IndisputableMonolith.Gravity.SevenGaps.Gap2PostingHistoryContinuumResidual

   1import IndisputableMonolith.Gravity.SevenGaps.Gap2PostingCocycleCarrier
   2import IndisputableMonolith.Gravity.SevenGaps.Gap2CertifiedFin8PhaseClose
   3import IndisputableMonolith.Gravity.SevenGaps.Gap2TickPhaseSubstrate
   4import IndisputableMonolith.Gravity.SevenGaps.Gap2TickPhaseTailBlocker
   5import IndisputableMonolith.Gravity.SevenGaps.Gap2TailAutFiberParityBlocker
   6import IndisputableMonolith.Gravity.SevenGaps.ZqContinuumBlocker
   7import IndisputableMonolith.Gravity.SevenGaps.ExactShellGaugeUV
   8
   9/-!
  10# Gap2 posting-history continuum residual (equal-strength API, OPEN)
  11
  12Codex-adjudicated next research surface after STOP A
  13(`Gap2PostingCocycleCarrier.carrier_forgets_posting_phase`): bank an
  14equal-strength continuum package on *actual posting histories*, not a
  15Gap2 flip and not a forgetful product enrichment.
  16
  17## Landed this module (API only)
  18
  19* `PostingHistoryContinuumData`: finite history carrier with forgetful
  20  map to `ExactPathClass`, Fin-8 phase, history mass `mu`, pushforward
  21  to `classMu`, and substantive Prop slots
  22  `generatedByActualPostingTransaction` / `historyGaugeInvariant`.
  23* `postingHistoryShellAmplitude`: history-level shell amplitude.
  24* `PostingHistoryExactCharacterBridge`: class-level tick with character
  25  pushforward, posting-transaction provenance, and `¬ ShellSigTick`.
  26* Named obligation `CharacterPushforwardOfForcedObligation` (OPEN).
  27* **OPEN** `TypedResidual_posting_history_attachment`.
  28* **OPEN** `TypedResidual_posting_history_continuum_close`.
  29* Adversarial gate: unconstrained `ExactPathClass × Fin 8` product is
  30  refused as continuum data without generation + pushforward discharge.
  31
  32## Explicit refusal
  33
  34Do **not** inhabit the continuum close by gluing
  35`PostingEnrichedPathClass` and setting generation to `True`. Do **not**
  36flip `gap2_continuum_and_measure`. No `sorry`, `admit`, new axiom, or
  37`native_decide`. Does **not** touch CPL.
  38
  39## S0 FREEZE (rev B plan H0, `D-qg-final-two-gates-plan-20260723`)
  40
  41Frozen 2026-07-23 BEFORE any mass or tick definition; changing any item
  42below is a new preregistration, not an edit.
  43
  44**Complete phase-blind history semantics.** For shell `n` and signature
  45`s : ShellSig n`, a posting construction history is a labeled exact
  46complex `K : ExactComplex (sigV s) (sigE s) (sigT s)` together with a
  47linear commit order on its posting alphabet
  48(`Fin (sigV s) ⊕ Fin (sigE s) ⊕ Fin (sigT s)`), i.e. a bijection from
  49`Fin (sigV s + sigE s + sigT s)`. The carrier contains ALL such pairs:
  50no phase-dependent, mass-dependent, or eligibility filtering of any
  51kind. Dual-entry state, where carried, is pinned to the canonical
  52balanced zero exactly as in `GaugeHistoryMeasure`.
  53
  54**Phase origin convention.** Every history initializes the eight-tick
  55transaction at Gray state `0` on the empty complex; the history phase is
  56the terminal transaction state after its commits
  57(`phaseBearingAdvance3 0 · `). A global origin shift rotates every
  58`tickRoot` by one fixed unit and preserves `OscillatoryTail`; ticks are
  59always compared at this fixed origin.
  60
  61**F1 ordered recipe list (finite; one verdict per entry; no shopping).**
  621. `commitCountTick`: every posted cell is one span-1 primitive commit;
  63   history phase = `phaseBearingAdvance3 0 (total commits)`. Registered
  64   risk at freeze time: may factor through `ShellSig` (decoy).
  652. `incidenceCoupledAdvance`: per-commit advance read from the posted
  66   cell's incidence relative to the current Gray-axis state; admissible
  67   ONLY if the coupling is a substrate theorem (new Foundation content),
  68   never a defined convenience.
  69After 2: list exhausted; STOP and re-plan with Jon.
