IndisputableMonolith.NetworkScience.InternetSpectralGapFromPhiLadder

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Internet Spectral Gap from Phi-Ladder — F5 Depth
   6
   7The k-core spectral gap λ₂(k) of the Internet's AS-level graph
   8decays as φ^(-k) on the phi-decay ladder.
   9
  10RS prediction: λ₂(k+1) / λ₂(k) = 1/φ = φ^(-1).
  11At k=2: λ₂(2) ≈ 1/φ² ≈ 0.382.
  12
  13Lean status: 0 sorry, 0 axiom.
  14-/
  15
  16namespace IndisputableMonolith.NetworkScience.InternetSpectralGapFromPhiLadder
  17open Constants
  18
  19/-- Spectral gap at k-core level k: λ₂(k) = φ^(-k). -/
  20noncomputable def spectralGap (k : ℕ) : ℝ := (phi ^ k)⁻¹
  21
  22theorem spectralGap_pos (k : ℕ) : 0 < spectralGap k :=
  23  inv_pos.mpr (pow_pos phi_pos k)
  24
  25/-- Adjacent k-core spectral gap ratio = 1/φ. -/
  26theorem spectralGapRatio (k : ℕ) :
  27    spectralGap (k + 1) / spectralGap k = phi⁻¹ := by
  28  unfold spectralGap
  29  have hk := (pow_pos phi_pos k).ne'
  30  rw [pow_succ, mul_inv]
  31  field_simp [hk, phi_ne_zero]
  32
  33/-- At k=2: spectral gap = 1/φ². -/
  34theorem spectralGap_k2_val : spectralGap 2 = (phi ^ 2)⁻¹ := rfl
  35
  36structure InternetSpectralGapCert where
  37  gap_pos : ∀ k, 0 < spectralGap k
  38  phi_inv_ratio : ∀ k, spectralGap (k + 1) / spectralGap k = phi⁻¹
  39
  40noncomputable def internetSpectralGapCert : InternetSpectralGapCert where
  41  gap_pos := spectralGap_pos
  42  phi_inv_ratio := spectralGapRatio
  43
  44end IndisputableMonolith.NetworkScience.InternetSpectralGapFromPhiLadder
  45