Residual Geometry Shapes Downstream Correction Work for Learned Warm Starts in Nonsmooth PDE Solvers

A learned warm start hands a solver a proposal that needs correction. We study how the residual metric used in training changes that downstream correction work in nonsmooth PDEs. In two-phase enthalpy diffusion, a network proposes a window of implicit steps, a linear decoder enforces balance and semismooth Newton corrects them; only the residual weighting changes. Branch structure selects the candidate metric: for phase-change diffusion the inverse stiffness K⁻¹ is, up to scale, the only quadratic metric under which no fixed-branch step amplifies differences, whereas a reaction term admits the identity but not K⁻¹. K⁻¹ weighting cuts Newton outer iterations by 37% at 4,000 updates, and the advantage persists at the longest tested budget. All four prospective placement tests where both metrics accept every request favor the predicted metric, and a preregistered obstacle test favors Euclidean weighting. Residual size cannot stand in for correction work: starts with smaller residuals in every vector norm can require more exact Newton steps, and reference-assisted repair of wrong-phase sites closes most of the outer-work gap. K⁻¹-trained starts help Newton but not a fixed-point corrector; continued on solver-accepted states, the learned route beats tuned FISTA and a coarse-grid classical route, while a training-free temperature-form FISTA is faster still. These results support selecting candidate metrics from branch structure and judging them by the deployed solver's work.