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

This paper gives a compact, constructive method for splitting a broad class of Eisenstein–Jacobi (EJ) interconnection network graphs into three edge-disjoint Hamiltonian cycles. EJ networks are hexagonal quotient-lattice graphs with three natural direction classes; a six-regular non-axis EJ network can contain at most three such cycles, so this decomposition is optimal. The method starts from the natural direction factors, applies elementary unit-parallelogram switches to the first two, and proves that the remaining edges form the third Hamiltonian cycle. Within the canonical ordered local-switch model, d=1 needs no switches, d=2 has optimal total cost four, and d=3 and d≥4 attain the component-counting lower bound in each modified factor. For d≥4, a two-valued alternating lift cancels the reduced-ratio dependence in the fine diagonal coordinate. A fine-incidence rank certificate then proves complement connectivity by showing that all diagonal arcs and released connectors form one cycle of length 4d−6. The compact certificate contains O(d) seed records and expands to the full O(N) edge lists only when required. Deterministic symbolic, full-quotient, and dictionary-free incidence audits are supplied as reproducibility files.