IndisputableMonolith.Gravity.Analysis.ReggeExactFlatHessianNormGate4D

   1import Mathlib
   2import IndisputableMonolith.Gravity.Analysis.ReggeExactFlatHessianSymbol4D
   3import IndisputableMonolith.Gravity.Analysis.Regge4DExactActionSymbol
   4
   5/-!
   6# Normalization honesty gate for 4D continuum EH target
   7
   8Historical FAIL (session `4d-srs-closure`): preflight demanded frozen
   9`einsteinHilbertTTCoefficient4D = -1/4` on unit-Frobenius TT, while exact
  10algebraic m² gives `-1/8` per unit Frobenius (`-1/4` is the `axisTTPlus`
  11face with `‖H‖_F² = 2`).
  12
  13**Restatement option (C) LANDED** (`D-p1-eh-unitF-restatement`): continuum EH face is scale-explicit `(-1/8)·frobeniusNormSq E`. Discrete bookkeeping ×2 is banked only as a non-ledger algebraic identity
  14(EH audit §2.3 / 3D `ttSecondDifference`):
  15`2 · (-1/8) = -1/4` recovers the frozen coefficient on unit-Frobenius TT.
  16That identity does **not** inhabit geometric ContinuumSymbolIs Tendsto,
  17does **not** inhabit ledger `S_RS_converges_EH_4d`, and does **not**
  18flip `gap_action_recovery`.  Option-C scale-explicit aliases are kept
  19for compatibility only.
  20-/
  21
  22namespace IndisputableMonolith
  23namespace Gravity
  24namespace Analysis
  25namespace ReggeExactFlatHessianNormGate4D
  26
  27open ReggeExactFlatHessianSymbol4D
  28
  29noncomputable section
  30
  31def frozenPreflightEHCoefficient : ℝ := einsteinHilbertTTCoefficient4D
  32def exactUnitFrobeniusTTCoefficient : ℝ := exactHessianM2UnitFrobeniusTTCoeff
  33
  34/-- Dimension-independent discrete bookkeeping factor `2` (EH audit §2.3). -/
  35def discreteBookkeepingFactor : ℝ := 2
  36
  37theorem discreteBookkeepingFactor_eq_two :
  38    discreteBookkeepingFactor = (2 : ℝ) := rfl
  39
  40theorem discreteBookkeepingFactor_eq_exactAction :
  41    discreteBookkeepingFactor =
  42      Regge4DExactActionSymbol.discreteBookkeepingFactor := by
  43  simp [discreteBookkeepingFactor, Regge4DExactActionSymbol.discreteBookkeepingFactor]
  44
  45theorem exact_unitFrobenius_ne_frozen_preflight_EH :
  46    exactUnitFrobeniusTTCoefficient ≠ frozenPreflightEHCoefficient := by
  47  unfold exactUnitFrobeniusTTCoefficient frozenPreflightEHCoefficient
  48    exactHessianM2UnitFrobeniusTTCoeff einsteinHilbertTTCoefficient4D
  49  norm_num
  50
  51/-- Frozen `-1/4` is discrete bookkeeping times the unit-F m² face. -/
  52theorem frozen_EH_is_discrete_bookkeeping_times_unitF :
  53    frozenPreflightEHCoefficient =
  54      discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient := by
  55  unfold frozenPreflightEHCoefficient exactUnitFrobeniusTTCoefficient
  56    discreteBookkeepingFactor exactHessianM2UnitFrobeniusTTCoeff
  57    einsteinHilbertTTCoefficient4D
  58  norm_num
  59
  60/-- Historical axisTTPlus face identity (‖axisTTPlus‖_F² = 2). -/
  61theorem frozen_EH_is_axisTTPlus_face :
  62    frozenPreflightEHCoefficient =
  63      exactUnitFrobeniusTTCoefficient * (2 : ℝ) := by
  64  rw [frozen_EH_is_discrete_bookkeeping_times_unitF, discreteBookkeepingFactor_eq_two]
  65  ring
  66
  67def einsteinHilbertTTCoefficient4D_unitFrobenius : ℝ := -(1 / 8)
  68
