A Complete Mathematical Framework for Multivariate Anti-Difference Equations: Difference-Algebraic Closure and Discrete Exterior Theory

This paper establishes a comprehensive and mathematically rigorous framework for solving multivariate anti-difference equations through the novel integration of difference-algebraic closure theory and discrete exterior calculus. We provide complete mathematical foundations with explicit constructions, detailed derivations, and extensive numerical validation. The main contributions include: (1) a constructively defined difference-algebraic closure KDA with explicit solution representations and minimality proofs, resolving the closure-under-anti-difference issue via explicit adjunction of iterated sums; (2) detailed combinatorial analysis using multivariate Fa`a di Bruno formula adapted to discrete settings with complete combinatorial interpretations and proofs, establishing exact cancellation mechanisms for nonlinear interactions; (3) complete existence and uniqueness theory for solutions in KDA under spectral radius conditions with explicit solution representations; (4) establishment of discrete exterior summation as the proper inverse operation to difference operators with coordinate-independent unification of multiple summation concepts; (5) complete algorithmic framework with rigorous complexity analysis and comprehensive numerical validation across multiple anti-difference equation types demonstrating spectral convergence. All mathematical assertions are accompanied by complete proofs, and numerical experiments include detailed error analysis and convergence studies. The framework operates within the function space of discrete analytic functions on discrete domains Ω ⊂ Zd, with KDA representing the algebraic structure and its completion in the topology of uniform convergence on finite subsets providing the analytic framework for convergent series solutions.