Event Fidelity Reorders Neural-Operator Selection at PDE Transitions

Average field error can favor a forecast that misses the shock, phase change, or pattern onset defining a PDE transition. We introduce Critical-Point-Bench (CPB), which evaluates event fidelity before ranking models by field error within the transition band. CPB rejects the entire candidate family when no model reproduces the event. A conditional-risk bound explains why global error loses control when transition events are rare. On the final Allen-Cahn holdout, CPB selects FNO over LinearNO and reduces critical normalized error from 0.00172 to 0.000414, a 75.9% decrease, while preserving phase recovery. On trajectory-held-out PDEBench, the model with the lowest aggregate error recovers only 21.0% of post-transition events, so CPB rejects it. Event fidelity therefore changes which model wins at a transition and reveals when no candidate has learned the defining physics.