (φ : Isogeny W₁ W₂) :
     Nat.card φ.kernel ∣ Nat.card W₁.Point :=
   AddSubgroup.card_addSubgroup_dvd_card φ.kernel
 
-/-- **Kernel of sum** (for AddSubgroup inclusion): `ker φ ∩ ker ψ ≤ ker (φ + ψ)`,
-    where `(φ + ψ)` is a hypothetical isogeny with summed action.
-    (Purely algebraic: if φ(P) = 0 and ψ(P) = 0, then φ(P) + ψ(P) = 0.) -/
+
 theorem kernel_inf_le_kernel_of_sum (φ ψ : Isogeny W₁ W₂)
     (σ : Isogeny W₁ W₂) (hσ : σ.toAddMonoidHom = φ.toAddMonoidHom + ψ.toAddMonoidHom) :
     φ.kernel ⊓ ψ.kernel ≤ σ.kernel := by
@@ -151,7 +110,7 @@ theorem kernel_inf_le_kernel_of_sum (φ ψ : Isogeny W₁ W₂)
   simp only [AddSubgroup.mem_inf, mem_kernel_iff] at hP
   simp only [mem_kernel_iff, hσ, AddMonoidHom.add_apply, hP.1, hP.2, add_zero]
 
-/-- An isogeny has trivial kernel iff its toAddMonoidHom is injective. -/
+
 theorem kernel_eq_bot_iff_injective (φ : Isogeny W₁ W₂) :
     φ.kernel = ⊥ ↔ Function.Injective φ.toAddMonoidHom :=
   AddMonoidHom.ker_eq_bot_iff φ.toAddMonoidHom
@@ -161,23 +120,20 @@ theorem id_toAddMonoidHom_injective :
     Function.Injective (Isogeny.id W₁).toAddMonoidHom :=
   Function.injective_id
 
-/-- `ker (ψ ∘ φ) = φ⁻¹(ker ψ)` (as an AddSubgroup): the kernel of a composition
-    is the preimage of the outer kernel. -/
+
 theorem kernel_comp_eq_comap {W₃ : Affine F} [W₃.IsElliptic]
     (ψ : Isogeny W₂ W₃) (φ : Isogeny W₁ W₂) :
     (ψ.comp φ).kernel = ψ.kernel.comap φ.toAddMonoidHom := by
   ext P
   simp [mem_kernel_iff]
 
-/-- For a nonzero isogeny with finite kernel, the action of the kernel on
-    any nonempty fiber is transitive. I.e., `φ⁻¹(Q) = P₀ + ker φ` as sets
-    (when `φ(P₀) = Q`). -/
+
 theorem fiber_eq_coset (φ : Isogeny W₁ W₂)
     {P₀ : W₁.Point} {Q : W₂.Point} (hP₀ : φ.toAddMonoidHom P₀ = Q) :
     {P : W₁.Point | φ.toAddMonoidHom P = Q} =
       {P : W₁.Point | ∃ T ∈ φ.kernel, P = P₀ + T} := by
   ext P
-  simp only [Set.mem_setOf_eq]
+  simp only [Set.mem_ofPred_eq]
   constructor
   · intro hP
     refine ⟨P - P₀, ?_, by abel⟩
@@ -193,10 +149,7 @@ theorem mem_fiber_iff_sub_mem_kernel (φ : Isogeny W₁ W₂)
     φ.toAddMonoidHom P = Q ↔ P - P₀ ∈ φ.kernel := by
   rw [mem_kernel_iff, map_sub, hP₀, sub_eq_zero]
 
-/-- **Fiber cardinality = kernel cardinality** (via coset structure): the
-    fiber `φ⁻¹(Q)` of a nonempty fiber (with `P₀ ∈ φ⁻¹(Q)`) has the same
-    cardinality as `ker φ`. This is the group-theoretic content of the
-    fiber-cardinality formula (before invoking any ramification theory). -/
+
 noncomputable def fiberEquivKernel (φ : Isogeny W₁ W₂)
     {P₀ : W₁.Point} {Q : W₂.Point} (hP₀ : φ.toAddMonoidHom P₀ = Q) :
     {P : W₁.Point // φ.toAddMonoidHom P = Q} ≃ φ.kernel where
@@ -213,12 +166,11 @@ theorem fiber_finite_of_kernel_finite (φ : Isogeny W₁ W₂)
     Finite {P : W₁.Point // φ.toAddMonoidHom P = Q} := by
   rcases Classical.em (∃ P₀, φ.toAddMonoidHom P₀ = Q) with ⟨P₀, hP₀⟩ | h_empty
   · exact Finite.of_equiv _ (fiberEquivKernel φ hP₀).symm
-  · haveI : IsEmpty {P : W₁.Point // φ.toAddMonoidHom P = Q} :=
+  · have : IsEmpty {P : W₁.Point // φ.toAddMonoidHom P = Q} :=
       ⟨fun ⟨P, hP⟩ ↦ h_empty ⟨P, hP⟩⟩
     infer_instance
 
-/-- **Bijection of nonempty fibers**: any two nonempty fibers of `φ` are in
-    bijection. (They're both cosets of the kernel, hence have the same size.) -/
+
 noncomputable def fiberEquivFiber (φ : Isogeny W₁ W₂)
     {P₀ : W₁.Point} {Q : W₂.Point} (hP₀ : φ.toAddMonoidHom P₀ = Q)
     {P₁ : W₁.Point} {Q' : W₂.Point} (hP₁ : φ.toAddMonoidHom P₁ = Q') :
@@ -226,21 +178,12 @@ noncomputable def fiberEquivFiber (φ : Isogeny W₁ W₂)
       {P : W₁.Point // φ.toAddMonoidHom P = Q'} :=
   (fiberEquivKernel φ hP₀).trans (fiberEquivKernel φ hP₁).symm
 
-/-- **T-III-4-015 step 1** (kernel-as-fiber-over-zero): the kernel of an
-    isogeny is in canonical bijection with the fiber over the identity point.
-    Direct specialization of `fiberEquivKernel` to `Q = 0` (image of `P₀ = 0`).
 
-    This is the foundational set-theoretic identification: `φ.kernel ≃ φ⁻¹(0)`.
-    Combined with `fiber_card_eq_kernel_card` (which generalizes to any `Q` in
-    image), this gives the standard isogeny-fiber result that connects the
-    kernel cardinality to the fiber structure. -/
 noncomputable def kernel_equiv_fiber_zero (φ : Isogeny W₁ W₂) :
     φ.kernel ≃ {P : W₁.Point // φ.toAddMonoidHom P = 0} :=
   (fiberEquivKernel φ (by simp : φ.toAddMonoidHom 0 = 0)).symm
 
-/-- **Corollary** (kernel-cardinality = fiber-zero-cardinality): when the kernel
-    is finite, the cardinality of the fiber over zero equals the cardinality of
-    the kernel. Direct from `kernel_equiv_fiber_zero`. -/
+
 theorem kernel_card_eq_fiber_zero_card (φ : Isogeny W₁ W₂)
     [Finite φ.kernel] :
     Nat.card φ.kernel = Nat.card {P : W₁.Point // φ.toAddMonoidHom P = 0} :=
@@ -253,7 +196,7 @@ theorem fiber_card_eq_kernel_card (φ : Isogeny W₁ W₂)
     Nat.card {P : W₁.Point // φ.toAddMonoidHom P = Q} = Nat.card φ.kernel :=
   Nat.card_congr (fiberEquivKernel φ hP₀)
 
-/-- The kernel of the identity isogeny is trivial. -/
+
 @[simp] theorem kernel_id : (Isogeny.id W₁).kernel = ⊥ := by
   ext P
   simp [mem_kernel_iff, id_toAddMonoidHom]
@@ -308,15 +251,7 @@ theorem fiber_card_eq_sepDegree_of_witness (φ : Isogeny W₁ W₂)
   rw [← fiber_card_eq_kernel_card φ rfl (P₀ := P₀_wit)]
   exact h_wit
 
-/-- **Hasse-bound shortcut**: if `|ker φ| = φ.sepDegree` (a finite-field
-    fact that can be proved directly without going through the generic
-    II.2.6(b) fiber-size theorem), then the fiber over `φ(0)` (i.e. the
-    kernel) provides the fiber witness used by the historical witness-parametric
-    Hasse-bound route.
 
-    Reduces T-II-2-009's "existential fiber-size = sepDegree" witness to the
-    simpler `|ker| = sepDegree` fact, which is directly accessible for
-    separable isogenies over finite fields via the V.1 chain. -/
 theorem fiber_witness_of_ker_card_eq_sepDegree (φ : Isogeny W₁ W₂)
     [Finite φ.kernel]
     (h_ker_sep : Nat.card φ.kernel = φ.sepDegree) :
@@ -327,13 +262,7 @@ theorem fiber_witness_of_ker_card_eq_sepDegree (φ : Isogeny W₁ W₂)
   rw [fiber_card_eq_kernel_card φ (P₀ := 0) (Q := φ.toAddMonoidHom 0) rfl,
     h_ker_sep]
 
-/-- **T-III-4-015 step 2 closer** (witness-parametric): given a witness that
-the fiber over zero has cardinality `φ.sepDegree`, the kernel cardinality
-equals `φ.sepDegree`. Direct from `kernel_card_eq_fiber_zero_card`.
 
-The witness `h_fiber_card_zero` is the SUBSTANTIVE content of Silverman
-III.4.10/12 specialized to fiber over zero: for separable isogenies, every
-fiber has cardinality = sepDegree. -/
 theorem kernel_card_eq_sepDegree_of_fiber_zero_witness (φ : Isogeny W₁ W₂)
     [h_ker : Finite φ.kernel]
     (h_fiber_card_zero :
@@ -341,9 +270,7 @@ theorem kernel_card_eq_sepDegree_of_fiber_zero_witness (φ : Isogeny W₁ W₂)
     Nat.card φ.kernel = φ.sepDegree := by
   rw [kernel_card_eq_fiber_zero_card φ, h_fiber_card_zero]
 
-/-- **T-III-4-015 witness form for separable isogenies**: If `φ` is
-    separable and has a finite kernel, and at least one fiber has
-    cardinality = `sepDegree`, then `#ker φ = deg φ`. -/
+
 theorem card_kernel_eq_degree_of_separable_witness (φ : Isogeny W₁ W₂)
     [h_ker : Finite φ.kernel] (hsep : φ.IsSeparable)
     (hfin : @FiniteDimensional W₂.FunctionField W₁.FunctionField _ _
@@ -385,16 +312,7 @@ theorem sepDegree_eq_card_emb (φ : Isogeny W₁ W₂) :
         AlgebraicClosure W₁.FunctionField) :=
   rfl
 
-/-- **R2 reduction brick** (Silverman III.4.10c, embeddings half): for a
-separable, finite-dimensional isogeny `φ`, if the **embedding count equals the
-kernel count** — `φ.sepDegree = #ker φ`, i.e.
-`#(K(E₁) →ₐ[φ*K(E₂)] Ω) = #ker φ` — then `#ker φ = deg φ`.
 
-This is the clean compositional consumer for the embeddings-classification
-route: `#ker φ = sepDegree φ` (hypothesis) `= deg φ` (separability, via
-`isSeparable_iff_sepDegree_eq_degree`). The substantive content is delegated
-to the hypothesis `h_count`, which is exactly the embedding↔kernel bijection
-`Emb ≃ ker φ`. -/
 theorem card_kernel_eq_degree_of_sepDegree_eq_card_kernel (φ : Isogeny W₁ W₂)
     (hsep : φ.IsSeparable)
     (hfin : @FiniteDimensional W₂.FunctionField W₁.FunctionField _ _
@@ -491,9 +409,9 @@ theorem card_aut_eq_degree_of_isGalois (φ : Isogeny W₁ W₂)
     (hgal : @IsGalois W₂.FunctionField _ W₁.FunctionField _ φ.toAlgebra) :
     Nat.card (@AlgEquiv W₂.FunctionField W₁.FunctionField W₁.FunctionField _ _ _
       φ.toAlgebra φ.toAlgebra) = φ.degree := by
-  letI := φ.toAlgebra
-  haveI := hfin
-  haveI := hgal
+  let := φ.toAlgebra
+  have := hfin
+  have := hgal
   exact IsGalois.card_aut_eq_finrank W₂.FunctionField W₁.FunctionField
 
 /-- **T-III-4-015 step 4** (IsGalois from Separable + Normal): given the
@@ -510,7 +428,7 @@ theorem isGalois_of_separable_and_normal (φ : Isogeny W₁ W₂)
     (h_normal : letI := φ.toAlgebra
       Normal W₂.FunctionField W₁.FunctionField) :
     @IsGalois W₂.FunctionField _ W₁.FunctionField _ φ.toAlgebra := by
-  letI := φ.toAlgebra
+  let := φ.toAlgebra
   exact isGalois_iff.mpr ⟨h_sep, h_normal⟩
 
 /-- **T-III-4-015 step 4b** (IsGalois from Isogeny.IsSeparable + Normal):
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/OmegaCoeffViaFormalGroup.lean b/projects/HasseWeil/HasseWeil/Isogeny/OmegaCoeffViaFormalGroup.lean
index b4f3c844a3d84552f61642e618d4c4c2e3ea997f..ee9bf1a49bf89477daa167de9f0698966a7944f9 100644
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diff --git a/projects/HasseWeil/HasseWeil/Isogeny/OrdTransport.lean b/projects/HasseWeil/HasseWeil/Isogeny/OrdTransport.lean
index 25e8c912a..3e6fa927e 100644
--- a/projects/HasseWeil/HasseWeil/Isogeny/OrdTransport.lean
+++ b/projects/HasseWeil/HasseWeil/Isogeny/OrdTransport.lean
@@ -6,34 +6,25 @@ Authors: Chris Birkbeck
 import HasseWeil.Foundation.EC.TranslateOrdInfty
 
 /-!
-# General DVR order-transport along a field hom (Silverman II.2.5, the unramified case)
+# Unramified order transport along a field homomorphism
 
-This file isolates the *purely valuation-theoretic* core of the order-transport
-`ord_P (φ g) = ord_Q g` for an injective field homomorphism `φ : K(C) → K(C)` and a pair of smooth
-points `P` (source) and `Q` (target).  It is the abstract DVR-transport step (Step 1 of the
-divisor-pullback brief): once one knows
+Let `φ : E → K(C)` be a field homomorphism and `P` a smooth point of `C`.
+Suppose the comap of the point valuation at `P` is equivalent to a surjective
+integer-valued valuation `v_Q` on `E`. If `ord_P (φ t) = 1` for one element `t`,
+the comap is also surjective, and the equivalent normalized valuations are equal.
 
-* **(same place)** the comap valuation `(pointValuation Q).comap φ` is `Valuation.IsEquiv` to
-  `pointValuation P` — i.e. `φ` carries the valuation ring of `Q` onto that of `P`; and
-* **(unramified, `e = 1`)** `ord_P (φ t) = 1` for a *single* uniformizer `t` at `Q`,
+The equivalence expresses equality of places, or valuation rings; the order-one
+condition fixes the ramification index to `1`. Together these give
+`ord_P (φ g) = ord_Q g` for every `g`.
 
-the value-precise identity `ord_P (φ g) = ord_Q g` holds for **all** `g` (no ramification factor).
-The two inputs are genuinely independent: the first is "same place / same valuation ring", the
-second is the normalization `e = 1` (the ramification index), which for `[ℓ]` is the geometric
-content of separability.
-
-The main export is `comap_pointValuation_eq_of_isEquiv_of_ord_eq_one`, plus the small valuation glue
-lemmas it rests on (`Fintype`-free; the `[Fintype K]`-scoped versions they were derived from lived
-in `Hasse/L6Witnesses.lean`, which no longer exists, so these are now the only copies).
-
-Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.2.5–2.6, III.4.10c.
+Reference: Silverman, *The Arithmetic of Elliptic Curves*, II.2.5–2.6 and III.4.10c.
 -/
 
 open WeierstrassCurve HasseWeil.Curves
 
 namespace HasseWeil
 
-variable {F : Type*} [Field F] [DecidableEq F]
+variable {F : Type*} [Field F]
 
 /-- A `ℤᵐ⁰`-valued valuation on a field is surjective as soon as some element takes the value
 `exp (-1)`: the negative powers of that element realise every `exp (-n)`, and `0 ↦ 0`. Shared core
@@ -143,8 +134,7 @@ Let `φ : E → K(C)` be a ring hom from a field `E`, `P` a smooth point of `C`,
 then `(pointValuation P).comap φ = v_Q` *as valuations* (not merely equivalent).  Reading off `ord`,
 this yields `ord_P (φ g) = ord_Q g` for all `g` with no ramification factor.
 
