A Chain- and Diagram-Level Semantics for Morphological Calculus Refinement, monodromy, and bivector orbit decompositions

What the math gives to ML

The paper’s transferable asset is a finite chain-level representation that preserves incidence information discarded by scalar counts or ordinary compositional algebra. Its identity \(\mathcal{M}(t)=\mathcal{P}(t)+(1+t)\mathcal{B}(t)\) explicitly separates genuine homological content from paired boundary or refinement overhead, while Smith labels distinguish removable unit pairs from torsion-bearing structure. This suggests neural modules whose intermediate representations are organized as chain complexes or cell diagrams rather than unconstrained feature tensors. The most practical first transfer is a refinement-invariant graph or hierarchical encoder that enforces \(\partial^2=0\), tracks boundary-rank metadata, and cancels unit-labelled pairs without changing the represented signal.