large_cabibbo_from_coupling_ratio

The module builds the cycle operator $\hat R$ on $\mathbb{C}^8$: the unitary permutation matrix induced by the 8-tick Gray-code Hamiltonian cycle on the cube $Q_3$. Vertices of $Q_3$ are the computational basis; each tick flips exactly one bit, and a full period is eight. Eigenvalues are eighth roots of unity; eigenmodes are DFT-8 modes that carry generation and mixing structure.

Each face-pair is identified with a generation and with a spatial axis. The generation flip count is the number of times that axis is flipped per Gray cycle (the bit-flip census). Upstream, the generation coupling asymmetry theorem states that axis 0 is driven twice as often as axes 1 and 2: $N_{\mathrm{flip}}(0)=2N_{\mathrm{flip}}(1)$ and $N_{\mathrm{flip}}(0)=2N_{\mathrm{flip}}(2)$. That double drive is the kinematic source of mass–weak misalignment.

One-line term proof: project the first conjunct of generation_coupling_asymmetry. That upstream theorem is itself discharged by native_decide on the finite flip census, so no further algebraic work is needed here.

In the Recognition framework the CKM matrix is meant to emerge from the cycle operator on the eight-tick Gray path (T7 octave, cube geometry forcing $D=3$). The doc-comment ties this 2:1 axis-0 vs axis-1 ratio directly to the large Cabibbo (1–2) mixing: generation 1 is driven twice as often, so mass and weak bases overlap strongly in the 1–2 sector and more weakly in 2–3. The declaration isolates that Cabibbo-facing half of the coupling asymmetry for later mixing-angle arguments. No downstream consumers are wired yet; it sits as a named kinematic lemma between the Gray chirality census and a future CKM extraction from $\hat R$.