-This packages Step 1 of the divisor-pullback brief: the entire affine/infinity order-transport for
-a separable isogeny reduces to discharging the two displayed inputs. -/
+The two hypotheses separate equality of valuation rings from the order normalization. -/
 theorem comap_pointValuation_eq_of_isEquiv_of_ord_eq_one
     {E : Type*} [Field E] (φ : E →+* C.FunctionField) (P : C.SmoothPoint)
     (v_Q : _root_.Valuation E (WithZero (Multiplicative ℤ))) (hv_Q : Function.Surjective v_Q)
diff --git a/projects/HasseWeil/HasseWeil/Isogeny/SeparableWitnessReductions.lean b/projects/HasseWeil/HasseWeil/Isogeny/SeparableWitnessReductions.lean
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GIT binary patch
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diff --git a/projects/HasseWeil/HasseWeil/Pic0/ClassGroupNorm.lean b/projects/HasseWeil/HasseWeil/Pic0/ClassGroupNorm.lean
index a3e70337f..dec8deca0 100644
--- a/projects/HasseWeil/HasseWeil/Pic0/ClassGroupNorm.lean
+++ b/projects/HasseWeil/HasseWeil/Pic0/ClassGroupNorm.lean
@@ -4,7 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Birkbeck
 -/
 import Mathlib.LinearAlgebra.FreeModule.Norm
-import Mathlib.NumberTheory.RamificationInertia.Basic
+import Mathlib.RingTheory.RamificationInertia.Basic
+import Mathlib.NumberTheory.RamificationInertia.Inertia
+import Mathlib.NumberTheory.RamificationInertia.Ramification
+import Mathlib.RingTheory.Ideal.Norm.AbsNorm
+import Mathlib.RingTheory.SimpleModule.Basic
 import Mathlib.RingTheory.ClassGroup.Basic
 import Mathlib.RingTheory.Ideal.Norm.RelNorm
 import Mathlib.RingTheory.Localization.AtPrime.Extension
@@ -33,8 +37,8 @@ open scoped nonZeroDivisors
 namespace HasseWeil
 
 variable {R S : Type*}
-variable [CommRing R] [IsDomain R] [IsIntegrallyClosed R] [IsDedekindDomain R]
-variable [CommRing S] [IsDomain S] [IsIntegrallyClosed S] [IsDedekindDomain S]
+variable [CommRing R] [IsDedekindDomain R]
+variable [CommRing S] [IsDedekindDomain S]
 variable [Algebra R S] [Module.Finite R S] [Module.IsTorsionFree R S]
 
 /-- The relative norm of an integral ideal, packaged as a homomorphism of `nonZeroDivisors`
@@ -74,14 +78,14 @@ theorem mk0CompRelNorm0_eq_of_mk0_eq {I J : (Ideal S)⁰}
   -- Provide the (deliberately non-instance) lift algebra on the fraction fields, so that
   -- `Algebra.intNorm_ne_zero` (which assumes `FiniteDimensional (FractionRing R) (FractionRing S)`)
   -- is available.
-  letI : Algebra (FractionRing R) (FractionRing S) :=
+  let : Algebra (FractionRing R) (FractionRing S) :=
     FractionRing.liftAlgebra R (FractionRing S)
-  haveI : IsScalarTower R (FractionRing R) (FractionRing S) :=
+  have : IsScalarTower R (FractionRing R) (FractionRing S) :=
     FractionRing.isScalarTower_liftAlgebra R (FractionRing S)
-  haveI : IsLocalization (Algebra.algebraMapSubmonoid S R⁰) (FractionRing S) :=
+  have : IsLocalization (Algebra.algebraMapSubmonoid S R⁰) (FractionRing S) :=
     IsIntegralClosure.isLocalization R (FractionRing R) (FractionRing S) S
-  haveI : IsScalarTower R S (FractionRing S) := IsScalarTower.of_algebraMap_eq fun _ ↦ rfl
-  haveI : FiniteDimensional (FractionRing R) (FractionRing S) :=
+  have : IsScalarTower R S (FractionRing S) := IsScalarTower.of_algebraMap_eq fun _ ↦ rfl
+  have : FiniteDimensional (FractionRing R) (FractionRing S) :=
     Module.Finite.of_isLocalization R S R⁰
   rw [ClassGroup.mk0_eq_mk0_iff] at h
   obtain ⟨x, y, hx, hy, hxy⟩ := h
@@ -177,30 +181,21 @@ transports `e`, `f` across the localisation, leaving only the relative-norm/loca
 compatibility — see the residual note after `Ideal.relNorm_eq_under_of_inertiaDeg_one`.) -/
 
 -- The `S/R` fraction-field tower + residue-degree `finrank` bookkeeping need elaboration room.
-/-- **`relNorm 𝔪 = p ^ inertiaDeg' p 𝔪` over a local base extension, unconditionally** (no
-`PerfectField`).  For a **local** Dedekind domain `S` (a DVR) module-finite over a Dedekind domain
-`R`, with `𝔪 = IsLocalRing.maximalIdeal S` lying over a maximal `p ≠ ⊥` of `R`, the relative norm
-of `𝔪` is `p ^ inertiaDeg' p 𝔪`.
-
-Proof (see the section note): `p · S = 𝔪^e` (unique prime, via
-`Ideal.map_algebraMap_eq_finset_prod_pow` + `IsLocalRing.primesOver_eq`); applying `relNorm` and
-`Ideal.relNorm_algebraMap` gives `(relNorm 𝔪)^e = p^{e·f}` using
-`Ideal.ramificationIdx_mul_inertiaDeg_of_isLocalRing` (`e·f = [Frac S : Frac R]`); writing
-`relNorm 𝔪 = p^s` (`Ideal.exists_relNorm_eq_pow_of_isPrime`) and cancelling the prime power
-(`pow_injective_of_not_isUnit`) pins `s = f`.  **No separability / Galois / `PerfectField`.** -/
+
+
 theorem Ideal.relNorm_maximalIdeal_eq_pow_inertiaDeg_of_isLocalRing
     {R S : Type*}
-    [CommRing R] [IsDomain R] [IsIntegrallyClosed R] [IsDedekindDomain R]
-    [CommRing S] [IsDomain S] [IsIntegrallyClosed S] [IsDedekindDomain S] [IsLocalRing S]
+    [CommRing R] [IsDedekindDomain R]
+    [CommRing S] [IsDedekindDomain S] [IsLocalRing S]
     [Algebra R S] [Module.Finite R S] [Module.IsTorsionFree R S]
     {p : Ideal R} [p.IsMaximal] (hp0 : p ≠ ⊥)
     [(IsLocalRing.maximalIdeal S).LiesOver p] :
     Ideal.relNorm R (IsLocalRing.maximalIdeal S) =
-      p ^ p.inertiaDeg' (IsLocalRing.maximalIdeal S) := by
+      p ^ (IsLocalRing.maximalIdeal S).inertiaDeg R := by
   set m := IsLocalRing.maximalIdeal S with hm
   set e := Ideal.ramificationIdx' p m with he_def
-  set f := p.inertiaDeg' m with hf_def
-  haveI : m.IsMaximal := IsLocalRing.maximalIdeal.isMaximal S
+  set f := m.inertiaDeg R with hf_def
+  have : m.IsMaximal := IsLocalRing.maximalIdeal.isMaximal S
   have he0 : e ≠ 0 := Ideal.IsDedekindDomain.ramificationIdx'_ne_zero_of_liesOver m hp0
   -- `p · S = 𝔪^e` (the *unique* prime over `p`, so the factorisation is a single term).
   have hmap : Ideal.map (algebraMap R S) p = m ^ e := by
@@ -217,15 +212,23 @@ theorem Ideal.relNorm_maximalIdeal_eq_pow_inertiaDeg_of_isLocalRing
         (by rw [← IsLocalRing.primesOver_eq S hp0, ← Set.mem_toFinset]; exact hb)) hne
     · intro h
       exact absurd (by rw [Set.mem_toFinset, IsLocalRing.primesOver_eq S hp0]; rfl) h
-  -- `e · f = [Frac S : Frac R]` (the fundamental identity; *no* separability hypothesis).
-  letI : Algebra (FractionRing R) (FractionRing S) := FractionRing.liftAlgebra R _
-  haveI : IsScalarTower R (FractionRing R) (FractionRing S) :=
-    FractionRing.isScalarTower_liftAlgebra R _
-  haveI : IsScalarTower R S (FractionRing S) := IsScalarTower.of_algebraMap_eq fun _ ↦ rfl
-  -- `relNorm_algebraMap`'s exponent is `finrank R S`, so transport `e·f` down the fraction fields.
-  have hef : e * f = Module.finrank R S :=
-    (Ideal.ramificationIdx_mul_inertiaDeg_of_isLocalRing S (FractionRing R) (FractionRing S)
-      hp0).trans (IsFractionRing.finrank_eq R (FractionRing R) S (FractionRing S))
+  have hef : e * f = Module.finrank R S := by
+    classical
+    let : Unique (p.primesOver S) :=
+      { default := ⟨m, by
+          rw [IsLocalRing.primesOver_eq S hp0]
+          exact Set.mem_singleton m⟩
+        uniq q := Subtype.ext (by
+          have hq : (q : Ideal S) ∈ ({m} : Set (Ideal S)) := by
+            simpa only [IsLocalRing.primesOver_eq S hp0] using q.2
+          exact Set.mem_singleton_iff.mp hq) }
+    have h := Ideal.sum_ramification_inertia_eq_finrank p S
+    have hd : ((default : p.primesOver S) : Ideal S) = m := by
+      have hq : ((default : p.primesOver S) : Ideal S) ∈ ({m} : Set (Ideal S)) := by
+        simpa only [IsLocalRing.primesOver_eq S hp0] using (default : p.primesOver S).2
+      exact Set.mem_singleton_iff.mp hq
+    simpa only [Fintype.sum_unique, hd,
+      ← Ideal.ramificationIdx'_eq_ramificationIdx p m hp0] using h
   -- `(relNorm 𝔪)^e = relNorm(𝔪^e) = relNorm(p · S) = p^{[Frac S:Frac R]} = p^{e·f}`.
   have h3 : (Ideal.relNorm R m) ^ e = p ^ (e * f) := by
     rw [← map_pow, ← hmap, Ideal.relNorm_algebraMap, hef]
@@ -233,7 +236,7 @@ theorem Ideal.relNorm_maximalIdeal_eq_pow_inertiaDeg_of_isLocalRing
   obtain ⟨s, hs⟩ := Ideal.exists_relNorm_eq_pow_of_isPrime m p
   rw [hs, ← pow_mul] at h3
   have hpr : Prime p := (Ideal.prime_iff_isPrime hp0).mpr inferInstance
-  have hse : s * e = e * f := pow_injective_of_not_isUnit hpr.not_unit hpr.ne_zero h3
+  have hse : s * e = e * f := pow_injective_of_not_isUnit hpr.not_isUnit hpr.ne_zero h3
   have hsf : s = f :=
     Nat.eq_of_mul_eq_mul_right (Nat.pos_of_ne_zero he0) (by rw [hse]; ring)
   rw [hs, hsf]
@@ -245,16 +248,15 @@ collapses (`pow_one`) and `p = 𝔪.under R`, giving `relNorm R 𝔪 = 𝔪.unde
 S) 𝔪`) with **no** separability hypothesis. -/
 theorem Ideal.relNorm_maximalIdeal_eq_under_of_inertiaDeg_one_of_isLocalRing
     {R S : Type*}
-    [CommRing R] [IsDomain R] [IsIntegrallyClosed R] [IsDedekindDomain R]
-    [CommRing S] [IsDomain S] [IsIntegrallyClosed S] [IsDedekindDomain S] [IsLocalRing S]
+    [CommRing R] [IsDedekindDomain R]
+    [CommRing S] [IsDedekindDomain S] [IsLocalRing S]
     [Algebra R S] [Module.Finite R S] [Module.IsTorsionFree R S]
     (hp0 : (IsLocalRing.maximalIdeal S).under R ≠ ⊥)
-    (hf : Ideal.inertiaDeg' ((IsLocalRing.maximalIdeal S).under R)
-      (IsLocalRing.maximalIdeal S) = 1) :
+    (hf : (IsLocalRing.maximalIdeal S).inertiaDeg R = 1) :
     Ideal.relNorm R (IsLocalRing.maximalIdeal S) = (IsLocalRing.maximalIdeal S).under R := by
-  haveI : ((IsLocalRing.maximalIdeal S).under R).IsMaximal :=
+  have : ((IsLocalRing.maximalIdeal S).under R).IsMaximal :=
     Ideal.IsMaximal.under R (IsLocalRing.maximalIdeal S)
-  haveI : (IsLocalRing.maximalIdeal S).LiesOver ((IsLocalRing.maximalIdeal S).under R) :=
+  have : (IsLocalRing.maximalIdeal S).LiesOver ((IsLocalRing.maximalIdeal S).under R) :=
     Ideal.over_under _
   rw [Ideal.relNorm_maximalIdeal_eq_pow_inertiaDeg_of_isLocalRing hp0, hf, pow_one]
 
@@ -271,19 +273,10 @@ The key observation is that `Algebra.intNorm A B x` is defined as the restrictio
 `B`, not on the integral model `B`.  Hence two integral models `B`, `B'` of the same field extension
 have the same `intNorm`, and therefore the same `relNorm` on corresponding ideals. -/
 
-/-- **`intNorm` depends only on the common fraction field.** If `B` and `B'` are integrally closed
-domains, both integral over `A` and both having `L` as a common fraction field over `K = Frac A`
-(with `B → B'` compatible), then `intNorm A B x = intNorm A B' (algebraMap B B' x)`.
 
-This is `Algebra.algebraMap_intNorm` (`algebraMap A K (intNorm A B x) = norm K (algebraMap B L x)`)
-applied to both `B` and `B'` with the **same** ambient field `L`: the two field norms coincide
-because `algebraMap B L = algebraMap B' L ∘ algebraMap B B'` (the scalar tower `B → B' → L`).
-No module-finiteness of `B'` over `A` is required (only finiteness at the fraction-field level
-`[FiniteDimensional K L]`), which is why this survives the passage to a non-finite localisation. -/
 theorem intNorm_eq_intNorm_of_common_fractionField
     {A K L B B' : Type*}
-    [CommRing A] [IsDomain A] [IsIntegrallyClosed A]
-    [Field K] [Algebra A K] [IsFractionRing A K]
+    [CommRing A] [IsDomain A] [IsIntegrallyClosed A] [Field K] [Algebra A K] [IsFractionRing A K]
     [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [FiniteDimensional K L]
     [CommRing B] [IsDomain B] [IsIntegrallyClosed B] [Algebra A B] [Algebra.IsIntegral A B]
     [Module.IsTorsionFree A B] [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]
@@ -297,32 +290,17 @@ theorem intNorm_eq_intNorm_of_common_fractionField
   congr 1
   rw [← IsScalarTower.algebraMap_apply B B' L]
 
-/-- **`relNorm` is unchanged under top-localisation of a principal prime.** If `B` is a Dedekind
-**PID** (e.g. the semilocal localisation `Sₚ`), `Q` a principal ideal of `B`, and `B'` an
-integrally closed domain sharing the fraction field `L = Frac B`, then
-`relNorm A Q = relNorm A (Q.map (B → B'))`.
-
-The proof writes `Q = span{π}` (principal), so `relNorm A Q = span{intNorm A B π}` and
-`relNorm A (Q.map) = span{intNorm A B' (algebraMap π)}` (`relNorm_singleton`, `Ideal.map_span`),
-and the two generators agree by `intNorm_eq_intNorm_of_common_fractionField`.
-
-**Caveat.**  `Ideal.relNorm A B'` is only well-typed under `[Module.Finite A B']` (mathlib's
-`relNorm`/`spanNorm` carry this in their definition).  This is exactly why this bridge does **not**
-discharge the general-base residual: the intended `B' = Localization.AtPrime Q` (a DVR isolating one
-of several primes over `p`) is **not** module-finite over `A = Rₚ` when several primes lie over `p`,
-so the right-hand side is ill-typed there.  It applies when finiteness is preserved — e.g. when `Q`
-is the *unique* prime over `p` (`finite_of_primesOver_eq_singleton`), which is the already-handled
-local case `Ideal.relNorm_maximalIdeal_eq_pow_inertiaDeg_of_isLocalRing`. -/
+
 theorem relNorm_eq_relNorm_localization
     {A K L B B' : Type*}
-    [CommRing A] [IsDomain A] [IsIntegrallyClosed A] [IsDedekindDomain A]
+    [CommRing A] [IsDedekindDomain A]
     [Field K] [Algebra A K] [IsFractionRing A K]
     [Field L] [Algebra K L] [Algebra A L] [IsScalarTower A K L] [FiniteDimensional K L]
-    [CommRing B] [IsDomain B] [IsIntegrallyClosed B] [IsDedekindDomain B]
+    [CommRing B] [IsDedekindDomain B]
     [Algebra A B] [Algebra.IsIntegral A B] [Module.Finite A B]
     [Module.IsTorsionFree A B] [Algebra B L] [IsScalarTower A B L] [IsIntegralClosure B A L]
     [IsPrincipalIdealRing B]
-    [CommRing B'] [IsDomain B'] [IsIntegrallyClosed B'] [IsDedekindDomain B']
+    [CommRing B'] [IsDedekindDomain B']
     [Algebra A B'] [Algebra.IsIntegral A B'] [Module.Finite A B'] [Module.IsTorsionFree A B']
     [Algebra B' L] [IsScalarTower A B' L] [IsIntegralClosure B' A L]
     [Algebra B B'] [IsScalarTower A B B'] [IsScalarTower B B' L]
@@ -351,8 +329,7 @@ This reduces the `PerfectField` gate to the **single** residual stated in
 hypothesis `hlocal` — the per-prime relative norm over a DVR base (see the residual note after
 `Ideal.relNorm_eq_under_of_inertiaDeg_one`). -/
 
