hypothesis5
plain-language theorem explainer
The declaration supplies a closed-form expression for the cosine of the Weinberg angle in terms of the golden ratio. Researchers deriving electroweak parameters from Recognition Science would reference this when testing φ-constrained gauge mixing against the measured W to Z mass ratio of 0.881. It consists of a direct definition that evaluates the square root expression without intermediate lemmas.
Claim. $cos(θ_W) = √(1 - 1/(2φ + 1))$
background
The module derives the W/Z mass ratio from φ-quantized gauge structure in Recognition Science. The Weinberg angle θ_W satisfies m_W / m_Z = cos(θ_W), with the observed ratio 0.8815. Upstream constants include the active edge count A = 1 per tick from IntegrationGap.A and Masses.Anchor.A, enforcing the φ-power balance φ^(A - gap) · φ^gap = φ at D = 3.
proof idea
The definition is a one-line wrapper that directly returns the real square root of 1 minus the reciprocal of 2φ + 1. No tactics or lemmas from the depends_on list are invoked; the body is the closed expression itself.
why it matters
This definition is used by bestPhiPrediction to select the closest match to the observed cos(θ_W) and by CKMMatrix.hypothesis5 in related mixing calculations. It advances the SM-003 goal of extracting the Weinberg angle from the φ-ladder and J-uniqueness in the forcing chain T5-T8. The module highlights its 0.8% error as promising for further refinement of the eight-tick octave constraints.
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