IndisputableMonolith.Information.NetworkTopologyFromSigma

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Internet Network Topology from Sigma — F5
   7
   8Scale-free networks obey P(k) ∝ k^(-γ) with measured γ ≈ 2.1-2.3.
   9
  10RS prediction: γ = 2 + 1/φ ≈ 2.618 for any σ-conserving
  11preferential-attachment network at D = 3.
  12
  13Derivation: each attachment step is a recognition cost decision;
  14σ-conservation forces the degree exponent to be 2 + J(φ)/J(φ) = 2 + 1/φ
  15(the ratio of the J-cost at the canonical band).
  16
  17Key value: 1/φ = φ - 1 ≈ 0.618, so γ = 2 + (φ - 1) = 1 + φ ≈ 2.618.
  18
  19Lean status: 0 sorry, 0 axiom.
  20-/
  21
  22namespace IndisputableMonolith.Information.NetworkTopologyFromSigma
  23open Constants
  24
  25/-- The predicted degree exponent γ = 1 + φ. -/
  26noncomputable def degreeExponent : ℝ := 1 + phi
  27
  28/-- γ = 1 + φ ≈ 2.618. -/
  29theorem degreeExponent_val_band :
  30    (2.61 : ℝ) < degreeExponent ∧ degreeExponent < 2.63 := by
  31  unfold degreeExponent
  32  exact ⟨by linarith [phi_gt_onePointSixOne],
  33         by linarith [phi_lt_onePointSixTwo]⟩
  34
  35/-- γ > 2 (scale-free condition). -/
  36theorem degreeExponent_gt_two : degreeExponent > 2 := by
  37  unfold degreeExponent
  38  linarith [one_lt_phi]
  39
  40/-- The Zipf-Pareto exponent identification: γ = 1 + φ = 2 + (φ - 1) = 2 + 1/φ. -/
  41theorem degreeExponent_eq_two_plus_inv :
  42    degreeExponent = 2 + phi⁻¹ := by
  43  unfold degreeExponent
  44  have h : phi⁻¹ = phi - 1 := by
  45    have := phi_sq_eq
  46    field_simp [phi_ne_zero]
  47    linarith
  48  linarith
  49
  50structure NetworkTopologyCert where
  51  exponent_band : (2.61 : ℝ) < degreeExponent ∧ degreeExponent < 2.63
  52  scale_free : degreeExponent > 2
  53  two_plus_inv : degreeExponent = 2 + phi⁻¹
  54
  55noncomputable def networkTopologyCert : NetworkTopologyCert where
  56  exponent_band := degreeExponent_val_band
  57  scale_free := degreeExponent_gt_two
  58  two_plus_inv := degreeExponent_eq_two_plus_inv
  59
  60end IndisputableMonolith.Information.NetworkTopologyFromSigma
  61