Weighted Homology and Cohomology of Weighted Polyhedra

What the math gives to ML

The paper introduces divisibility-constrained integer weights on simplices and uses them to modify boundary and coboundary operators while preserving the chain-complex identity \(\partial^w\partial^w=0\). This yields incidence operators whose coefficients encode hierarchical multiplicities rather than merely adjacency, together with integral torsion groups that distinguish weighted complexes having the same ordinary topology. A transferable construction is a weighted simplicial message-passing or Hodge layer whose up/down operators use these ratio-valued incidences, with a chain-consistency diagnostic enforcing exact algebraic cancellation. The most realistic first target is graph- or mesh-based learning, where simplex weights represent confidence, multiplicity, scale, or physical material classes.