Three-fault re-rooting in dense Eisenstein–Jacobi networks: an obstructed fault set at every diameter t ≥ 3, the exact census for 3 ≤ t ≤ 8, and the threshold triple {0,2,4}

Abstract

The dense Eisenstein–Jacobi network of diameter t is the degree-six circulant Γₜ=C_N(t,t+1,2t+1) on N=3t²+3t+1 vertices, the Cayley graph of ℤ[ω]/(α), α=(t+1)+tω, on the six units. Albader (arXiv:2606.18712) relocates the source of a one-to-all broadcast so that every faulty vertex lies at distance exactly t from the new source; a fault set F admits such a source exactly when bigcap_(f ∈ F)(f+Bₜ) ≠ emptyset, where Bₜ is the set of vertices at distance t from 0. He proves that every one- or two-vertex fault set admits a re-rooted source, exhibits one three-vertex fault set at t=3 that does not, and stops. We show that an obstructed three-vertex fault set exists at every diameter t ≥ 3: for t ≥ 22 by a counting argument, |Bₜ|³+3N<N², fed by the bound |Bₜ| ≤ 6t, which we derive from a two-sided characterisation of hop distance by the hexagonal norm max{|x|,|y|,|x+y|}; for 3 ≤ t ≤ 21 by explicit witnesses. The single triple {0,2,4} is obstructed at every 6 ≤ t ≤ 21, while, by computation, it admits a re-rooted source at t=3,4,5. The exact number Tthree(t) of obstructed three-vertex fault sets is determined for 3 ≤ t ≤ 8 (407, 4,392, 22,932, 82,931, 238,797, 587,202), and computations extend the count to t ≤ 25, locating t=13 as the first diameter at which obstructed triples are the majority. We also record that the source's printed vertex label tx+(2t+1)y is not a function on ℤ[ω]/(α) — the same slip as in a second preprint of the author — and that tx-(t+1)y is. The formal development is in Lean 4: its general-t argument is proved by the kernel with no native evaluation, its finite statements by compiled evaluation; the computations that reach beyond it are labelled as such.

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