When Do Physical Constraints Correct Neural PDE Rollouts?

Autoregressive neural PDE surrogates are routinely regularized and corrected with physical constraints, yet the relationship between enforcing a constraint and reducing prediction error remains poorly understood. We study this relationship for gradient-flow rollouts, where the free energy must decrease along the trajectory. Enforcing this temporal condition on fixed pretrained surrogates yields a discrete Lyapunov certificate, and the same certificate produces sharply different field correction: on Allen–Cahn it removes 99.9% of the structural error and 75.6% of the field error; on Cahn–Hilliard it removes 96.5% of the structural error and 1.1% of the field error. We explain this contrast through the certificate-to-error ratio $R=\Vert {\delta }_{\mathrm{cert}}\Vert /\Vert e\Vert$. The same-step field benefit obeys $B_{\mathrm{local}}\le R$, and for small corrections $B_{\mathrm{local}}\approx R\cos \theta$, where $\cos \theta$ is the correction–error alignment. The bound holds exactly across 15,412 activated steps, and the alignment sign predicts whether enforcement corrects or degrades a rollout across systems, distribution shifts, backbones, and independent training seeds. Autoregressive feedback can amplify a small local correction into a complete long-horizon rescue, while lower field error alone does not ensure correct dynamics. Constraint satisfaction is therefore an incomplete proxy for prediction quality: certification, field correction, and dynamical fidelity are distinct measurable properties of a learned rollout.