IndisputableMonolith.Gravity.SevenGaps.Gap2EnrichedCarrierPhase

   1import IndisputableMonolith.Gravity.SevenGaps.Gap2TickPhaseTailBlocker
   2import IndisputableMonolith.Gravity.SevenGaps.Gap2SignatureBlockerAttack
   3
   4/-!
   5# Wave C R5 attack: enriched-carrier phase below `ExactPathClass`
   6
   7Implements decision `D-qg-c1-r4-enriched-carrier-20260722` on the
   8continuum R5 residual
   9
  10    TypedResidual_continuum_substrate_oscillatoryTail
  11      := ∃ phase, OscillatoryTail phase ∧ ¬ OscillatoryTail zeroPhase
  12
  13after the signature-level Fin-8 attack stalled (mesoscopic-only cube
  14dominance; `SignatureFin8OscillatoryTailBlocker` DEFINED unproved).
  15
  16## Choice this session: A attempted → C terminal
  17
  18Route A (enriched carrier forcing eventual mass balance / identical-zero
  19late amplitudes) is the design-correct attack surface. This module banks
  20the carrier API, descent, and a concrete quotient-internal tick that
  21escapes `ShellSigTick`. Proving eventual fiber-mass balance (or
  22identical-zero late amplitudes) for that tick refused in one Elmo
  23session: no honest partition of Burnside masses is available from the
  24self-loop invariant alone.
  25
  26Route B (`SignatureFin8OscillatoryTailBlocker`) refused: late-shell mass
  27fragmentation makes the blocker less plausible, not more; the prior
  28session already killed dominance as a proof strategy.
  29
  30Therefore the credit-bearing terminal is route C: a sharper typed
  31residual naming the enriched-carrier obligation, plus characterization
  32and bridge lemmas. R5 itself stays OPEN (uninhabited).
  33
  34Does NOT flip `gap2_continuum_and_measure`. No `sorry`, `admit`, new
  35axiom, or `native_decide`. Does NOT edit `Gap2ContinuumMeasureResidualDAG`.
  36-/
  37
  38namespace IndisputableMonolith
  39namespace Gravity
  40namespace SevenGaps
  41namespace Gap2EnrichedCarrierPhase
  42
  43open ExactShellGaugeUV
  44open ZqContinuumBlocker
  45open Gap2TickPhaseSubstrate
  46open Gap2TickPhaseTailBlocker
  47open Gap2SignatureBlockerAttack
  48
  49noncomputable section
  50
  51/-! ## §1. Enriched labeled tick (GlobalEquivalent-invariant) -/
  52
  53/-- A Fin-8 tick on labeled exact complexes. -/
  54abbrev LabeledTick := ∀ v e t : ℕ, ExactComplex v e t → Fin 8
  55
  56/-- **Enrichment hyp.** Tick is constant on `GlobalEquivalent` orbits. -/
  57def GlobalEquivalentInvariant (lab : LabeledTick) : Prop :=
  58  ∀ (v e t : ℕ) (K K' : ExactComplex v e t),
  59    GlobalEquivalent K K' → lab v e t K = lab v e t K'
  60
  61/-- Descent of an invariant labeled tick to exact path classes. -/
  62noncomputable def descendedTick
  63    (lab : LabeledTick) (hInv : GlobalEquivalentInvariant lab) :
  64    ∀ n : ℕ, ExactPathClass n → Fin 8 :=
  65  fun _n c =>
  66    Quotient.lift (lab (sigV c.1) (sigE c.1) (sigT c.1))
  67      (fun K K' h => hInv _ _ _ K K' h) c.2
  68
  69theorem descendedTick_mk
  70    (lab : LabeledTick) (hInv : GlobalEquivalentInvariant lab)
  71    {n : ℕ} (s : ShellSig n) (K : ExactComplex (sigV s) (sigE s) (sigT s)) :
