IndisputableMonolith.Cosmology.EtaBExactRungDerivation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Foundation.IntegrationGap
   4
   5/-!
   6# η_B Rung Integer: Three Combinatorial Witnesses
   7
   8The integer `−44` that pins the baryon-to-photon ratio's φ-rung is
   9realized via three combinatorial witnesses, all reducing to
  10`integrationGap D − 1 = 44` at `D = 3`. The three witnesses are:
  11
  12* **Gap-from-dimension:** `1 − D²(D+2)` at `D = 3` gives `1 − 45 = −44`.
  13* **Chirality × torsion:** `−(bitFlipCount(0) × |torsionGap(0,1)|) = −(4 × 11) = −44`,
  14  with the factors coming from the canonical Gray code on `Q_3` and
  15  the CW-filtration torsion spectrum `{0, 11, 17}`.
  16* **Fermionic-DOF doubling:** `1 − fermionicDOF / 2 = 1 − 90/2 = −44`,
  17  where `fermionicDOF = 2 × integrationGap D = 90` via matter-antimatter
  18  doubling.
  19
  20We are explicit: these are NOT three independent forcing routes. They
  21are three distinct combinatorial reparameterizations of the same
  22underlying integer `D²(D+2) − 1 = 44` at `D = 3`. The structural
  23unification across three combinatorial settings is the genuine content,
  24not statistical independence.
  25
  26## Status
  27
  28* `0 sorry`
  29* No theory-specific axioms; standard kernel only.
  30-/
  31
  32namespace IndisputableMonolith
  33namespace Cosmology
  34namespace EtaBExactRungDerivation
  35
  36open Constants
  37open Foundation.IntegrationGap
  38
  39/-! ## Combinatorial integers -/
  40
  41/-- The Gray code chirality flip count on the 3-cube, axis 0.
  42    On the canonical Gray cycle `[0,1,3,2,6,7,5,4]`, bit 0 flips four times. -/
  43def bitFlipCount0 : ℕ := 4
  44
  45/-- The CW-filtration torsion gap between generations 0 and 1.
  46    From the torsion spectrum `{0, 11, 17}`, the gap is `|τ₁ − τ₀| = 11`. -/
  47def torsionGap01 : ℕ := 11
  48
  49/-- Matter-antimatter doubling of the integration-gap-worth of fermionic DOF. -/
  50def fermionicDOF : ℕ := 2 * integrationGap D
  51
  52theorem fermionicDOF_eq : fermionicDOF = 90 := by
  53  unfold fermionicDOF
  54  rw [integrationGap_at_D3]
  55
  56/-! ## Three combinatorial witnesses for −44 -/
  57
  58/-- Witness A (gap-from-dimension): `eta_B_rung_from_dimension D := A − integrationGap D`. -/
  59def eta_B_rung_from_dimension (d : ℕ) : ℤ :=
  60  (A : ℤ) - (integrationGap d : ℤ)
  61
  62theorem eta_B_rung_from_dimension_at_D3 : eta_B_rung_from_dimension D = -44 := by
  63  unfold eta_B_rung_from_dimension
  64  have hgap : (integrationGap D : ℤ) = 45 := by exact_mod_cast integrationGap_at_D3
  65  rw [hgap]
  66  decide
  67
  68/-- Witness B (chirality × torsion): `−(bitFlipCount0 × torsionGap01)`. -/
  69def eta_B_rung_from_chirality : ℤ :=
  70  -((bitFlipCount0 : ℤ) * (torsionGap01 : ℤ))
  71
  72theorem eta_B_rung_from_chirality_eq : eta_B_rung_from_chirality = -44 := by
  73  unfold eta_B_rung_from_chirality bitFlipCount0 torsionGap01
  74  decide
  75
  76/-- Witness C (fermionic-DOF doubling): `A − fermionicDOF / 2`. -/
  77def eta_B_rung_from_fermionic : ℤ :=
  78  (A : ℤ) - ((fermionicDOF / 2 : ℕ) : ℤ)
  79
  80theorem eta_B_rung_from_fermionic_eq : eta_B_rung_from_fermionic = -44 := by
  81  unfold eta_B_rung_from_fermionic
  82  rw [fermionicDOF_eq]
  83  decide
  84
  85/-! ## Convergence of the three witnesses -/
  86
  87/-- The gap-from-dimension and chirality × torsion witnesses agree at `D = 3`. -/
  88theorem witnesses_AB_agree :
  89    eta_B_rung_from_dimension D = eta_B_rung_from_chirality := by
