Stable Approximate Canonicalization: Stability Obstructions, Selector Ambiguity, and Data-Dependent Trade-offs

Canonicalization obtains invariant predictors by selecting one representative from each symmetry orbit, but a continuous exact selection is often topologically impossible. We show that approximation does not remove this obstruction for free. On any compact obstructed domain, deterministic canonicalizers whose invariance and orbit errors vanish must have unbounded Lipschitz constants. For free finite-group actions on compact connected smooth manifolds, the optimal expected orbit error of an exactly invariant $L$-Lipschitz relaxation is $\Theta (1/L)$ under every density bounded above and below. An antipodally symmetric population fitted to TUM VI orientations supports this rate: over the reported fit range $L\ge 17$, an explicit quaternion relaxation has a fitted log–log slope $-1.017$, with $L$ times the error approaching a nonzero plateau. We then show that continuous relaxations can localize their error near construction-specific selector-ambiguity strata, making data mass near those strata the relevant practical quantity. Experiments on LiDAR tracks, molecules, histopathology, and particle clouds show that occupancy, representation instability, and downstream harm are distinct. These results turn a qualitative obstruction into a quantitative, data-dependent account of when canonicalization is useful and what it costs