The Limits of Exact Symmetry as a Transfer Prior: Dimensionless Invariants Give No Durable Advantage for Neural Operators

A central promise of physics-aware representations is transfer: an embedding respecting a system's governing structure should extrapolate into regimes never seen in training. Fluid equations offer a clean test, admitting an exactly correct structural representation—dynamical similarity makes the solution operator depend on the dimensional parameters only through the dimensionless groups Re and Ma. We compare two operators with identical architectures and budgets: a naive operator conditioned on raw parameters (μ, L, c₀) and a similarity operator conditioned on the invariants, both evaluated strictly out of distribution—on length scales outside the training band and on (Re,Ma) configurations disjoint from training. The exact representation improves out-of-band transfer only when training covers a single length scale (4.2×); the advantage collapses once training contains modest parameter diversity, by which point the naive operator has recovered the invariants itself. All results are means over five seeds; the collapse is present in every seed and under both an MLP and a Fourier Neural Operator backbone. A positive control then manipulates the proposed cause directly: re-parameterizing so that recovering the invariant requires a nonlinearity—holding feature conditioning or the extrapolation axis fixed—restores a durable 2.3–2.6× advantage. What governs the value of an exact symmetry prior is therefore not its correctness but the recoverability of the invariant it encodes.