A_structural_value
plain-language theorem explainer
The structural Wolfenstein A parameter equals exactly 6/11. Anyone deriving CKM hierarchy or Jarlskog CP measures from generation torsion cites this identity. The proof unfolds the ratio of absolute torsion gaps Δτ₂₃/Δτ₁₂ and evaluates the integers 6 and 11 by norm_num.
Claim. The structural Wolfenstein parameter $A$ equals $6/11$. Explicitly, if generation torsions are $\tau=(0,11,17)$, then $A_{\mathrm{struct}}=|\tau_2-\tau_1|/|\tau_1-\tau_0|=6/11$.
background
In the CKM-from-cube module, quark mixing is read off the Q₃ hypercube together with generation torsions ${0,11,17}$ and Gray-code chirality. Mass eigenstates carry those torsion levels; off-diagonal mass overlaps are suppressed by $\varphi^{-|\Delta\tau_{ij}|}$.
The Wolfenstein $A$ parameter controls $|V_{cb}|\sim A\lambda^2$. Its structural proxy is defined as the absolute torsion-gap ratio $|\Delta\tau_{23}|/|\Delta\tau_{12}|$, i.e. $(\mathrm{torsionGap},1,2).\mathrm{natAbs}/(\mathrm{torsionGap},0,1).\mathrm{natAbs}$. With $\tau_0=0$, $\tau_1=11$, $\tau_2=17$ one has $\Delta\tau_{23}=6$ and $\Delta\tau_{12}=11$, so the bare ratio is $6/11\approx 0.545$ (observed $A\approx 0.82$ after flip corrections).
Torsion gaps themselves are the integer differences $\tau_j-\tau_i$ on Fin 3; they also drive the hierarchy $|V_{us}|\gg|V_{cb}|\gg|V_{ub}|$ via $\varphi$-ladder suppressions $\varphi^{-11}$, $\varphi^{-6}$, $\varphi^{-17}$.
proof idea
Term-mode proof by unfolding. Simplify with the definitions of the structural $A$, of torsionGap, and of the torsion vector $\tau$. The goal reduces to the concrete rational equality $6/11=6/11$, discharged by norm_num. No external lemmas beyond those definitional expansions are required.
why it matters
Fixes the bare structural value of Wolfenstein $A$ that enters every subsequent CKM amplitude built from cube geometry. Downstream, jarlskog_positive rewrites this identity to obtain $A_{\mathrm{struct}}>0$, and cp_small_but_nonzero uses the same positive $A$ inside $J\sim A^2\lambda^6$ to show CP violation is small but nonzero without fine-tuning (the smallness is $\varphi$-suppression, not a free parameter).
In the broader RS chain the integer gaps 6 and 11 are forced by the generation torsion set ${0,11,17}$ on the eight-tick/Q₃ structure (T7 octave and cube geometry), so $A=6/11$ is a pure combinatorial prediction before flip-weight corrections. The residual gap to the experimental $A\approx 0.82$ is explicitly left to those corrections, not absorbed into a free fit.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.