Does Known Physics Help Neural PDE Models? Only When It's Non-Redundant and Resolvable

Structural priors - equation residual penalties, cross-family pretraining and conditioning, and discretization-invariant architectures are the default prescription for neural PDE surrogates, yet they are rarely tested against baselines with comparable tuning. Under a fixed architecture family and evaluation protocol, we ask which priors beat a strong baseline: a from-scratch neural operator with multi-step pushforward training. Only one does, and we characterize it mechanistically. A known-equation residual outperforms the best generic regularizer (weight decay, input noise, or data-only training, each tuned at matched budget) at every observation density, including full supervision. The advantage is capacity-robust only for nonlinear operators: as model width grows it persists for Burgers, KdV, and Allen-Cahn while collapsing to or below parity for linear families (heat, advection-diffusion), which serve as controls. We explain this dissociation with a measured quantity, the non-redundant content of the one-step map provably zero iff the operator is linear, which predicts which families retain a durable advantage. We further prove and verify the boundary of the effect: on under-resolved grids the enforced residual equals the spectral-truncation closure term, a bias irreducible by data or capacity, and the penalty reverses sign and harms nonlinear models in proportion to the unresolved tail energy. A pre-registered hypothesis that the benefit is sparsity-gated is falsified and reported as such. In the same regime, cross-family pretraining and in-context conditioning fail to beat the from-scratch baseline across three backbones and all model widths tested. The result is a falsification-tested account of when encoding known physics pays: nonlinear dynamics and resolvable evaluation.