Threshold Comparison & Retention
Accurate regression need not yield the correct equation after support selection. For a median/sample-standard-deviation threshold in neural PDE discovery, we derive an exact two-signal retention boundary and a complete-vector perturbation bound. Exact advection-diffusion libraries lose diffusion despite coefficient recovery to numerical precision.
All-fold correct-support candidates are rejected by conditional validation; post hoc energy pooling preserves their higher minimum loss in all sixty states. Supplied-support estimates outperform unchanged selected equations in all 240 physical-residual comparisons on four unused analytic trajectories.
Diagnostics & Conclusions
True-support refits of forty saved sparse states reduce diffusion error without restoring all derivative accuracy. A fixed sequential threshold ridge comparator recovers exact support in sixty ideal controls but selects extra terms in every learned state.
With coefficients, observation points and weights fixed, exact replacement of the learned library and target reverses their learned-row ordering in 59/60 states. Weighting both losses also fails the specified Burgers diffusion-benefit criterion at every tested budget.
These finite diagnostics distinguish threshold incompatibility, refit damage and score rejection; they establish neither automatic correction nor general recovery or predictive stability.