IndisputableMonolith.Foundation.GapDerivation

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Foundation.NineParities
   4
   5/-!
   6# Gap-45 Derivation from Spatial Dimension
   7
   8Closes boundary item B-22: the coherence energy exponent = D + 2
   9(configuration dimension of a recognition event), giving E_coh = φ^{−5}
  10at D = 3.
  11
  12## B-22 Resolution
  13
  14A recognition event has D + 2 independent degrees of freedom:
  15D spatial (from the lattice, T8), 1 temporal (tick advance, T2),
  161 balance (ledger neutrality J(x)=J(x⁻¹), T3).  The coherence
  17energy is φ^{−1} per degree of freedom, so E_coh = φ^{−(D+2)}.
  18At D = 3 this gives φ^{−5}, matching `Constants.E_coh`.
  19
  20## Main Results
  21
  22- `gap_at_D3`: D²(D+2) = 9 × 5 = 45
  23- `coprimality_odd`: gcd(2^D, D²(D+2)) = 1 for all odd D
  24- `coprimality_even_fails`: gcd(2^D, D²(D+2)) > 1 for all even D ≥ 2
  25- `gap_balance`: φ^{1−gap} × φ^{gap} = φ  (matter-coherence link)
  26
  27The coprimality result provides a fourth argument that D must be odd.
  28Combined with Alexander duality (selecting D = 3), gap-45 follows
  29from D = 3 alone.
  30
  31## Status: 0 sorry, 0 axiom
  32-/
  33
  34namespace IndisputableMonolith.Foundation.GapDerivation
  35
  36open Constants
  37
  38/-! ## Definitions -/
  39
  40/-- Spatial dimension, forced by T8. -/
  41def D : ℕ := 3
  42
  43/-- Configuration dimension of a recognition event:
  44    D spatial + 1 temporal (T2) + 1 balance (T3). -/
  45def configDim (d : ℕ) : ℕ := d + 2
  46
  47/-- Number of independent ledger parities: D².
  48    At D = 3 the linear formula 3D coincides with D². -/
  49def parityCount (d : ℕ) : ℕ := d ^ 2
  50
  51/-- Dimension gap: (parity count) × (coherence exponent) = D²(D+2). -/
  52def dimensionGap (d : ℕ) : ℕ := parityCount d * configDim d
  53
  54/-! ## B-22: Configuration Dimension -/
  55
  56theorem configDim_at_D3 : configDim D = 5 := by native_decide
  57
  58/-- The Fibonacci route (2^D − D) and configuration route (D + 2) agree
  59    at D = 3.  The identity 2^D − D = D + 2 characterizes D = 3. -/
  60theorem dual_routes : 2 ^ D - D = configDim D := by native_decide
  61
  62/-! ## Parity Count -/
  63
  64theorem parityCount_at_D3 : parityCount D = 9 := by native_decide
  65
  66/-- 3D = D² at D = 3 (this identity holds only at D = 0, 3). -/
  67theorem three_D_eq_D_sq : 3 * D = D ^ 2 := by native_decide
  68
  69/-- The parametric parity count matches the NineParities enumeration. -/
  70theorem parityCount_matches_enumeration :
  71    parityCount D = Fintype.card NineParities.ParityIndex := by
  72  rw [parityCount_at_D3, NineParities.parity_count_eq_nine]
  73
  74/-! ## Gap = 45 -/
  75
  76theorem gap_at_D3 : dimensionGap D = 45 := by native_decide
  77
  78theorem gap_factors : dimensionGap D = 9 * 5 := by native_decide
  79
  80theorem gap_is_lcm : Nat.lcm 9 5 = 45 := by native_decide
  81
  82/-! ## Coprimality Forces Odd Dimension -/
  83
  84/-- For odd D = 2k+1, D²(D+2) is odd (product of odd numbers),
  85    hence coprime with any power of 2. -/
  86theorem coprimality_odd (k : ℕ) :
  87    Nat.Coprime (2 ^ (2 * k + 1)) ((2 * k + 1) ^ 2 * (2 * k + 3)) := by
  88  suffices h : Nat.Coprime 2 ((2 * k + 1) ^ 2 * (2 * k + 3)) from h.pow_left _
  89  show Nat.gcd 2 ((2 * k + 1) ^ 2 * (2 * k + 3)) = 1
  90  have hodd : (2 * k + 1) ^ 2 * (2 * k + 3) =
  91      2 * (4 * k ^ 3 + 10 * k ^ 2 + 7 * k + 1) + 1 := by ring
  92  rw [hodd]
  93  set n := 4 * k ^ 3 + 10 * k ^ 2 + 7 * k + 1
  94  rw [Nat.gcd_rec]
  95  have : (2 * n + 1) % 2 = 1 := by omega
  96  rw [this]
  97  decide
  98
  99/-- For even D = 2k (k ≥ 1), D²(D+2) is even, so gcd(2^D, D²(D+2)) > 1. -/
 100theorem coprimality_even_fails (k : ℕ) (hk : 0 < k) :
 101    ¬ Nat.Coprime (2 ^ (2 * k)) ((2 * k) ^ 2 * (2 * k + 2)) := by
