Encoder Topology Determines the Sign of Latent Jacobian Eigenvalues in Neural PDE Surrogates

Neural surrogates for partial differential equations are evaluated almost exclusively throughheld-out error, a statement about outputs that is silent about mechanism. We introduce twomechanistic measurements computed on an already-trained model, without retraining or modification: a dictionary-completeness estimator that quantifies how much of the latent representation is explained by named physical observables, evaluated out-of-sample with an explicitsample-to-basis guard, and JARS, a regime diagnostic given by the statistical sign of the dominant eigenvalue of the latent dynamics Jacobian restricted to the subspace that those observablesfail to explain — computed at a cost independent of latent dimension. Across a corpus of 34 surrogates spanning five PDE families in one and two spatial dimensions, the diagnostic separatescompressive from token-preserving encoders in 21 of 21 converged cases over three independent data seeds (Cohen’s d = +2.09, Mann–Whitney p = 0.029), and is invariant to arbitraryupstream rescaling, out-of-distribution extrapolation and additive noise up to 20%. We thenproduce the effect rather than observe it: toggling a single architectural switch on a 2D Navier–Stokes surrogate, with data, seed, parameter count and optimisation schedule held fixed, flipsthe regime from +0.064 to −0.167, reproducibly across three data seeds. The diagnostic doesnot reduce to a standard stability measure — it disagrees with the unprojected spectral radiuson part of the corpus — and we state without mitigation the eight limitations that bound theseclaims, including a mediation effect in two dimensions that we were unable to eliminate.