  70-/
  71
  72namespace IndisputableMonolith
  73namespace Gravity
  74namespace SevenGaps
  75namespace Gap2PostingHistoryContinuumResidual
  76
  77open ExactShellGaugeUV
  78open ZqContinuumBlocker
  79open ZqShellBalanceBlocker
  80open Gap2TickPhaseSubstrate
  81open Gap2TickPhaseTailBlocker
  82open Gap2TailAutFiberParityBlocker
  83open Gap2PostingCocycleCarrier
  84open Gap2CertifiedFin8PhaseClose
  85open Classical
  86
  87noncomputable section
  88
  89/-! ## §1. Posting-history continuum data -/
  90
  91/-- Finite posting-history carrier with mass pushforward onto exact path
  92classes. The fields `generatedByActualPostingTransaction` and
  93`historyGaugeInvariant` are substantive Prop slots: inhabiting them is
  94the attachment work of a later session, not a `True`-shell. -/
  95structure PostingHistoryContinuumData where
  96  History : ℕ → Type
  97  instFintype : ∀ n, Fintype (History n)
  98  forget : ∀ {n}, History n → ExactPathClass n
  99  phase : ∀ {n}, History n → Fin 8
 100  mu : ∀ {n}, History n → ℝ
 101  /-- Histories are generated by an actual period-8 recognition posting
 102  transaction (not an external Fin-8 glue on classes). -/
 103  generatedByActualPostingTransaction : Prop
 104  /-- History labels / masses are gauge-invariant under the history
 105  equivalence appropriate to the posting substrate. -/
 106  historyGaugeInvariant : Prop
 107  mu_nonnegative : ∀ n (h : History n), 0 ≤ mu h
 108  /-- Pushforward of history mass recovers Burnside class mass. -/
 109  pushforward_mu :
 110    ∀ n (c : ExactPathClass n),
 111      (∑ h ∈ @Finset.univ (History n) (instFintype n),
 112        if forget h = c then mu h else (0 : ℝ)) = classMu c
 113
 114/-- History-level shell amplitude: mass-weighted 8th-root character sum. -/
 115def postingHistoryShellAmplitude (P : PostingHistoryContinuumData) (n : ℕ) :
 116    ℂ :=
 117  ∑ h ∈ @Finset.univ (P.History n) (P.instFintype n),
 118    (P.mu h : ℂ) * tickRoot (P.phase h)
 119
 120/-! ## §2. Character bridge to exact-path classes -/
 121
 122/-- Provenance: the class-level tick is forced by the posting phase on
 123histories that forget to that class. -/
 124def PostingTransactionForcedExactTick (P : PostingHistoryContinuumData)
 125    (tau : ∀ n, ExactPathClass n → Fin 8) : Prop :=
 126  ∀ n (h : P.History n), P.phase h = tau n (P.forget h)
 127
 128/-- Bridge from history continuum data to a class-level Fin-8 tick with
 129character pushforward and decoy escape. Uninhabited in this module. -/
 130structure PostingHistoryExactCharacterBridge
 131    (P : PostingHistoryContinuumData) where
 132  tau : ∀ n, ExactPathClass n → Fin 8
 133  /-- Fiberwise history character amplitude equals
 134  `classMu c * tickRoot (tau n c)`. -/
 135  character_pushforward :
 136    ∀ n (c : ExactPathClass n),
 137      (∑ h ∈ @Finset.univ (P.History n) (P.instFintype n),
 138          if P.forget h = c then (P.mu h : ℂ) * tickRoot (P.phase h)
 139          else (0 : ℂ)) =
 140        (classMu c : ℂ) * tickRoot (tau n c)
 141  provenance : PostingTransactionForcedExactTick P tau
 142  not_shellSig : ¬ ShellSigTick tau
 143
 144/-- Discharge helper (named obligation): forced posting tick plus mass
 145pushforward imply the bridge `character_pushforward` field. Banked as a
 146Prop so later sessions can prove it without widening this API surface;
 147not inhabited here. -/
 148def CharacterPushforwardOfForcedObligation
 149    (P : PostingHistoryContinuumData)
 150    (tau : ∀ n, ExactPathClass n → Fin 8) : Prop :=
 151  PostingTransactionForcedExactTick P tau →
 152    ∀ n (c : ExactPathClass n),
 153      (∑ h ∈ @Finset.univ (P.History n) (P.instFintype n),
 154          if P.forget h = c then (P.mu h : ℂ) * tickRoot (P.phase h)
 155          else (0 : ℂ)) =
 156        (classMu c : ℂ) * tickRoot (tau n c)
 157
 158/-- Packaging note: given `PostingHistoryExactCharacterBridge P`, the
 159identity
 160`postingHistoryShellAmplitude P n = exactShellAmplitude (tickDerivedPhase B.tau) n`
 161follows from fiber partition plus `character_pushforward`. Left as an
 162inhabitation obligation for a later session (not required to bank the
 163OPEN residuals). -/
 164def postingHistoryAmplitudeMatchesExactShell
 165    (P : PostingHistoryContinuumData)
 166    (B : PostingHistoryExactCharacterBridge P) : Prop :=
 167  ∀ n : ℕ,
 168    postingHistoryShellAmplitude P n =
 169      exactShellAmplitude (tickDerivedPhase B.tau) n
 170
 171/-! ## §3. OPEN residuals -/
 172
 173/-- **OPEN residual.** Missing substrate attachment: actual posting
 174histories related to `ExactComplex` / exact classes so that
 175`PostingHistoryContinuumData` is inhabited with
 176`generatedByActualPostingTransaction` and `historyGaugeInvariant`.
