  96    √(1 - 1/3.618) = √(1 - 0.276) = √0.724 ≈ 0.851
  97
  98    Same as hypothesis 3. -/
  99noncomputable def hypothesis4 : ℝ := Real.sqrt (1 - 1/(phi^2 + 1))
 100
 101/-- Hypothesis 5: cos(θ_W) = √(1 - 1/(2φ + 1))
 102
 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

papers checked against this theorem

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