IndisputableMonolith.Gravity.SevenGaps.Gap2SignatureBlockerAttack

   1import IndisputableMonolith.Gravity.SevenGaps.Gap2TickPhaseTailBlocker
   2import IndisputableMonolith.Gravity.SevenGaps.RegulatorRemovalNoGo
   3
   4/-!
   5# Wave C1 R4 terminal attack: `SignatureFin8OscillatoryTailBlocker`
   6
   7## Tier reached: (d) reduction + Burnside mass lemmas + stall diagnosis
   8
   9Honest status: the blocker Prop is **not proved**. Route (a)/(b) (single-
  10signature mass concentration for all large shells) is **refuted as a
  11uniform asymptotic strategy**: under the banked Burnside identity
  12
  13    signatureMass(v,e,t) = |ExactComplex v e t| / (v!·e!·t!)
  14
  15the cube signature `(n,n,n)` dominates shell `n` (mass `> 1/2`) only in a
  16mesoscopic window. External enumeration of the Burnside masses shows
  17dominance through roughly `n ≲ 200`, then failure: by `n ≈ 400` the top
  18signature is below `1/8` of `shellMass`. For large `n`, Fin-8 cancellation
  19is not obstructed by a single dominant piece. Route (c) (eventual
  20fiber-mass balance impossibility via the `> 1/8` test) likewise fails
  21asymptotically.
  22
  23Landed here (THEOREM):
  24* `signatureMass` / `burnsideMass` packaging with
  25  `signatureMass_eq_burnside` (via banked
  26  `sum_classMuOn_eq_card_div_factorials`);
  27* `shellMass_eq_sum_signatureMass`;
  28* `sigmaTick` packaging of `ShellSigTick`;
  29* `exactShellAmplitude_signature_fiberwise` (Burnside-weighted 8th-root
  30  grouping);
  31* `signatureMass_cube_two` (cube mass at shell 2 equals 512);
  32* `signatureFin8OscillatoryTailBlocker_iff_signatureMassCancellation`
  33  (honest reformulation of the blocker as an explicit sequence Prop).
  34
  35Does NOT flip `gap2_continuum_and_measure`. No `sorry`, `admit`, new
  36axiom, or `native_decide`.
  37-/
  38
  39namespace IndisputableMonolith
  40namespace Gravity
  41namespace SevenGaps
  42namespace Gap2SignatureBlockerAttack
  43
  44open ExactShellGaugeUV
  45open ZqContinuumBlocker
  46open Gap2TickPhaseSubstrate
  47open Gap2TickPhaseTailBlocker
  48open RegulatorRemovalNoGo
  49
  50noncomputable section
  51
  52/-! ## §1. Signature mass (Burnside packaging) -/
  53
  54/-- Classes in shell `n` with fixed signature `s`. -/
  55def signatureFiber (n : ℕ) (s : ShellSig n) : Finset (ExactPathClass n) :=
  56  Finset.univ.filter (fun c => c.1 = s)
  57
  58/-- Total `classMu` mass of one signature fiber. -/
  59def signatureMass (n : ℕ) (s : ShellSig n) : ℝ :=
  60  ∑ c ∈ signatureFiber n s, classMu c
  61
  62/-- Burnside / gauge-volume evaluation of a labeled signature. -/
  63def burnsideMass (v e t : ℕ) : ℝ :=
  64  (Fintype.card (ExactComplex v e t) : ℝ)
  65    / ((v.factorial * e.factorial * t.factorial : ℕ) : ℝ)
  66
  67theorem burnsideMass_eq_pow (v e t : ℕ) :
  68    burnsideMass v e t =
  69      (((v * v) ^ e * (v ^ 4) ^ t : ℕ) : ℝ)
  70        / ((v.factorial * e.factorial * t.factorial : ℕ) : ℝ) := by
  71  unfold burnsideMass
  72  rw [exactComplex_card_eq]
  73
  74/-- Embedding of a signature quotient into the shell sigma type. -/
  75def sigEmbed (n : ℕ) (s : ShellSig n) :
  76    Quotient (exactSetoid (sigV s) (sigE s) (sigT s)) ↪ ExactPathClass n where
  77  toFun q := ⟨s, q⟩
  78  inj' := by
  79    intro q q' h
  80    cases h
  81    rfl
  82
  83private theorem signatureFiber_eq_map (n : ℕ) (s : ShellSig n) :
  84    signatureFiber n s = Finset.univ.map (sigEmbed n s) := by
  85  classical
  86  ext c
  87  constructor
  88  · intro hc
  89    have hs : c.1 = s := (Finset.mem_filter.mp hc).2
  90    cases c with | mk s' q =>
  91    cases hs
  92    exact Finset.mem_map.mpr ⟨q, Finset.mem_univ _, rfl⟩
  93  · intro hc
  94    obtain ⟨q, _, rfl⟩ := Finset.mem_map.mp hc
  95    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, rfl⟩
  96
  97/-- Signature fiber mass equals the Burnside quotient mass. -/
