Wrong Operator or Blind Design? A Post-Training Diagnostic for Physics-Informed Coefficient Learning

arXiv:2608.16925v3 Announce Type: replace Abstract: Physics-informed and hybrid models integrate data and physics, and read learned coefficients as physical. Yet a small prediction error guarantees neither that the postulated operator is right, nor that the design identifies the coefficient, nor that training reached its minimum. On one parabolic benchmark where a sensor aggregates several diffusion rates and the truth is known, a correct operator recovers the coefficient to machine precision. A wrong one leaves it $30\%$ off while the in-domain error is 1.36 times the noise, and drifts $27\%$ across windows. At $7\%$ noise the error falls below the noise; the coefficient stays $30\%$ off. Without the truth, a post-training coefficient diagnostic questions the trained model with three classical statistics, without oracle; the first two need no refit. Does it explain the data? Can the design resolve the named physics? Does the coefficient depend on the window? The specification test keeps its pre-registered false-alarm ceiling and rejects every replicate of both wrong operators tested; a network frozen at its minimum returns the same verdicts. Where the design is blind, a wrong operator confined to the unseen rate leaves the test mute; the design reading flags it beforehand. A network stopped short of its minimum mimics a wrong operator; an analytic second fit separates the two. The test has a notch against a missing reaction term, miscalibrates under correlated noise and loses power beyond $7\%$ noise. Each warning names its remedy (operator, experiment or training); silence certifies nothing.