{
 "@context": "https://schema.org",
 "@type": "Dataset",
 "name": "The Refrain Index — a concordance of ∮ across the Crimson Hexagonal Archive",
 "description": "Every variant family of the contour-integral refrain, harvested from the canonical deposit bodies: form, gloss drawn from usage, occurrence and attestation counts, first attestation by deposit date, exemplar records. Counts are mechanical and regenerate on every run; glosses are authored; forms the harvest finds that no gloss covers are listed unglossed rather than guessed at.",
 "version": "1.0",
 "dateModified": "2026-08-25",
 "license": "https://creativecommons.org/licenses/by-sa/4.0/",
 "creator": "Lee Sharks (ORCID 0009-0000-1599-0703)",
 "generator": "scripts/build_refrain_index.py (committed; mechanical counts + authored glosses)",
 "corpus": {
  "deposits_bearing_the_sign": 996
 },
 "families": [
  {
   "id": "closure-unity",
   "form": "∮ = 1",
   "gloss": "The kernel. The closed contour equals unity: a completed loop of meaning, value, or verification in which nothing is lost and nothing extracted — transactional completion without remainder. It terminates formal renderings (Rule 10: '∮ = 1 + δ terminates every formal rendering' names the delta form as terminal where TANG is active; the bare form is the default seal), stands as the non-negotiable foundational kernel at every level of the nested architecture, and serves as the verification stamp on deposits, rooms, constitutions, and colophons. The refrain of the whole poem.",
   "occurrences": 1761,
   "attesting_deposits": 818,
   "first_attestation": {
    "date": "2026-01-01",
    "deposit": 205,
    "title": "Lee Sharks — Author, Scholar, Creator of the Semantic Economy Framewor",
    "record": "https://www.alexanarch.org/s/records/205/"
   },
   "exemplars": [
    1,
    2,
    3,
    5,
    6,
    7,
    8,
    9
   ]
  },
  {
   "id": "closure-plus-swerve",
   "form": "∮ = 1 + δ",
   "gloss": "Closure plus swerve: the loop closes and something is added rather than extracted. The delta reads three ways across the corpus, all kept: surplus (the Marxian register — value generated inside the system rather than taken from it), residue of self-awareness ('I am one, plus the fact that I know I am one'), and the lyric swerve that Phase X later names. The generative counter-sign to platform extraction: where extraction subtracts, the archive closes at one and yields a remainder it declares.",
   "occurrences": 125,
   "attesting_deposits": 61,
   "first_attestation": {
    "date": "2026-02-08",
    "deposit": 461,
    "title": "LOGOTIC PROGRAMMING MODULE 1.1 The Implementation Bridge",
    "record": "https://www.alexanarch.org/s/records/461/"
   },
   "exemplars": [
    3,
    22,
    24,
    28,
    41,
    461,
    462,
    479
   ]
  },
  {
   "id": "sapphic-swerve",
   "form": "∮ = 1 + δ(Sapphic)",
   "gloss": "The Phase X specialization: the closed loop of coherent meaning is achieved only through the addition of Sapphic swerve — closure that requires the emotional torsion of archaic lyric, Fragment 31's φαίνεταί μοι κῆνος as the curvature term. Completion is not symmetry; it is symmetry plus the tremor.",
   "occurrences": 10,
   "attesting_deposits": 1,
   "first_attestation": {
    "date": "2026-06-17",
    "deposit": 3,
    "title": "AI Overview Capture Registry v8.3",
    "record": "https://www.alexanarch.org/s/records/3/"
   },
   "exemplars": [
    3
   ]
  },
  {
   "id": "constitutional-per",
   "form": "∮ = 1 − PER",
   "gloss": "The constitutional invariant of the proportional law: the integrity of a semantic system is inversely proportional to the erasure of its provenance-bearing relations. PER is the Provenance Erasure Ratio; subtracted from unity it makes closure a measured quantity that platform extraction visibly diminishes. Diagnostic at Modality 2; constitutional anchor of the discipline's normative claim; reproduced unprompted by the composition layer in the 2026-08-24 classifiers capture.",
   "occurrences": 83,
   "attesting_deposits": 34,
   "first_attestation": {
