Multi-Orientation Edge-Minimum Repair for Non-Redundant Fault-Tolerant Broadcasting in Dense Eisenstein–Jacobi Networks

Dense Eisenstein–Jacobi (EJ) networks are degreesix algebraic interconnection networks whose finite quotient geometry is naturally represented by a hexagonal axial-coordinate ball. This paper studies non-redundant one-to-all broadcast repair in the dense EJ network generated by α = (t + 1) + tω, where t is the network diameter. We propose EJ-MOEM, a multi-orientation edge-minimum repair method that evaluates a constant-size family of hexagonal broadcast-tree orientations, selects a fault-aware candidate, contracts the fault-pruned tree into healthy components, and reconnects these components using external component-crossing repair edges. The resulting structure is a rooted spanning tree of the healthy subgraph: every healthy node receives the message exactly once, no faulty node is used, and the original healthy tree components are preserved. We prove that, for a chosen orientation whose fault-pruned component graph is connected, exactly c − 1 external repair edges are necessary and sufficient, where c is the number of healthy components. We also prove a depth-certificate theorem for EJ coordinate-reduction trees: every one-fault placement admits a repair of depth at most t + 1, and every two-fault placement admits a repair of depth at most t + 2. The proof uses the three-strip representation of EJ hexagons, a sector-suffix attachment lemma, a non-adjacent-sector separation lemma, and a six-direction shielding classification for paired cuts. Extended validation includes exhaustive one- and two-fault enumeration for t = 2, . . . , 12, 14, 16, 18 (up to N = 1027 and 525,825 two-fault placements at t = 18), structured theorem-critical tests through t = 30, and large random tests through t = 200, all with 100% success and no violation of the theorem. The results show that post-fault local broadcast repair in EJ networks is distinct from precomputed tree-diversity approaches and from the Gaussian degree-four case.