  72    descendedTick lab hInv n ⟨s, Quotient.mk _ K⟩ =
  73      lab (sigV s) (sigE s) (sigT s) K :=
  74  rfl
  75
  76/-- Derived real phase of an enriched labeled tick. -/
  77def enrichedPhase
  78    (lab : LabeledTick) (hInv : GlobalEquivalentInvariant lab) :
  79    ∀ n : ℕ, ExactPathClass n → ℝ :=
  80  tickDerivedPhase (descendedTick lab hInv)
  81
  82/-! ## §2. Concrete enriched invariant: self-loop count mod 8 -/
  83
  84/-- Number of loop edges (both endpoints equal). -/
  85noncomputable def selfLoopCount {v e t : ℕ} (K : ExactComplex v e t) : ℕ := by
  86  classical
  87  exact (Finset.univ.filter
  88    (fun i : Fin e => (K.edgeVerts i).1 = (K.edgeVerts i).2)).card
  89
  90theorem selfLoopCount_congr {v e t : ℕ} {K K' : ExactComplex v e t}
  91    (r : ExactRelabel K K') :
  92    selfLoopCount K = selfLoopCount K' := by
  93  classical
  94  unfold selfLoopCount
  95  let ε := r.eEquiv
  96  have hiff : ∀ i : Fin e,
  97      ((K.edgeVerts i).1 = (K.edgeVerts i).2) ↔
  98        ((K'.edgeVerts (ε i)).1 = (K'.edgeVerts (ε i)).2) := by
  99    intro i
 100    have h := r.edge_comm i
 101    constructor
 102    · intro hloop
 103      have : r.vEquiv (K.edgeVerts i).1 = r.vEquiv (K.edgeVerts i).2 := by
 104        simp [hloop]
 105      have hK' : K'.edgeVerts (ε i) =
 106          (r.vEquiv (K.edgeVerts i).1, r.vEquiv (K.edgeVerts i).2) := by
 107        simpa [Prod.map] using h
 108      simpa [hK'] using this
 109    · intro hloop
 110      have hK' : K'.edgeVerts (ε i) =
 111          (r.vEquiv (K.edgeVerts i).1, r.vEquiv (K.edgeVerts i).2) := by
 112        simpa [Prod.map] using h
 113      have : r.vEquiv (K.edgeVerts i).1 = r.vEquiv (K.edgeVerts i).2 := by
 114        simpa [hK'] using hloop
 115      exact r.vEquiv.injective this
 116  have himg :
 117      Finset.univ.filter
 118          (fun i : Fin e => (K'.edgeVerts i).1 = (K'.edgeVerts i).2) =
 119        (Finset.univ.filter
 120          (fun i : Fin e => (K.edgeVerts i).1 = (K.edgeVerts i).2)).image ε := by
 121    ext j
 122    simp only [Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_image]
 123    constructor
 124    · intro hj
 125      refine ⟨ε.symm j, ?_, ε.apply_symm_apply j⟩
 126      exact (hiff (ε.symm j)).mpr (by simpa [ε.apply_symm_apply] using hj)
 127    · rintro ⟨i, hi, rfl⟩
 128      exact (hiff i).mp hi
 129  rw [himg, Finset.card_image_of_injective _ ε.injective]
 130
 131theorem selfLoopCount_ge_invariant {v e t : ℕ} {K K' : ExactComplex v e t}
 132    (h : GlobalEquivalent K K') :
 133    selfLoopCount K = selfLoopCount K' := by
 134  obtain ⟨r⟩ := h
 135  exact selfLoopCount_congr r
 136
 137/-- Labeled tick: self-loop count mod 8. -/
 138noncomputable def selfLoopTick : LabeledTick :=
 139  fun _v _e _t K =>
 140    ⟨selfLoopCount K % 8, Nat.mod_lt _ (by norm_num : (0 : ℕ) < 8)⟩
 141
 142theorem selfLoopTick_invariant : GlobalEquivalentInvariant selfLoopTick := by
 143  intro v e t K K' h
 144  apply Fin.ext
 145  change selfLoopCount K % 8 = selfLoopCount K' % 8
 146  rw [selfLoopCount_ge_invariant h]
 147