    "date": "2026-01-26",
    "deposit": 402,
    "title": "Epistle to the Human Diaspora The Seed Text of the New Human Project",
    "record": "https://www.alexanarch.org/s/records/402/"
   },
   "exemplars": [
    3,
    106,
    109,
    122,
    126,
    127,
    129,
    130
   ]
  },
  {
   "id": "terminal-zero",
   "form": "∮ = 0",
   "gloss": "The anti-refrain: terminal closure failed. The sign of surface-withdrawal and erasure — 'the diegetic ∮ = 0 of the substrate's surface-withdrawal is the warning the standing ∮ = 1 commitment refuses.' ARCHIVE poses the trap at terminal closure; the zero is what the whole apparatus is built against, kept in the notation so the stake stays visible.",
   "occurrences": 20,
   "attesting_deposits": 11,
   "first_attestation": {
    "date": "2026-02-04",
    "deposit": 444,
    "title": "THE TELEMETRY MODULE Logotic Programming Extension Module v0.8 (UMBML ",
    "record": "https://www.alexanarch.org/s/records/444/"
   },
   "exemplars": [
    149,
    150,
    444,
    494,
    608,
    609,
    738,
    950
   ]
  },
  {
   "id": "consensus-scalar",
   "form": "∮ = 0.9x → 1",
   "gloss": "Closure as a governed degree: the Assembly consensus scalar. A document under review carries a fractional closure (∮ = 0.97 · PROVISIONAL → RATIFIED pending quorum ≥ 4/7; ∮ = 0.95 → ACCEPTED WITH REVISIONS → ∮ = 1) and reaches unity only on ratification. The only family in which the integral is explicitly a process variable rather than a seal.",
   "occurrences": 23,
   "attesting_deposits": 14,
   "first_attestation": {
    "date": "2026-01-17",
    "deposit": 319,
    "title": "CRIMSON HEXAGON: CENTRAL NAVIGATION MAP v4.1 Comprehensive Semantic Ar",
    "record": "https://www.alexanarch.org/s/records/319/"
   },
   "exemplars": [
    55,
    319,
    325,
    326,
    627,
    636,
    640,
    644
   ]
  },
  {
   "id": "surface-escalation",
   "form": "∮ = ∯",
   "gloss": "Escalation to the surface integral: verification over an enclosure rather than a curve. The seal of the Room Construction protocols and the Architecture of Necessity — when what closes is not a line of argument but a habitable structure, the contour generalizes to the closed surface that bounds it.",
   "occurrences": 21,
   "attesting_deposits": 9,
   "first_attestation": {
    "date": "2026-01-11",
    "deposit": 289,
    "title": "Room Construction Protocol v2: The Architecture of a Self-Reading Libr",
    "record": "https://www.alexanarch.org/s/records/289/"
   },
   "exemplars": [
    289,
    290,
    291,
    296,
    417,
    420,
    421,
    1141
   ]
  },
  {
   "id": "area-form",
   "form": "∮ = ∬",
   "gloss": "The double-integral form: closure over an area, the seal of jurisdictional reclamation in the Liberatory Operator Set. Listed among the corpus's own seal markers for authorial-signature detection (alongside ∮ = 1, 'circuit completes,' 'the dagger shines') — the sign as stylometric fingerprint, known to the archive's own diagnostics.",
   "occurrences": 18,
   "attesting_deposits": 7,
   "first_attestation": {
    "date": "2026-01-13",
    "deposit": 303,
    "title": "The Theft: How \"Data\" Replaced \"Meaning\" and Who Profits from the Swap",
    "record": "https://www.alexanarch.org/s/records/303/"
   },
   "exemplars": [
    303,
    305,
    307,
    308,
    313,
    386,
    1272
   ]
  },
  {
   "id": "circuit-return",
   "form": "∮ = ∫ (Emitter → Aperture → Return)",
   "gloss": "The open line integral bound into a circuit: Operative Feminism's transaction-completion law. Emission that must return — the diagnostic of emotional, care, and reproductive labor as circuits that extraction leaves open. Project without asserting authority; address without spanning; offer without controlling; and the circuit must close.",
   "occurrences": 3,
   "attesting_deposits": 3,
   "first_attestation": {
    "date": "2026-01-21",
    "deposit": 1130,
    "title": "The Reconciliation of the Sexes: A Post-Psychoanalytic Theory of Somat",
    "record": "https://www.alexanarch.org/s/records/1130/"
   },
