Gauge-energy preservation under congestion-controlled network repair

Abstract

We study preservation of finite gauge-energy flows under local network repair after Bernoulli edge failures. A macroscopic demand network records terminal pairs to be routed, while a microscopic physical network contains local backup routes, bypasses and shared corridors. We prove a deterministic gauge-energy repair theorem: if the usable demand network $\mathcal{B}^{\sharp}$ carries a finite $Φ$-energy flow $θ$, then the repaired physical network $H$ carries a lifted finite $Φ$-energy flow $Θ$ with $$ \mathcal{E}^Φ_H(Θ) \le L\,β_Φ(K)\,\mathcal{E}^Φ_{\mathcal{B}^{\sharp}}(θ), $$ where $L$ bounds route length, $K$ bounds routing congestion, $\mathcal{E}$ marks energy, and $β_Φ$ is the gauge dilation constant. We then convert this comparison into probabilistic repair criteria: finite-dependent local repair is handled via domination by product measures, and random repair lengths via a variable-cost formulation compatible with chemical-distance estimates. As a main application, we prove a finite-dependent local bypass theorem: any macroscopic network whose supercritical percolation cluster supports a finite gauge-energy flow remains gauge-energy stable after bounded-range local reinforcement, provided the local repair probability is sufficiently high. This yields reinforced lattice and wedge-type examples and provides a potential-theoretic framework for random network repair beyond tree-like or edge-disjoint constructions.