 148/-- Descended class-level tick from self-loop counts. -/
 149def selfLoopClassTick : ∀ n : ℕ, ExactPathClass n → Fin 8 :=
 150  descendedTick selfLoopTick selfLoopTick_invariant
 151
 152def selfLoopPhase : ∀ n : ℕ, ExactPathClass n → ℝ :=
 153  tickDerivedPhase selfLoopClassTick
 154
 155/-! ## §3. Witness: self-loop tick is NOT a `ShellSigTick` -/
 156
 157/-- Two loops at vertex 0, signature `(2,2,0)`. -/
 158def twoLoopsComplex : ExactComplex 2 2 0 where
 159  edgeVerts := fun _ => (0, 0)
 160  tetVerts := fun i => i.elim0
 161
 162/-- Two parallel non-loop edges `(0,1)`, signature `(2,2,0)`. -/
 163def twoBridgesComplex : ExactComplex 2 2 0 where
 164  edgeVerts := fun _ => (0, 1)
 165  tetVerts := fun i => i.elim0
 166
 167theorem selfLoopCount_twoLoops : selfLoopCount twoLoopsComplex = 2 := by
 168  classical
 169  unfold selfLoopCount twoLoopsComplex
 170  have h :
 171      (Finset.univ.filter
 172        (fun i : Fin 2 =>
 173          (((fun _ : Fin 2 => ((0 : Fin 2), (0 : Fin 2))) i).1 =
 174            ((fun _ : Fin 2 => ((0 : Fin 2), (0 : Fin 2))) i).2))) =
 175        (Finset.univ : Finset (Fin 2)) := by
 176    ext i
 177    simp
 178  rw [h]
 179  simp
 180
 181theorem selfLoopCount_twoBridges : selfLoopCount twoBridgesComplex = 0 := by
 182  classical
 183  unfold selfLoopCount twoBridgesComplex
 184  have h :
 185      (Finset.univ.filter
 186        (fun i : Fin 2 =>
 187          (((fun _ : Fin 2 => ((0 : Fin 2), (1 : Fin 2))) i).1 =
 188            ((fun _ : Fin 2 => ((0 : Fin 2), (1 : Fin 2))) i).2))) =
 189        (∅ : Finset (Fin 2)) := by
 190    ext i
 191    simp
 192  rw [h]
 193  simp
 194
 195theorem not_ge_twoLoops_twoBridges :
 196    ¬ GlobalEquivalent twoLoopsComplex twoBridgesComplex := by
 197  intro h
 198  have := selfLoopCount_ge_invariant h
 199  rw [selfLoopCount_twoLoops, selfLoopCount_twoBridges] at this
 200  exact (by decide : ¬ (2 = 0)) this
 201
 202/-- Shell signature `(2,2,0)`. -/
 203def doubleEdgeSig : ShellSig 2 :=
 204  ⟨(⟨2, by norm_num⟩, ⟨2, by norm_num⟩, ⟨0, by norm_num⟩), by
 205    change max (2 : ℕ) (max 2 0) = 2
 206    rw [Nat.max_zero, max_self]⟩
 207
 208def twoLoopsClass : ExactPathClass 2 :=
 209  ⟨doubleEdgeSig, Quotient.mk _ twoLoopsComplex⟩
 210
 211def twoBridgesClass : ExactPathClass 2 :=
 212  ⟨doubleEdgeSig, Quotient.mk _ twoBridgesComplex⟩
 213
 214theorem selfLoopClassTick_twoLoops :
 215    selfLoopClassTick 2 twoLoopsClass = ⟨2, by norm_num⟩ := by
 216  simp only [selfLoopClassTick, twoLoopsClass, descendedTick_mk, selfLoopTick,
 217    selfLoopCount_twoLoops]
 218
 219theorem selfLoopClassTick_twoBridges :
 220    selfLoopClassTick 2 twoBridgesClass = ⟨0, by norm_num⟩ := by
 221  simp only [selfLoopClassTick, twoBridgesClass, descendedTick_mk, selfLoopTick,
 222    selfLoopCount_twoBridges]
 223
 224/-- **THEOREM.** The self-loop tick uses quotient-internal incidence data:
 225it does not factor through shell signature alone. -/
 226theorem selfLoopClassTick_not_ShellSigTick :
 227    ¬ ShellSigTick selfLoopClassTick := by
 228  rintro ⟨sigma, hσ⟩
 229  have hL := hσ 2 twoLoopsClass