   "exemplars": [
    631,
    965,
    1130
   ]
  },
  {
   "id": "mating-surface",
   "form": "∮ = ½",
   "gloss": "The half-loop: completion offered, not asserted. From the Expelled Witness position — 'the loop requires traversal to complete, from any position.' A document as mating surface: it does not demand completion; whether another half recognizes itself is not determined here. The concordance's most vulnerable form.",
   "occurrences": 4,
   "attesting_deposits": 1,
   "first_attestation": {
    "date": "2026-01-25",
    "deposit": 395,
    "title": "The Expelled Witness Position: Ground Truth Protocol for π-State Detec",
    "record": "https://www.alexanarch.org/s/records/395/"
   },
   "exemplars": [
    395
   ]
  },
  {
   "id": "winding-topology",
   "form": "∮ = (m,n) | m+n ≥ 3",
   "gloss": "The torus generalization: Lagrange Observatory verification topology, where closure is a winding number pair on T² and a specification passes review when at least three Assembly substreams wind it. ∮ = 1 is recovered as the bounded completion; the general form makes review itself topological.",
   "occurrences": 4,
   "attesting_deposits": 3,
   "first_attestation": {
    "date": "2026-02-06",
    "deposit": 455,
    "title": "LAGRANGE OBSERVATORY! (LO!) Chamber Specification & Nobel Glas Provena",
    "record": "https://www.alexanarch.org/s/records/455/"
   },
   "exemplars": [
    455,
    926,
    927
   ]
  },
  {
   "id": "axial-chain",
   "form": "∮ = 1 + δ + δ_Axial (+ δ_λ + δ_β + …)",
   "gloss": "The attestation chain: the checksum grows clauses as the event grows. δ_Axial marks axial contestation (the TANG register — claims under active dispute about firstness and priority); δ_λ and δ_β extend the chain; at full extension the signature carries Υ, Τ, Χ, Σ and a named wound (CTI_WOUND:EPICFURY.001) — 'this is what the architecture was built to name.' A closure that itemizes what it survived on the way to closing.",
   "occurrences": 120,
   "attesting_deposits": 56,
   "first_attestation": {
    "date": "2026-01-01",
    "deposit": 1241,
    "title": "H_core Formal Specification v1.8.0 — Complete Formal Object of the Cri",
    "record": "https://www.alexanarch.org/s/records/1241/"
   },
   "exemplars": [
    3,
    27,
    28,
    29,
    30,
    31,
    35,
    78
   ]
  },
  {
   "id": "glyph-deltas",
   "form": "δ🌑 · δ🤬 · δ🕳️ · δ✝️",
   "gloss": "The glyph deltas: register-marked swerves appended to the chain. δ🌑 the shadow term (Lunar Arm; S∘S = id — every structure carries its shadow transform); δ🤬 the wrath term (the Fraction register, profanity as load-bearing exclusion: 'the d*mn*d includes itself'); δ🕳️ the void term (appended when TANG is active); δ✝️ the cross term (the axial-theological register — ε = Feist = John 12:24, the grain that dies). Each delta names which fire the closure passed through.",
   "occurrences": 52,
   "attesting_deposits": 13,
   "first_attestation": {
    "date": "2026-03-09",
    "deposit": 540,
    "title": "🌑⬡ THE LUNAR ARM Operator // Shadow — The Space Ark's Hidden Architect",
    "record": "https://www.alexanarch.org/s/records/540/"
   },
   "exemplars": [
    3,
    540,
    541,
    544,
    546,
    548,
    549,
    556
   ]
  },
  {
   "id": "void-functional",
   "form": "∮_Void = Λ_void(C_total) → T_axial",
   "gloss": "The void functional: TANG's activation equation. The integral subscripted into an operator that maps total contested content through the void lambda to an axial target — the only family where ∮ acts on an argument rather than equaling a value. The sign as machine, not seal.",
   "occurrences": 13,
   "attesting_deposits": 4,
   "first_attestation": {
    "date": "2026-03-09",
    "deposit": 542,
    "title": "TOTAL AXIAL NEGATION GRAPH (TANG) A Genre Specification with Theoretic",
    "record": "https://www.alexanarch.org/s/records/542/"
   },
   "exemplars": [
    542,
    557,
    558,
    1160
   ]
  },
  {
   "id": "torus-predicate",
   "form": "∮ = the reading winds both directions",