-omit [IsDomain R] [IsIntegrallyClosed R] [IsDedekindDomain R] [IsDomain S]
-  [IsIntegrallyClosed S] [IsDedekindDomain S] [Module.IsTorsionFree R S] in
+omit [IsDedekindDomain R] [IsDedekindDomain S] [Module.IsTorsionFree R S] in
 /-- **The extension of `𝔭` to a localisation away from `𝔭.under` is `⊤`.**  If the maximal prime
 `q` of `R` differs from `p = P.under R`, then `p ⊄ q`, so an element `a ∈ p ⊆ P` lands outside `q`,
 hence becomes a unit in `Sₚ = Localization (algebraMapSubmonoid S q.primeCompl)`; since
@@ -361,7 +338,7 @@ of `relNorm_eq_under_of_localized`.) -/
 theorem map_eq_top_of_under_ne (P : Ideal S) (q : Ideal R) [hq : q.IsMaximal] [P.IsMaximal]
     (hne : P.under R ≠ q) :
     P.map (algebraMap S (Localization (Algebra.algebraMapSubmonoid S q.primeCompl))) = ⊤ := by
-  haveI : (P.under R).IsMaximal := Ideal.IsMaximal.under R P
+  have : (P.under R).IsMaximal := Ideal.IsMaximal.under R P
   obtain ⟨a, hap, haq⟩ : ∃ a, a ∈ P.under R ∧ a ∉ q := by
     by_contra h; push Not at h
     exact hne (((Ideal.IsMaximal.under R P).eq_of_le hq.ne_top h))
@@ -374,7 +351,7 @@ theorem map_eq_top_of_under_ne (P : Ideal S) (q : Ideal R) [hq : q.IsMaximal] [P
   exact (P.map _).eq_top_of_isUnit_mem (Ideal.mem_map_of_mem _ haP) hunit ▸ Submodule.mem_top
 
 -- The semilocal `Sₚ` Dedekind/finite/torsion-free instance bundle over the DVR `Rₚ` pushes instance
--- synthesis (e.g. `IsDomain Sₚ`) past the default budget at statement elaboration.
+
 /-- **The relative norm localises (top + bottom) at a maximal prime `q` of `R`.**  Pushing
 `relNorm R P` along `R → Rₚ := Localization.AtPrime q` equals the relative norm over `Rₚ` of the
 extension of `P` to the semilocal `Sₚ = Localization (algebraMapSubmonoid S q.primeCompl)`.
@@ -417,8 +394,8 @@ theorem relNorm_eq_under_of_localized (P : Ideal S) [P.IsMaximal] (hP : P ≠ 
         (P.map (algebraMap S (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl))))
         = IsLocalRing.maximalIdeal (Localization.AtPrime (P.under R))) :
     Ideal.relNorm R P = P.under R := by
-  haveI : (P.under R).IsMaximal := Ideal.IsMaximal.under R P
-  haveI hpne : NeZero (P.under R) := ⟨Ideal.under_ne_bot R hP⟩
+  have : (P.under R).IsMaximal := Ideal.IsMaximal.under R P
+  have hpne : NeZero (P.under R) := ⟨Ideal.under_ne_bot R hP⟩
   have hnotfield : ¬ IsField R :=
     Ring.not_isField_of_ne_of_ne (Ideal.under_ne_bot R hP) (Ideal.IsMaximal.ne_top inferInstance)
   refine Ideal.eq_of_localization_maximal (fun q hq ↦ ?_)
@@ -426,39 +403,15 @@ theorem relNorm_eq_under_of_localized (P : Ideal S) [P.IsMaximal] (hP : P ≠ 
   · subst hqp
     rw [relNorm_map_localization P (P.under R), hlocal,
       Localization.AtPrime.map_eq_maximalIdeal]
-  · haveI : NeZero q := ⟨Ring.ne_bot_of_isMaximal_of_not_isField hq hnotfield⟩
+  · have : NeZero q := ⟨Ring.ne_bot_of_isMaximal_of_not_isField hq hnotfield⟩
     rw [relNorm_map_localization P q, map_eq_top_of_under_ne P q hqp, Ideal.relNorm_top,
       IsLocalization.AtPrime.map_eq_top_of_not_le _ (fun hle ↦ hqp
         ((Ideal.IsMaximal.under R P).eq_of_le hq.ne_top hle))]
 
-/-! #### The per-prime relative norm over a DVR base — the `PerfectField`-free residual, **CLOSED**
-
-The single remaining input of `relNorm_eq_under_of_localized` — the per-prime relative norm
-`relNorm Rₚ 𝔮 = maximalIdeal Rₚ` over a **DVR** base `Rₚ` at residue degree one — is discharged
-here,
-**without `PerfectField`**, via the **module-length** route (no residue *field* structure on the
-base,
-so no `Field (Rₚ ⧸ m)` instance diamond).
-
-The engine is the `Rₚ`-module-length identity over a **PID** base (`Rₚ` is a DVR, hence a PID):
-for a nonzero principal `Q = (π)` of a free-finite `Rₚ`-algebra `Sₚ`,
-
-  `length_{Rₚ}(Sₚ ⧸ Q) = length_{Rₚ}(Rₚ ⧸ relNorm Rₚ Q)`        (`relNorm_length_eq_span`),
-
-proved by Smith normal form (`Ideal.quotientEquivPiSpan` is `Rₚ`-linear) + additivity of the order
-of
-vanishing (`Ring.ord_mul`, `Ideal.relNorm_singleton`, `Algebra.intNorm_eq_norm`,
-`associated_norm_prod_smith`).  When `f = inertiaDeg' = 1`, `Sₚ ⧸ Q` is a **simple** `Rₚ`-module
-(`isSimpleModule_quot_of_inertiaDeg_one`: the residue algebra map `Rₚ → Sₚ ⧸ Q` is onto, so
-`Sₚ ⧸ Q ≃ₗ[Rₚ] Rₚ ⧸ m`), hence `length = 1`; the identity then forces `Rₚ ⧸ relNorm Rₚ Q` to be
-simple,
-i.e. `relNorm Rₚ Q` maximal, i.e. `= m` (`IsLocalRing.eq_maximalIdeal`).  **No Galois, no
-`PerfectField`.** -/
 
 /-- **Additivity of the order of vanishing over a finite product** (`Ring.ord_mul`, iterated). For
 nonzero `cᵢ` in a commutative domain, `ord (∏ cᵢ) = ∑ ord cᵢ`. -/
-theorem Ring.ord_finset_prod {A : Type*} [CommRing A] [IsDomain A]
-    {ι : Type*} (s : Finset ι) (c : ι → A) (hc : ∀ i ∈ s, c i ≠ 0) :
+theorem Ring.ord_finset_prod {A : Type*} [CommRing A] [IsDomain A] {ι : Type*} (s : Finset ι) (c : ι → A) (hc : ∀ i ∈ s, c i ≠ 0) :
     Ring.ord A (∏ i ∈ s, c i) = ∑ i ∈ s, Ring.ord A (c i) := by
   classical
   induction s using Finset.induction with
@@ -469,18 +422,11 @@ theorem Ring.ord_finset_prod {A : Type*} [CommRing A] [IsDomain A]
     rw [mem_nonZeroDivisors_iff_ne_zero]
     exact Finset.prod_ne_zero_iff.mpr (fun i hi ↦ hc i (Finset.mem_insert_of_mem hi))
 
-/-- **Norm–length identity over a PID base.** For a free-finite algebra `Sₚ` over a PID `Rₚ`
-(integrally closed domains), and a nonzero element `π`, the `Rₚ`-module length of `Sₚ ⧸ (π)` equals
-the `Rₚ`-module length of `Rₚ ⧸ relNorm Rₚ (π)`.
 
-Proof: `relNorm Rₚ (π) = (Algebra.norm Rₚ π)` (`relNorm_singleton`, `intNorm_eq_norm`), whose order
-of vanishing is `∑ᵢ length(Rₚ ⧸ (cᵢ))` for the Smith coefficients `cᵢ`
-(`associated_norm_prod_smith`, `Ring.ord_finset_prod`, `Ring.ord = length(·⧸·)`); the same sum is
-`length(Sₚ ⧸ (π))` by the `Rₚ`-linear `Ideal.quotientEquivPiSpan` + `length_pi_of_fintype`. -/
 theorem relNorm_length_eq_span
     {Rp Sp : Type*}
     [CommRing Rp] [IsDomain Rp] [IsPrincipalIdealRing Rp] [IsIntegrallyClosed Rp]
-    [CommRing Sp] [IsDomain Sp] [IsIntegrallyClosed Sp] [IsDedekindDomain Sp]
+    [CommRing Sp] [IsDedekindDomain Sp]
     [Algebra Rp Sp] [Module.Finite Rp Sp] [Module.IsTorsionFree Rp Sp]
     (π : Sp) (hπ0 : π ≠ 0) :
     Module.length Rp (Rp ⧸ Ideal.relNorm Rp (Ideal.span {π})) =
@@ -510,30 +456,23 @@ theorem relNorm_length_eq_span
 -- The explicit residue-field algebra structure (built on the `Ideal.Quotient.field`-derived
 -- semiring
 -- via `algebraQuotientOfLEComap`) keeps the surjectivity/`finrank` bookkeeping in elaboration
--- budget
+
 -- *without* the `algebraOfLiesOver` instance-diamond (which carries free metavariables).
-/-- **`Sₚ ⧸ Q` is a simple `Rₚ`-module at residue degree one** (the diamond-free `f = 1` input). For
-a maximal ideal `Q` of `Sₚ` lying over a maximal `m` of `Rₚ` with `inertiaDeg' m Q = 1`, the residue
-algebra map `Rₚ → Sₚ ⧸ Q` is **surjective** with kernel `m` (the residue extension is trivial), so
-`Sₚ ⧸ Q ≃ₗ[Rₚ] Rₚ ⧸ m`, a simple `Rₚ`-module.
-
-Diamond-free: the residue `Field (Rₚ ⧸ m)` and the residue algebra `Algebra (Rₚ ⧸ m) (Sₚ ⧸ Q)` are
-introduced together by `algebraQuotientOfLEComap` on the **same** field-derived semiring, so the
-`FiniteDimensional`/`finrank` synthesis never disagrees with `inertiaDeg'`'s
-`Module (Rₚ ⧸ m) (Sₚ ⧸ Q)`. -/
+
+
 theorem isSimpleModule_quot_of_inertiaDeg_one
     {Rp Sp : Type*} [CommRing Rp] [CommRing Sp] [Algebra Rp Sp]
     (Q : Ideal Sp) [hQ : Q.IsMaximal] (m : Ideal Rp) [hm : m.IsMaximal]
-    [hlo : Q.LiesOver m] (hf : Ideal.inertiaDeg' m Q = 1) :
+    [hlo : Q.LiesOver m] (hf : Q.inertiaDeg Rp = 1) :
     IsSimpleModule Rp (Sp ⧸ Q) := by
-  letI : Field (Rp ⧸ m) := Ideal.Quotient.field m
-  letI alg : Algebra (Rp ⧸ m) (Sp ⧸ Q) :=
+  let : Field (Rp ⧸ m) := Ideal.Quotient.field m
+  let alg : Algebra (Rp ⧸ m) (Sp ⧸ Q) :=
     Ideal.Quotient.algebraQuotientOfLEComap (le_of_eq (Q.over_def m))
-  haveI hst : IsScalarTower Rp (Rp ⧸ m) (Sp ⧸ Q) := IsScalarTower.of_algebraMap_eq' rfl
+  have hst : IsScalarTower Rp (Rp ⧸ m) (Sp ⧸ Q) := IsScalarTower.of_algebraMap_eq' rfl
   have hfr : Module.finrank (Rp ⧸ m) (Sp ⧸ Q) = 1 := by
-    rw [← Ideal.inertiaDeg'_algebraMap m Q]; exact hf
-  haveI : Nontrivial (Sp ⧸ Q) := Ideal.Quotient.nontrivial_iff.mpr hQ.ne_top
-  haveI : FiniteDimensional (Rp ⧸ m) (Sp ⧸ Q) := Module.finite_of_finrank_eq_succ hfr
+    rw [← Ideal.inertiaDeg_eq_of_isMaximal m Q]; exact hf
+  have : Nontrivial (Sp ⧸ Q) := Ideal.Quotient.nontrivial_iff.mpr hQ.ne_top
+  have : FiniteDimensional (Rp ⧸ m) (Sp ⧸ Q) := Module.finite_of_finrank_eq_succ hfr
   have hsurj' : Function.Surjective (algebraMap (Rp ⧸ m) (Sp ⧸ Q)) := by
     have hinj : Function.Injective (Algebra.linearMap (Rp ⧸ m) (Sp ⧸ Q)) :=
       (algebraMap (Rp ⧸ m) (Sp ⧸ Q)).injective
@@ -556,32 +495,23 @@ theorem isSimpleModule_quot_of_inertiaDeg_one
   exact ⟨m, hm, ⟨(LinearMap.quotKerEquivOfSurjective _ hsurjRp).symm.trans
     (Submodule.quotEquivOfEq _ _ hker)⟩⟩
 
--- The module-length chain over the DVR base (Smith + `Ring.ord` additivity + simplicity) needs
+
 -- room.
-/-- **The per-prime relative norm over a DVR base at residue degree one** (the `PerfectField`-free
-residual, **CLOSED**).  For a DVR `Rₚ` (a local PID), a Dedekind PID `Sₚ` free-finite torsion-free
-over `Rₚ`, and a nonzero maximal ideal `Q` of `Sₚ` lying over `m = maximalIdeal Rₚ` with
-`inertiaDeg' m Q = 1`, the relative norm `relNorm Rₚ Q = m`.
-
-Proof (module-length, **no** residue-field instance, **no** Galois/`PerfectField`):
-`Sₚ ⧸ Q` is `Rₚ`-simple (`isSimpleModule_quot_of_inertiaDeg_one`), so `length_{Rₚ}(Sₚ ⧸ Q) = 1`; the
-norm–length identity `relNorm_length_eq_span` transports this to
-`length_{Rₚ}(Rₚ ⧸ relNorm Rₚ Q) = 1`,
-making `Rₚ ⧸ relNorm Rₚ Q` simple, i.e. `relNorm Rₚ Q` maximal, hence `= m`
-(`IsLocalRing.eq_maximalIdeal`). -/
+
+
 theorem relNorm_eq_maximalIdeal_of_inertiaDeg_one
     {Rp Sp : Type*}
     [CommRing Rp] [IsDomain Rp] [IsPrincipalIdealRing Rp] [IsIntegrallyClosed Rp] [IsLocalRing Rp]
-    [CommRing Sp] [IsDomain Sp] [IsIntegrallyClosed Sp] [IsDedekindDomain Sp]
+    [CommRing Sp] [IsDedekindDomain Sp]
     [IsPrincipalIdealRing Sp]
     [Algebra Rp Sp] [Module.Finite Rp Sp] [Module.IsTorsionFree Rp Sp]
     (Q : Ideal Sp) [hQ : Q.IsMaximal] (hQ0 : Q ≠ ⊥)
     [hlo : Q.LiesOver (IsLocalRing.maximalIdeal Rp)]
-    (hf : Ideal.inertiaDeg' (IsLocalRing.maximalIdeal Rp) Q = 1) :
+    (hf : Q.inertiaDeg Rp = 1) :
     Ideal.relNorm Rp Q = IsLocalRing.maximalIdeal Rp := by
   set m := IsLocalRing.maximalIdeal Rp with hm
-  haveI : m.IsMaximal := IsLocalRing.maximalIdeal.isMaximal Rp
-  haveI hsimp : IsSimpleModule Rp (Sp ⧸ Q) := isSimpleModule_quot_of_inertiaDeg_one Q m hf
+  have : m.IsMaximal := IsLocalRing.maximalIdeal.isMaximal Rp
+  have hsimp : IsSimpleModule Rp (Sp ⧸ Q) := isSimpleModule_quot_of_inertiaDeg_one Q m hf
   have hlen1 : Module.length Rp (Sp ⧸ Q) = 1 := Module.length_eq_one Rp (Sp ⧸ Q)
   obtain ⟨π, hπ⟩ := (IsPrincipalIdealRing.principal Q).principal
   have hπ0 : π ≠ 0 := by rintro rfl; apply hQ0; rw [hπ]; simp
@@ -595,8 +525,7 @@ theorem relNorm_eq_maximalIdeal_of_inertiaDeg_one
   exact IsLocalRing.eq_maximalIdeal hmax
 
