Reduced characteristic number criteria for equivariant bordism of $T^k$- and $(\mathbb{Z}_2)^k$-manifolds with isolated fixed points

What the math gives to ML

The paper shows that a finite collection of equivariant fixed-point contributions can be distinguished by evaluating powers of one carefully chosen scalar polynomial, because the resulting coefficient matrix is Vandermonde. The transferable asset is not equivariant bordism itself, but the finite-support algebra: distinct scalar scores make low-order moments identify individual components, while coincident scores create an explicit degeneracy. This suggests a principled anti-collapse regularizer for routers, prototypes, or latent modes that maximizes the conditioning of a Vandermonde feature matrix rather than merely adding pairwise repulsion. The first target should be MoE routing, where expert collapse is measurable and the scalar score can be taken directly from router logits or learned expert features.