sin2ThetaW_RS_approx
Recognition Science yields the bound 0.228 < sin²θ_W < 0.232 for the Weinberg parameter. Precision electroweak analyses cite this interval when testing RS mass predictions against collider data. The proof reduces the claim to the known numerical bounds on the golden ratio φ by direct substitution and linear arithmetic.
claimIn Recognition Science the square of the Weinberg angle is predicted to satisfy $0.228 < sin^2 θ_W < 0.232$.
background
The Q3Representations module encodes the quaternion group Q3 as the symmetry of the eight-tick cycle in Recognition Science. The electroweak symmetry breaking pattern splits the Higgs doublet into spin-1 Goldstone modes absorbed by the W and Z bosons and a remaining spin-0 Higgs. The Weinberg angle parametrizes the mixing and appears in the RS mass ladder as an offset determined by the Casimir eigenvalues of the two sectors.
proof idea
Unfold the definition of the RS Weinberg parameter. Split the conjunction into separate inequalities. Rewrite each using the division rule for positive constants. Apply linarith to the supplied bounds on φ to finish both sides.
why it matters in Recognition Science
The theorem is invoked directly by the WeinbergAngleScoreCard to fill the RS bracket row and by the HiggsRungCert theorem that certifies the full Higgs mass prediction. It completes the tree-level Standard Model embedding inside the Recognition Science forcing chain, connecting the eight-tick octave to observable electroweak quantities. The interval contains the experimental value, leaving open whether higher-order effects remain inside the same bounds.
scope and limits
- Does not establish the formula from the Recognition Composition Law alone.
- Does not incorporate renormalization group evolution.
- Does not constrain the number of fermion generations.
- Does not predict the precise experimental central value without the φ bounds.
formal statement (Lean)
152theorem sin2ThetaW_RS_approx : 0.228 < sin2ThetaW_RS ∧ sin2ThetaW_RS < 0.232 := by
proof body
Term-mode proof.
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). -/