sin2ThetaW_RS_val
plain-language theorem explainer
The theorem fixes the Recognition Science value of the weak mixing angle squared at the algebraic expression (3 - φ)/6. Electroweak model builders inside the Recognition framework cite this equality when placing W, Z, and Higgs masses on the phi-ladder. The proof is a one-line reflexivity that matches the theorem statement to the body of the upstream definition sin2ThetaW_RS.
Claim. $sin^2 θ_W^{RS} = (3 - φ)/6$, where φ is the golden-ratio fixed point of the Recognition Composition Law.
background
The Q3Representations module encodes the quaternion group Q3 as the symmetry of the 8-tick cycle that governs the Recognition operator. Under electroweak breaking the Higgs doublet splits into three spin-1 Goldstone modes and one spin-0 physical Higgs; the Casimir eigenvalues of these sectors, together with the curvature J''(1) = 1, fix the quartic coupling λ_RS = 1/2 and the SU(2) coupling g. The upstream definition supplies the explicit real number sin2ThetaW_RS := (3 - φ)/6 that enters the relation g = 2 sin θ_W · m_Z / v.
proof idea
The proof is a one-line reflexivity that applies directly to the defining equation of sin2ThetaW_RS.
why it matters
The result supplies the concrete algebraic anchor for sin²θ_W^RS required by the mass-ladder construction in the Standard Model sector. It closes the step that converts the forced curvature J''(1) = 1 into the ratio m_H / m_W after the loop-correction factor sin²θ_W / (1 - sin²θ_W) is inserted. The value is consistent with the T7 eight-tick octave and the alpha-band constraints of the framework.
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