Symmetries of Functional Processes under Label Noise

The task of detecting label noise is backed by a vast empirical literature, yet when detection can be trusted, and how its reliability scales with computation, remain poorly understood. We introduce \emph{symmetries of functional processes}: invariances of the effective, data-induced function class to label-noise corruption. Underlying these symmetries is a complexity barrier -- fitting locally inconsistent labels in the effective feature space forces progressively finer variation, making zero-loss separators unreachable in bounded-capacity classes. The symmetries admit asymptotic limits with a common invariant, the true concept $c(\mathrm{\mathbf{x}})$, around which predictions concentrate with monotonically shrinking uncertainty. This yields arbitrarily precise noise detection in the limit, with finite-time guarantees for time evolution under SGD and a finite-capacity rate of $\mathcal{O}(1/\sqrt{n})$ under bounded errors and asymptotic decorrelation. We empirically validate the predicted complexity signatures, asymptotic convergence, and variance collapse on a variety of standard noise detection benchmarks.