Topological Fault Tolerance, Fiedler Routability, and Partially Bridged Percolation of the Interior Chrysene Lattice

Abstract

We study directional port compatibility, finite bridge reliability and graph-theoretic fault tolerance in the interior chrysene lattice. In the fixed-port model, the disubstituted family $D=\{6,12\}$ supports only the $y$-axis, while the admitted trisubstituted and tetrasubstituted families produce exact bridge--family counts. On a $3\times3$ layer, the fixed-port and orientation-admissible family sums are $4{,}410{,}944$ and $10{,}648{,}000$, compared with the corrected scalar-degree sum $60{,}454{,}593$. Under independent lateral-edge installation on a compatible scaffold with intact vertical contacts, retention of at least one complete site in each prescribed daughter half has probability $[1-(1-p_b^4)^{h-2}]^2$. Within the $3\times3\times2h$ box family, the $95\%$ retention thresholds are $90$, $108$ and $162$ nodes at $p_b=0.95$, $0.90$ and $0.80$; the stated containment model gives corresponding length parameters $239$, $243$ and $259$ bp, with $233$ bp at saturation. An edge lies on an elementary plaquette exactly when its endpoints admit a three-edge bypass. This yields a unit-metric distortion bound and preserves connectivity after deletion. Plaquette completeness directly excludes the seven- and fourteen-vertex crosses but has a sharp $19$-vertex local minimum; the $27$-vertex box requires an additional corner-closure hypothesis. We give the cycle ranks, boundary-map ranks, Laplacian gaps and single- and double-edge resilience of the two benchmark boxes, separating graph connectivity from mechanical stability and chemical repair. Reproducible verification and finite Lean certificates accompany the conditional molecular interpretation.