Exterior Antidifference: A Systematic Theory of Discrete Integration on Simplicial Complexes and Cellular Sheaves

This paper establishes a comprehensive and rigorous theory of exterior antidifference as the discrete counterpart to exterior integration in differential geometry. Building systematically upon the well-developed theory of exterior integration on smooth fiber bundles, we construct a complete discrete analog through summation over fibers in discrete fiber bundles. We provide precise mathematical definitions for discrete differential forms, discrete exterior derivatives, and the exterior antidifference operator, with careful attention to orientation conventions and combinatorial structures. The theory is developed through four main pillars: (1) a constructive definition of exterior antidifference via fiber summation in simplicial and cellular bundles; (2) proof of fundamental properties including linearity, degree reduction, commutation with the discrete exterior derivative (discrete Stokes theorem),naturality, and projection formulas; (3) establishment of a complete discrete de Rham cohomology and Hodge theory framework; and (4) development of computational methods and applications in geometric analysis and mathematical physics. All results are presented with complete mathematical proofs and verified through computational examples.