Beyond Boundary Agreement: Sharp Repair Costs for Hard-Routed Mixtures of Experts

Can a hard-routed mixture of experts be made $L$-Lipschitz without changing its selected expert set, while preserving its original function? We show that boundary agreement and transition-layer representation impose distinct architectural costs. A weighted projection inequality transfers approximation error of a forced, rescaled transition profile into excess repair risk. For an explicit two-expert Top-1 model, the original experts and the optimal boundary value use at most one ReLU, yet every width-$Q$ shallow repair has an explicit leading-risk gap of at least $c_0{\Lambda }_0^{-2Q}$. A self-contained differential certificate gives this bound without restrictions on network parameters. Consequently, asymptotically optimal shallow repair requires diverging width; matching the $O(L^{-2})$ relative excess of a two-stage, two-ReLU repair requires width $\Omega (\log L)$. In contrast, scalar one-dimensional repair admits exact convex endpoint optimization when expert Lipschitz constants are below $L$, with at most six additional units per shallow ReLU branch. We also establish finite-budget scalar and vector trace oracles, a quantitative Top-$k$ trace-width obstruction, and a scalar polyhedral expansion with a sharp-order $O(L^{-2})$ remainder. Selected expert calls, evaluated nonlinear units, and auxiliary computation are accounted for separately.