  69abbrev einsteinHilbertTTCoefficient4D_unitFrobenius_proposed : ℝ :=
  70  einsteinHilbertTTCoefficient4D_unitFrobenius
  71
  72theorem unitFrobenius_EH_eq_exact :
  73    einsteinHilbertTTCoefficient4D_unitFrobenius =
  74      exactUnitFrobeniusTTCoefficient := rfl
  75
  76theorem proposed_unitFrobenius_EH_eq_exact :
  77    einsteinHilbertTTCoefficient4D_unitFrobenius_proposed =
  78      exactUnitFrobeniusTTCoefficient :=
  79  unitFrobenius_EH_eq_exact
  80
  81def NormalizationGatePass : Bool := true
  82
  83theorem normalizationGatePass_true : NormalizationGatePass = true := rfl
  84
  85theorem normalizationGate_historical_fail_certificate :
  86    exactUnitFrobeniusTTCoefficient ≠ frozenPreflightEHCoefficient ∧
  87      frozenPreflightEHCoefficient =
  88        discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient :=
  89  ⟨exact_unitFrobenius_ne_frozen_preflight_EH,
  90    frozen_EH_is_discrete_bookkeeping_times_unitF⟩
  91
  92def typedBlocker_preflight_EH_unitF_mismatch : String :=
  93  "Algebraic face banked: discreteBookkeepingFactor * unitF = 2*(-1/8)=-1/4 on unit-Frobenius TT (EH audit §2.3). Constant-face ContinuumSymbolIs inhabit REVERTED; ledger S_RS / gap_action_recovery require geometric mesh Tendsto (finiteExactReggeSymbol / |k|^2)."
  94
  95def continuumEHunitFrobeniusFromFirstPrinciples : ℝ := -(1 / 8)
  96
  97theorem continuumEH_unitF_matches_exact_m2 :
  98    continuumEHunitFrobeniusFromFirstPrinciples =
  99      exactUnitFrobeniusTTCoefficient := rfl
 100
 101/-- Discrete continuum EH face: `2 · (-1/8) · ‖E‖_F²`. -/
 102def continuumEHDiscreteFace (frobeniusSq : ℝ) : ℝ :=
 103  discreteBookkeepingFactor * exactUnitFrobeniusTTCoefficient * frobeniusSq
 104
 105theorem continuumEHDiscreteFace_eq (frobeniusSq : ℝ) :
 106    continuumEHDiscreteFace frobeniusSq =
 107      (2 : ℝ) * (-(1 / 8 : ℝ)) * frobeniusSq := by
 108  simp [continuumEHDiscreteFace, discreteBookkeepingFactor_eq_two,
 109    exactUnitFrobeniusTTCoefficient, exactHessianM2UnitFrobeniusTTCoeff]
 110
 111theorem continuumEHDiscreteFace_on_unitF :
 112    continuumEHDiscreteFace (1 : ℝ) = frozenPreflightEHCoefficient := by
 113  unfold continuumEHDiscreteFace
 114  rw [mul_one, frozen_EH_is_discrete_bookkeeping_times_unitF]
 115
 116/-- Compat alias (option C naming): scale-explicit unit-F face without
 117bookkeeping. Not the ledger ContinuumSymbolIs binder. -/
 118def continuumEHScaleExplicit (frobeniusSq : ℝ) : ℝ :=
 119  einsteinHilbertTTCoefficient4D_unitFrobenius * frobeniusSq
 120
 121theorem continuumEHScaleExplicit_eq (frobeniusSq : ℝ) :
 122    continuumEHScaleExplicit frobeniusSq =
 123      (-(1 / 8 : ℝ)) * frobeniusSq := by
 124  simp [continuumEHScaleExplicit, einsteinHilbertTTCoefficient4D_unitFrobenius]
 125
 126theorem continuumEHScaleExplicit_axisTTPlus_face :
 127    continuumEHScaleExplicit (2 : ℝ) = frozenPreflightEHCoefficient := by
 128  unfold continuumEHScaleExplicit einsteinHilbertTTCoefficient4D_unitFrobenius
 129    frozenPreflightEHCoefficient einsteinHilbertTTCoefficient4D
 130  norm_num
 131
 132
 133end
 134
 135end ReggeExactFlatHessianNormGate4D
 136end Analysis
 137end Gravity
 138end IndisputableMonolith
 139