 230  have hB := hσ 2 twoBridgesClass
 231  have hσEq : sigma 2 twoLoopsClass.1 = sigma 2 twoBridgesClass.1 := rfl
 232  have hne : selfLoopClassTick 2 twoLoopsClass ≠
 233      selfLoopClassTick 2 twoBridgesClass := by
 234    rw [selfLoopClassTick_twoLoops, selfLoopClassTick_twoBridges]
 235    decide
 236  exact hne (hL.trans (hσEq.trans hB.symm))
 237
 238/-! ## §4. Bridges toward R5 (do not inhabit R5) -/
 239
 240/-- Enriched eventual mass balance closes the bare continuum R5 residual
 241shape. Primary wiring target when a future session inhabits the hyp. -/
 242theorem oscillatoryTail_of_enriched_eventual_balance
 243    (lab : LabeledTick) (hInv : GlobalEquivalentInvariant lab)
 244    (hbal : EventuallyTickFiberMassBalanced (descendedTick lab hInv)) :
 245    OscillatoryTail (enrichedPhase lab hInv) ∧
 246      ¬ OscillatoryTail zeroPhase :=
 247  ⟨eventuallyTickFiberMassBalanced_implies_oscillatoryTail
 248      (descendedTick lab hInv) hbal,
 249    zeroPhase_not_oscillatoryTail⟩
 250
 251/-- Identical-zero amplitudes also close the bare R5 shape. -/
 252theorem oscillatoryTail_of_enriched_identically_zero
 253    (lab : LabeledTick) (hInv : GlobalEquivalentInvariant lab)
 254    (hzero : ∀ n : ℕ, exactShellAmplitude (enrichedPhase lab hInv) n = 0) :
 255    OscillatoryTail (enrichedPhase lab hInv) ∧
 256      ¬ OscillatoryTail zeroPhase := by
 257  refine ⟨?_, zeroPhase_not_oscillatoryTail⟩
 258  intro ε hε
 259  refine ⟨0, fun m n _hm _hmn => ?_⟩
 260  have hamp :
 261      ∀ k ∈ Finset.Ico m n, exactShellAmplitude (enrichedPhase lab hInv) k = 0 :=
 262    fun k _ => hzero k
 263  rw [Finset.sum_eq_zero hamp, norm_zero]
 264  exact hε
 265
 266/-! ## §5. Sharper typed residual (credit-bearing C terminal) -/
 267
 268/-- **Sharpened R5 residual.** A `GlobalEquivalent`-invariant labeled tick
 269whose descent has `OscillatoryTail` and is not a bare signature coloring.
 270
 271Strictly stronger than bare
 272`TypedResidual_continuum_substrate_oscillatoryTail`. Decision
 273`D-qg-c1-r4-enriched-carrier-20260722` redirects the attack here. -/
 274def TypedResidual_enriched_carrier_oscillatoryTail : Prop :=
 275  ∃ (lab : LabeledTick) (hInv : GlobalEquivalentInvariant lab),
 276    OscillatoryTail (enrichedPhase lab hInv) ∧
 277      ¬ ShellSigTick (descendedTick lab hInv)
 278
 279/-- Bare R5 shape (DAG-owned; quoted here for wiring docs only). -/
 280def TypedResidual_continuum_substrate_oscillatoryTail : Prop :=
 281  ∃ phase : ∀ n : ℕ, ExactPathClass n → ℝ,
 282    OscillatoryTail phase ∧ ¬ OscillatoryTail zeroPhase
 283
 284/-- Enriched residual implies the bare DAG R5 residual. -/
 285theorem bare_r5_of_enriched_carrier_oscillatoryTail
 286    (h : TypedResidual_enriched_carrier_oscillatoryTail) :
 287    TypedResidual_continuum_substrate_oscillatoryTail := by
 288  obtain ⟨lab, hInv, htail, _⟩ := h
 289  exact ⟨enrichedPhase lab hInv, htail, zeroPhase_not_oscillatoryTail⟩
 290
 291/-- Primary close theorem name to wire into the DAG when inhabited. -/
 292theorem typedResidual_continuum_substrate_oscillatoryTail_of_enriched
