A_structural
plain-language theorem explainer
The structural Wolfenstein A is the ratio of successive generation-torsion gaps, Δτ₂₃/Δτ₁₂, equal to 6/11. Anyone deriving CKM parameters from Q₃ geometry cites this as the uncorrected leading-order A before the Berry face-flux factor. It is a one-line real quotient of the two torsion differences.
Claim. Define the structural Wolfenstein parameter by $A_{\mathrm{struct}} := \Delta\tau_{23}/\Delta\tau_{12}\in\mathbb{R}$, where the generation torsion values are $\tau=(0,11,17)$, so $\Delta\tau_{12}=\tau_1-\tau_0=11$ and $\Delta\tau_{23}=\tau_2-\tau_1=6$.
background
The module derives the Wolfenstein A parameter of the CKM matrix from first principles of the 3-cube Q₃. Gray-code traversal of the eight vertices yields flip counts (4,2,2) on the three axes; that chirality pairs with a generation torsion ladder τ ∈ {0, 11, 17} on the three fermion generations.
The successive torsion gaps are Δτ₁₂ := τ₁ − τ₀ and Δτ₂₃ := τ₂ − τ₁. Numerically these are 11 and 6. Their ratio is the structural (uncorrected) A that later receives a Berry face-flux correction of 3/2.
The same integers reappear in the fine-structure and baryon-asymmetry formulae via the product 4 × 11 = 44, so the torsion gaps are not ad hoc CKM inputs but shared Q₃ data.
proof idea
Pure definition: cast the two natural-number gaps Δτ₂₃ and Δτ₁₂ to ℝ and form their quotient. No tactics or lemmas are required at the definition site; the equality A_struct = 6/11 is proved immediately afterward by unfolding τ and norm_num.
why it matters
This is step 3 of the five-line derivation in the module header: A_struct = 6/11 is the leading-order Wolfenstein A before the face-flux correction. Downstream, A_corrected multiplies it by the Berry factor 3/2 to obtain the exact value 9/11, which sits inside the PDG 1σ band and closes 97% of the raw gap (gap_nearly_closed).
It is packaged into CKMExactCert (a_structural field) and exported to CKMFromCube as wolfenstein_A_structural. The same torsion integers feed the 44-connection linking α⁻¹, η_B ≈ φ⁻⁴⁴, and CKM A, all governed by Q₃ chirality [4,2,2] × generation torsion. Within the broader RS chain this sits after the eight-tick octave (T7) and three spatial dimensions (T8), which fix the cube whose Gray code supplies the flip counts.
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