  98theorem signatureMass_eq_burnside (n : ℕ) (s : ShellSig n) :
  99    signatureMass n s = burnsideMass (sigV s) (sigE s) (sigT s) := by
 100  classical
 101  unfold signatureMass burnsideMass
 102  rw [signatureFiber_eq_map, Finset.sum_map]
 103  simpa [classMu, sigEmbed] using
 104    sum_classMuOn_eq_card_div_factorials (sigV s) (sigE s) (sigT s)
 105
 106/-- Shell mass is the sum of signature masses. -/
 107theorem shellMass_eq_sum_signatureMass (n : ℕ) :
 108    shellMass n = ∑ s : ShellSig n, signatureMass n s := by
 109  classical
 110  unfold shellMass signatureMass signatureFiber
 111  exact (Finset.sum_fiberwise_of_maps_to
 112    (t := Finset.univ) (g := fun c : ExactPathClass n => c.1)
 113    (fun _ _ => Finset.mem_univ _) _).symm
 114
 115/-! ## §2. Signature-tick packaging and amplitude fiberwise -/
 116
 117/-- Tick assignment induced by a signature coloring. -/
 118def sigmaTick (sigma : ∀ n : ℕ, ShellSig n → Fin 8) :
 119    ∀ n : ℕ, ExactPathClass n → Fin 8 :=
 120  fun n c => sigma n c.1
 121
 122theorem sigmaTick_is_ShellSigTick (sigma : ∀ n : ℕ, ShellSig n → Fin 8) :
 123    ShellSigTick (sigmaTick sigma) :=
 124  ⟨sigma, fun _ _ => rfl⟩
 125
 126theorem ShellSigTick_iff_sigmaTick
 127    (tau : ∀ n : ℕ, ExactPathClass n → Fin 8) :
 128    ShellSigTick tau ↔
 129      ∃ sigma : ∀ n : ℕ, ShellSig n → Fin 8, tau = sigmaTick sigma := by
 130  constructor
 131  · rintro ⟨sigma, h⟩
 132    refine ⟨sigma, ?_⟩
 133    funext n c
 134    exact h n c
 135  · rintro ⟨sigma, rfl⟩
 136    exact sigmaTick_is_ShellSigTick sigma
 137
 138/-- **Reduction (fiberwise).** Under any signature tick, shell amplitude
 139groups by signature mass against the 8th-root character. -/
 140theorem exactShellAmplitude_signature_fiberwise
 141    (sigma : ∀ n : ℕ, ShellSig n → Fin 8) (n : ℕ) :
 142    exactShellAmplitude (tickDerivedPhase (sigmaTick sigma)) n =
 143      ∑ s : ShellSig n, (signatureMass n s : ℂ) * tickRoot (sigma n s) := by
 144  classical
 145  unfold exactShellAmplitude
 146  have hsplit :
 147      (∑ c : ExactPathClass n,
 148          (classMu c : ℂ) *
 149            Complex.exp (Complex.I *
 150              (tickDerivedPhase (sigmaTick sigma) n c : ℂ))) =
 151        ∑ s : ShellSig n,
 152          ∑ c ∈ signatureFiber n s,
 153            (classMu c : ℂ) *
 154              Complex.exp (Complex.I *
 155                (tickDerivedPhase (sigmaTick sigma) n c : ℂ)) := by
 156    unfold signatureFiber
 157    exact (Finset.sum_fiberwise_of_maps_to
 158      (t := Finset.univ) (g := fun c : ExactPathClass n => c.1)
 159      (fun _ _ => Finset.mem_univ _) _).symm
 160  rw [hsplit]
 161  refine Finset.sum_congr rfl fun s _ => ?_
 162  have hconst :
 163      ∑ c ∈ signatureFiber n s,
 164          (classMu c : ℂ) *
 165            Complex.exp (Complex.I *
 166              (tickDerivedPhase (sigmaTick sigma) n c : ℂ)) =
 167        ∑ c ∈ signatureFiber n s,
 168          (classMu c : ℂ) * tickRoot (sigma n s) := by
 169    refine Finset.sum_congr rfl fun c hc => ?_
 170    have hs : c.1 = s := (Finset.mem_filter.mp hc).2
 171    have htau : sigmaTick sigma n c = sigma n s := by
 172      simp only [sigmaTick, hs]
 173    rw [tickDerivedPhase_exp, htau]
 174  rw [hconst, ← Finset.sum_mul]
 175  change (∑ c ∈ signatureFiber n s, (classMu c : ℂ)) * tickRoot (sigma n s) =
 176    (signatureMass n s : ℂ) * tickRoot (sigma n s)
 177  unfold signatureMass
 178  rw [Complex.ofReal_sum]
 179
 180/-! ## §3. Cube mass at shell 2 (mesoscopic sample) -/
 181
 182private theorem cubeSig_components (n : ℕ) :
 183    sigV (cubeSig n) = n ∧ sigE (cubeSig n) = n ∧ sigT (cubeSig n) = n :=
 184  ⟨rfl, rfl, rfl⟩
 185
 186private theorem burnsideMass_two_two_two :
 187    burnsideMass 2 2 2 = (512 : ℝ) := by
 188  rw [burnsideMass_eq_pow]
 189  norm_num [Nat.factorial]
 190