 293    (h : TypedResidual_enriched_carrier_oscillatoryTail) :
 294    TypedResidual_continuum_substrate_oscillatoryTail :=
 295  bare_r5_of_enriched_carrier_oscillatoryTail h
 296
 297/-- Primary close theorem from eventual balance on any enriched tick. -/
 298theorem typedResidual_continuum_substrate_oscillatoryTail_of_enriched_eventual_balance
 299    (lab : LabeledTick) (hInv : GlobalEquivalentInvariant lab)
 300    (hbal : EventuallyTickFiberMassBalanced (descendedTick lab hInv)) :
 301    TypedResidual_continuum_substrate_oscillatoryTail :=
 302  ⟨enrichedPhase lab hInv,
 303    oscillatoryTail_of_enriched_eventual_balance lab hInv hbal⟩
 304
 305/-- Schema package: enriched API inhabited; analytic tail OPEN. -/
 306structure EnrichedCarrierPhaseSubstrate where
 307  lab : LabeledTick
 308  invariant : GlobalEquivalentInvariant lab
 309  not_shellSigTick : ¬ ShellSigTick (descendedTick lab invariant)
 310
 311def selfLoopEnrichedSubstrate : EnrichedCarrierPhaseSubstrate where
 312  lab := selfLoopTick
 313  invariant := selfLoopTick_invariant
 314  not_shellSigTick := selfLoopClassTick_not_ShellSigTick
 315
 316theorem enrichedCarrierPhaseSubstrate_nonempty :
 317    Nonempty EnrichedCarrierPhaseSubstrate :=
 318  ⟨selfLoopEnrichedSubstrate⟩
 319
 320theorem signatureBlocker_iff_no_shellSig_oscillatoryTail :
 321    SignatureFin8OscillatoryTailBlocker ↔
 322      ¬ ∃ tau : ∀ n : ℕ, ExactPathClass n → Fin 8,
 323          ShellSigTick tau ∧ OscillatoryTail (tickDerivedPhase tau) :=
 324  Iff.rfl
 325
 326/-! ## §6. Status (R5 open; gap2 unflipped) -/
 327
 328structure Gap2EnrichedCarrierPhaseStatus where
 329  enrichedApiLanded : Bool
 330  selfLoopWitnessEscapesShellSig : Bool
 331  r5BridgeLanded : Bool
 332  enrichedResidualDefinedUninhabited : Bool
 333  eventualBalanceForWitnessProved : Bool
 334  signatureBlockerProved : Bool
 335  bareR5Closed : Bool
 336  gap2ContinuumAndMeasure : Bool
 337
 338def gap2EnrichedCarrierPhaseStatus : Gap2EnrichedCarrierPhaseStatus where
 339  enrichedApiLanded := true
 340  selfLoopWitnessEscapesShellSig := true
 341  r5BridgeLanded := true
 342  enrichedResidualDefinedUninhabited := true
 343  eventualBalanceForWitnessProved := false
 344  signatureBlockerProved := false
 345  bareR5Closed := false
 346  gap2ContinuumAndMeasure := false
 347
 348theorem gap2EnrichedCarrierPhaseStatus_flags :
 349    gap2EnrichedCarrierPhaseStatus.enrichedApiLanded = true ∧
 350      gap2EnrichedCarrierPhaseStatus.selfLoopWitnessEscapesShellSig = true ∧
 351      gap2EnrichedCarrierPhaseStatus.r5BridgeLanded = true ∧
 352      gap2EnrichedCarrierPhaseStatus.enrichedResidualDefinedUninhabited =
 353        true ∧
 354      gap2EnrichedCarrierPhaseStatus.eventualBalanceForWitnessProved =
 355        false ∧
 356      gap2EnrichedCarrierPhaseStatus.signatureBlockerProved = false ∧
 357      gap2EnrichedCarrierPhaseStatus.bareR5Closed = false ∧
 358      gap2EnrichedCarrierPhaseStatus.gap2ContinuumAndMeasure = false := by
 359  decide
 360
 361end
 362
 363end Gap2EnrichedCarrierPhase
 364end SevenGaps
 365end Gravity
 366end IndisputableMonolith
 367