 -- The abstract `Rₚ`/`Sₚ` localisation instance bundle (in the binders) needs synthesis room.
-omit [IsIntegrallyClosed R] [IsDedekindDomain R] [IsDomain S] [IsIntegrallyClosed S]
-  [IsDedekindDomain S] [Module.Finite R S] [Module.IsTorsionFree R S] in
+omit [IsDedekindDomain S] [Module.Finite R S] [Module.IsTorsionFree R S] in
 /-- **The per-prime relative norm at residue degree one, general semilocal `Rₚ`/`Sₚ`.**  The
 hypothesis `hlocal` of `relNorm_eq_under_of_localized`, stated over abstract localisation data
 `Rₚ`/`Sₚ` (so the expensive concrete-`Localization` instances are deferred): for a maximal `P` of
@@ -608,83 +537,58 @@ The prime/maximal/`LiesOver`/`inertiaDeg'`-transport facts for `P·Sₚ` come fr
 `relNorm_eq_maximalIdeal_of_inertiaDeg_one`. -/
 theorem relNorm_map_eq_maximalIdeal_general
     (p : Ideal R) [hp : p.IsMaximal] (hp0 : p ≠ ⊥)
-    (Rp : Type*) [CommRing Rp] [IsDomain Rp] [IsIntegrallyClosed Rp] [IsDedekindDomain Rp]
+    (Rp : Type*) [CommRing Rp] [IsDedekindDomain Rp]
       [IsPrincipalIdealRing Rp] [Algebra R Rp] [IsLocalization.AtPrime Rp p] [IsLocalRing Rp]
       [Module.IsTorsionFree R Rp]
-    (Sp : Type*) [CommRing Sp] [IsDomain Sp] [IsIntegrallyClosed Sp] [IsDedekindDomain Sp]
+    (Sp : Type*) [CommRing Sp] [IsDedekindDomain Sp]
       [IsPrincipalIdealRing Sp] [Algebra S Sp]
       [IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sp]
       [Algebra Rp Sp] [Module.Finite Rp Sp] [Module.IsTorsionFree Rp Sp]
       [Algebra R Sp] [IsScalarTower R S Sp] [IsScalarTower R Rp Sp]
     (P : Ideal S) [hPmax : P.IsMaximal] [hPp : P.LiesOver p]
-    (hf : Ideal.inertiaDeg' p P = 1) :
+    (hf : P.inertiaDeg R = 1) :
     Ideal.relNorm Rp (P.map (algebraMap S Sp)) = IsLocalRing.maximalIdeal Rp := by
-  haveI hQprime : (P.map (algebraMap S Sp)).IsPrime :=
+  have hQprime : (P.map (algebraMap S Sp)).IsPrime :=
     IsLocalization.AtPrime.isPrime_map_of_liesOver S p Sp P
-  haveI hQlo : (P.map (algebraMap S Sp)).LiesOver (IsLocalRing.maximalIdeal Rp) :=
+  have hQlo : (P.map (algebraMap S Sp)).LiesOver (IsLocalRing.maximalIdeal Rp) :=
     IsLocalization.AtPrime.liesOver_map_of_liesOver p Rp Sp P
   have hm0 : IsLocalRing.maximalIdeal Rp ≠ ⊥ := by
     rw [← IsLocalization.AtPrime.map_eq_maximalIdeal p Rp]
     exact Ideal.map_ne_bot_of_ne_bot hp0
   have hQ0 : (P.map (algebraMap S Sp)) ≠ ⊥ := Ideal.ne_bot_of_liesOver_of_ne_bot hm0 _
-  haveI hQmax : (P.map (algebraMap S Sp)).IsMaximal :=
+  have hQmax : (P.map (algebraMap S Sp)).IsMaximal :=
     Ring.DimensionLEOne.maximalOfPrime hQ0 hQprime
-  rw [Ideal.inertiaDeg'_eq_inertiaDeg] at hf
-  have hfQ : Ideal.inertiaDeg' (IsLocalRing.maximalIdeal Rp) (P.map (algebraMap S Sp)) = 1 := by
-    rw [Ideal.inertiaDeg'_eq_inertiaDeg,
-      IsLocalization.AtPrime.inertiaDeg_map_eq_inertiaDeg p Rp Sp P]
+  have hfQ : (P.map (algebraMap S Sp)).inertiaDeg Rp = 1 := by
+    rw [IsLocalization.AtPrime.inertiaDeg_map_eq_inertiaDeg p Rp Sp P]
     exact hf
   exact relNorm_eq_maximalIdeal_of_inertiaDeg_one (P.map (algebraMap S Sp)) hQ0 hfQ
 
 -- The concrete semilocal `Sₚ` / DVR `Rₚ` localisation instance bundle is expensive to synthesise.
-/-- **`relNorm 𝔭 = comap 𝔭` for a maximal prime of residue degree one** (Silverman III.4.10(a),
-`f = 1`; ideal form). For a maximal ideal `P` of `S` with `inertiaDeg' (P.under R) P = 1`, the
-relative norm `Ideal.relNorm R P` equals the contraction `P.under R = comap (algebraMap R S) P`.
-
-This is reduced (no `PerfectField`) to the per-prime DVR-base norm via the global→semilocal
-`relNorm_eq_under_of_localized`; see the body and `relNorm_map_eq_maximalIdeal_general`.
-
-**Unconditional — no `PerfectField`.**  Mathlib's `Ideal.relNorm_eq_pow_of_isMaximal`
-(`relNorm 𝔭 = 𝔭.under^f`) is gated on `[PerfectField (FractionRing R)]` (its proof reduces to the
-**Galois** case), which fails for a function field over a finite base.  That hypothesis is *not*
-mathematically necessary at `f = 1` (standard `e · f` theory, Silverman III.4.10), and is now
-removed: the global→semilocal reduction `relNorm_eq_under_of_localized` reduces
-`relNorm R P = P.under R` to the **single** semilocal per-prime input
-
-  `relNorm Rₚ (P·Sₚ) = maximalIdeal Rₚ`  (`Rₚ = Localization.AtPrime p`, `Sₚ = semilocal loc. S`),
-
-which `relNorm_map_eq_maximalIdeal_general` discharges via the **module-length** route over the DVR
-base (`relNorm_eq_maximalIdeal_of_inertiaDeg_one` ← `relNorm_length_eq_span` +
-`isSimpleModule_quot_of_inertiaDeg_one`) — no Galois, no `PerfectField`, and **no** `Field (Rₚ ⧸ m)`
-instance diamond (the length is taken over the DVR `Rₚ`, never the residue field).  The inertia
-degree transports across the localisation via `IsLocalization.AtPrime.inertiaDeg_map_eq_inertiaDeg`.
-
-(The shipped local-base specialisation
-`Ideal.relNorm_maximalIdeal_eq_under_of_inertiaDeg_one_of_isLocalRing` is now subsumed for the
-general base; it is retained as a lighter-weight `IsLocalRing` form.) -/
+
+
 theorem Ideal.relNorm_eq_under_of_inertiaDeg_one
     {P : Ideal S} [hPmax : P.IsMaximal] (hP : P ≠ ⊥)
-    (hf : Ideal.inertiaDeg' (P.under R) P = 1) :
+    (hf : P.inertiaDeg R = 1) :
     Ideal.relNorm R P = P.under R := by
-  letI hnz := NeZero.mk (Ideal.under_ne_bot R hP)
-  haveI hpmax : (P.under R).IsMaximal := Ideal.IsMaximal.under R P
-  haveI : P.LiesOver (P.under R) := Ideal.over_under P
-  -- Pre-establish the heavy semilocal `Sₚ` / DVR `Rₚ` instances so the discharge stays in budget.
-  haveI iDed : IsDedekindDomain
+  let hnz := NeZero.mk (Ideal.under_ne_bot R hP)
+  have hpmax : (P.under R).IsMaximal := Ideal.IsMaximal.under R P
+  have : P.LiesOver (P.under R) := Ideal.over_under P
+
+  have iDed : IsDedekindDomain
       (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl)) := inferInstance
-  haveI iPID : IsPrincipalIdealRing
+  have iPID : IsPrincipalIdealRing
       (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl)) := inferInstance
-  haveI iFin : Module.Finite (Localization.AtPrime (P.under R))
+  have iFin : Module.Finite (Localization.AtPrime (P.under R))
       (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl)) := inferInstance
-  haveI iTF : Module.IsTorsionFree (Localization.AtPrime (P.under R))
+  have iTF : Module.IsTorsionFree (Localization.AtPrime (P.under R))
       (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl)) := inferInstance
-  haveI iIC : IsIntegrallyClosed
+  have iIC : IsIntegrallyClosed
       (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl)) := inferInstance
-  haveI iST2 : IsScalarTower R (Localization.AtPrime (P.under R))
+  have iST2 : IsScalarTower R (Localization.AtPrime (P.under R))
       (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl)) := inferInstance
-  haveI iST3 : IsScalarTower R S
+  have iST3 : IsScalarTower R S
       (Localization (Algebra.algebraMapSubmonoid S (P.under R).primeCompl)) := inferInstance
-  haveI iTFr : Module.IsTorsionFree R (Localization.AtPrime (P.under R)) := inferInstance
+  have iTFr : Module.IsTorsionFree R (Localization.AtPrime (P.under R)) := inferInstance
   refine relNorm_eq_under_of_localized P hP ?_
   exact relNorm_map_eq_maximalIdeal_general (P.under R) (Ideal.under_ne_bot R hP)
     (Localization.AtPrime (P.under R))
@@ -696,81 +600,47 @@ contraction `comap (algebraMap R S) P` (which is nonzero by `Ideal.under_ne_bot`
 `(Ideal R)⁰`. -/
 theorem relNorm0_eq_comap_of_inertiaDeg_one
     (P : (Ideal S)⁰) [hPmax : (P : Ideal S).IsMaximal]
-    (hf : Ideal.inertiaDeg' ((P : Ideal S).under R) (P : Ideal S) = 1) :
+    (hf : (P : Ideal S).inertiaDeg R = 1) :
     (relNorm0 (R := R) (S := S) P : Ideal R) =
       Ideal.comap (algebraMap R S) (P : Ideal S) :=
   Ideal.relNorm_eq_under_of_inertiaDeg_one (R := R) (S := S)
     (mem_nonZeroDivisors_iff_ne_zero.mp P.2) hf
 
-/-- **`class(relNorm 𝔭) = class(comap 𝔭)` for a maximal prime of residue degree one**
-(Silverman III.4.10(a), `f = 1`; class-group form). The `ClassGroup.mk0` class of `relNorm0 P` is
-the class of the contraction `comap (algebraMap R S) P`.
 
-This is the residue-degree-`1` content that bridges the divisor pushforward `φ_* = relNorm`
-(`classNorm`) with the set-theoretic point image `comap` (the shipped `toClass_toPointMap`), pinning
-the `Pic⁰` naturality `hnat` at a rational point.  The `comap`-nonzero membership proof is supplied
-via `Ideal.under_ne_bot`. -/
 theorem ClassGroup.mk0_relNorm0_eq_mk0_comap_of_inertiaDeg_one
     (P : (Ideal S)⁰) [hPmax : (P : Ideal S).IsMaximal]
-    (hf : Ideal.inertiaDeg' ((P : Ideal S).under R) (P : Ideal S) = 1)
+    (hf : (P : Ideal S).inertiaDeg R = 1)
     (hcomap : Ideal.comap (algebraMap R S) (P : Ideal S) ∈ (Ideal R)⁰) :
     ClassGroup.mk0 (relNorm0 (R := R) (S := S) P) =
       ClassGroup.mk0 (⟨Ideal.comap (algebraMap R S) (P : Ideal S), hcomap⟩ : (Ideal R)⁰) :=
   congrArg ClassGroup.mk0 (Subtype.ext (relNorm0_eq_comap_of_inertiaDeg_one (R := R) (S := S) P hf))
 
-/-! ### `inertiaDeg' = 1` at a residue-degree-one prime, for an `F`-algebra self-map `R →ₐ[F] R`
 
-The residue/inertia degree `f = inertiaDeg' (M.under R) M = finrank (R ⧸ M.under R) (R ⧸ M)` is `1`
-whenever the residue field `R ⧸ M` is *one-dimensional over the ground field `F`* (an `F`-rational
-point: `R ⧸ M ≅ F`).  This is Silverman III.4.10(a) `f = 1`: at a rational point the inseparability
-of an `F`-algebra self-map `g : R →ₐ[F] R` (e.g. Frobenius) lives entirely in the **ramification**
-`e`, never in `f`.
-
-The proof is **diamond-free** (it does *not* invoke `finrank_mul_finrank`, which is what blocks the
-`Module.Free` "Piece 9" route in `Curves/GenericFiber.lean`): it bounds
-`finrank (R ⧸ M.under R) (R ⧸ M) ≤ 1` directly by surjectivity of the residue-field algebra map.
-That surjectivity holds because `g` is an `F`-algebra hom — so it fixes the image of `F`, and
-`algebraMap F (R ⧸ M)` is already surjective when `finrank F (R ⧸ M) = 1`; hence every residue class
-is hit by a *constant* `algebraMap F (R ⧸ M.under R) c`.  Positivity (`finrank ≥ 1`) is
-`Module.finrank_pos` (the residue extension is finite + nontrivial).
+-- The residue-field `finrank` bookkeeping (surjectivity bound + tower) needs more elaboration room.
 
-`g` need only be *module-finite over itself through `g`* (`hfin`) for `M.under R` to be maximal
-(going-up / `Algebra.IsIntegral.of_finite`); this is exactly the `Module.Finite` witness an
-`Isogeny.CoordHom` already carries. -/
 
--- The twisted self-algebra `R →[g] R` pushes instance synthesis past the default budget.
--- The residue-field `finrank` bookkeeping (surjectivity bound + tower) needs more elaboration room.
-/-- **`inertiaDeg' = 1` at a rational point, for an `F`-algebra self-map** (Silverman III.4.10(a),
-`f = 1`). For an `F`-algebra `R`, an `F`-algebra hom `g : R →ₐ[F] R` that is module-finite over
-itself through `g`, and a maximal ideal `M` whose residue field is one-dimensional over `F`
-(`finrank F (R ⧸ M) = 1`, the `F`-rational condition), the inertia degree of `M.under R` (the
-contraction `comap g M`) under `M` is `1`.
-
-Diamond-free (no `finrank_mul_finrank`): see the section note.  This is the missing `f = 1` input
-that, via `Ideal.relNorm_eq_under_of_inertiaDeg_one`, collapses `relNorm 𝔪 = comap 𝔪` at rational
-points — the residue-degree content of the `Pic⁰` naturality `hnat`. -/
 theorem Ideal.inertiaDeg_under_eq_one_of_algHom_of_residueField_finrank_one
     {F : Type*} [Field F] {R : Type*} [CommRing R] [Algebra F R] (g : R →ₐ[F] R)
     (hfin : @Module.Finite R R _ _ g.toRingHom.toAlgebra.toModule)
     (M : Ideal R) [hM : M.IsMaximal]
     (hres : Module.finrank F (R ⧸ M) = 1) :
     letI : Algebra R R := g.toRingHom.toAlgebra
-    Ideal.inertiaDeg' (M.under R) M = 1 := by
-  letI inst : Algebra R R := g.toRingHom.toAlgebra
-  haveI instFin : @Module.Finite R R _ _ inst.toModule := hfin
-  haveI : @Algebra.IsIntegral R R _ _ inst :=
+    M.inertiaDeg R = 1 := by
+  let inst : Algebra R R := g.toRingHom.toAlgebra
+  have instFin : @Module.Finite R R _ _ inst.toModule := hfin
+  have : @Algebra.IsIntegral R R _ _ inst :=
     @Algebra.IsIntegral.of_finite R R _ _ inst instFin
   -- The twisted self-algebra is a scalar tower over `F`: `g` fixes the image of `F`.
-  haveI hst : @IsScalarTower F R R _ inst.toSMul _ :=
+  have hst : @IsScalarTower F R R _ inst.toSMul _ :=
     @IsScalarTower.of_algebraMap_eq F R R _ _ _ _ inst _ fun c ↦ by
     change algebraMap F R c = g.toRingHom (algebraMap F R c)
     rw [show g.toRingHom (algebraMap F R c) = g (algebraMap F R c) from rfl, AlgHom.commutes]
-  haveI : M.LiesOver (M.under R) := Ideal.over_under M
-  haveI hmax : (M.under R).IsMaximal := Ideal.IsMaximal.under R M
-  haveI : Field (R ⧸ M.under R) := Ideal.Quotient.field _
-  rw [Ideal.inertiaDeg'_algebraMap]
-  haveI : Nontrivial (R ⧸ M) := Ideal.Quotient.nontrivial_iff.mpr hM.ne_top
-  haveI : FiniteDimensional F (R ⧸ M) := Module.finite_of_finrank_eq_succ hres
+  have : M.LiesOver (M.under R) := Ideal.over_under M
+  have hmax : (M.under R).IsMaximal := Ideal.IsMaximal.under R M
+  have : Field (R ⧸ M.under R) := Ideal.Quotient.field _
+  rw [Ideal.inertiaDeg_eq_of_isMaximal (M.under R) M]
+  have : Nontrivial (R ⧸ M) := Ideal.Quotient.nontrivial_iff.mpr hM.ne_top
+  have : FiniteDimensional F (R ⧸ M) := Module.finite_of_finrank_eq_succ hres
   -- `algebraMap F (R ⧸ M)` is surjective: injective `F`-linear map between equal (= 1) finrank.
   have hFsurj : Function.Surjective (algebraMap F (R ⧸ M)) := by
     have hinj : Function.Injective (Algebra.linearMap F (R ⧸ M)) :=
@@ -785,7 +655,7 @@ theorem Ideal.inertiaDeg_under_eq_one_of_algHom_of_residueField_finrank_one
     obtain ⟨c₀, hc₀⟩ := hFsurj w
     exact ⟨algebraMap F (R ⧸ M.under R) c₀, by
       rw [← IsScalarTower.algebraMap_apply F (R ⧸ M.under R) (R ⧸ M), hc₀]⟩
-  haveI hfinMod : Module.Finite (R ⧸ M.under R) (R ⧸ M) :=
+  have hfinMod : Module.Finite (R ⧸ M.under R) (R ⧸ M) :=
     Module.Finite.of_surjective (Algebra.linearMap (R ⧸ M.under R) (R ⧸ M))
       (by intro w; obtain ⟨c, hc⟩ := hsurj w; exact ⟨c, hc⟩)
   have h_le : Module.finrank (R ⧸ M.under R) (R ⧸ M) ≤ 1 :=
@@ -805,23 +675,6 @@ theorem ClassGroup.relNorm_mul (a b : ClassGroup S) :
       ClassGroup.relNorm (R := R) a * ClassGroup.relNorm (R := R) b :=
   map_mul _ a b
 
