flip_axis0

The module derives the Wolfenstein CKM parameter $A$ from first principles of $Q_3$ geometry, with zero sorry and zero axioms. The Gray code supplies a Hamiltonian path on the $3$-cube; along that path each coordinate axis flips a fixed number of times. Those three integers form the chirality triple $[4,2,2]$.

The local helper flipCount is the piecewise map $\mathrm{Fin},3\to\mathbb{N}$ sending axis $0$ to $4$ and axes $1,2$ to $2$. The present statement simply records the axis-$0$ clause. Together with the companion axis statements it is step 1 of the five-line derivation that ends at $A_{\mathrm{corrected}}=9/11$.

The same factor $4$ reappears in the product $44=4\times 11$ that also governs the fine-structure and baryon-to-photon formulae, tying CKM geometry to the eight-tick/$D=3$ forcing chain.

One-line reflexivity. The left-hand side is the application of the piecewise definition of flip count at $\langle 0,_\rangle$, which reduces by the first clause to the numeral $4$; rfl closes.

This is the first bullet of the module's five-line derivation of $A_{\mathrm{corrected}}=9/11\approx 0.818$ (PDG $0.826\pm 0.013$, within $0.6\sigma$). The axis-$0$ count $4$ multiplies generation torsion $\Delta\tau_{12}=11$ to give the universal $44$ that also appears in $\alpha^{-1}$ and $\eta_B\approx\varphi^{-44}$. Sibling statements for axes $1$ and $2$, total flips, and Gray asymmetry build the rest of the $[4,2,2]$ package that feeds the structural ratio $6/11$ and the face-flux correction $3/2$. The result sits inside the Standard Model sector of Recognition Science and inherits the $Q_3$ geometry forced by the eight-tick octave and $D=3$ (T7–T8).