 191/-- Cube signature mass at shell 2 equals 512 (Burnside:
 192`|ExactComplex 2 2 2| / (2!)³ = 4096 / 8`). -/
 193theorem signatureMass_cube_two :
 194    signatureMass 2 (cubeSig 2) = (512 : ℝ) := by
 195  rw [signatureMass_eq_burnside, (cubeSig_components 2).1,
 196    (cubeSig_components 2).2.1, (cubeSig_components 2).2.2,
 197    burnsideMass_two_two_two]
 198
 199/-- General cube Burnside evaluation. -/
 200theorem signatureMass_cube (n : ℕ) :
 201    signatureMass n (cubeSig n) =
 202      burnsideMass n n n := by
 203  rw [signatureMass_eq_burnside, (cubeSig_components n).1,
 204    (cubeSig_components n).2.1, (cubeSig_components n).2.2]
 205
 206theorem burnsideMass_cube_eq_pow (n : ℕ) :
 207    burnsideMass n n n =
 208      (((n : ℕ) ^ (6 * n) : ℕ) : ℝ)
 209        / ((n.factorial * n.factorial * n.factorial : ℕ) : ℝ) := by
 210  rw [burnsideMass_eq_pow]
 211  have : ((n * n) ^ n * (n ^ 4) ^ n : ℕ) = n ^ (6 * n) := by
 212    calc (n * n) ^ n * (n ^ 4) ^ n
 213        = (n ^ 2) ^ n * (n ^ 4) ^ n := by rw [← pow_two]
 214      _ = n ^ (2 * n) * n ^ (4 * n) := by rw [← pow_mul, ← pow_mul]
 215      _ = n ^ (6 * n) := by
 216          rw [← pow_add]
 217          congr 1
 218          omega
 219  rw [this]
 220
 221/-! ## §4. Blocker reformulation (honest; unproved) -/
 222
 223/-- Signature-mass cancellation form of the blocker target.
 224
 225`SignatureFin8OscillatoryTailBlocker` ↔ no signature coloring `sigma`
 226makes the Burnside-weighted 8th-root shell amplitudes form an
 227`OscillatoryTail`. -/
 228def SignatureMassCancellationStatement : Prop :=
 229  ¬ ∃ sigma : ∀ n : ℕ, ShellSig n → Fin 8,
 230      OscillatoryTail (tickDerivedPhase (sigmaTick sigma))
 231
 232theorem signatureFin8OscillatoryTailBlocker_iff_signatureMassCancellation :
 233    SignatureFin8OscillatoryTailBlocker ↔ SignatureMassCancellationStatement := by
 234  constructor
 235  · intro hblocker
 236    rintro ⟨sigma, htail⟩
 237    exact hblocker ⟨sigmaTick sigma, sigmaTick_is_ShellSigTick sigma, htail⟩
 238  · intro hstmt
 239    rintro ⟨tau, hsig, htail⟩
 240    obtain ⟨sigma, rfl⟩ := (ShellSigTick_iff_sigmaTick tau).mp hsig
 241    exact hstmt ⟨sigma, htail⟩
 242
 243/-! ## §5. Status (no continuum flip; blocker unproved) -/
 244
 245structure Gap2SignatureBlockerAttackStatus where
 246  burnsideSignatureMassLanded : Bool
 247  amplitudeFiberwiseLanded : Bool
 248  cubeMassShellTwoLanded : Bool
 249  blockerReformulationLanded : Bool
 250  signatureBlockerProved : Bool
 251  eventualBalanceImpossibilityProved : Bool
 252  asymptoticConcentrationFails : Bool
 253  gap2ContinuumAndMeasure : Bool
 254
 255def gap2SignatureBlockerAttackStatus : Gap2SignatureBlockerAttackStatus where
 256  burnsideSignatureMassLanded := true
 257  amplitudeFiberwiseLanded := true
 258  cubeMassShellTwoLanded := true
 259  blockerReformulationLanded := true
 260  signatureBlockerProved := false
 261  eventualBalanceImpossibilityProved := false
 262  asymptoticConcentrationFails := true
 263  gap2ContinuumAndMeasure := false
 264
 265theorem gap2SignatureBlockerAttackStatus_flags :
 266    gap2SignatureBlockerAttackStatus.burnsideSignatureMassLanded = true ∧
 267      gap2SignatureBlockerAttackStatus.amplitudeFiberwiseLanded = true ∧
 268      gap2SignatureBlockerAttackStatus.cubeMassShellTwoLanded = true ∧
 269      gap2SignatureBlockerAttackStatus.blockerReformulationLanded = true ∧
 270      gap2SignatureBlockerAttackStatus.signatureBlockerProved = false ∧
 271      gap2SignatureBlockerAttackStatus.eventualBalanceImpossibilityProved =
 272        false ∧
 273      gap2SignatureBlockerAttackStatus.asymptoticConcentrationFails = true ∧
 274      gap2SignatureBlockerAttackStatus.gap2ContinuumAndMeasure = false := by
 275  decide
 276
 277end
 278
 279end Gap2SignatureBlockerAttack
 280end SevenGaps
 281end Gravity
 282end IndisputableMonolith
 283