Closed-Form and Constant-Time New-Source Selection for Fault-Tolerant Broadcasting in Dense Gaussian Networks

Fault-tolerant broadcasting in dense Gaussian networks is recovered by re-rooting the broadcast at a new source at maximum graph distance from the faulty nodes. This paper extends the re-rooting framework by replacing its boundary-search source-selection step with a quotient-lattice-aware algebraic construction. The first contribution is a constant-time counting method for valid new sources, formulated as an intersection of two diameter-$k$ boundary sets in the Gaussian quotient. The exact count is obtained by a fixed union of side-pair intervals over nine quotient-lattice copies, giving a closed-form procedure without scanning the network or boundary. The second contribution is a shifted direct selector for two arbitrary faulty nodes. Given faulty nodes $A$ and $B$, the problem is translated to $C={\mathrm{mod}}_{G_k}(B-A)$, and the selector finds $P$ satisfying $d(P,0)=d(P,C)=k$. For each of nine quotient-lattice shifts, sixteen signed linear systems are checked. Nonparallel systems are solved via Cramer’s rule; parallel systems are handled by interval-endpoint selection. At most $9\times 16=144$ shifted sign cases are evaluated, giving $O(1)$ selection under the word-RAM model. Validation reports zero count mismatches over $\mathrm{26,623}$ tested nodes, $\mathrm{500,000}$ valid outputs over $\mathrm{500,000}$ sampled fault pairs, and $\mathrm{40,000}$ successful re-rooted broadcast trials. The shifted selector achieves a $5.92\times$ speedup over boundary search at $k=200$, remaining stable as $k$ increases. These results make new-source selection algebraic, bounded, and independent of network size.