Too Symmetric to See: Diagnosing Over-Invariance in Finite-Symmetry Geometric Learning

<title>Abstract</title> <p>Invariant models can fail silently when their representation imposes more symmetry than the target. For finite phase symmetries, replacing complex coordinates with magnitudes imposes a continuous phase symmetry and can discard target-relevant relative phase without triggering an invariance-error warning. We study this over-invariance failure mode in projective geometric learning. A controlled benchmark gives the irreducible error of every magnitude-only predictor in closed form, separating persistent information loss from finite-budget optimization. We evaluate projective character pooling (PCP), one orbit-free repair that is exactly invariant for declared diagonal root-phase actions. On the Dwork quintic, matched-capacity, wrong-charge, generic-phase and checkpoint-intervention controls associate restored phase channels with a lower determinant-ratio flatness diagnostic. On a bicubic threefold, an absolutely validated residue sampler supports evaluation beyond a single projective geometry. A boundary audit shows that exact action invariance and low local loss do not ensure smooth global gluing. A full-orbit reference shows a finite-budget trade-off: PCP has the lower mean diagnostic after 30 updates, whereas exact averaging has the lower mean at 200 and 1,000 updates. The results make over-invariance measurable and provide controlled instruments for detecting it. PCP is a scoped construction, not a universal alternative to orbit averaging.</p>