cos2_theta_W
plain-language theorem explainer
The definition sets the square of the cosine of the Weinberg angle to one minus the sine squared, using the golden-ratio expression for the sine term. Electroweak mass-ratio checks in Recognition Science cite this when confirming the W-to-Z ratio matches the phi-derived cosine. It is a direct algebraic substitution from the upstream sine definition.
Claim. $cos^2 θ_W := 1 - sin^2 θ_W$, where $sin^2 θ_W = (3 - φ)/6$ and $φ$ is the golden ratio.
background
In the electroweak sector the model places W, Z, and Higgs bosons on phi-ladder rungs with B_pow = 1 and r0 = 55. The Weinberg angle satisfies sin²θ_W = (3 - φ)/6 ≈ 0.2303, which produces the mass relation M_Z = M_W / cos θ_W. This supplies the complementary cos²θ_W needed to write the mass ratio directly as cos²θ_W.
proof idea
The declaration is a one-line definition that subtracts the value of sin2_theta_W from unity.
why it matters
This definition is invoked by BosonVerificationCert to certify the interval (0.7696, 0.7698) for cos²θ_W and the equality wz_mass_ratio_sq = cos2_theta_W. It also supports the theorem wz_ratio_eq_cos2 that equates the mass ratio to this term. Within the framework it closes the link between the phi-derived Weinberg angle and the observed W/Z ratio, consistent with the eight-tick octave.
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