asCoreGap_pos

formal source

  51/-- The AS-level spectral gap at k=2 (the observed CAIDA value ≈ 0.382 ≈ 1/φ²). -/
  52def asCoreGap : ℝ := spectralGap 2
  53
  54theorem asCoreGap_pos : 0 < asCoreGap := spectralGap_pos 2
  55
  56/-- Adjacent k-core spectral gaps ratio by 1/φ. -/
  57theorem spectralGap_ratio (k : ℕ) :
  58    spectralGap (k + 1) / spectralGap k = phi⁻¹ := by
  59  unfold spectralGap
  60  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  61  have h : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
  62    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
  63    rw [zpow_add₀ hphi_ne]; simp
  64  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  65  rw [hcast, h]
  66  have hk_pos : 0 < phi ^ (-(k : ℤ)) := zpow_pos Constants.phi_pos _
  67  field_simp [hk_pos.ne']
  68
  69structure InternetSpectralGapCert where
  70  gap_pos : ∀ k, 0 < spectralGap k
  71  strictly_decreasing : ∀ k, spectralGap (k + 1) < spectralGap k
  72  ratio : ∀ k, spectralGap (k + 1) / spectralGap k = phi⁻¹
  73  as_core_pos : 0 < asCoreGap
  74
  75/-- Internet spectral-gap certificate. -/
  76def internetSpectralGapCert : InternetSpectralGapCert where
  77  gap_pos := spectralGap_pos
  78  strictly_decreasing := spectralGap_strictly_decreasing
  79  ratio := spectralGap_ratio
  80  as_core_pos := asCoreGap_pos
  81
  82end
  83end InternetSpectralGap
  84end NetworkScience