canonical_returns

This module derives the CKM CP phase $\delta_{\mathrm{CKM}}$ as a Berry phase accumulated by generation eigenstates transported around the directed 8-tick Gray-code cycle on the cube. The cycle is chiral: different axes flip different numbers of times (the flip-count pattern $[4,2,2]$), so generation-dependent geometric phases appear, while $\theta_{\mathrm{QCD}}$ remains an energetic parameter driven to zero by $J$-cost.

The cycle permutation $\sigma$ sends each vertex to the next vertex in the Gray order: if $v$ is visited at tick $t$, $\sigma(v)$ is the vertex at tick $t+1$. Explicitly it acts as $0\mapsto 1$, $1\mapsto 3$, $2\mapsto 6$, $3\mapsto 2$, $4\mapsto 0$, $5\mapsto 4$, $6\mapsto 7$, $7\mapsto 5$. Upstream, cyclePerm_period states that this map has period exactly eight on every vertex (and is not the identity for fewer iterations). The eight-tick phases themselves are $k\pi/4$ for $k=0,\ldots,7$, matching the T7 octave landmark of the forcing chain.

A closed transport path is the prerequisite for a well-defined discrete Berry phase $\gamma_{\mathrm{Berry}}(g)=\sum_{t=0}^{7}\arg\langle\psi_g(t)|\psi_g(t+1)\rangle$.

One-line term wrapper: instantiate the upstream theorem that the cycle permutation has period exactly eight at the given vertex $v$. No extra case analysis or arithmetic is performed here; the finitary native decision already lives in that period lemma.

Path closure after one full octave is the geometric fact that lets generation eigenstates accumulate a single-valued Berry phase rather than an open-path holonomy. In this module that phase feeds the generation-dependent increments (each axis-$k$ flip contributes $\pi/4$, total $\mathrm{bitFlipCount}(k)\cdot\pi/4$) and ultimately the combination $\delta_{\mathrm{CKM}}=\gamma_{\mathrm{Berry}}(3)-\gamma_{\mathrm{Berry}}(2)-\gamma_{\mathrm{Berry}}(1)$ plus corrections. The nonzero CP phase is then forced by Gray-code chirality and directed traversal, while Strong CP stays resolved because $\theta_{\mathrm{QCD}}$ is energetic, not topological. The result sits on the T7 eight-tick octave ($2^3$ period) of the forcing chain and underwrites the sibling claims that Berry phases differ by generation and that the raw CP phase is nonzero. No downstream dependents are recorded yet; the immediate consumers are the closed-path and phase-accumulation constructions in the same file.