Continuous Equivariant Learning with Linearly Many Frames

Weighted frame averaging makes an ordinary network equivariant by evaluating it in several input-dependent coordinate systems. How many frames are needed to preserve continuity? For $n$ labeled vectors spanning ${\mathbb{R}}^d$, with $d\ge 2$ fixed, the minimum worst-case support of a continuous $\mathrm{SO}(d)$- or $\mathrm{O}(d)$-equivariant weighted frame grows linearly in $n$. The lower bound follows from characteristic classes of Grassmannians. An explicit family of $d(n-d)+1$ probes, adapted from Wronskian rank condensers, gives the upper bound. Determinant weights of order $2k$ preserve $C^{2k-1}$ regularity across probe failures, and this guarantee is sharp. For planar rotations the minimum support is exactly $n-1$. In synthetic force-learning experiments, disagreement between transported predictions produces jumps under hard frame selection, while continuous averaging removes them. At $24$ particles and a common update budget, trained Wronskian hard selection reduces force error by $42.2\%$ relative to a Gram-matrix model; averaging over $61$ probes reduces the hard model's error by a further $21.4\%$, using $6.5$ times its recorded worker time.