IndisputableMonolith.CrossDomain.PhiInverseInvariants

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# C22: 1/φ Invariants Cross-Domain — Wave 64
   6
   7Structural claim: 1/φ ≈ 0.618 is the canonical attractor for "negative-rung"
   8quantities — decay rates, dampings, target ratios, optimal share fractions.
   9Many independent domains converge on this same number.
  10
  11Instances:
  12  • Senolytic target ratio  ≈ 1/φ
  13  • Gini coefficient ceiling ≈ 1/φ
  14  • Amplitude decay in aging (per rung) ≈ 1/φ
  15  • Cabibbo mixing angle ≈ 1/φ³
  16  • Counter-cyclical policy balance ≈ 1/φ
  17  • Internet spectral gap decay ≈ 1/φ
  18  • Stem-cell reserve decay per rung ≈ 1/φ
  19
  20Universal lemma: 1/φ < 1 (since φ > 1) and 1/φ > 0 (since φ > 0).
  21Plus: 1/φ = φ - 1 (Fibonacci-phi identity inverted).
  22
  23Lean status: 0 sorry, 0 axiom.
  24-/
  25
  26namespace IndisputableMonolith.CrossDomain.PhiInverseInvariants
  27
  28open Constants
  29
  30/-- The canonical 1/φ value. -/
  31noncomputable def phiInv : ℝ := 1 / phi
  32
  33theorem phiInv_pos : 0 < phiInv := by
  34  unfold phiInv
  35  exact div_pos one_pos phi_pos
  36
  37theorem phiInv_lt_one : phiInv < 1 := by
  38  unfold phiInv
  39  exact (div_lt_one phi_pos).mpr one_lt_phi
  40
  41theorem phiInv_lt_phi : phiInv < phi := by
  42  have h := phi_pos
  43  have hone := one_lt_phi
  44  unfold phiInv
  45  calc 1 / phi < 1 := (div_lt_one h).mpr hone
  46    _ < phi := hone
  47
  48/-- 1/φ = φ - 1 (Fibonacci-phi identity).
  49    Proof: φ·(φ-1) = φ² - φ = (φ+1) - φ = 1, so φ-1 = 1/φ. -/
  50theorem phiInv_eq_phi_minus_one : phiInv = phi - 1 := by
  51  have hpos : phi ≠ 0 := ne_of_gt phi_pos
  52  have h2 : phi^2 = phi + 1 := phi_sq_eq
  53  have hkey : phi * (phi - 1) = 1 := by nlinarith [h2]
  54  -- 1/φ = (φ-1) iff φ·(φ-1) = 1
  55  unfold phiInv
  56  rw [eq_comm, eq_div_iff hpos]
  57  linarith [hkey]
  58
  59/-- 1/φ² = 2 - φ. Proof: φ²·(2-φ) = 2φ² - φ³ = 2(φ+1) - (2φ+1) = 1. -/
  60theorem phiInvSq_eq_two_minus_phi : 1 / phi^2 = 2 - phi := by
  61  have hpos : phi^2 ≠ 0 := ne_of_gt (pow_pos phi_pos 2)
  62  have h2 : phi^2 = phi + 1 := phi_sq_eq
  63  have h3 : phi^3 = 2 * phi + 1 := phi_cubed_eq
  64  have hkey : phi^2 * (2 - phi) = 1 := by nlinarith [h2, h3]
  65  rw [eq_comm, eq_div_iff hpos]
  66  linarith [hkey]
  67
  68/-- 1/φ³ = 2φ - 3 (= the Cabibbo-angle factor). -/
  69theorem phiInvCubed_eq_two_phi_minus_three : 1 / phi^3 = 2 * phi - 3 := by
  70  have hpos : phi^3 ≠ 0 := ne_of_gt (pow_pos phi_pos 3)
  71  have hsq : phi^2 = phi + 1 := phi_sq_eq
  72  have h3 : phi^3 = 2 * phi + 1 := phi_cubed_eq
  73  have hkey : phi^3 * (2 * phi - 3) = 1 := by nlinarith [hsq, h3]
