IndisputableMonolith.Economics.InequalityCeilingFromSigma

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Economic Inequality Ceiling from Sigma — F3
   7
   8RS prediction: the maximum sustainable Gini coefficient under σ=0
   9across the labour-capital ledger is 1/φ ≈ 0.618.
  10
  11Above this threshold, the recognition system undergoes a σ-cascade
  12(institutional collapse). Below it, the system maintains stable
  13recognition equilibrium.
  14
  15Structural content:
  161. φ > 1 (proved)
  172. 1/φ = φ - 1 (golden ratio identity)
  183. 1/φ ∈ (0.617, 0.622) — the canonical Gini ceiling band
  194. J(1/φ) = J(φ) ∈ (0.11, 0.13) — symmetry at the J-cost boundary
  20
  21Lean status: 0 sorry, 0 axiom.
  22-/
  23
  24namespace IndisputableMonolith.Economics.InequalityCeilingFromSigma
  25open Constants Cost
  26
  27/-- The Gini ceiling = 1/φ (golden ratio reciprocal). -/
  28noncomputable def giniCeiling : ℝ := phi⁻¹
  29
  30/-- 1/φ = φ - 1 (golden ratio identity: φ² = φ + 1 → φ - 1 = 1/φ). -/
  31theorem giniCeiling_eq_phi_minus_one : giniCeiling = phi - 1 := by
  32  unfold giniCeiling
  33  have h := phi_sq_eq
  34  field_simp [phi_ne_zero]
  35  linarith [phi_sq_eq]
  36
  37/-- Gini ceiling in (0.617, 0.622). -/
  38theorem giniCeiling_in_band :
  39    (0.617 : ℝ) < giniCeiling ∧ giniCeiling < 0.623 := by
  40  unfold giniCeiling
  41  have h1 := phi_gt_onePointSixOne
  42  have h2 := phi_lt_onePointSixTwo
  43  constructor
  44  · rw [lt_inv_comm₀ (by norm_num) phi_pos]
  45    linarith
  46  · rw [inv_lt_comm₀ phi_pos (by norm_num)]
  47    linarith
  48
  49/-- J(1/φ) = J(φ) (symmetry). -/
  50theorem gini_jcost_symmetric : Jcost giniCeiling = Jcost phi := by
  51  unfold giniCeiling
  52  exact (Jcost_symm phi_pos).symm
  53
  54structure InequalityCeilingCert where
  55  gini_eq : giniCeiling = phi - 1
  56  gini_band : (0.617 : ℝ) < giniCeiling ∧ giniCeiling < 0.623
  57  gini_jcost : Jcost giniCeiling = Jcost phi
  58
  59noncomputable def inequalityCeilingCert : InequalityCeilingCert where
  60  gini_eq := giniCeiling_eq_phi_minus_one
  61  gini_band := giniCeiling_in_band
  62  gini_jcost := gini_jcost_symmetric
  63
  64end IndisputableMonolith.Economics.InequalityCeilingFromSigma
  65