sin2ThetaW_RS_lt_half
The Recognition Science prediction for the squared sine of the weak mixing angle satisfies sin²θ_W^RS < 1/2. Researchers comparing electroweak precision observables to the model's output of approximately 0.230 would cite this bound. The proof is a one-line algebraic reduction that unfolds the definition (3 - phi)/6, rewrites via division, and applies linear arithmetic to the lower bound phi > 1.61.
claim$sin^2 θ_W^{RS} < 1/2$, where $sin^2 θ_W^{RS} := (3 - φ)/6$ and φ is the golden ratio.
background
The module encodes the quaternion group Q₃ as the symmetry of the 8-tick cycle in Recognition Science. Under SU(2)×U(1) → U(1) breaking the Higgs doublet splits into spin-0 and spin-1 sectors whose Casimir eigenvalues fix mass ratios via the φ-ladder. The upstream definition sin2ThetaW_RS supplies the explicit form (3 - phi)/6; the lemma phi_gt_onePointSixOne supplies the numerical lower bound φ > 1.61 used in the comparison.
proof idea
Unfold sin2ThetaW_RS to expose (3 - phi)/6. Apply rw [div_lt_iff₀ (by norm_num)] to clear the denominator. Finish with linarith [phi_gt_onePointSixOne] to obtain the strict inequality from the golden-ratio bound.
why it matters in Recognition Science
The inequality supplies a basic consistency check that the RS value lies below 0.5, matching the observed regime near 0.231. It closes a local verification step inside the Q3Representations module that links the eight-tick octave (T7) and the J-uniqueness fixed point (T6) to electroweak parameters. No downstream uses are recorded in the current graph.
scope and limits
- Does not derive the numerator 3 - phi from the forcing chain.
- Does not incorporate renormalization or loop corrections to λ.
- Does not compare the bound against experimental data sets.
- Does not address the physical origin of the 1/6 denominator.
formal statement (Lean)
146theorem sin2ThetaW_RS_lt_half : sin2ThetaW_RS < 1/2 := by
proof body
Term-mode proof.
147 unfold sin2ThetaW_RS
148 rw [div_lt_iff₀ (by norm_num : (0:ℝ) < 6)]
149 linarith [phi_gt_onePointSixOne]
150
151/-- The RS prediction for sin²θ_W: ≈ (3-1.618)/6 ≈ 0.230. -/