Certified Euclidean-Residue Minimal-Alignment Switch Decompositions for Three Edge-Disjoint Hamiltonian Cycles in Eisenstein–Jacobi Networks

Eisenstein–Jacobi (EJ) networks are degree-six quotient-lattice interconnection networks. For a generator α =a + bρ, let N = a^2 + ab + b^2 and d = gcd(a, b). If d = 1, the three natural unit directions already give three edge-disjoint Hamiltonian cycles. If d > 1, each unit direction splits into d cycles and the EDHC problem becomes a cycle-splicing problem. Existing non-coprime EJ decompositions prove existence by using a rectangular representation and exchange schedules. This paper develops a different, local switch calculus in the natural Cayley geometry. The first two Hamiltonian cycles are built using the minimum possible d − 1 intercomponent switches each, and the third factor is obtained as the unused edge complement. The contribution is deliberately not a new existence theorem for all non-coprime EJ networks; rather, it is a compact, formula-driven, minimal-switch decomposition for Euclidean-residue families whose complement incidence is proved symbolically. The proof separates four ingredients: componentlabel collapse, anchor cancellation, noncollision of lifted switch representatives, and connected complement incidence. No infinitefamily theorem in this manuscript is proved by finite witnesses or by computational enumeration. The theorem scope is stated for the parameter ranges where an algebraic complement-incidence certificate is written down. Tables and CSV data are used only to verify and reproduce the formulas, never as proof of an infinitefamily theorem.