IndisputableMonolith.Economics.WealthDistributionFromSigma

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Wealth Distribution from Sigma Conservation — F3 Economics Depth
   6
   7Pareto distribution of wealth: P(W > w) ∝ w^(-α) with α ≈ 1.5-2.
   8
   9RS prediction: the Pareto exponent α = 1 + 1/φ ≈ 1.618.
  10(Same as the network degree exponent γ = 1 + φ for scale-free networks,
  11but here the exponent for the complementary CDF is α = 1/φ + 1.)
  12
  13Actually from Zipf/Pareto duality: the Zipf exponent = α - 1 = 1/φ ≈ 0.618.
  14
  15The Gini ceiling = 1/φ ≈ 0.618 (proved in InequalityCeilingFromSigma.lean).
  16
  17Lean: prove 1 + 1/φ ∈ (1.61, 1.63).
  18
  19Lean status: 0 sorry, 0 axiom.
  20-/
  21
  22namespace IndisputableMonolith.Economics.WealthDistributionFromSigma
  23open Constants
  24
  25/-- Pareto exponent α = 1 + 1/φ = 1 + φ - 1 = φ (from φ² = φ+1). -/
  26noncomputable def paretoExponent : ℝ := 1 + phi⁻¹
  27
  28/-- 1/φ = φ - 1. -/
  29theorem inv_phi_eq : phi⁻¹ = phi - 1 := by
  30  have h := phi_sq_eq
  31  field_simp [phi_ne_zero]
  32  linarith
  33
  34/-- Pareto exponent = φ. -/
  35theorem paretoExponent_eq_phi : paretoExponent = phi := by
  36  unfold paretoExponent
  37  rw [inv_phi_eq]
  38  ring
  39
  40/-- Pareto exponent ∈ (1.61, 1.62). -/
  41theorem paretoExponent_band :
  42    (1.61 : ℝ) < paretoExponent ∧ paretoExponent < 1.62 := by
  43  rw [paretoExponent_eq_phi]
  44  exact ⟨phi_gt_onePointSixOne, phi_lt_onePointSixTwo⟩
  45
  46structure WealthDistributionCert where
  47  exponent_eq_phi : paretoExponent = phi
  48  exponent_band : (1.61 : ℝ) < paretoExponent ∧ paretoExponent < 1.62
  49  inv_phi : phi⁻¹ = phi - 1
  50
  51noncomputable def wealthDistributionCert : WealthDistributionCert where
  52  exponent_eq_phi := paretoExponent_eq_phi
  53  exponent_band := paretoExponent_band
  54  inv_phi := inv_phi_eq
  55
  56end IndisputableMonolith.Economics.WealthDistributionFromSigma
  57