Multi-Orientation Edge-Minimum Repair for Non-Redundant Fault-Tolerant Broadcasting in Dense Gaussian Networks

Dense Gaussian networks are degree-four algebraic interconnection networks with compact diameter and simple modular routing. This paper studies non-redundant one-to-all broadcast repair in the dense Gaussian network generated by α = k + (k + 1)i. We propose multi-orientation edge-minimum repair (MOEM), which evaluates a constant-size family of Gaussian broadcast-tree orientations, selects a fault-aware orientation, contracts the fault-pruned tree into healthy components, and reconnects those components using external component-crossing repair edges. The resulting structure is a rooted spanning tree of the healthy subgraph, so each healthy node receives the message exactly once and no faulty node is used. We prove that, for a chosen orientation with c fault-pruned components and a connected healthy component graph, the repair step is non-redundant and uses the minimum possible number c − 1 of external component-repair edges. We also prove that, forevery one- or two-fault placement, the MOEM orientation family contains a repair with depth at most k + 2. The depth proof combines a certificate framework, an explicit four-case offaxis analysis, and a five-component orthogonal-axis certificate. Exhaustive validation for k = 5, . . . , 10 and large-scale validation through k = 200 confirm the implementation and show that random two-fault repairs use approximately two external repair edges.