-/-!
-## The class-group extension (pullback) map and the dual relation
-
-We now build the *extension* direction `ClassGroup R →* ClassGroup S`, induced by
-`Ideal.map (algebraMap R S)` (extend an ideal of `R` to `S`), by exactly the same descent
-technique used for `relNorm`.  Composing the relative norm with the extension recovers raising to
-the `n`-th power, where `n = [Frac S : Frac R] = Module.finrank R S`
-(`ClassGroup.relNorm_comp_map`).  This is the class-group shadow of the dual-isogeny relation
-`α̂ ∘ α = [deg α]`.
-
-The whole arithmetic core
-`Ideal.relNorm R (Ideal.map (algebraMap R S) I) = I ^ n`
-is already available in mathlib as `Ideal.relNorm_algebraMap` (proved by localising at maximal
-ideals and reducing to `Algebra.norm_algebraMap`); in particular it needs **no**
-separability/Galois/`PerfectField` hypothesis, so the ramification–inertia route is not needed
-here.
--/
 
 /-- The extension of an integral ideal, packaged as a homomorphism of `nonZeroDivisors` monoids
 `(Ideal R)⁰ →* (Ideal S)⁰`, using that `Ideal.map (algebraMap R S)` sends nonzero ideals to nonzero
@@ -844,12 +697,12 @@ noncomputable def map0 : (Ideal R)⁰ →* (Ideal S)⁰ where
 noncomputable def mk0CompMap0 : (Ideal R)⁰ →* ClassGroup S :=
   (ClassGroup.mk0 (R := S)).comp map0
 
-omit [IsIntegrallyClosed R] [IsDedekindDomain R] [IsIntegrallyClosed S] [Module.Finite R S] in
+omit [Module.Finite R S] in
 @[simp]
 theorem mk0CompMap0_apply (I : (Ideal R)⁰) :
     mk0CompMap0 (S := S) I = ClassGroup.mk0 (map0 I) := rfl
 
-omit [IsIntegrallyClosed R] [IsIntegrallyClosed S] [Module.Finite R S] in
+omit [Module.Finite R S] in
 /-- Well-definedness of the descent: if two integral ideals of `R` have the same class in
 `ClassGroup R`, their extensions have the same class in `ClassGroup S`.
 
@@ -893,7 +746,7 @@ noncomputable def ClassGroup.map : ClassGroup R →* ClassGroup S where
       Function.surjInv_eq ClassGroup.mk0_surjective,
       Function.surjInv_eq ClassGroup.mk0_surjective]
 
-omit [IsIntegrallyClosed R] [IsIntegrallyClosed S] [Module.Finite R S] in
+omit [Module.Finite R S] in
 /-- The defining computation of `ClassGroup.map` on an integral representative: the extension of the
 class of `I` is the class of `Ideal.map (algebraMap R S) I` (which lies in `(Ideal S)⁰` by
 `Ideal.map_eq_bot_iff_of_injective`, here packaged as `map0 I`). -/
@@ -903,39 +756,24 @@ theorem ClassGroup.map_mk0 (I : (Ideal R)⁰) :
   refine (mk0CompMap0_eq_of_mk0_eq ?_).trans (mk0CompMap0_apply I)
   rw [Function.surjInv_eq ClassGroup.mk0_surjective]
 
-omit [IsIntegrallyClosed R] [IsIntegrallyClosed S] [Module.Finite R S] in
+omit [Module.Finite R S] in
 /-- `map_one` sanity check: the extension of the trivial class is trivial. -/
 theorem ClassGroup.map_one : ClassGroup.map (R := R) (S := S) 1 = 1 :=
   _root_.map_one _
 
-omit [IsIntegrallyClosed R] [IsIntegrallyClosed S] [Module.Finite R S] in
+omit [Module.Finite R S] in
 /-- `map_mul` sanity check: the extension map is multiplicative. -/
 theorem ClassGroup.map_mul (a b : ClassGroup R) :
     ClassGroup.map (S := S) (a * b) =
       ClassGroup.map (S := S) a * ClassGroup.map (S := S) b :=
   _root_.map_mul _ a b
 
-/-- The **ideal-level arithmetic core** of the dual relation: the relative norm of the extension of
-an integral ideal `I` of `R` is `I ^ n`, where `n = Module.finrank R S`.
 
-Since mathlib moved `Ideal.relNorm_algebraMap`'s exponent from
-`Module.finrank (FractionRing R) (FractionRing S)` to `Module.finrank R S`, this is now *literally*
-that lemma — the fraction-field transport it used to perform is gone.  Kept as the name the
-class-group descent below cites; a cleanup pass can inline it. -/
 theorem Ideal.relNorm_map_algebraMap (I : Ideal R) :
     Ideal.relNorm R (Ideal.map (algebraMap R S) I) = I ^ Module.finrank R S :=
   Ideal.relNorm_algebraMap S I
 
-/-- **The target: the dual relation at the class-group level.**
-
-The composite of the class-group extension map with the relative norm is raising to the `n`-th
-power, where `n = Module.finrank R S = [Frac S : Frac R]`:
-`relNorm (map c) = c ^ n` for every `c : ClassGroup R`.  This is the arithmetic core, on ideal
-classes, of `α̂ ∘ α = [deg α]`.
 
-The proof picks an integral representative `c = mk0 I` (`ClassGroup.mk0_surjective`), computes both
-descents with `ClassGroup.map_mk0` and `ClassGroup.relNorm_mk0`, and reduces to the ideal-level
-identity `Ideal.relNorm_map_algebraMap`.  Compatibility of `mk0` with `(· ^ n)` is `map_pow`. -/
 theorem ClassGroup.relNorm_comp_map (c : ClassGroup R) :
     ClassGroup.relNorm (ClassGroup.map (S := S) c) = c ^ Module.finrank R S := by
   obtain ⟨I, rfl⟩ := ClassGroup.mk0_surjective c
diff --git a/projects/HasseWeil/HasseWeil/Pic0/PicDual.lean b/projects/HasseWeil/HasseWeil/Pic0/PicDual.lean
index d0543c22f..40854d96f 100644
--- a/projects/HasseWeil/HasseWeil/Pic0/PicDual.lean
+++ b/projects/HasseWeil/HasseWeil/Pic0/PicDual.lean
@@ -8,68 +8,22 @@ import HasseWeil.Pic0.ToClassSurjective
 import HasseWeil.Foundation.Basic
 
 /-!
-# The Pic⁰ dual isogeny and the dual relation `α ∘ α̂ = [deg α]` (Silverman III.6.1)
-
-For an endomorphism `α : Isogeny E E` of an elliptic curve `E`, equipped with a coordinate-ring
-restriction witness `ch : α.CoordHom` (the comorphism `α* : R → R` on `R := E.CoordinateRing`),
-this file builds the **Pic⁰ dual** as a point endomorphism and proves the dual relation, following
-**Silverman, *The Arithmetic of Elliptic Curves*, III.6.1** *to the letter*.
-
-## Silverman III.6.1 (verified against the in-repo PDF, book p.80–82)
-
-Let `κ : E ≅ Pic⁰(E)`, `P ↦ class of `(P) − (O)``.  Silverman III.6.1(b) **defines** the dual via
-the **divisor pullback** `φ*` (II.3.6/II.3.7):
-
-```
-            κ            φ*              κ⁻¹
-  φ̂ :  E ───→ Pic⁰(E) ───→ Pic⁰(E) ───→ E ,
-```
-
-i.e. `φ̂ = κ⁻¹ ∘ φ* ∘ κ`, where `φ* : Pic⁰(E) → Pic⁰(E)` is the **pullback** of divisor classes
-`(Q) ↦ Σ_{P ↦ Q} e_φ(P)·(P)` (Silverman II.3, book p.29).  At the level of the affine ideal class
-group `Pic⁰(E) ≅ ClassGroup R`, this pullback is the **ideal extension** `𝔪_Q ↦ 𝔪_Q·𝒪 = ∏ 𝔓^{e}`,
-which is exactly the shipped `HasseWeil.Isogeny.classMap` (= `HasseWeil.ClassGroup.map`).
-
-The companion **divisor pushforward** `φ_*` (II.3.6/II.3.7), `(P) ↦ (φP)`, is at the ideal level the
-**relative norm** `𝔓 ↦ 𝔮^{f}`, the shipped `HasseWeil.Isogeny.classNorm` (= `ClassGroup.relNorm`).
-Silverman II.3.6(e) gives `φ_* ∘ φ* = [deg φ]` on `Pic⁰`; its shipped shadow is
-`HasseWeil.Isogeny.classNorm_comp_classMap` (`classNorm (classMap c) = c ^ finrank R R`).
-
-## Main definitions
-
-* `HasseWeil.Isogeny.picDual` — `φ̂ = κ⁻¹ ∘ classMap ∘ κ` as a point endomorphism `E.Point →+
-  E.Point` (the III.6.1(b) construction; `classMap` = extension = the divisor pullback `φ*`).
-* `HasseWeil.Isogeny.picPushforward` — `κ⁻¹ ∘ classNorm ∘ κ`, the κ-transport of the divisor
-  pushforward `φ_*` (= `classNorm` = relative norm).
-
-## Main results
-
-* `HasseWeil.Isogeny.picPushforward_comp_picDual` / `..._degree` — **unconditional** (modulo the
-  carried `CoordHom`/`Module.Finite`/tower witnesses): the κ-transport of Silverman II.3.6(e),
-  `picPushforward ∘ picDual = [finrank R R]` (resp. `= [α.degree]`), as `AddMonoidHom`s.
-* `HasseWeil.Isogeny.toAddMonoidHom_comp_picDual` — **the III.6.1 target** `α ∘ α̂ = [deg α]`
-  (`α.toAddMonoidHom.comp α.picDual = [α.degree]`), carrying the III.3.4 **naturality** of `κ` on
-  the point map (`hnat : κ ∘ α = classNorm ∘ κ`) as a witness-parametric hypothesis.
-* `HasseWeil.Isogeny.picDual_comp_toAddMonoidHom` — the companion `α̂ ∘ α = [deg α]`
-  (III.6.2(a) order), under the same naturality witness.
-
-## Why the III.3.4 naturality is a *carried hypothesis*, not derived
-
-The unconditional results above tie `picDual` to `picPushforward` (both κ-transports), with **no**
-extra input.  Identifying `picPushforward` with the *actual* point map `α.toAddMonoidHom` is the
-**III.3.4 naturality** `κ ∘ α = classNorm ∘ κ` (i.e. `κ₂(αP) = φ_*κ₁(P)`).  As documented in
-`HasseWeil/Pic0/ToClassFunctorial.lean`, the point-map ↔ ideal-map link the shipped API exposes is
-the **`comap`** (set-theoretic point image), which on a prime `𝔓 / 𝔮` gives `𝔮`, whereas the
-pushforward `φ_* = relNorm` gives `𝔮^{f}` — they differ by the **inertia/residue degree** `f`.
-Bridging them is exactly the ramification bookkeeping `relNorm (comap …) = (·)^?`
-(`relNorm_eq_pow_of_isMaximal`, needing `PerfectField (FractionRing R)`), the III.3.4 content the
-`Isogeny` carries as *independent data*.  It is therefore taken as a hypothesis here (dischargeable
-per isogeny: Frobenius, multiplication-by-`n`), keeping every theorem `#print axioms`-clean.
+# The Picard dual isogeny
+
+For an elliptic-curve endomorphism `α` with a coordinate-ring comorphism,
+transporting ideal extension along `κ : E ≃ Pic⁰(E)` defines the Picard dual.
+Transporting the relative ideal norm defines the point-group pushforward.
+The norm–extension identity gives `picPushforward ∘ picDual = [deg α]`.
+Under the naturality relation between the point map and `κ`, this yields
+`α ∘ α̂ = [deg α]`; surjectivity gives the companion relation by cancellation.
+
+At rational points, residue degree one identifies the relative ideal norm
+with contraction. This proves the corresponding naturality reduction
+without a perfectness assumption on the function field.
 
 ## References
 
-* [Silverman, *The Arithmetic of Elliptic Curves*], II.3.6–3.7 (divisor pullback/pushforward,
-  `φ_*φ* = [deg]`), III.3.4 (functoriality of `E ≅ Pic⁰(E)`), III.6.1 (the dual isogeny).
+* [Silverman, *The Arithmetic of Elliptic Curves*], II.3.6–3.7, III.3.4, III.6.1.
 -/
 
 open WeierstrassCurve Polynomial
@@ -114,7 +68,7 @@ Silverman's **divisor pullback** `φ* : Pic⁰(E) → Pic⁰(E)`.  This is *exac
 definition `φ̂ = κ₁⁻¹ ∘ φ* ∘ κ₂`.
 
 The `CoordHom` witness `ch`, injectivity `hinj`, and finiteness `hfin` are the data/hypotheses the
-shipped `classMap` requires (carried, not discharged universally — instantiable per isogeny). -/
+`classMap` requires (carried, not discharged universally — instantiable per isogeny). -/
 noncomputable def picDual (ch : α.CoordHom) (hinj : Function.Injective ch.toAlgHom)
     (hfin : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule) :
     E.Point →+ E.Point :=
@@ -188,7 +142,7 @@ theorem classTransport_comp_eq
 
 /-! ### The dual relation `φ_* ∘ φ* = [deg]`, transported to the point group
 
-Silverman II.3.6(e): `φ_* ∘ φ* = [deg φ]` on `Pic⁰(E)`.  Its shipped class-group shadow is
+Silverman II.3.6(e): `φ_* ∘ φ* = [deg φ]` on `Pic⁰(E)`.  Its class-group shadow is
 `classNorm_comp_classMap` (`classNorm (classMap c) = c ^ finrank R R`).  We transport it across `κ`
 to obtain `picPushforward ∘ picDual = [finrank R R]` as point endomorphisms — **unconditionally**
 (beyond the carried `CoordHom`/`Module.Finite` witnesses the `classMap`/`classNorm` already need).
@@ -196,7 +150,7 @@ The `α.degree` form follows from the degree bridge `classNorm_comp_classMap_deg
 
 /-- **κ-transport of `φ_* ∘ φ* = [deg]` (coordinate-ring `finrank` form).**
 
-`picPushforward ∘ picDual = [finrank R R]` as point endomorphisms, the κ-conjugate of the shipped
+`picPushforward ∘ picDual = [finrank R R]` as point endomorphisms, the κ-conjugate of the
 class-group identity `classNorm (classMap c) = c ^ finrank R R` (= Silverman II.3.6(e)
 `φ_* ∘ φ* = [deg]`).  Unconditional beyond the carried `CoordHom`/`Module.Finite` witnesses.
 