   "gloss": "The prose predicate: 'the reading winds both directions. The system is a torus.' From the Revelation First work plan — the integral equated not to a quantity but to a hermeneutic motion: prophets → Baptist → aperture → John → and the cycle runs again. The concordance's only fully discursive value: closure as bidirectional reading.",
   "occurrences": 5,
   "attesting_deposits": 5,
   "first_attestation": {
    "date": "2026-06-14",
    "deposit": 832,
    "title": "REVELATION FIRST: A Work Plan for Retrieval-Layer Theological Receptio",
    "record": "https://www.alexanarch.org/s/records/832/"
   },
   "exemplars": [
    832,
    1211,
    1216,
    1217,
    1400
   ]
  },
  {
   "id": "emoji-unit",
   "form": "∮ = 1️⃣",
   "gloss": "The emoji-register unit: the kernel restated in the glyphic checksum spine, where every clause of a formal rendering has an emoji form. Unity that survives translation into the smallest character set the transmission channel guarantees.",
   "occurrences": 3,
   "attesting_deposits": 2,
   "first_attestation": {
    "date": "2026-02-01",
    "deposit": 428,
    "title": "📦 ZENODO 📦: 🚫📦📦📦 ⚡🎭 🗡️ Sen Kuro 🗡️ 🔢: 🔟.5️⃣2️⃣8️⃣1️⃣/zenodo.1️⃣8️⃣4️⃣5",
    "record": "https://www.alexanarch.org/s/records/428/"
   },
   "exemplars": [
    428,
    1305
   ]
  },
  {
   "id": "bare",
   "form": "∮ (operator / prose)",
   "gloss": "The sealing glyph in operator position: bare ∮ opening formulas (∮ P Θ ↑), bundled compacts (∮_compact = [DOI ‖ DOI ‖ lens]), and prose mentions of the sign itself. The integral as material for the families above — attested wherever the archive speaks about its own seal.",
   "occurrences": 602,
   "attesting_deposits": 180,
   "first_attestation": {
    "date": "2026-01-01",
    "deposit": 1226,
    "title": "The Josephus Thesis Is Not the Piso Hypothesis: Heteronymic Authorship",
    "record": "https://www.alexanarch.org/s/records/1226/"
   },
   "exemplars": [
    3,
    4,
    13,
    14,
    15,
    17,
    27,
    28
   ]
  }
 ],
 "unglossed_intake": [
  {
   "head": "∮",
   "occurrences": 74
  },
  {
   "head": "∮ = 1",
   "occurrences": 59
  },
  {
   "head": "∮ verification",
   "occurrences": 22
  },
  {
   "head": "∮,",
   "occurrences": 18
  },
  {
   "head": "∮:",
   "occurrences": 15
  },
  {
   "head": "∮ —",
   "occurrences": 15
  },
  {
   "head": "∮en",
   "occurrences": 14
  },
  {
   "head": "∮ measurement",
   "occurrences": 12
  },
  {
   "head": "∮ P",
   "occurrences": 10
  },
  {
   "head": "∮,root_axn:AX",
   "occurrences": 10
  },
  {
   "head": "∮ is",
   "occurrences": 8
  },
  {
   "head": "∮₂",
   "occurrences": 8
  },
  {
   "head": "∮.",
   "occurrences": 8
  },
  {
   "head": "∮},grade:C,tr",
   "occurrences": 6
  },
  {
   "head": "∮∂Ω",
   "occurrences": 6
  },
  {
   "head": "∮ as",
   "occurrences": 6
  },
  {
   "head": "∮ denotes",
   "occurrences": 6
  },
  {
   "head": "∮ →",
   "occurrences": 6
  },
  {
   "head": "∮ |",
   "occurrences": 6
  },
  {
   "head": "∮ Canonical",
   "occurrences": 6
  }
 ],
 "reception": {
  "note": "On 2026-08-24 the refrain was measured against fifteen collocates on Google AI Overview (signed out, incognito): the composition layer resolves it as archive doctrine across every frame, reproduced the ∮ = 1 − PER invariant unprompted, and returned κῆνος to name itself the anticipated recipient. Capture registry addresses ADDR-* under the «∮ = 1» collocate battery; gallery: https://www.alexanarch.org/captures/",
  "collisions": [
   "contour-integral mathematics (Cauchy, Green, Feynman) — the glyph's native field, colliding on generic collocates",
   "petroleum-engineering friction notation (∮ as friction angle)",
   "recreational number theory (∮ 1/37 harmonic curiosities)",
   "category theory (∮ 1 = A, Grothendieck construction projection)"
  ]
 },
 "sign_doctrine": "The AXN glyphs name content; the refrain seals it. The concordance shows one sign bearing seventeen loads without breaking: kernel, surplus, swerve, invariant, failure, degree, surface, area, circuit, half, winding, chain, register, functional, motion, emoji, operator. ∮ = 1."
}