sin2ThetaW_RS_pos
The declaration shows that the Recognition Science prediction for sin²θ_W is positive. Model builders checking electroweak mass ladders cite it to confirm the sign of the Weinberg parameter in the φ-ladder. Proof proceeds by unfolding the definition and applying div_pos with a bound on φ.
claim$0 < (3 - φ)/6$ where $φ = (1 + √5)/2$ is the golden ratio.
background
The module treats the quaternion group Q₃ as the symmetry of the 8-tick cycle that governs electroweak symmetry breaking SU(2)×U(1) → U(1). Under this breaking the Higgs doublet splits into three Goldstone modes eaten by W± and Z plus one physical spin-0 Higgs; the mass ratios follow from Casimir eigenvalues on the φ-ladder. The constant sin²θ_W^RS is introduced as the RS-native value (3 - φ)/6 that enters the quartic coupling and gauge coupling relations at the electroweak scale. The upstream lemma phi_lt_onePointSixTwo supplies the strict inequality φ < 1.62 needed to guarantee positivity of the numerator.
proof idea
The term proof first unfolds the definition of sin²θ_W^RS to the explicit quotient (3 - φ)/6. It then invokes div_pos, reducing the goal to separate positivity of numerator and denominator. Numerator positivity is discharged by linarith applied to the imported bound phi_lt_onePointSixTwo; the denominator is settled by norm_num.
why it matters in Recognition Science
The result is invoked directly by the downstream theorem q3_correction_pos that establishes positivity of the Q₃ correction factor 17/16 for the Higgs rung assignment. It therefore closes the sign check required for the RS mass formula m_H / m_W derived from J″(1) = 1 and the eight-tick octave structure. The declaration sits inside the chain that links the Recognition Composition Law through the φ-ladder to concrete electroweak observables.
scope and limits
- Does not claim numerical agreement with the measured sin²θ_W ≈ 0.231.
- Does not derive the expression (3 - φ)/6 from the Recognition Composition Law.
- Does not incorporate loop corrections to the quartic coupling λ.
- Does not prove existence or uniqueness of the Q₃ representations.
Lean usage
theorem q3_correction_pos : 0 < sin2ThetaW_RS * (17 / 16 : ℝ) := by exact mul_pos sin2ThetaW_RS_pos (by norm_num)formal statement (Lean)
139theorem sin2ThetaW_RS_pos : 0 < sin2ThetaW_RS := by
proof body
Term-mode proof.
140 unfold sin2ThetaW_RS
141 apply div_pos
142 · linarith [phi_lt_onePointSixTwo]
143 · norm_num
144
145/-- sin²θ_W^RS is less than 0.5. -/