sin2ThetaW_RS_approx
plain-language theorem explainer
Recognition Science predicts the Weinberg angle from the golden ratio via the J-cost potential on the φ-ladder. The theorem certifies that the RS value satisfies the strict numerical bracket 0.228 < sin²θ_W < 0.232. Electroweak model builders and precision-test analyses would cite this interval. The proof is a short tactic sequence that unfolds the definition and closes both sides of the conjunction with linear arithmetic against the established bounds on φ.
Claim. $0.228 < (3 - φ)/6 < 0.232$, where φ denotes the golden ratio.
background
The Q₃ Representations module encodes the quaternion group Q₃ as the symmetry of the eight-tick cycle that appears in the Recognition operator. Under electroweak breaking the Higgs doublet splits into spin-1 Goldstone modes (eaten by W and Z) and a spin-0 physical Higgs; the Casimir eigenvalues of these sectors fix the mass ratios once the bosons are placed on the φ-ladder. The Weinberg angle enters the mass relation m_Z / m_W = 1 / cos θ_W, which is rewritten in RS units as an offset on the ladder controlled by the J-cost curvature J″(1) = 1.
proof idea
The proof unfolds sin2ThetaW_RS to expose the explicit expression (3 - φ)/6. It splits the conjunction with constructor, rewrites each half using lt_div_iff₀ or div_lt_iff₀ (with the positive-denominator side condition), and finishes both goals by linarith against the upstream lemmas phi_gt_onePointSixOne and phi_lt_onePointSixTwo.
why it matters
The result is invoked directly by row_sin2_thetaW_RS_bracket in the WeinbergAngleScoreCard and by higgsRungCert in the HiggsRungAssignment module. It supplies the numerical anchor for the electroweak mixing angle inside the Recognition Science chain that begins from the forced value of φ (T6) and the eight-tick octave (T7). The bound therefore closes one concrete link between the abstract forcing chain and observable Standard-Model parameters.
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