Coupled Equivariance Tests Cannot Certify What Action Labels Mean

Equivariant outputs guide orientations and actions in downstream programs, but passing a coupled test need not establish that a public label invokes its declared action. Both test branches can execute a 50-degree turn under a label declared to mean 10 degrees. Applying the same group automorphism to the input and expected output transformations reindexes every equivariance residual, preserving the complete residual multiset and reindexing-invariant verdicts. On frozen REKD orientation and MACE force-prediction checkpoints, conventional coupled tests miss all 34 injected shared relabelings but detect all 68 one-sided controls; independent label–state probes detect all 34 shared relabelings. MACE’s mean normalized force equivariance error is 4.85×10⁻¹⁶ even as fixed-label response error increases. In a constructed REKD/HPatches matcher, multiplying rotation labels by five in an optional orientation filter lowers held-out F1@3 px for selecting fixed candidate matches from 0.691 to 0.419 across 24 scenes. A probe-derived boundary correction restores 0.691 without retraining. Coupled consistency alone does not identify the binding needed to use an externally named action.