The kernel of the Birman-Craggs-Johnson homomorphism

What the math gives to ML

The paper develops a finite-dimensional Boolean-function algebra built from quadratic refinements of a symplectic vector space over \(\mathbb{F}_2\), together with an explicit degree filtration and an \(\operatorname{Sp}_{2g}(\mathbb{F}_2)\)-action. The transferable asset is not the mapping-class-group application itself, but the combination of paired coordinates, a nontrivial bilinear cocycle, and symmetry-preserving low-degree feature spaces. This suggests a compact equivariant module for binary or discretized representations: represent interactions using the paper's quadratic relation rather than unconstrained pairwise features, and train with random symplectic augmentations. The idea is most plausible for models whose inputs already have natural paired channels, such as graph edge orientations, error-correcting codes, spin variables, or learned binary latent codes.