@@ -210,7 +164,7 @@ theorem picPushforward_comp_picDual (ch : α.CoordHom) (hinj : Function.Injectiv
         ((@Module.finrank E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule : ℕ) :
           ℤ))).toAddMonoidHom :=
   -- `picPushforward ∘ picDual = classTransport classNorm ∘ classTransport classMap`, and
-  -- `classNorm (classMap c) = c ^ finrank` is the shipped `classNorm_comp_classMap`.
+  -- `classNorm (classMap c) = c ^ finrank` is the `classNorm_comp_classMap`.
   classTransport_comp_eq (α.classMap ch hinj hfin) (α.classNorm ch hinj hfin) _
     (α.classNorm_comp_classMap ch hinj hfin)
 
@@ -238,19 +192,12 @@ theorem picPushforward_comp_picDual_degree (ch : α.CoordHom)
   congr 2
   exact_mod_cast (α.degree_eq_finrank_coordinateRing_of_tower_eq ch S S' hSR hS'FF).symm
 
-/-! ### The III.6.1 target `α ∘ α̂ = [deg α]` on the point group
+/-! ### The dual relation on the point group
 
-The unconditional `picPushforward_comp_picDual` ties `picDual` to `picPushforward`.  To reach the
-brief's target `α ∘ α̂ = [deg α]` — with the *actual* point map `α.toAddMonoidHom` — we additionally
-need the **Silverman III.3.4 naturality** of `κ` on the point map: that under `κ`, the point map
-`α.toAddMonoidHom` corresponds to the divisor pushforward `φ_*` (= `classNorm`).  In the κ-conjugate
-form, this is exactly `α.toAddMonoidHom = α.picPushforward …`.
-
-As documented in `HasseWeil/Pic0/ToClassFunctorial.lean`, the shipped point-map ↔ ideal-map link is
-the **`comap`** (set-theoretic point image), which differs from `φ_* = relNorm` by the residue
-degree; bridging them is the III.3.4 ramification bookkeeping the `Isogeny` carries as independent
-data.  We therefore take the naturality as a witness-parametric hypothesis (dischargeable per
-isogeny), keeping the result `#print axioms`-clean. -/
+The relation between the Picard dual and Picard pushforward yields the
+point-group dual relation under the naturality condition
+`κ (α P) = classNorm (κ P)`. The relative norm differs from contraction by the
+residue degree; for rational points this degree is one. -/
 
 /-- **The III.3.4 naturality witness, raw form**: under `κ = toClassEquiv'`, the point map of `α`
 corresponds to the relative-norm class map `classNorm` (the divisor pushforward `φ_*`):
@@ -279,44 +226,14 @@ theorem toAddMonoidHom_eq_picPushforward_iff (ch : α.CoordHom)
     rw [picPushforward_apply, ← h P,
       (WeierstrassCurve.Affine.Point.toClassEquiv' (W := E)).symm_apply_apply]
 
-/-! #### Isolating the `hnat` residual to the rational-point ideal identity (`comap = relNorm`)
-
-`Naturality` is structurally a **carried datum**: an abstract `Isogeny E E` stores its `pullback`
-and `toAddMonoidHom` as *independent* fields (`HasseWeil/Basic.lean`), with no derivation linking
-the point map to the comorphism `ch` — so `hnat` cannot be proved in general, only verified per
-isogeny where both fields are explicit (Frobenius, mult-by-`n`).
-
-What we *can* ship axiom-clean is the exact reduction of the **right-hand side** of `Naturality` to
-a concrete ideal class.  The RHS `classNorm (κ P)` for a rational point `P = (x, y)` is the `mk0`
-class of the **relative norm** `relNorm (maximalIdealAt P) = relNorm ⟨X − x, Y − y⟩` (Silverman's
-divisor pushforward `φ_*`).  Combined with the shipped `ToClassFunctorial.toClass_toPointMap`
-(whose ideal is the **`comap`** `α*⁻¹(maximalIdealAt P)`), this pins the *entire* remaining content
-of `hnat` to the single residue-degree identity
-
-```
-class (relNorm (maximalIdealAt P)) = class (comap α* (maximalIdealAt P))     (rational P)
-```
-
-which holds because a rational point has residue degree `f = 1`, so `relNorm 𝔪_P = 𝔪_P ∩ R =
-comap α* 𝔪_P` (for `f = 1`, `relNorm 𝔭 = (comap 𝔭)^f = comap 𝔭`).  Mathlib has no
-`relNorm = (comap)^{inertiaDeg'}` lemma (and the project's `inertiaDeg' = 1`-at-smooth-points
-computation is `Module.Free`-diamond-blocked, see `Curves/GenericFiber.lean` Piece 9), so this last
-step stays a per-isogeny obligation; the lemma below is the bridge that *reduces to it*. -/
-
-/-- **The `classNorm`/`κ` side of `Naturality`, evaluated at a rational point (axiom-clean
-bridge).**
-
-For a rational point `P = (x, y)` (a mathlib `Point.some`), the value of the relative-norm class map
-`classNorm` on `κ P = toClassEquiv' P` is the `mk0` class of the **relative ideal norm** of the
-maximal ideal `XYIdeal E x (C y) = ⟨X − x, Y − y⟩` at `P` (= Silverman's divisor pushforward `φ_*`,
-packaged as `relNorm0`).  This unfolds the *right-hand side* of the III.3.4 naturality predicate
-`Naturality` at `Point.some`, reducing the whole `hnat` obligation to the ideal-class identity
-`class (relNorm 𝔪_P) = class (comap α* 𝔪_P)` against the shipped `toClass_toPointMap` (the
-`comap` form) — exactly the residue-degree-`1` content (see the section note).
-
-Proof: `κ (some) = toClass (some) = mk (XYIdeal' h) = mk0 ⟨XYIdeal, _⟩` (the project's
-`mk0_eq_mk_XYIdeal'` bridge), then `ClassGroup.relNorm_mk0` computes the norm on the integral
-representative. -/
+/-! #### Naturality at rational points
+
+For a rational point `P`, `classNorm (κ P)` is represented by the relative
+norm of its maximal ideal. Residue degree one identifies this norm with the
+contraction under the coordinate-ring comorphism. Thus naturality reduces
+to agreement between the given point map and the contracted ideal class. -/
+
+/-- The relative-norm class map on a rational point is represented by the relative norm of its maximal ideal. -/
 theorem classNorm_toClassEquiv'_some (ch : α.CoordHom) (hinj : Function.Injective ch.toAlgHom)
     (hfin : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule)
     {x y : F} (h : E.Nonsingular x y)
@@ -331,9 +248,9 @@ theorem classNorm_toClassEquiv'_some (ch : α.CoordHom) (hinj : Function.Injecti
         ch.isTorsionFree hinj
       ClassGroup.mk0 (HasseWeil.relNorm0 (R := E.CoordinateRing) (S := E.CoordinateRing)
         ⟨WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y), hmem⟩) := by
-  letI := ch.toAlgebra
-  haveI : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
-  haveI : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
+  let := ch.toAlgebra
+  have : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
+  have : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
         ch.isTorsionFree hinj
   rw [WeierstrassCurve.Affine.Point.toClassEquiv'_apply,
     WeierstrassCurve.Affine.Point.toClass_some,
@@ -341,23 +258,11 @@ theorem classNorm_toClassEquiv'_some (ch : α.CoordHom) (hinj : Function.Injecti
   show HasseWeil.ClassGroup.relNorm (ClassGroup.mk0 _) = _
   rw [HasseWeil.ClassGroup.relNorm_mk0]
 
-/-! #### Discharging `hnat`: the residue-degree-`1` (`comap = relNorm`) step at rational points
-
-Combining the shipped `classNorm_toClassEquiv'_some` (the `relNorm0` form of the `Naturality` RHS)
-with the residue-degree bridge of `ClassGroupNorm` (`relNorm 𝔪 = comap 𝔪` at `inertiaDeg' = 1`, the
-`inertiaDeg' = 1` itself supplied **diamond-free** by
-`Ideal`.`inertiaDeg_under_eq_one_of_algHom_…`),
-we reduce the entire `hnat`/`Naturality` obligation to the **point-map ↔ `comap` agreement** — that
-the actual point map `α.toAddMonoidHom` sends `(x, y)` to a point whose `κ`-class is the contraction
-`comap α* 𝔪_{(x,y)}` (Silverman III.3.4, the `comap` form shipped as `toClass_toPointMap`).
+/-! #### The residue-degree-one norm identity
 
-The residue-degree step needs `PerfectField (FractionRing R) = PerfectField K(E)` (the hypothesis of
-mathlib's `Ideal.relNorm_eq_pow_of_isMaximal`).  This holds for perfect base fields (e.g. `char 0`,
-or algebraically closed of characteristic `0`).  **Honest residual:** for a function field over a
-*finite* field `K(E)` is *imperfect*, so this last `relNorm = comap` step is not covered by the
-mathlib lemma there; pinning the exponent without `PerfectField`/Galois would require a degree-count
-adaptation of the project's `relNorm_maximalIdealAt` (`Curves/NormValuation.lean`) to the twisted
-`R →[α*] R` extension.  Everything *below* the `PerfectField` hypothesis is unconditional. -/
+The rational-point residue field has rank one over the base field. The
+class-group norm identity at residue degree one then identifies relative
+norm with contraction, without assuming the function field is perfect. -/
 
 /-- **The `comap α* 𝔪_P` ideal is a nonzero divisor** (carried side condition for the `mk0` class).
 For a rational point `P = (x, y)`, the contraction `comap α* (XYIdeal E x (C y))` of the (nonzero)
@@ -371,36 +276,16 @@ theorem comap_XYIdeal_mem_nonZeroDivisors (ch : α.CoordHom)
     Ideal.comap ch.toAlgHom.toRingHom
       (WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y)) ∈
     (Ideal E.CoordinateRing)⁰ := by
-  letI := ch.toAlgebra
-  haveI : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
-  haveI : Algebra.IsIntegral E.CoordinateRing E.CoordinateRing :=
+  let := ch.toAlgebra
+  have : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
+  have : Algebra.IsIntegral E.CoordinateRing E.CoordinateRing :=
     Algebra.IsIntegral.of_finite _ _
   have hbot : WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y) ≠ ⊥ :=
     mem_nonZeroDivisors_iff_ne_zero.mp hmem
   exact mem_nonZeroDivisors_iff_ne_zero.mpr
     (Ideal.under_ne_bot (A := E.CoordinateRing) (B := E.CoordinateRing) hbot)
 
-/-- **The `classNorm`/`comap` identity at a rational point (residue degree `1`) — UNCONDITIONAL.**
-
-For a rational point `P = (x, y)` and a coordinate-ring restriction `ch`, the relative-norm class
-`classNorm0 (XYIdeal E x (C y))` (the `Naturality` RHS, via `classNorm_toClassEquiv'_some`) equals
-the class of the **contraction** `comap α* (XYIdeal E x (C y))` — Silverman's divisor pushforward
-`φ_*` collapses to the set-theoretic point image at a rational point, where the residue degree
-`f = 1`.
-
-**`PerfectField`-free.**  Mathlib's `Ideal.relNorm_eq_pow_of_isMaximal` is gated on
-`[PerfectField (FractionRing R)]` (its proof reduces to the Galois case, which fails for a function
-field over a finite base), but that hypothesis is *not* needed at `f = 1`: the shipped
-`ClassGroup.mk0_relNorm0_eq_mk0_comap_of_inertiaDeg_one` (`ClassGroupNorm.lean`) discharges
-`relNorm 𝔪 = comap 𝔪` from `inertiaDeg' = 1` alone via the module-length / DVR-base route, with
-**no**
-`PerfectField`.  This removes the previous `[PerfectField E.FunctionField]` hypothesis.
-
-Proof: `XYIdeal` is maximal (`quotientXYIdealEquiv`) with residue field `F` (so `finrank F (R/𝔪) =
-1`), hence `inertiaDeg' (𝔪.under R) 𝔪 = 1`
-(`Ideal.inertiaDeg_under_eq_one_of_algHom_of_residueField_finrank_one`, diamond-free); then
-`ClassGroup.mk0_relNorm0_eq_mk0_comap_of_inertiaDeg_one` (`relNorm 𝔪 = comap 𝔪` under `f = 1`,
-unconditional) finishes. -/
+/-- At a rational point, residue degree one identifies the relative-norm ideal class with the contracted ideal class. -/
 theorem mk0_relNorm0_XYIdeal_eq_mk0_comap
     (ch : α.CoordHom) (hinj : Function.Injective ch.toAlgHom)
     (hfin : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule)
@@ -419,18 +304,16 @@ theorem mk0_relNorm0_XYIdeal_eq_mk0_comap
       ClassGroup.mk0 (⟨Ideal.comap ch.toAlgHom.toRingHom
         (WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y)), hcomap⟩ :
         (Ideal E.CoordinateRing)⁰) := by
-  letI := ch.toAlgebra
-  haveI : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
-  haveI : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
+  let := ch.toAlgebra
+  have : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule := hfin
+  have : @Module.IsTorsionFree E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule :=
         ch.isTorsionFree hinj
-  haveI hMmax : (WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y)).IsMaximal :=
+  have hMmax : (WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y)).IsMaximal :=
     Ideal.Quotient.maximal_of_isField _
       ((WeierstrassCurve.Affine.CoordinateRing.quotientXYIdealEquiv h.1).toRingEquiv.isField
         (Field.toIsField F))
-  have hf : Ideal.inertiaDeg'
-      ((WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y)).under
-        E.CoordinateRing)
-      (WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x (Polynomial.C y)) = 1 :=
+  have hf : (WeierstrassCurve.Affine.CoordinateRing.XYIdeal E x
+      (Polynomial.C y)).inertiaDeg E.CoordinateRing = 1 :=
     Ideal.inertiaDeg_under_eq_one_of_algHom_of_residueField_finrank_one ch.toAlgHom hfin _
       (WeierstrassCurve.Affine.Point.finrank_quotient_XYIdeal_eq_one h.1)
   exact ClassGroup.mk0_relNorm0_eq_mk0_comap_of_inertiaDeg_one
@@ -484,21 +367,7 @@ theorem Naturality_of_toClassEquiv'_some_eq_relNorm0
     rw [classNorm_toClassEquiv'_some ch hinj hfin h hmem]
     exact hpoint x y h hmem
 
-/-- **Discharging `hnat` from point-map ↔ `comap` agreement (residue degree `1`) — UNCONDITIONAL.**
-
-The `Naturality` predicate `hnat` holds whenever, for every rational point `P = (x, y)`, the actual
-point map `α.toAddMonoidHom` sends `(x, y)` to a point whose `κ`-class is the contraction
-`comap α* 𝔪_{(x,y)}` (the `comap` form of Silverman III.3.4 — exactly what `toClass_toPointMap`
-supplies for a *geometric* isogeny whose point map is `toPointMap`).  This is the precise reduction
-of `hnat` to its remaining content: the `relNorm`-vs-`comap` gap is closed here by the residue
-identity `mk0_relNorm0_XYIdeal_eq_mk0_comap`, which is now **`PerfectField`-free** (it routes
-through
-the shipped `ClassGroup.mk0_relNorm0_eq_mk0_comap_of_inertiaDeg_one`, valid at `f = 1` with no
-`PerfectField`).  **No `PerfectField E.FunctionField` hypothesis** — the only residual is the
-point-map ↔ `comap` agreement `hpoint`, i.e. Silverman III.3.4 for the *actual* point map (supplied
-per isogeny by `toClass_toPointMap` whenever the point map is the geometric `toPointMap`).
-
-The basepoint case is automatic (`κ 0 = 0 = ofMul (classNorm 1)`). -/
+/-- Agreement of the point map with contracted rational-point ideal classes implies naturality of the Picard identification. -/
 theorem Naturality_of_toClassEquiv'_some_eq_comap
     (ch : α.CoordHom) (hinj : Function.Injective ch.toAlgHom)
     (hfin : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule)
@@ -536,25 +405,7 @@ theorem Naturality_of_toClassEquiv'_some_eq_comap
       mk0_relNorm0_XYIdeal_eq_mk0_comap ch hinj hfin h hmem hcomap]
     exact hpoint x y h hcomap
 
-/-- **UNCONDITIONAL `Naturality` from a `CoordHom` (Silverman III.3.4) — the `hnat` discharge.**
-
-Given a coordinate-ring restriction `ch : α.CoordHom` and the **single genuine residual** that the
-actual point map `α.toAddMonoidHom` realises the III.3.4 *set-theoretic point image* — i.e. for each
-rational `(x, y)` the `κ`-class of `α.toAddMonoidHom (x, y)` is the contraction `comap α* 𝔪_{(x,y)}`
-(`hpoint`, exactly what the shipped `toClass_toPointMap` supplies for an isogeny whose point map is
-the geometric `toPointMap`) — the full naturality `α.Naturality ch hinj hfin` holds.
-
-This is the **unconditional** wiring requested for the Route-C `hnat` discharge: it threads the
-divisor-pushforward RHS (`classNorm_toClassEquiv'_some`) through the residue-degree-`1`
-identity `relNorm 𝔪 = comap 𝔪`, now **`PerfectField`-free** (`mk0_relNorm0_XYIdeal_eq_mk0_comap` →
-`ClassGroup.mk0_relNorm0_eq_mk0_comap_of_inertiaDeg_one`, `inertiaDeg' = 1` from
-`Ideal.inertiaDeg_under_eq_one_of_algHom_of_residueField_finrank_one`).  The `relNorm`-vs-`comap`
-residue bookkeeping is *fully discharged*; the only carried datum is `hpoint`, the point-map ↔
-`comap` agreement (Silverman III.3.4 for the actual point map), which is the genuine per-isogeny /
-base-change content (CoordHom data) — **not** a `PerfectField`/Galois obligation.
-
-Identical statement to `Naturality_of_toClassEquiv'_some_eq_comap`; provided under the
-`naturality_of_coordHom` name the Route-C assembly consumes. -/
+/-- Naturality follows from the coordinate-ring comorphism and its agreement with the rational-point map. -/
 theorem naturality_of_coordHom
     (ch : α.CoordHom) (hinj : Function.Injective ch.toAlgHom)
     (hfin : @Module.Finite E.CoordinateRing E.CoordinateRing _ _ ch.toAlgebra.toModule)
@@ -617,7 +468,7 @@ for `β = φ + ψ`.  At the point level, with the III.3.4 naturality
 composite `picDual ∘ α.toAddMonoidHom` is `κ⁻¹ ∘ (classMap ∘ classNorm) ∘ κ`.
 
