w_rung

formal source

 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
 183    The correct derivation requires the full one-loop EW correction. -/
 184noncomputable def higgs_rung_prediction : ℝ :=
 185  -- m_H/m_W = φ^Δ where Δ = log_φ(2·sin²θ_W · (v/m_W)^2 / (1 - 2·sin²θ_W))
 186  -- Using v = 246 GeV, m_W = 80.4 GeV: v/m_W ≈ 3.06
 187  let s2 := sin2ThetaW_RS
 188  let ratio_sq := 2 * s2 * (246 / 80.4)^2 / (1 - 2 * s2)
 189  Real.log ratio_sq / Real.log phi
 190
 191theorem higgs_rung_prediction_pos : 0 < higgs_rung_prediction := by
 192  unfold higgs_rung_prediction sin2ThetaW_RS
 193  apply div_pos