A Reference-Free Lower Bound on the Symmetry Error of 3D Generative Models

Scalable 3D generative models increasingly use free-form backbones even when the data or downstream task has a known symmetry, learning rotation and permutation invariance from augmentation rather than enforcing it architecturally. Augmentation teaches symmetry only approximately, so deployed models carry a residual symmetry-breaking error that is hard to measure: the reference distribution is unavailable after training, and existing equivariance diagnostics quantify the network's mismatch with the group action rather than the distance between the generated distribution and the target. We define the symmetry defect as the Kullback–Leibler divergence between a model and its own group-orbit average, computable from the model likelihood alone. A classical information-geometric Pythagorean identity makes this model-only quantity a lower bound on the error to any symmetric target, and the same identity is constructive: applying one uniformly random group element to each sample removes exactly the certified component without retraining. We separate the population quantity that carries the guarantee from its finite Monte-Carlo estimate, and exact enumeration of a finite subgroup, itself a valid lower bound, from sampled continuous-group averaging with its Jensen bias. Across known-density targets, a free-form diffusion model with a probability-flow likelihood, real ShapeNet point clouds, and pretrained conformer and point-cloud generators, finite-sample estimates respect the population bound, non-equivariant backbones read a measurable defect while equivariant counterparts read near zero, symmetrization markedly improves likelihoods, and the score ranks candidate models in strong agreement with hidden reference orderings. Because the defect is a lower bound, a near-zero reading certifies near-invariance rather than closeness to the target: the certificate falsifies rather than validates.