Verification Limits for Physics-Informed Solvers: Internal Signals Fails to Detect Failure Modes

Physics-informed solvers are based on training to minimize a residual loss, represented in terms of physical differential equations. That loss would typically be the only signal available to an automated pipeline ranking qualities of runs. In this study, we show that for the violation of the solver's own modeling assumption, a crucial failure class, no computable quantity from the model's internal signals can act as a verifier. Using an enriched neural solver for crack propagation, we sweep loading until the linear-elastic assumption fails. The physical quantity indicates degradation by a factor of $15.7$ while the training loss contains no potential signal to identify the decline (Spearman $\rho=-0.41$, $n{=}200$). Then, we conclude two further impossibility results. Any post-hoc read of the converged field returns a solution function determined by a single scalar and therefore cannot encode that scalar's own error with other degrees of freedom, confirmed by a fit insisting on the elastic exponent despite letting a near-tip exponent change. The natural matched-asymptotic overlap residual is exactly independent of the load parameter, because the inner and outer fields are commonly controlled by the same reported scalar and would cancel identically. Detection can only be possible once information absent from the solution is available externally. Combining the yield stress and the plastic near-tip field will track the true error ($\rho=+0.95$) and separate from baseline noise earlier than the error does. Thus, we argue that this generalizes that the verification of a single-form solver must come from external sources.