wolfenstein_lambda_structural
plain-language theorem explainer
Structural Wolfenstein Cabibbo parameter fixed at $(\varphi-1)^2/\varphi$, algebraically identical to $\varphi^{-3}\approx 0.236$. Anyone deriving CKM angles or the Jarlskog invariant from Q$_3$ torsion and Gray-code flip counts cites this constant. It is a bare real definition, not a proved equality to experiment.
Claim. The structural Wolfenstein parameter is the real number $\lambda_{\mathrm{struct}} = (\varphi - 1)^2 / \varphi$, where $\varphi$ is the golden ratio. Equivalently $\lambda_{\mathrm{struct}} = \varphi^{-3}$, since $\varphi - 1 = \varphi^{-1}$.
background
The ambient module builds the CKM matrix as the overlap between mass and weak eigenstates on the Q$3$ cube. Mass eigenstates carry generation torsions ${0,11,17}$; weak eigenstates are even-sign-flip SU(2) generators. Off-diagonal mass overlaps are J-cost suppressed by $\varphi^{-|\Delta\tau{ij}|}$, while flip-count ratios from Gray-code chirality $[4,2,2]$ rescale the mixing angles.
The Cabibbo angle (Wolfenstein $\lambda$) is the 1–2 mixing. Pure torsion gap $\Delta\tau_{12}=11$ would give $\varphi^{-11}\sim 5\times 10^{-5}$, far below the observed $\lambda\approx 0.225$. The module therefore treats $\lambda$ as an effective parameter that folds in flip-count asymmetry and an 8-tick overlap, not bare torsion suppression.
Recognition Science forces $\varphi$ as the unique self-similar fixed point (forcing chain T6). The elementary identity $\varphi-1=\varphi^{-1}$ then converts the compact algebraic form $(\varphi-1)^2/\varphi$ into $\varphi^{-3}$.
proof idea
No proof: the declaration is a noncomputable real definition. The right-hand side is the closed-form expression $(\varphi-1)^2/\varphi$. Downstream lemmas unfold it and rewrite via $\varphi-1=\varphi^{-1}$ to obtain $\varphi^{-3}$.
why it matters
This constant is the RS-native stand-in for Wolfenstein $\lambda$ inside the CKM-from-cube program. It is the input to lambda_structural_bounds, which sandwiches it between $\varphi^{-3}$ and $\varphi^{-2}$, and to the structural Jarlskog invariant $J\propto A^2\lambda^6\sin\delta$. Hierarchy and positivity theorems for $J$ reduce immediately to $\lambda_{\mathrm{struct}}\in(0,1)$ and the sixth-power suppression $\lambda^6\ll 1$, explaining small CP violation without fine-tuning.
The numerical value $\varphi^{-3}\approx 0.236$ sits about 4% above the measured $0.2243$; the doc-comment flags that the exact match needs the Phase-3 Berry-phase overlap. Within the forcing chain it sits downstream of T6 ($\varphi$) and the eight-tick octave (T7), and upstream of all CKM angle and Jarlskog claims in this module family.
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