Classifying Three-Fault Re-Rootability in Dense Gaussian and Eisenstein–Jacobi Networks

Dense Gaussian and dense Eisenstein–Jacobi networks admit lightweight fault-tolerant broadcasting by re-rooting the broadcast at a new source whose distance from every faulty node is equal to the network diameter. Existing one- and two-fault re-rooting results show that such a source always exists for at most two faulty nodes, while explicit counterexamples show that arbitrary three-fault configurations cannot always be handled by pure re-rooting. This paper turns that limitation into a high-order classification problem. For faults F1, F2, F3, pure three-fault rerooting is shown to be equivalent to the nonemptiness of the triple diameter-boundary intersection B(F1) ∩ B(F2) ∩ B(F3). By translation invariance, every instance reduces to a normalized pair of displacements (A,B) with faults 0, A,B. For dense Gaussian networks, we partition the distance-k boundary into four directed sides of a quotient diamond; for dense Eisenstein–Jacobi networks, we partition the distance-t boundary into six irected sides of a quotient hexagon. We then give self-contained finite algebraic certificates for deciding triple re-rootability in both families by enumerating side triples, quotient-lattice shifts, and integer feasibility systems. The resulting framework separates three-fault configurations into fully re-rootable triples and nonfull triples that remain two-of-three re-rootable by the universal two-fault theorem. Exact enumeration for Gaussian k = 2, . . . , 8 and EJ t = 2, . . . , 8 shows that pure three-fault re-rootability decreases with diameter, reaching 70.22% in G8 and 65.04% in EJ8. A larger sampled evaluation with 200,000 triples per setting for k, t ∈ {10, 25, 50, 100, 200} estimates the full-rerootability density as 4.43% ± 0.09% for Gaussian k = 200 and 4.23% ± 0.09% for EJ t = 200 at the 95% binomial confidence level, while zero pairwise failures occur in all tested settings. In addition, we give closed parametric positive and obstruction families, improve the visualization of representative Gaussian and EJ triples, and separate formal theorems from computational evidence through an explicit theorem-status audit. We also prove a rigorous density-zero upper bound: the fully rerootable density is O(1/k) in the Gaussian family and O(1/t) in the EJ family. The sampled density trend is therefore no longer only empirical; the experiments quantify the finite-diameter rate while the counting theorem proves that the density tends to zero. Together, the formal certificates, closed obstruction families, asymptotic density bound, and reproducible enumeration results provide a structured foundation for high-order re-rooting and identify the boundary between pure source relocation and hybrid repair.