gray_asymmetry
plain-language theorem explainer
On the Q₃ cube, the Gray-code flip count along spatial axis 0 is twice that along axis 1 (4 = 2×2). Anyone assembling the chirality pattern [4,2,2] for the Wolfenstein A correction cites this identity. The proof is pure definitional reflexivity on the piecewise flip-count table.
Claim. The Gray-code bit-flip count on spatial axis $0$ equals twice the count on axis $1$: $\mathrm{flips}(0)=2\,\mathrm{flips}(1)$.
background
The module derives the Wolfenstein parameter $A$ from first principles of $Q_3$ geometry. The Gray-code traversal of the three-cube assigns a flip count to each spatial axis: four flips on axis 0 and two each on axes 1 and 2, written $[4,2,2]$.
Those counts are packaged by a piecewise map sending $\mathrm{Fin},3$ indices to natural numbers (4, 2, 2). The present statement isolates the factor-of-two imbalance between the distinguished axis and either of the other two. Downstream, generation torsion ${\Delta\tau_{12}=11,\Delta\tau_{23}=6}$ and a face-flux Berry correction combine with this chirality to give $A_{\mathrm{corrected}}=9/11$.
proof idea
One-line definitional proof. Unfolding the piecewise flip-count map on $\langle 0\rangle$ and $\langle 1\rangle$ yields the numerals 4 and 2; the equality $4=2\cdot 2$ is then closed by rfl. No lemmas beyond the definition itself are required.
why it matters
This is step 1 of the five-line CKMExact derivation: the Gray-code pattern $[4,2,2]$. It feeds the certificate bundle ckmExactCert, which packages $A_{\mathrm{structural}}=6/11$, the Berry face-flux factor $3/2$, and the exact value $A=9/11$ against the PDG band $0.826\pm 0.013$.
The same axis-0 count of 4 multiplies $\Delta\tau_{12}=11$ to produce the integer 44 that also governs $\alpha^{-1}$ and the baryon-to-photon ratio $\eta_B\approx\varphi^{-44}$. Thus the asymmetry is not a local CKM detail; it is the chirality seed shared by three independent RS observables. Framework landmarks: eight-tick $Q_3$ geometry (T7) and the three spatial axes forced at T8.
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