 **Honest scope note.** This is `φ* ∘ φ_*` (extension after norm) — the **opposite** order to the
-shipped `classNorm_comp_classMap` (`classNorm ∘ classMap = φ_* ∘ φ*`).  The two orders are *not*
+`classNorm_comp_classMap` (`classNorm ∘ classMap = φ_* ∘ φ*`).  The two orders are *not*
 interchangeable from commutativity of `ClassGroup` alone (`f(g c) = g(f c)` needs `f ∘ g = g ∘ f`
 as maps, not a commutative codomain), and mathlib provides only `relNorm_algebraMap`
 (`relNorm ∘ map`), **not** `map ∘ relNorm = (·)^n`.  The reverse-order class identity
@@ -664,6 +515,7 @@ theorem comp_eq_mulByInt_of_comp_eq_of_surjective
   -- `g (f (g Q)) = g (m • Q) = m • g Q`, and the RHS `[m] (g Q) = m • g Q`.
   rw [hf, g.map_zsmul, mulByInt_apply]
 
+omit [WeierstrassCurve.IsElliptic E] in
 /-- **Right-cancellation of a surjective additive point endomorphism (Silverman II.2.3, equality
 form).**  If `g : E.Point →+ E.Point` is **surjective** and two maps `f₁, f₂` agree after
 post-composition with `g` (i.e. `f₁ ∘ g = f₂ ∘ g`), then `f₁ = f₂`.  This is the abstract content of
@@ -730,9 +582,9 @@ theorem picDual_comp_toAddMonoidHom_of_surjective_degree (ch : α.CoordHom)
 Silverman III.6.1(a)/III.6.2 states the dual is **unique**: `φ̂` is the *only* `δ` with
 `δφ = [deg φ]`.  The proof is right-cancellation of the surjective (nonconstant) `φ`.  We give the
 point-map form:
-from `(picDual α) ∘ α = [deg α]` (the shipped `picDual_comp_toAddMonoidHom_of_surjective`) and any
+from `(picDual α) ∘ α = [deg α]` (the `picDual_comp_toAddMonoidHom_of_surjective`) and any
 hypothetical `β` with `β ∘ α = [deg α]`, the surjection `α` cancels on the right to force
-`picDual α = β`.  Specialising `β` to the κ-transport of a *shipped* dual isogeny (e.g. the
+`picDual α = β`.  Specialising `β` to the κ-transport of a dual isogeny (e.g. the
 Verschiebung `V` of Frobenius `π`, with `V ∘ π = [q]` from `IsDualOf V π`) **identifies `picDual π`
 with `V`** — the milestone "Pic⁰ dual of Frobenius = Verschiebung". -/
 
@@ -796,26 +648,23 @@ theorem picDual_eq_of_comp_toAddMonoidHom_eq_degree (ch : α.CoordHom)
     ((picDual_comp_toAddMonoidHom_of_surjective_degree ch hinj hfin hnat hsurjDual
       S S' hSR hS'FF).trans hβ.symm)
 
-/-! #### Non-circular `picDual` of a *single* isogeny whose degree is independently known
+/-! #### Duals of isogenies with independently known degrees
 
-The III.6.1(a) uniqueness `picDual_eq_of_comp_toAddMonoidHom_eq_degree` identifies `picDual α` with
-any point map `δ` satisfying `δ ∘ α = [deg α]` — **non-circularly whenever `deg α` is known
-independently** of the Pic⁰ chain.  Two such cases, used to seed the dual algebra for `rπ − s`:
+The uniqueness statement in Silverman III.6.1(a) identifies `picDual α` with
+any point map `δ` satisfying `δ ∘ α = [deg α]`. For Frobenius, its degree
+is the field cardinality `q` and the composition with Verschiebung is `[q]`.
+For scalar multiplication `[n]`, the degree is `n²` and self-composition is
+`[n²]`.
 
-* **`picDual π = V`** (Frobenius dual = Verschiebung), where `deg π = #K = q` is shipped
-  (`frobeniusIsog_degree`) and `V ∘ π = [q]` comes from `IsDualOf V π`.
-* **`picDual [n] = [n]`** (scalar self-dual, III.6.2(b)/(d)), where `deg [n] = n²` and
-  `[n] ∘ [n] = [n²]` are shipped (`mulByInt_degree`, `mulByInt_comp_eq_mul`).
-
-Both are genuinely non-circular: the degree of a *single* Frobenius / scalar is shipped, unlike
-`deg(rπ − s)` (which is the Route-C conclusion).  They are the seeds for the dual-additivity step
-`picDual(rπ − s) = rV − s` — whose *combination* nonetheless needs III.6.2(c) (see the note in
-`RouteCGeometric.lean`). -/
+These degree calculations identify the individual duals without assuming
+the degree formula for `rπ − s`. The further formula
+`picDual(rπ − s) = rV − s` requires dual additivity, Silverman III.6.2(c).
+-/
 
 /-- **`picDual α = δ` when `δ ∘ α = [d]` and `α.degree = d` (degree given as data, non-circular).**
 
-The III.6.1(a) uniqueness specialised to an *explicit integer degree* `d`: if `α.degree = d` (an
-**independently-shipped** degree, e.g. `frobeniusIsog_degree`, `mulByInt_degree`) and a point map
+The III.6.1(a) uniqueness specialised to an *explicit integer degree* `d`: if `α.degree = d` (a
+**known** degree, e.g. `frobeniusIsog_degree`, `mulByInt_degree`) and a point map
 `δ` satisfies `δ ∘ α = [d]`, then `picDual α = δ`.  This is the non-circular seed form (it does not
 route through the Pic⁰ push-pull degree, only through the *given* `d`). -/
 theorem picDual_eq_of_comp_toAddMonoidHom_eq_of_degree_eq (ch : α.CoordHom)
@@ -847,11 +696,11 @@ The III.6.2(b) dual-of-composition `(φ∘ψ)^ = ψ̂∘φ̂` specialised to `r
 `picDual_eq_of_comp_toAddMonoidHom_eq_of_degree_eq` with the **independently-known** degree
 `deg(r • α) = deg α · r²` (`Isogeny.zsmul_degree` / `mulByInt_degree`) and the composite
 `(r • δ) ∘ (r • α) = r² • (δ ∘ α) = r² • [deg α] = [r² · deg α]`, where `δ` is the dual value of `α`
-(any point map with `δ ∘ α = [deg α]`, e.g. the κ-transport of a shipped `IsDualOf` partner).
+(any point map with `δ ∘ α = [deg α]`, e.g. the κ-transport of a `IsDualOf` partner).
 
-This is **non-circular**: it routes only through the *given* degree `deg α` (independently shipped
+This is **non-circular**: it routes only through the *given* degree `deg α` (independently
 for a single Frobenius / scalar / `α`), never through the Pic⁰ push-pull degree of `r • α`.  It is
-the generic engine behind the Route-C seed `picDual(rπ) = rV` (take `α = π`, `δ = V`,
+the generic engine behind the dual-additivity seed `picDual(rπ) = rV` (take `α = π`, `δ = V`,
 `deg π = #K = q`).
 
 Residuals: the per-`(r • α)` CoordHom data (`chr` + the two surjectivities + tower `(S, S')`),
@@ -898,10 +747,10 @@ theorem picDual_zsmul_eq_zsmul_of_comp_eq
   exact picDual_eq_of_comp_toAddMonoidHom_eq_of_degree_eq chr hinjr hfinr hnatr hsurjDualr
     hsurjr S S' hSR hS'FF hdeg hcomp
 
-/-- **`picDual(r • α) = r • α̂` from a shipped `IsDualOf` partner (route-2 seed, `IsDualOf` form).**
+/-- **`picDual(r • α) = r • α̂` from a `IsDualOf` partner (route-2 seed, `IsDualOf` form).**
 
 As `picDual_zsmul_eq_zsmul_of_comp_eq`, but taking the dual value of `α` in the *full-isogeny*
-`IsDualOf δ_isog α` form (`δ_isog ∘ α = [deg α]`), which is how the Route-C Vieta bundle supplies it
+`IsDualOf δ_isog α` form (`δ_isog ∘ α = [deg α]`), which is how the dual-additivity Vieta bundle supplies it
 (e.g. `IsDualOf V π`).  Concludes `picDual(r • α) = r • δ_isog` at the point-map level. -/
 theorem picDual_zsmul_eq_zsmul_of_isDual
     {α δ_isog : Isogeny E E} (r : ℤ) (hr : r ≠ 0)
@@ -932,10 +781,10 @@ The Silverman III.8 trace relation `α + α̂ = [tr α]` *determines* `α̂` onc
 `δ` satisfies the *same* relation `α + δ = [tr α]`: subtract `α` to get `α̂ = [tr α] − α = δ`.  This
 is a pure point-group cancellation (no degree, no uniqueness, **non-circular**).
 
-It is the final algebraic step of the Route-C dual-additivity output `picDual(rπ−s) = rV − s`: with
+It is the final algebraic step of the dual-additivity dual-additivity output `picDual(rπ−s) = rV − s`: with
 `α = rπ − s`, the irreducible input `htrace_dual` is III.8 for `α` (equivalently III.6.2(c)
 additivity, since `tr(rπ−s) = r·t − 2s`), while the candidate identity `htrace_delta`
-(`α + (rV − s) = [r·t − 2s]`) is derived *non-circularly* from the shipped `π + V = [t]`.  This
+(`α + (rV − s) = [r·t − 2s]`) is derived *non-circularly* from the `π + V = [t]`.  This
 lemma performs the subtraction generically. -/
 theorem picDual_eq_of_trace_relations (ch : α.CoordHom)
     (hinj : Function.Injective ch.toAlgHom)
@@ -958,7 +807,7 @@ maps).**
 The point-map identity `(r·π − s) + (r·V − s) = [r·t − 2s]` for abstract endomorphisms `π, V` of
 `E.Point` satisfying `π + V = [t]` (the Frobenius trace relation).  This is the **non-circular**
 half
-of the Route-C dual-additivity output: pointwise
+of the dual-additivity dual-additivity output: pointwise
 `r·π P − s·P + r·V P − s·P = r·(π P + V P) − 2s·P = r·(t·P) − 2s·P = (r·t − 2s)·P`.
 
 It carries no `picDual` and no degree — pure point-group algebra from `hsum` — and is the candidate
@@ -990,8 +839,8 @@ Pic⁰ dual `picDual α` equals `r·V − s` — *provided* the Silverman III.8
 `htrace_dual` is the **single irreducible residual** (Silverman III.8 for `α`, equivalently
 III.6.2(c) additivity since `tr α = r·t − 2s`); everything else — the candidate trace half
 `smul_sub_add_smul_sub_eq_mulByInt` — is **non-circular** from `hsum`.  This is the engine that
-discharges the Route-C `hpicval` (take `α = rπ − s`, `π = π.toAddMonoidHom`, `V = V.toAddMonoidHom`;
-`hbeta` is the `rfl`-true `genuineIsogSmulSub_toAddMonoidHom`-shape and `hsum` the shipped
+discharges the dual-additivity `hpicval` (take `α = rπ − s`, `π = π.toAddMonoidHom`, `V = V.toAddMonoidHom`;
+`hbeta` is the `rfl`-true `genuineIsogSmulSub_toAddMonoidHom`-shape and `hsum` the
 `π + V = [t]`).  The per-piece `picDual` values `picDual π = V`, `picDual(rπ) = rV`,
 `picDual [n] = [n]` (the non-circular seeds) are exactly what `htrace_dual` decomposes into. -/
 theorem picDual_eq_smul_sub_of_sum_trace (ch : α.CoordHom)
diff --git a/projects/HasseWeil/HasseWeil/Pic0/ToClassSurjective.lean b/projects/HasseWeil/HasseWeil/Pic0/ToClassSurjective.lean
index 31ceaa04d..e7193a4aa 100644
--- a/projects/HasseWeil/HasseWeil/Pic0/ToClassSurjective.lean
+++ b/projects/HasseWeil/HasseWeil/Pic0/ToClassSurjective.lean
@@ -9,88 +9,6 @@ import Mathlib.RingTheory.OrderOfVanishing.Basic
 import Mathlib.RingTheory.Polynomial.DegreeLT
 import HasseWeil.Foundation.Ramification
 
-/-!
-# Surjectivity of `Point.toClass` and the isomorphism `E ≅ Pic⁰(E)` (affine model)
-
-For a Weierstrass curve `W : WeierstrassCurve.Affine F` over a field `F`, mathlib provides the
-group homomorphism
-
-```
-WeierstrassCurve.Affine.Point.toClass : W.Point →+ Additive (ClassGroup W.CoordinateRing)
-```
-
-(`Mathlib/AlgebraicGeometry/EllipticCurve/Affine/Point.lean`) sending the point at infinity to `0`
-and an affine point `P = (x, y)` to the class of the fractional ideal `⟨X - x, Y - y⟩`. Mathlib
-proves this is **injective** (`WeierstrassCurve.Affine.Point.toClass_injective`), realising
-`W.Point` as a subgroup of the affine ideal class group. It does **not** prove surjectivity.
-
-This file packages the surjectivity. The statement that every ideal class is the class of some
-`XYIdeal'` (or trivial) is the genus-1 divisor-reduction theorem: on a smooth genus-1 curve every
-degree-0 divisor class is represented by `(P) - (O)`. We isolate that statement as the
-predicate `ClassRepresentableByPoints W` and prove that it is *equivalent* to surjectivity of
-`toClass`, and that surjectivity packages into the group isomorphism
-`W.Point ≃+ Additive (ClassGroup W.CoordinateRing)`.
-
-## Main definitions
-
-* `WeierstrassCurve.Affine.Point.ClassRepresentableByPoints`: the predicate that every element of
-  `ClassGroup W.CoordinateRing` is trivial or the class of `XYIdeal' h` for a nonsingular point.
-* `WeierstrassCurve.Affine.Point.ClassReducesToCodimLEOne`: the genus-1 reduction input — every
-  nonzero integral ideal class has a representative of `F`-codimension `≤ 1`. This is now **proved
-  unconditionally** (`classReducesToCodimLEOne_holds`).
-* `WeierstrassCurve.Affine.Point.toClassEquiv'`: the **unconditional** group isomorphism
-  `W.Point ≃+ Additive (ClassGroup W.CoordinateRing)` for `[W.IsElliptic]`, built from injectivity +
-  the unconditional surjectivity `toClass_surjective'`.
-
-## Main results
-
-The **norm-degree dictionary** (the surjectivity counterpart of mathlib's `natDegree_norm_ne_one`,
-which powers `toClass_injective`), all axiom-clean:
-
-* `eq_XYIdeal_of_finrank_quotient_eq_one`: every nonzero ideal `I` with `finrank F (R/I) = 1` is
-  the point ideal `XYIdeal W x (C y)` of a (necessarily nonsingular, under `[W.IsElliptic]`) point.
-* `mk0_eq_one_of_finrank_quotient_eq_zero` and
-  `mk0_eq_mk_XYIdeal'_of_finrank_quotient_eq_one`: the two codimension endpoints, identifying
-  codimension `0` (resp. `1`) ideals with the trivial class (resp. a point class).
-* `integralIdealRepresentableByPoints_of_classReducesToCodimLEOne` and
-  `toClass_surjective_of_classReducesToCodimLEOne`: the codimension reduction predicate implies the
-  integral-ideal reduction, hence surjectivity of `toClass` (using the now axiom-clean Dedekind
-  instance `HasseWeil.coordinateRing_isDedekindDomain`).
-
-and the unconditional headline results:
-
-* `toClass_surjective'`: **unconditional** surjectivity of `toClass` for `[W.IsElliptic]`.
-* `toClassEquiv'`: the **unconditional** isomorphism `W.Point ≃+ Pic⁰(E)`.
-
-## Implementation notes
-
-The full unconditional surjectivity reduces, via three landed and axiom-clean ingredients, to the
-genus-1 codimension reduction `ClassReducesToCodimLEOne`, which is **now also proved**.
-
-1. `IsDedekindDomain W.CoordinateRing` is **available and axiom-clean** via
-   `HasseWeil.coordinateRing_isDedekindDomain` (imported here), which fires under `[W.IsElliptic]`.
-
-2. **The ring-theoretic codimension infrastructure** (field-agnostic, no `[Fintype F]`):
-   * `finiteDimensional_quotient_of_ne_bot`: every nonzero ideal has finite `F`-codimension.
-   * `finrank_quotient_smul`: codimension additivity for multiplication by a *principal* ideal.
-   * `finrank_quotient_mul`: the **general** codimension additivity
-     `finrank F (R ⧸ I·J) = finrank F (R ⧸ I) + finrank F (R ⧸ J)` for arbitrary nonzero `I, J`,
-     via the short exact sequence `0 → I/(I·J) → R/(I·J) → R/I → 0`
-     (`Submodule.quotientQuotientEquivQuotient` + `Submodule.finrank_quotient_add_finrank`) and the
-     invertible-ideal kernel isomorphism `quotIdealMulEquiv : I/(I·J) ≃ R/J` built from mathlib's
-     `FractionalIdeal.quotientEquiv`.
-
-3. **The genus-1 reduction proper**, `classReducesToCodimLEOne_holds`: every nonzero integral ideal
-   class has a representative of `F`-codimension `≤ 1` (Riemann–Roch for `g = 1`). Proved from the
-   concrete Riemann–Roch inequality `exists_mem_norm_natDegree_le` (every nonzero ideal `I` has a
-   nonzero element of norm degree `≤ finrank F (R ⧸ I) + 1`, by an exact `F`-dimension count using
-   `CoordinateRing.degree_norm_smul_basis` and rank–nullity) plus the general additivity (2): from a
-   small-norm element of a representative of the inverse class, the Dedekind factorisation
-   `(a) = I'·J` yields a codimension-`≤ 1` representative of the class.
-
-Everything in this file is `#print axioms`-clean (`[propext, Classical.choice, Quot.sound]`),
-including the unconditional `toClass_surjective'` and `toClassEquiv'`.
--/
 
 open Polynomial Module
 
@@ -240,10 +158,8 @@ theorem eq_of_le_of_finrank_quotient_eq {I J : Ideal W.CoordinateRing} (hJI : J
   exact (Submodule.Quotient.mk_eq_zero _).mp hr0
 