 102  intro h
 103  have h1 : 2 ∣ 2 ^ (2 * k) := dvd_pow (dvd_refl 2) (by omega)
 104  have h2 : 2 ∣ (2 * k) ^ 2 * (2 * k + 2) := ⟨2 * k ^ 2 * (2 * k + 2), by ring⟩
 105  have h3 := Nat.dvd_gcd h1 h2
 106  rw [h] at h3
 107  exact absurd h3 (by norm_num)
 108
 109/-- At D = 3: gcd(8, 45) = 1. -/
 110theorem coprime_at_D3 : Nat.Coprime (2 ^ D) (dimensionGap D) := by native_decide
 111
 112/-! ## φ-Dependent Results -/
 113
 114noncomputable section
 115
 116/-- B-22: E_coh = φ^{−(D+2)} at D = 3. -/
 117def E_coh_gap : ℝ := phi ^ (-(configDim D : ℤ))
 118
 119theorem E_coh_gap_eq : E_coh_gap = phi ^ (-5 : ℤ) := by
 120  unfold E_coh_gap configDim D; norm_num
 121
 122/-! ### Bridge to the runtime constant
 123
 124The two theorems below upgrade the prose "matching `Constants.E_coh`" to
 125machine-checked identities: the coherence exponent of the *actual* runtime
 126constants `Constants.E_coh` and `Constants.hbar` IS the configuration
 127dimension `D + 2`. This is the part of the `ℏ = φ⁻⁵` story that is more than a
 128unit choice. The forced content is the count `configDim D = D + 2 = 5` (`D = 3`
 129from T8, `+1` tick from T2, `+1` balance from T3); the one modeling input is
 130`φ⁻¹` per configuration degree of freedom. The absolute SI value of `ℏ` still
 131needs a dimensional anchor (`Constants.NativeDimensionalBoundary`). -/
 132
 133/-- The RS-native coherence energy equals `φ` to the minus configuration
 134dimension: `Constants.E_coh = φ^(-(D+2))`. -/
 135theorem Constants_E_coh_eq_configDim :
 136    Constants.E_coh = phi ^ (-(configDim D : ℤ)) := by
 137  have hcfg : (-(configDim D : ℤ)) = (-5 : ℤ) := by
 138    have := configDim_at_D3; omega
 139  rw [hcfg, ← Real.rpow_intCast phi (-5 : ℤ)]
 140  unfold Constants.E_coh Constants.cLagLock
 141  norm_num
 142
 143/-- The RS-native action quantum has exponent equal to the configuration
 144dimension: `Constants.hbar = φ^(-(D+2))`. The exponent `5` is the forced
 145`D + 2`, not a free parameter. -/
 146theorem hbar_exponent_eq_configDim :
 147    Constants.hbar = phi ^ (-(configDim D : ℤ)) := by
 148  have hcfg : (-(configDim D : ℤ)) = (-5 : ℤ) := by
 149    have := configDim_at_D3; omega
 150  rw [hcfg, Constants.hbar_eq_phi_inv_fifth, ← Real.rpow_intCast phi (-5 : ℤ)]
 151  norm_num
 152
 153/-- Active edge count per tick. -/
 154def A : ℤ := 1
 155
 156/-- η_B · Θ_crit = φ^A = φ, where η_B = φ^{A−gap} and Θ_crit = φ^{gap}. -/
 157theorem gap_balance :
 158    phi ^ (A - ↑(dimensionGap D)) * phi ^ (↑(dimensionGap D) : ℤ) = phi := by
 159  have hg : (↑(dimensionGap D) : ℤ) = 45 := by exact_mod_cast gap_at_D3
 160  rw [hg, show A = (1 : ℤ) from rfl, ← zpow_add₀ (ne_of_gt phi_pos)]
 161  have : (1 : ℤ) - 45 + 45 = 1 := by norm_num
 162  rw [this, zpow_one]
 163
 164end
 165
 166/-! ## Master Certificate -/
 167
 168structure Gap45Cert where
 169  config_dim : configDim D = 5
 170  parity_count : parityCount D = 9
 171  parity_matches : parityCount D = Fintype.card NineParities.ParityIndex
 172  gap : dimensionGap D = 45
 173  coprime : Nat.Coprime (2 ^ D) (dimensionGap D)
 174  ecoh : E_coh_gap = phi ^ (-5 : ℤ)
 175  balance : phi ^ (A - ↑(dimensionGap D)) * phi ^ (↑(dimensionGap D) : ℤ) = phi
 176  odd_coprime : ∀ k, Nat.Coprime (2 ^ (2*k+1)) ((2*k+1)^2 * (2*k+3))
 177  even_not_coprime : ∀ k, 0 < k → ¬ Nat.Coprime (2^(2*k)) ((2*k)^2 * (2*k+2))
 178
 179noncomputable def gap45_cert : Gap45Cert where
 180  config_dim := configDim_at_D3
 181  parity_count := parityCount_at_D3
 182  parity_matches := parityCount_matches_enumeration
 183  gap := gap_at_D3
 184  coprime := coprime_at_D3
 185  ecoh := E_coh_gap_eq
 186  balance := gap_balance
 187  odd_coprime := coprimality_odd
 188  even_not_coprime := coprimality_even_fails
 189
 190end IndisputableMonolith.Foundation.GapDerivation
 191