  90  rw [eta_B_rung_from_dimension_at_D3, eta_B_rung_from_chirality_eq]
  91
  92/-- The gap-from-dimension and fermionic-DOF witnesses agree at `D = 3`. -/
  93theorem witnesses_AC_agree :
  94    eta_B_rung_from_dimension D = eta_B_rung_from_fermionic := by
  95  rw [eta_B_rung_from_dimension_at_D3, eta_B_rung_from_fermionic_eq]
  96
  97/-- The chirality × torsion and fermionic-DOF witnesses agree at `D = 3`. -/
  98theorem witnesses_BC_agree :
  99    eta_B_rung_from_chirality = eta_B_rung_from_fermionic := by
 100  rw [eta_B_rung_from_chirality_eq, eta_B_rung_from_fermionic_eq]
 101
 102/-! ## Bridge identities (the three reparameterizations as one integer) -/
 103
 104/-- The chirality × torsion product equals the integration gap minus one,
 105    at `D = 3`: `4 × 11 = 45 − 1`. -/
 106theorem chirality_product_equals_gap_minus_one :
 107    ((bitFlipCount0 : ℤ) * (torsionGap01 : ℤ))
 108      = (integrationGap D : ℤ) - (A : ℤ) := by
 109  have hgap : (integrationGap D : ℤ) = 45 := by exact_mod_cast integrationGap_at_D3
 110  unfold bitFlipCount0 torsionGap01 A
 111  rw [hgap]
 112  decide
 113
 114/-- The fermionic-DOF half equals the integration gap, at `D = 3`: `90/2 = 45`. -/
 115theorem fermionic_half_equals_gap :
 116    ((fermionicDOF / 2 : ℕ) : ℤ) = (integrationGap D : ℤ) := by
 117  rw [fermionicDOF_eq]
 118  have : (integrationGap D : ℤ) = 45 := by exact_mod_cast integrationGap_at_D3
 119  rw [this]
 120  decide
 121
 122/-! ## Counterfactual rungs at other dimensions -/
 123
 124theorem D1_counterfactual_rung : eta_B_rung_from_dimension 1 = -2 := by
 125  unfold eta_B_rung_from_dimension integrationGap parityCount configDim A
 126  decide
 127
 128theorem D2_counterfactual_rung : eta_B_rung_from_dimension 2 = -15 := by
 129  unfold eta_B_rung_from_dimension integrationGap parityCount configDim A
 130  decide
 131
 132theorem D5_counterfactual_rung : eta_B_rung_from_dimension 5 = -174 := by
 133  unfold eta_B_rung_from_dimension integrationGap parityCount configDim A
 134  decide
 135
 136/-! ## Master certificate -/
 137
 138structure EtaBExactRungCert where
 139  /-- Witness A yields `−44`. -/
 140  witness_A : eta_B_rung_from_dimension D = -44
 141  /-- Witness B yields `−44`. -/
 142  witness_B : eta_B_rung_from_chirality = -44
 143  /-- Witness C yields `−44`. -/
 144  witness_C : eta_B_rung_from_fermionic = -44
 145  /-- A and B agree. -/
 146  AB_agree : eta_B_rung_from_dimension D = eta_B_rung_from_chirality
 147  /-- A and C agree. -/
 148  AC_agree : eta_B_rung_from_dimension D = eta_B_rung_from_fermionic
 149  /-- B and C agree. -/
 150  BC_agree : eta_B_rung_from_chirality = eta_B_rung_from_fermionic
 151  /-- Bridge identity 1: `4 × 11 = 45 − 1`. -/
 152  chirality_gap_bridge :
 153    ((bitFlipCount0 : ℤ) * (torsionGap01 : ℤ))
 154      = (integrationGap D : ℤ) - (A : ℤ)
 155  /-- Bridge identity 2: `90/2 = 45`. -/
 156  fermionic_gap_bridge :
 157    ((fermionicDOF / 2 : ℕ) : ℤ) = (integrationGap D : ℤ)
 158
 159theorem etaBExactRungCert : EtaBExactRungCert where
 160  witness_A := eta_B_rung_from_dimension_at_D3
 161  witness_B := eta_B_rung_from_chirality_eq
 162  witness_C := eta_B_rung_from_fermionic_eq
 163  AB_agree := witnesses_AB_agree
 164  AC_agree := witnesses_AC_agree
 165  BC_agree := witnesses_BC_agree
 166  chirality_gap_bridge := chirality_product_equals_gap_minus_one
 167  fermionic_gap_bridge := fermionic_half_equals_gap
 168
 169end EtaBExactRungDerivation
 170end Cosmology
 171end IndisputableMonolith
 172