 177STOP A product enrichment does not discharge this. -/
 178def TypedResidual_posting_history_attachment : Prop :=
 179  ∃ P : PostingHistoryContinuumData,
 180    P.generatedByActualPostingTransaction ∧ P.historyGaugeInvariant
 181
 182/-- Packaged continuum close: continuum data + character bridge +
 183oscillatory tail on the derived tick phase. -/
 184structure PostingHistoryContinuumClose where
 185  data : PostingHistoryContinuumData
 186  bridge : PostingHistoryExactCharacterBridge data
 187  tail : OscillatoryTail (tickDerivedPhase bridge.tau)
 188
 189/-- **OPEN residual.** Nonempty posting-history continuum close
 190(equal-strength terminal toward
 191`TypedResidual_continuum_substrate_oscillatoryTail` /
 192`CertifiedGap2Fin8PhaseClose`). Deliberately uninhabited here. -/
 193def TypedResidual_posting_history_continuum_close : Prop :=
 194  Nonempty PostingHistoryContinuumClose
 195
 196/-- Contingent discharge into bare continuum R5 shape. Does not inhabit
 197either residual. -/
 198theorem bare_r5_of_posting_history_continuum_close
 199    (h : TypedResidual_posting_history_continuum_close) :
 200    BareR5ResidualShape := by
 201  obtain ⟨pack⟩ := h
 202  exact ⟨tickDerivedPhase pack.bridge.tau, pack.tail,
 203    zeroPhase_not_oscillatoryTail⟩
 204
 205/-! ## §3b. V2 certified close (H0, rev B: fake close type-impossible) -/
 206
 207/-- **V2 certified posting-history close.** Strengthens
 208`PostingHistoryContinuumClose` per the rev B plan H0: the package must
 209PROVE the data's own provenance Props (a `True`-shell instantiation
 210still has to prove whatever Prop the data declared, and an honest data
 211declares the substantive generation/gauge statements), carry the
 212amplitude equality with the exact shell sum, and carry the decoy
 213escapes. This is the ONLY close shape a gap2 flip binding may cite. -/
 214structure PostingHistoryCertifiedCloseV2 where
 215  data : PostingHistoryContinuumData
 216  generated : data.generatedByActualPostingTransaction
 217  gaugeInvariant : data.historyGaugeInvariant
 218  bridge : PostingHistoryExactCharacterBridge data
 219  amplitude_matches : postingHistoryAmplitudeMatchesExactShell data bridge
 220  not_shellConstant : ¬ ShellConstant (tickDerivedPhase bridge.tau)
 221  not_eventuallyZero : ¬ EventuallyZeroPhase (tickDerivedPhase bridge.tau)
 222  tail : OscillatoryTail (tickDerivedPhase bridge.tau)
 223
 224/-- **OPEN residual.** Nonempty V2 certified close. Deliberately
 225uninhabited here. -/
 226def TypedResidual_posting_history_certified_close_v2 : Prop :=
 227  Nonempty PostingHistoryCertifiedCloseV2
 228
 229/-- V2 discharges the weaker packaged close (contingent; does not
 230inhabit). -/
 231theorem posting_history_continuum_close_of_v2
 232    (h : TypedResidual_posting_history_certified_close_v2) :
 233    TypedResidual_posting_history_continuum_close := by
 234  obtain ⟨pack⟩ := h
 235  exact ⟨⟨pack.data, pack.bridge, pack.tail⟩⟩
 236
 237/-- V2 discharges the certified Fin-8 close under the
 238`postingHistoryForced` recipe kind (contingent; does not inhabit). -/
 239theorem certified_fin8_phase_close_of_v2
 240    (h : TypedResidual_posting_history_certified_close_v2) :
 241    TypedResidual_certified_fin8_phase_close := by
 242  obtain ⟨pack⟩ := h
 243  exact ⟨{ tau := pack.bridge.tau
 244           provenance :=
 245             { kind := CertifiedTickRecipeKind.postingHistoryForced
 246               globalEquivalentProvenance := trivial
 247               notAmplitudeOracle := trivial
 248               notShellwiseClassicalAssembly := trivial
 249               tickDerived := trivial }
 250           not_shellConstant := pack.not_shellConstant
 251           not_eventuallyZero := pack.not_eventuallyZero
 252           not_decoy := pack.bridge.not_shellSig
 253           tail := pack.tail }⟩
 254
 255/-- V2 discharges the bare continuum R5 shape (contingent). -/
 256theorem bare_r5_of_v2
 257    (h : TypedResidual_posting_history_certified_close_v2) :
 258    BareR5ResidualShape :=
 259  bare_r5_of_posting_history_continuum_close
 260    (posting_history_continuum_close_of_v2 h)
 261
 262/-! ## §4. Adversarial gate (product enrichment refused) -/
 263
 264/-- Named refusal: unconstrained product enrichment
 265(`PostingEnrichedPathClass` = `ExactPathClass × Fin 8`) is **not** an
 266inhabitant of `PostingHistoryContinuumData` for continuum-close purposes
 267without discharging `generatedByActualPostingTransaction` and
 268`pushforward_mu` from actual posting-transaction provenance.
