Symmetry-enforced scalar insensitivity and finite state response

This preprint examines how an invariant scalar observation can conceal a finite change in a nonlinear dynamical state. Under stated smoothness and symmetry assumptions, odd perturbations at a symmetry-fixed center have zero first-order sensitivity in an invariant readout. Equivariance additionally makes exact symmetry partners indistinguishable through their invariant output histories. A second-variation formulation separates curvature inherited from the initial preparation, curvature generated during evolution, and curvature introduced by the readout. The application reanalyzes archived experiments in a shearing, self-gravitating magnetofluid. Among 96 prepared directions across four selected families, the 48 numerically odd directions coincide with the archived scalar-null classification while retaining nonzero complete-state contrasts. Their raw even scalar responses substantially exceed their antisymmetric responses. A separate six-center comparison gives participation ratios of 15.50-16.70 for the complete state and 2.45-5.22 for joint stress-channel histories. These are metric-dependent measures of response-spectrum concentration, not universal counts of missing physical degrees of freedom. The symmetry identities are established mathematics. The contribution is their explicit organization around the observation map and the reproducible interpretation of the archived comparisons. Raw even responses include preparation and readout curvature; they are not isolated measures of nonlinear dynamical generation. The results concern selected preparations and finite retained resolution, without establishing spatial convergence or validated sensor reconstruction. The deposit contains the paper, detailed supplement, editable sources, figures, compact evidence, provenance, and saved-data reproduction scripts. It reuses evidence from the parent study (https://doi.org/10.5281/zenodo.23633307); the companion analyses are not independent replications. No new trajectories are integrated in this reanalysis. OpenAI Codex assisted with source inspection, mathematical discussion, archived-data reductions, plotting code, and manuscript preparation. Jake Campbell directed the research and accepts responsibility for the manuscript.