Tensor Algebra Toolkit for Folded Mixture Models: Symmetry-Aware Moments, Orbit-Space Estimation, and Poly-LAN Rates

We develop a symmetry-aware toolkit for finite mixtures whose components are only identifiable up to a finite \emph{folding} group action. The correct estimand is the multiset of parameter orbits in the quotient space, not an ordered list of raw parameters. We design invariant tensor summaries via the Reynolds projector, show that mixtures become convex combinations in a low-dimensional invariant feature space, and prove identifiability, stability, and asymptotic normality \emph{on the quotient}. Our loss is a Hausdorff distance on orbit multisets; we prove it coincides with a bottleneck assignment metric and is thus computable in polynomial time. We give finite-sample Hausdorff bounds, a two-step efficient GMM formulation, consistent selection of the number of components, robustness to contamination, and minimax lower bounds that certify Poly-LAN rates $n^{-1/D}$ when the first nonzero invariant curvature appears at order $D$. The framework is illustrated for the hyperoctahedral group (signed permutations) and dihedral symmetries in the plane.