A_structural_pos
plain-language theorem explainer
The structural Wolfenstein ratio A_structural = Δτ₂₃/Δτ₁₂ equals 6/11 and is strictly positive. Anyone deriving the CKM A parameter from Q₃ generation torsion cites this positivity before multiplying by the Berry face-flux correction. The proof rewrites via the exact 6/11 identity and finishes by numerical comparison.
Claim. The structural CKM ratio $A_{\mathrm{structural}} := \Delta\tau_{23}/\Delta\tau_{12}$ satisfies $0 < A_{\mathrm{structural}}$. Equivalently, since $A_{\mathrm{structural}} = 6/11$, one has $0 < 6/11$.
background
In the CKMExact module the Wolfenstein parameter $A$ is built from Q₃ geometry alone. Generation torsion on the three axes is ${\tau_1,\tau_2,\tau_3}={0,11,17}$, so the successive gaps are $\Delta\tau_{12}=11$ and $\Delta\tau_{23}=6$. The structural ratio is defined by dividing those gaps: $A_{\mathrm{structural}}:=\Delta\tau_{23}/\Delta\tau_{12}$.
The same module records the Gray-code flip counts $(4,2,2)$ on axes $(0,1,2)$ and the face-flux Berry correction that later multiplies this ratio to give $A_{\mathrm{corrected}}=9/11$. Positivity of the structural piece is the elementary gate before that correction is applied.
Upstream constants fix the eight-tick octave ($\tau_0=1$) in which the Gray code and generation labels live; the torsion integers themselves come from the particle-generation ladder already proved in the foundation layer.
proof idea
One-line tactic proof. Rewrite the goal with the already-proved identity $A_{\mathrm{structural}}=6/11$, then norm_num discharges $0<6/11$ over the reals. No further lemmas are required.
why it matters
This is step 3 of the five-line derivation that produces $A_{\mathrm{corrected}}=9/11\approx0.818$ from first principles, matching the PDG value $0.826\pm0.013$ to $0.6\sigma$. The structural ratio $6/11$ is the pure torsion contribution; the subsequent face-flux factor $3/2$ supplies the Berry correction.
The same integers reappear in the "44 connection": $44=4\times11$ governs $\alpha^{-1}$, the baryon-to-photon ratio $\eta_B\approx\varphi^{-44}$, and the CKM $A$ parameter, all tied to Q₃ chirality (Gray code $[4,2,2]$ times generation torsion). The theorem sits inside the Standard Model sector of Recognition Science and closes the positivity obligation for the structural piece; the module reports zero sorry and zero axioms for the full chain through $A_{\mathrm{corrected}}$.
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