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

Invariant representations can discard target-relevant information while passing every symmetry test. We diagnose this over-invariance through conditional variance on the fibres collapsed by a representation. A synthetic projective benchmark gives the exact magnitude-only error floor, and a local H₃⁺ control separates a persistent radial deficit from optimization error. Projective character pooling (PCP) retains selected phase information without orbit enumeration. On the Dwork quintic, matched-capacity, charge and intervention controls connect this information to lower determinant-ratio flatness error. On a bicubic threefold, intersection-volume checks validate the residue sampler before representation comparison. Exact orbit averaging matches or surpasses PCP at longer tested budgets. A canonicalization control exposes a separate failure: exact action invariance and low local loss coexist with discontinuous boundary gluing. The study provides a workflow for distinguishing information loss, optimization error and geometric validity in finite-symmetry learning.