sin2ThetaW_RS_approx

formal source

 149  linarith [phi_gt_onePointSixOne]
 150
 151/-- The RS prediction for sin²θ_W: ≈ (3-1.618)/6 ≈ 0.230. -/
 152theorem sin2ThetaW_RS_approx : 0.228 < sin2ThetaW_RS ∧ sin2ThetaW_RS < 0.232 := by
 153  unfold sin2ThetaW_RS
 154  constructor
 155  · rw [lt_div_iff₀ (by norm_num : (0:ℝ) < 6)]
 156    linarith [phi_lt_onePointSixTwo]
 157  · rw [div_lt_iff₀ (by norm_num : (0:ℝ) < 6)]
 158    linarith [phi_gt_onePointSixOne]
 159
 160/-! ## The Higgs Rung Assignment -/
 161
 162/-- The W-boson rung on the φ-ladder (from the mass law). -/
 163def w_rung : ℤ := 21
 164
 165/-- The Higgs rung: rung_W + Δ where Δ is derived from the Q₃ geometry.
 166
 167    The Q₃ analysis gives: the physical Higgs is the singlet (spin-0) mode
 168    of the broken SU(2). Its coupling to the Higgs mechanism is:
 169    m_H² = 2λv² where λ = sin²θ_W/(1-sin²θ_W) · m_W²/v² × (something).
 170
 171    The rung offset Δ satisfies: φ^Δ = m_H/m_W.
 172    Observed: m_H/m_W ≈ 125.2/80.4 ≈ 1.557.
 173    φ^0 = 1, φ^1 = 1.618.
 174    Since 1 < 1.557 < φ, the offset Δ ∈ (0,1).
 175
 176    The fractional rung offset is not an integer — it reflects that the Higgs
 177    mass has a radiative correction of order α/(4π).
 178
 179    The RS approximation: Δ ≈ 2 sin²θ_W · log_φ(m_Z/m_W) + 1.
 180    With sin²θ_W = 0.231, m_Z/m_W = 1.134:
 181    Δ ≈ 2·0.231·0.28 + 1 ≈ 1.13 → m_H/m_W ≈ φ^1.13 ≈ 1.72 (still ~10% off)
 182