 omit [DecidableEq F] in
-/-- **An `F`-algebra hom `R → F` from a codimension-1 quotient.** If `R/I` is 1-dimensional over the
-field `F`, then `algebraMap F (R/I)` is an isomorphism, and composing its inverse with the quotient
-map gives an `F`-algebra hom `φ : R →ₐ[F] F` whose kernel is exactly `I`. Evaluating `φ` at the
-images of `X` and `Y` will produce the point. -/
+
+
 theorem exists_algHom_ker_eq_of_finrank_quotient_eq_one (I : Ideal W.CoordinateRing)
     [Nontrivial (W.CoordinateRing ⧸ I)]
     (hfin : Module.finrank F (W.CoordinateRing ⧸ I) = 1) :
@@ -289,8 +205,8 @@ theorem equation_of_algHom (φ : W.CoordinateRing →ₐ[F] F) :
   exact heq.symm
 
 omit [DecidableEq F] in
-/-- The images of `X - x` and `Y - y` in `R` lie in the kernel of `φ` whenever
-`x = φ(X̄)` and `y = φ(Ȳ)`. -/
+
+
 theorem algHom_XClass_eq_zero (φ : W.CoordinateRing →ₐ[F] F) :
     φ (CoordinateRing.XClass W (φ (CoordinateRing.mk W (C X)))) = 0 := by
   set x := φ (CoordinateRing.mk W (C X)) with hx
@@ -303,7 +219,7 @@ theorem algHom_XClass_eq_zero (φ : W.CoordinateRing →ₐ[F] F) :
   rw [Algebra.algebraMap_self_apply, sub_self]
 
 omit [DecidableEq F] in
-/-- The image of `Y - y` in `R` lies in the kernel of `φ` whenever `y = φ(Ȳ)`. -/
+
 theorem algHom_YClass_eq_zero (φ : W.CoordinateRing →ₐ[F] F) :
     φ (CoordinateRing.YClass W (C (φ (CoordinateRing.mk W Y)))) = 0 := by
   set y := φ (CoordinateRing.mk W Y) with hy
@@ -328,9 +244,9 @@ theorem eq_XYIdeal_of_finrank_quotient_eq_one [W.IsElliptic] (I : Ideal W.Coordi
   set y := φ (CoordinateRing.mk W Y) with hy
   have hEq : W.Equation x y := equation_of_algHom φ
   have hNS : W.Nonsingular x y := (equation_iff_nonsingular).mp hEq
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ CoordinateRing.XYIdeal W x (C y)) :=
+  have : FiniteDimensional F (W.CoordinateRing ⧸ CoordinateRing.XYIdeal W x (C y)) :=
     finiteDimensional_quotient_XYIdeal hEq
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ I) :=
+  have : FiniteDimensional F (W.CoordinateRing ⧸ I) :=
     Module.finite_of_finrank_eq_succ hfin
   refine ⟨x, y, hNS, ?_⟩
   have hsub : CoordinateRing.XYIdeal W x (C y) ≤ I := by
@@ -352,9 +268,7 @@ theorem finiteDimensional_quotient_of_ne_bot [W.IsElliptic] (I : Ideal W.Coordin
   classical
   exact Module.Finite.equiv (Ideal.quotientEquivDirectSum F (CoordinateRing.basis W) hI).symm
 
-/-- **The degree of an ideal of `F[X]`.** For an ideal `J` of the PID `F[X]`, the natural degree of
-its monic generator. On `⊥` it is `0`; on a nonzero ideal `⟨g⟩` it is `g.natDegree`. This packages
-the codimension `finrank F (F[X] ⧸ J)` purely polynomially. -/
+
 noncomputable def idealNatDegree (J : Ideal F[X]) : ℕ :=
   (Submodule.IsPrincipal.generator J).natDegree
 
@@ -422,10 +336,10 @@ theorem finrank_quotient_eq_sum_smithCoeffs [W.IsElliptic] (I : Ideal W.Coordina
     Module.finrank F (W.CoordinateRing ⧸ I)
       = ∑ i, (Ideal.smithCoeffs (CoordinateRing.basis W) I hI i).natDegree := by
   classical
-  haveI hFree : ∀ i, Module.Free F
+  have hFree : ∀ i, Module.Free F
       (F[X] ⧸ Ideal.span ({Ideal.smithCoeffs (CoordinateRing.basis W) I hI i} : Set F[X])) :=
     fun i ↦ inferInstance
-  haveI hFin : ∀ i, Module.Finite F
+  have hFin : ∀ i, Module.Finite F
       (F[X] ⧸ Ideal.span ({Ideal.smithCoeffs (CoordinateRing.basis W) I hI i} : Set F[X])) :=
     fun i ↦ inferInstance
   rw [Ideal.finrank_quotient_eq_sum (I := I) (hI := hI) F (CoordinateRing.basis W)]
@@ -443,12 +357,8 @@ theorem smul_ideal_eq_span_mul (a : W.CoordinateRing) (I : Ideal W.CoordinateRin
   · rintro ⟨z, hz, rfl⟩; exact ⟨z, hz, by rw [smul_eq_mul]⟩
 
 omit [DecidableEq F] in
-/-- **Codimension additivity for multiplication by a principal ideal.** For a nonzero `a ∈ R` and a
-nonzero ideal `I`, `finrank F (R ⧸ ⟨a⟩·I) = finrank F (R ⧸ I) + finrank F (R ⧸ ⟨a⟩)`. This is the
-additivity of `F`-codimension along the multiplication-by-`a` short exact sequence
-`R ⧸ I → R ⧸ (a•I) → R ⧸ ⟨a⟩` (`Ideal.mulQuot`/`Ideal.quotOfMul`, restricted to `F`-linear maps and
-fed to `Module.length_eq_add_of_exact`, with `Module.length F = finrank F` over the field `F`).
-It is the special case of the genus-1 norm-degree additivity for a principal factor. -/
+
+
 theorem finrank_quotient_smul [W.IsElliptic] {a : W.CoordinateRing} (ha : a ≠ 0)
     (I : Ideal W.CoordinateRing) (hI : I ≠ ⊥) :
     Module.finrank F (W.CoordinateRing ⧸ (a • I)) =
@@ -459,17 +369,17 @@ theorem finrank_quotient_smul [W.IsElliptic] {a : W.CoordinateRing} (ha : a ≠
   have haspan : Ideal.span ({a} : Set W.CoordinateRing) ≠ ⊥ := by
     rwa [Ne, Ideal.span_singleton_eq_bot]
   have haI : a • I ≠ ⊥ := by rw [smul_ideal_eq_span_mul]; exact mul_ne_zero haspan hI
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ I) := finiteDimensional_quotient_of_ne_bot _ hI
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ Ideal.span ({a} : Set W.CoordinateRing)) :=
+  have : FiniteDimensional F (W.CoordinateRing ⧸ I) := finiteDimensional_quotient_of_ne_bot _ hI
+  have : FiniteDimensional F (W.CoordinateRing ⧸ Ideal.span ({a} : Set W.CoordinateRing)) :=
     finiteDimensional_quotient_of_ne_bot _ haspan
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ (a • I)) :=
+  have : FiniteDimensional F (W.CoordinateRing ⧸ (a • I)) :=
     finiteDimensional_quotient_of_ne_bot _ haI
   have hlen := Module.length_eq_add_of_exact
     ((Ideal.mulQuot a I).restrictScalars F) ((Ideal.quotOfMul a I).restrictScalars F)
     (Ideal.mulQuot_injective I hanz) (Ideal.quotOfMul_surjective I)
     (Ideal.exact_mulQuot_quotOfMul I)
   rw [Module.length_eq_finrank, Module.length_eq_finrank, Module.length_eq_finrank,
-    ← Nat.cast_add, ENat.coe_inj] at hlen
+    ← Nat.cast_add, ENat.natCast_inj] at hlen
   exact hlen
 
 omit [DecidableEq F] in
@@ -554,10 +464,8 @@ private noncomputable def quotEquivOneCoeIdealQuot_aux [W.IsElliptic] {J : Ideal
     exact hry
 
 omit [DecidableEq F] in
-/-- **The invertible-ideal quotient isomorphism `I/(I·J) ≃ R/J`.** For nonzero ideals `I, J` of the
-(Dedekind) coordinate ring `R`, the `R`-module quotient of the ideal `I` (viewed as a submodule of
-`R`) by its submodule `I·J` is `R`-linearly isomorphic to `R/J`; this holds because `I` is
-invertible. -/
+
+
 noncomputable def quotIdealMulEquiv [W.IsElliptic] {I J : Ideal W.CoordinateRing}
     (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
     ((I : Submodule W.CoordinateRing W.CoordinateRing) ⧸
@@ -582,23 +490,17 @@ noncomputable def quotIdealMulEquiv [W.IsElliptic] {I J : Ideal W.CoordinateRing
     (eqe.trans (quotEquivOneCoeIdealQuot_aux hinj).symm)
 
 omit [DecidableEq F] in
-/-- **General codimension additivity.** For arbitrary nonzero ideals `I, J` of
-`R := W.CoordinateRing`,
-`finrank F (R ⧸ I·J) = finrank F (R ⧸ I) + finrank F (R ⧸ J)`. This upgrades `finrank_quotient_smul`
-(the principal-factor case) to all factors, via the short exact sequence
-`0 → I/(I·J) → R/(I·J) → R/I → 0` (`Submodule.quotientQuotientEquivQuotient` +
-`Submodule.finrank_quotient_add_finrank`) together with the invertible-ideal kernel identification
-`I/(I·J) ≃ R/J` (`quotIdealMulEquiv`). It is the genus-1 norm-degree additivity in `F`-codimension
-form. -/
+
+
 theorem finrank_quotient_mul [W.IsElliptic] {I J : Ideal W.CoordinateRing}
     (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
     Module.finrank F (W.CoordinateRing ⧸ (I * J)) =
       Module.finrank F (W.CoordinateRing ⧸ I) + Module.finrank F (W.CoordinateRing ⧸ J) := by
   have hIJ : I * J ≠ ⊥ := mul_ne_zero hI hJ
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ (I * J)) :=
+  have : FiniteDimensional F (W.CoordinateRing ⧸ (I * J)) :=
     finiteDimensional_quotient_of_ne_bot _ hIJ
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ I) := finiteDimensional_quotient_of_ne_bot _ hI
-  haveI : FiniteDimensional F (W.CoordinateRing ⧸ J) := finiteDimensional_quotient_of_ne_bot _ hJ
+  have : FiniteDimensional F (W.CoordinateRing ⧸ I) := finiteDimensional_quotient_of_ne_bot _ hI
+  have : FiniteDimensional F (W.CoordinateRing ⧸ J) := finiteDimensional_quotient_of_ne_bot _ hJ
   have hIJ_le : I * J ≤ I := Ideal.mul_le_left
   set M : Submodule (W.CoordinateRing) (W.CoordinateRing ⧸ (I * J)) :=
     Submodule.map (Submodule.mkQ (I * J))
@@ -620,7 +522,7 @@ theorem finrank_quotient_mul [W.IsElliptic] {I J : Ideal W.CoordinateRing}
     have e3 : (LinearMap.range g) ≃ₗ[R] M := LinearEquiv.ofEq _ _ hrange
     exact ((Submodule.quotEquivOfEq _ _ hker.symm).trans (g.quotKerEquivRange.trans e3)).symm
   have eM : M ≃ₗ[W.CoordinateRing] (W.CoordinateRing ⧸ J) := e2a.trans (quotIdealMulEquiv hI hJ)
-  haveI : FiniteDimensional F M := (eM.restrictScalars F).symm.finiteDimensional
+  have : FiniteDimensional F M := (eM.restrictScalars F).symm.finiteDimensional
   have key : Module.finrank F ((W.CoordinateRing ⧸ (I * J)) ⧸ (M.restrictScalars F))
       + Module.finrank F (M.restrictScalars F) = Module.finrank F (W.CoordinateRing ⧸ (I * J)) :=
     Submodule.finrank_quotient_add_finrank (M.restrictScalars F)
@@ -666,7 +568,7 @@ theorem mk0_eq_mk_XYIdeal'_of_finrank_quotient_eq_one [W.IsElliptic] (J : Ideal
     (hfin : Module.finrank F (W.CoordinateRing ⧸ J) = 1) :
     ∃ (x y : F) (h : W.Nonsingular x y),
       ClassGroup.mk0 ⟨J, hmem⟩ = ClassGroup.mk W.FunctionField (CoordinateRing.XYIdeal' h) := by
-  haveI : Nontrivial (W.CoordinateRing ⧸ J) := Module.nontrivial_of_finrank_eq_succ hfin
+  have : Nontrivial (W.CoordinateRing ⧸ J) := Module.nontrivial_of_finrank_eq_succ hfin
   obtain ⟨x, y, h, hJ⟩ := eq_XYIdeal_of_finrank_quotient_eq_one J hfin
   have hmem' : CoordinateRing.XYIdeal W x (C y) ∈ (Ideal W.CoordinateRing)⁰ := hJ ▸ hmem
   refine ⟨x, y, h, ?_⟩
@@ -723,10 +625,7 @@ private theorem two_nsmul_degree_le' {p : F[X]} {n : ℕ} (hp : p.degree < (n :
         rw [nsmul_eq_mul]; push_cast; ring, Nat.cast_le]
     lia
 
-/-- **The basis-combination map** `(p, q) ↦ p · 1 + q · Ȳ`, restricted to polynomials of bounded
-degree (`degreeLT F a × degreeLT F b`), as an `F`-linear map into `R`. Its image is the space of
-elements whose norm degree we can control; its kernel intersected with the codimension count gives
-the Riemann–Roch element. -/
+
 noncomputable def basisCombMap (W : WeierstrassCurve.Affine F) (a b : ℕ) :
     (Polynomial.degreeLT F a × Polynomial.degreeLT F b) →ₗ[F] W.CoordinateRing where
   toFun pq := (pq.1 : F[X]) • (1 : W.CoordinateRing) + (pq.2 : F[X]) • (CoordinateRing.basis W 1)
@@ -780,19 +679,15 @@ theorem basisCombMap_ne_zero {a b : ℕ} (pq : Polynomial.degreeLT F a × Polyno
   exact Prod.ext (Subtype.ext hp) (Subtype.ext hq)
 
 omit [DecidableEq F] in
-/-- **The genus-1 Riemann–Roch inequality (concrete form).** Every nonzero ideal `I` of
-`R := W.CoordinateRing` contains a nonzero element `a` with `(Algebra.norm F[X] a).natDegree ≤
-finrank F (R ⧸ I) + 1`. This is `ℓ(D) ≥ deg D` for `g = 1` made explicit, proved by the exact
-`F`-dimension count: the bounded-degree basis combinations form an `F`-subspace of dimension
-`ℓ + 1 > ℓ = finrank F (R ⧸ I)`, so rank–nullity for its composite into `R ⧸ I` forces a nonzero
-kernel element. No finiteness of `F` is needed (`F[X]` is Euclidean). -/
+
+
 theorem exists_mem_norm_natDegree_le [W.IsElliptic] (I : Ideal W.CoordinateRing) (hI : I ≠ ⊥) :
     ∃ a ∈ I, a ≠ 0 ∧
       (Algebra.norm F[X] a).natDegree ≤ Module.finrank F (W.CoordinateRing ⧸ I) + 1 := by
   classical
   set R := W.CoordinateRing
   set ℓ := Module.finrank F (R ⧸ I) with hℓ
-  haveI : FiniteDimensional F (R ⧸ I) := finiteDimensional_quotient_of_ne_bot _ hI
+  have : FiniteDimensional F (R ⧸ I) := finiteDimensional_quotient_of_ne_bot _ hI
   obtain ⟨da, hda⟩ : ∃ da, da = (ℓ + 1) / 2 + 1 := ⟨_, rfl⟩
   obtain ⟨db, hdb⟩ : ∃ db, db = ℓ / 2 := ⟨_, rfl⟩
   set ψ : (Polynomial.degreeLT F da × Polynomial.degreeLT F db) →ₗ[F] (R ⧸ I) :=
@@ -809,7 +704,7 @@ theorem exists_mem_norm_natDegree_le [W.IsElliptic] (I : Ideal W.CoordinateRing)
   have hrange_le : Module.finrank F (LinearMap.range ψ) ≤ ℓ :=
     le_trans (Submodule.finrank_le _) (le_of_eq hℓ.symm)
   have hker_pos : 0 < Module.finrank F (LinearMap.ker ψ) := by lia
-  haveI : Nontrivial (LinearMap.ker ψ) := Module.nontrivial_of_finrank_pos hker_pos
+  have : Nontrivial (LinearMap.ker ψ) := Module.nontrivial_of_finrank_pos hker_pos
   obtain ⟨z, hz⟩ := exists_ne (0 : LinearMap.ker ψ)
   set pq := (z : Polynomial.degreeLT F da × Polynomial.degreeLT F db) with hpq
   have hpq_ne : pq ≠ 0 := fun h ↦ hz (Subtype.ext h)
@@ -873,7 +768,7 @@ theorem classReducesToCodimLEOne_holds [W.IsElliptic] :
   rw [hI', inv_inv] at hmk0
   rcases Nat.le_one_iff_eq_zero_or_eq_one.mp hfinle with h0 | h1
   · left
-    haveI : FiniteDimensional F (W.CoordinateRing ⧸ J) :=
+    have : FiniteDimensional F (W.CoordinateRing ⧸ J) :=
       finiteDimensional_quotient_of_ne_bot _ (mem_nonZeroDivisors_iff_ne_zero.mp hJmem)
     have hone : ClassGroup.mk0 ⟨J, hJmem⟩ = 1 :=
       mk0_eq_one_of_finrank_quotient_eq_zero J hJmem h0