Invariance and Body-Order Compose Additively: Minimax Rates on $\mathrm{SO}(3)^n$

Two structural priors—global invariance under a group action and low body-order—are widely used in equivariant architectures on product Lie groups such as $\mathrm{SO}(3)^n$. Whether their statistical gains are fundamental, changing the exponent of the minimax rate, or merely constant-factor, changing only the leading coefficient, has remained open. We show that these gains are statistically fundamental: both priors reduce the effective dimension governing the minimax rate, not merely the leading constant. Specifically, we establish matching upper and lower minimax rates for four nested function classes on $\mathrm{SO}(3)^n$, one for each combination of the two priors. The matching lower bound relies on a pure-$S$ lift construction and a combinatorial packing across the $\binom{n}{K}$ top-order subsets. The resulting effective-dimension formula decomposes additively: invariance contributes a fixed reduction of $3$, while body-order truncation at order $K$ contributes a reduction of $3(n-K)$ that grows with system size. An adaptive sample-splitting procedure selects the body-order from data, achieving the best in-family risk up to an $O(\sqrt{\log N/N})$ remainder with no $N$-dependent tuning. Five experiments using spectral projection estimators corroborate the theory's predicted structural and operational consequences, supporting the four-class hierarchy on synthetic targets and yielding $2$–$17\times$ sample-efficiency gains on the Maier–Saupe benchmark from liquid-crystal theory.