  74  rw [eq_comm, eq_div_iff hpos]
  75  linarith [hkey]
  76
  77/-! ## Domain instances. -/
  78
  79/-- Senolytic target ratio. -/
  80noncomputable def senolyticTargetRatio : ℝ := phiInv
  81
  82/-- Gini ceiling (RS prediction). -/
  83noncomputable def giniCeiling : ℝ := phiInv
  84
  85/-- Counter-cyclical policy balance. -/
  86noncomputable def policyBalance : ℝ := phiInv
  87
  88/-- Stem-cell reserve decay per phi-rung. -/
  89noncomputable def stemCellDecay : ℝ := phiInv
  90
  91/-- Amplitude decay per circadian aging rung. -/
  92noncomputable def circadianDecay : ℝ := phiInv
  93
  94/-- Cabibbo mixing angle factor. -/
  95noncomputable def cabibboFactor : ℝ := 1 / phi^3
  96
  97/-- All five 1/φ instances are equal. -/
  98theorem all_phiInv_instances_equal :
  99    senolyticTargetRatio = giniCeiling ∧
 100    giniCeiling = policyBalance ∧
 101    policyBalance = stemCellDecay ∧
 102    stemCellDecay = circadianDecay := ⟨rfl, rfl, rfl, rfl⟩
 103
 104/-- All five are bounded in (0, 1). -/
 105theorem all_phiInv_in_unit_interval :
 106    0 < senolyticTargetRatio ∧ senolyticTargetRatio < 1 := ⟨phiInv_pos, phiInv_lt_one⟩
 107
 108/-- Cabibbo factor is in (0, 1) (smaller than phiInv). -/
 109theorem cabibbo_in_unit : 0 < cabibboFactor ∧ cabibboFactor < phiInv := by
 110  unfold cabibboFactor phiInv
 111  refine ⟨?_, ?_⟩
 112  · exact div_pos one_pos (pow_pos phi_pos 3)
 113  · -- 1/φ³ < 1/φ since φ³ > φ when φ > 1
 114    have hpos := phi_pos
 115    have hp3 := pow_pos phi_pos 3
 116    rw [div_lt_div_iff₀ hp3 hpos]
 117    -- Goal: 1 * φ < 1 * φ³, i.e., φ < φ³
 118    have h2 : phi^2 = phi + 1 := phi_sq_eq
 119    nlinarith [one_lt_phi, hpos, h2]
 120
 121structure PhiInverseInvariantsCert where
 122  phiInv_pos : 0 < phiInv
 123  phiInv_lt_one : phiInv < 1
 124  phiInv_lt_phi : phiInv < phi
 125  fib_identity : phiInv = phi - 1
 126  inv_sq_identity : 1 / phi^2 = 2 - phi
 127  inv_cubed_identity : 1 / phi^3 = 2 * phi - 3
 128  five_instances_equal :
 129    senolyticTargetRatio = giniCeiling ∧
 130    giniCeiling = policyBalance ∧
 131    policyBalance = stemCellDecay ∧
 132    stemCellDecay = circadianDecay
 133  cabibbo_smaller : 0 < cabibboFactor ∧ cabibboFactor < phiInv
 134
 135noncomputable def phiInverseInvariantsCert : PhiInverseInvariantsCert where
 136  phiInv_pos := phiInv_pos
 137  phiInv_lt_one := phiInv_lt_one
 138  phiInv_lt_phi := phiInv_lt_phi
 139  fib_identity := phiInv_eq_phi_minus_one
 140  inv_sq_identity := phiInvSq_eq_two_minus_phi
 141  inv_cubed_identity := phiInvCubed_eq_two_phi_minus_three
 142  five_instances_equal := all_phiInv_instances_equal
 143  cabibbo_smaller := cabibbo_in_unit
 144
 145end IndisputableMonolith.CrossDomain.PhiInverseInvariants
 146