lambda_RS
plain-language theorem explainer
Defines the Recognition Science Cabibbo parameter as λ_RS = (φ − 1)²/φ, which equals φ⁻³ on the golden-ratio ladder. CKM and gauge-bridge developments cite it as the structural seed for the Wolfenstein λ. The body is a one-line closed-form definition in terms of the forced constant φ.
Claim. The RS Cabibbo (Wolfenstein) parameter is the real number $\lambda_{\mathrm{RS}} = (\varphi - 1)^2 / \varphi$, equivalently $\varphi^{-3}$.
background
Module CKMExact derives Wolfenstein parameters from Q₃ geometry: Gray-code flip counts (4,2,2), generation torsion gaps Δτ₁₂ = 11 and Δτ₂₃ = 6, and a Berry face-flux correction that lifts A_structural = 6/11 to A_corrected = 9/11. Section 8 turns to the Cabibbo angle λ itself.
Here φ is the unique self-similar fixed point forced by the Recognition Composition Law (T5–T6). Because φ − 1 = φ⁻¹, the displayed formula collapses to φ⁻³, the same dream-fraction scale that appears elsewhere in the framework. An upstream namesake in Q3Representations is the unrelated Higgs quartic λ = J″(1)/2 = 1/2; the present definition is the CKM ladder quantity, not that coupling.
proof idea
Pure definition: the real is introduced by the closed form (φ − 1)²/φ. No tactics or lemmas are invoked at the declaration site. Downstream positivity and interval proofs unfold this def and apply the standard φ bounds (φ > 1.61, φ < 1.62, φ² = φ + 1).
why it matters
Supplies the φ-ladder seed for the Cabibbo sector of the exact CKM certificate (CKMExactCert) and for the T8-to-gauge/Standard-Model bridge. Immediate children prove 0 < λ_RS, the tight numerical window 0.234 < λ_RS < 0.238, and that |λ_RS − λ_PDG|/λ_PDG < 0.06, confirming the φ⁻³ origin with only a small residual geometric correction pending. The same φ⁻³ scale links the CKM ladder to the broader Recognition landmarks (eight-tick octave, D = 3 cube, and the 44-governed α and η_B identities already used for A).
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