Certificates of Joint Training and Learnability Preservation in Resistive Networks

We give sufficient initial-state certificates for jointly training several output voltages of a finite passive resistive network. All original conductances follow the projected Euclidean gradient flow of a joint quadratic voltage loss. A constructive current-preserving lift converts separation of the output levels into a quantitative lower bound on physical sensitivity, including regular faces with zero conductances. Coupling that bound to a minimum-cut distance from output isolation yields task continuation and exponential convergence under an explicit rule for harmless component detachment. A state-dependent admission policy preserves a positive reserve along locally finite histories of accepted targets without requiring finite total target variation. With two positive thresholds, the reserve certifies a ball of one next target around the current response and an explicit settling-time bound independent of the length of the accepted history. The certificate is sufficient, not a test of all reachable targets. It remains in one strict output-order chamber, may reject feasible requests, does not guarantee acceptance of an external request stream, and does not guarantee replenishment of reserve after a lower-threshold task. Its summary uses output voltages, a conductance-cut distance and the original conductance norm; the cut distance is not an output-only measurement. The contribution is the physical sensitivity estimate and its combination with structural safety and future-task admission, not a new abstract gradient-flow or reference-governor principle. This is a preprint, not a claim of journal peer review.