Repairing Nonlinear Model Merges: A Rank–Width Separation for GELU Networks

Parameter addition need not add the functions implemented by task updates. We characterize the capacity of every local repair for nondegenerate two-layer exact-GELU networks with fixed hidden biases. The optimal population error has exactly four regimes: first order when final adapter rank cannot contain the summed first-order update; second order when an explicit neuronwise compatibility condition fails; third order when compatibility holds but exact additivity fails; or zero. A unique quadratic repair rescales rows and preserves the actual summed-update ranks, so additional rank cannot improve these error orders. A quantitative extension characterizes near-compatibility: uniformly over bounded direction families, third-order local accuracy is possible exactly when a coefficient obstruction is of the order of the update scale, and a correction within the concatenated adapter budget achieves it. Globally, we determine the minimum exact hidden width. The exact-additivity condition is also necessary and sufficient for any same-width representation, even at distant parameters and with an affine output branch. Rank-one tasks can instead require width 3m. A calibration interpolation threshold and reproducible synthetic experiments distinguish population expressivity from finite-sample fitting and task preservation.