Ratio-Independent Three-Cycle Decomposition with Optimal Ordered Local-Switch Cost in Six-Regular Non-Axis Eisenstein--Jacobi Networks

Six-regular simple Eisenstein--Jacobi (EJ) networks are degree-six quotient-lattice interconnection networks. This paper gives a ratio-independent decomposition of every six-regular simple non-axis EJ network into three edge-disjoint Hamiltonian cycles using a canonical ordered local-switch model based on unit-parallelogram exchanges. The admitted $d=1$ branch needs no switches; $d=2$ has optimal total cost four; and for $d=3$ and d≥4 both modified factors attain the component-counting lower bound $d-1$. Factor-local switches commute, so chronological interleaving does not alter the final factors or cost within the model. Orbit normalization identifies the exact domain and excludes the unique normalized non-axis norm-three degeneration. For d≥4, an equal-coordinate alternating lift removes reduced-ratio dependence from the fine diagonal coordinate. A block-chain invariant, exhaustive interior-template lemma, and parity-specific successor permutations certify the unused complement: rank advances by one modulo $4d-6$, and arc and connector bijections prove complete coverage. The certificate uses $O(d)$ seed records and expands to the full edge lists in $O(N)$ time. Deterministic symbolic and full-quotient audits, including a dictionary-free fine-incidence check for every 4≤ d≤201, are provided in the accompanying reproducibility package and are not proof premises.