faceFlux_13

Module CKMExact extracts the Wolfenstein parameter $A$ from $Q_3$ geometry. The cube's Gray-code Hamiltonian path has flip counts $(4,2,2)$ on axes $(0,1,2)$: axis 0 flips four times, axes 1 and 2 twice each. Face flux on a coordinate face is the sum of the two axis flip counts that bound it.

Generation mixing is read off faces of this cube. The $1\to 2$ face uses axes $0,1$; the $2\to 3$ face uses axes $1,2$; the $1\to 3$ face uses axes $0,2$. Because axis 0 is the high-flip (dominant) axis, any face that includes it inherits flux $6$, while the pure subdominant face has flux $4$.

The module's five-line derivation multiplies structural torsion $\Delta\tau_{23}/\Delta\tau_{12}=6/11$ by the Berry face-flux ratio $6/4=3/2$ to obtain $A_{\mathrm{corrected}}=9/11$.

One-line term proof by rfl. Once faceFlux is defined as the sum of the two Gray-code flip counts on the spanning axes, the numerals ⟨0,_⟩ and ⟨2,_⟩ reduce definitionally to $4+2$, which is $6$ by closed arithmetic. No lemmas are invoked beyond the definitional unfolding of faceFlux and the already-proved flip counts on axes 0 and 2.

This pins the high-flux value that enters the Berry correction factor in the CKMExact chain. The module doc records berryCorrection = faceFlux(12)/faceFlux(23) = 6/4 = 3/2, and the $1\to 3$ face shares the same flux $6$ as $1\to 2$ because both include dominant axis 0. That common value is why $V_{us}$ and the $1\to 3$ channel see the high-flux face while $V_{cb}$ sees the low-flux face.

Downstream the corrected Wolfenstein $A$ is $A_{\mathrm{corrected}}=(6/11)\times(3/2)=9/11\approx 0.818$, within $0.6\sigma$ of PDG $0.826\pm 0.013$. The same $Q_3$ chirality $[4,2,2]$ and the factor $44=4\times 11$ reappear in the RS expressions for $\alpha^{-1}$ and $\eta_B\approx\varphi^{-44}$, tying CKM geometry to the eight-tick / $\varphi$-ladder constants. No used_by edges are recorded yet; the immediate consumers are the in-module equalities berry_correction_eq and A_corrected_exact.