Cofilling Shattering: A Syndrome-Support Hierarchy for Check Erasures

What the math gives to ML

The paper defines a two-sided coding-theoretic hierarchy: a syndrome subspace must be supported on few check coordinates while every nonzero linear combination requires a high-weight preimage in the variable coordinates. This is more informative than checking only individual latent directions, because it detects an easy sparse combination hidden inside an apparently robust basis. A transferable neural-network use is a coding regularizer for sparse-adversarial robustness or error-correcting latent bottlenecks, where arbitrary combinations of q learned feature directions should require at least s coordinated feature changes. The exact binary objective is combinatorial, but small-subspace enumeration and differentiable relaxed minimum-weight decoding make a practical training loss.