 269
 270This is an API honesty gate linked to STOP A
 271(`carrier_forgets_posting_phase`): the forgetful product erases posting
 272phase under descent and cannot be laundered into continuum data by
 273setting the generation Prop to `True`. -/
 274def AdversarialGate_unconstrainedProductRefused : Prop :=
 275  ¬ FactorsThroughPostingForget fun _n H => postingPhase H
 276
 277/-- The adversarial gate holds by STOP A (carrier forgets posting phase). -/
 278theorem adversarialGate_unconstrainedProductRefused :
 279    AdversarialGate_unconstrainedProductRefused :=
 280  postingPhase_not_factors_through_forget
 281
 282/-- Packaging note: any continuum-data candidate built from product
 283enrichment must still face the generation + pushforward fields; the
 284STOP A forgetfulness theorem remains the discriminator against fake
 285descent. -/
 286theorem continuumData_refuses_forgetful_product_descent :
 287    TypedResidual_carrier_forgets_posting_phase :=
 288  typedResidual_carrier_forgets_posting_phase
 289
 290/-! ## §5. Status (API banked; attachment/close OPEN; gap2 unflipped) -/
 291
 292structure Gap2PostingHistoryContinuumResidualStatus where
 293  apiBanked : Bool
 294  s0FreezeBanked : Bool
 295  v2CertifiedCloseBanked : Bool
 296  attachmentOpen : Bool
 297  continuumCloseOpen : Bool
 298  v2CloseOpen : Bool
 299  adversarialGateBanked : Bool
 300  certifiedCloseInhabited : Bool
 301  gap2ContinuumAndMeasure : Bool
 302
 303def gap2PostingHistoryContinuumResidualStatus :
 304    Gap2PostingHistoryContinuumResidualStatus where
 305  apiBanked := true
 306  s0FreezeBanked := true
 307  v2CertifiedCloseBanked := true
 308  attachmentOpen := true
 309  continuumCloseOpen := true
 310  v2CloseOpen := true
 311  adversarialGateBanked := true
 312  certifiedCloseInhabited := false
 313  gap2ContinuumAndMeasure := false
 314
 315theorem gap2PostingHistoryContinuumResidualStatus_flags :
 316    gap2PostingHistoryContinuumResidualStatus.apiBanked = true ∧
 317      gap2PostingHistoryContinuumResidualStatus.s0FreezeBanked = true ∧
 318      gap2PostingHistoryContinuumResidualStatus.v2CertifiedCloseBanked =
 319        true ∧
 320      gap2PostingHistoryContinuumResidualStatus.attachmentOpen = true ∧
 321        gap2PostingHistoryContinuumResidualStatus.continuumCloseOpen =
 322          true ∧
 323          gap2PostingHistoryContinuumResidualStatus.v2CloseOpen = true ∧
 324          gap2PostingHistoryContinuumResidualStatus.adversarialGateBanked =
 325            true ∧
 326            gap2PostingHistoryContinuumResidualStatus.certifiedCloseInhabited =
 327              false ∧
 328              gap2PostingHistoryContinuumResidualStatus.gap2ContinuumAndMeasure =
 329                false := by
 330  decide
 331
 332end
 333
 334end Gap2PostingHistoryContinuumResidual
 335end SevenGaps
 336end Gravity
 337end IndisputableMonolith
 338