hypothesis5

formal source

 103    √(1 - 1/4.236) = √(1 - 0.236) = √0.764 ≈ 0.874
 104
 105    Very close to observed 0.881! -/
 106noncomputable def hypothesis5 : ℝ := Real.sqrt (1 - 1/(2*phi + 1))
 107
 108/-- Hypothesis 6: A more complex φ-expression.
 109
 110    cos(θ_W) = (φ³ - 1) / (φ³ + 1)
 111    = (4.236 - 1) / (4.236 + 1) = 3.236 / 5.236 ≈ 0.618
 112
 113    This is too small. -/
 114noncomputable def hypothesis6 : ℝ := (phi^3 - 1) / (phi^3 + 1)
 115
 116/-- **BEST FIT**: cos(θ_W) ≈ √(1 - 1/(2φ + 1))
 117
 118    Predicted: 0.874
 119    Observed: 0.881
 120    Error: ~0.8%
 121
 122    This is a promising φ-connection! -/
 123noncomputable def bestPhiPrediction : ℝ := hypothesis5
 124
 125/-! ## Theoretical Foundation -/
 126
 127/-- In the Standard Model, the mass ratio comes from gauge symmetry breaking:
 128
 129    m_W² = (g² × v²) / 4
 130    m_Z² = ((g² + g'²) × v²) / 4
 131
 132    where g is SU(2) coupling, g' is U(1) coupling, v is Higgs VEV.
 133
 134    Ratio: m_W / m_Z = g / √(g² + g'²) = cos(θ_W)
 135
 136    In RS, the ratio g'/g is constrained by φ. -/