alpha_coupling_derived

formal source

  77    **Formula**: α = 1/alphaInv where alphaInv = 4π·11·exp(f_gap/(4π·11))
  78    
  79    **Proved**: α > 0 (derived from geometric seed, always positive). -/
  80theorem alpha_coupling_derived : alpha > 0 := by
  81  unfold alpha alphaInv alpha_seed
  82  positivity
  83
  84/-- **C-014.2**: Strong coupling α_s (at M_Z).
  85
  86    Derived from wallpaper groups: α_s = 2/17 ≈ 0.1176
  87    Matches PDG 2022: 0.1179 ± 0.0009
  88
  89    **Formula**: α_s = 2/W where W = 17 -/
  90theorem alpha_s_coupling_derived : alpha_s_pred = 2 / 17 := by
  91  simp only [alpha_s_pred, alpha_s_geom]
  92  norm_num
  93
  94/-- **C-014.3**: Weak mixing angle sin²θ_w (from φ-structure).
  95
  96    Best φ-based prediction: sin²θ_w = (3 - φ) / 6 ≈ 0.230
  97    Observed value: 0.2229 ± 0.0003
  98    Match: Within ~3%
  99
 100    **Formula**: sin²θ_w = (3 - φ) / 6 -/
 101theorem weak_mixing_phi_based : bestPrediction = (3 - phi) / 6 := by
 102  unfold bestPrediction prediction3
 103  rfl
 104
 105/-! ## C-014: Structural Origins -/
 106
 107/-- The geometric factors that determine all three couplings:
 108
 109    1. α: 4π·11 = 44π (cube passive edges)
 110    2. α_s: 2/17 = 2/W (wallpaper groups)