On the Theory of Nonlinear Exterior Anti-Difference: A Constructive Difference Algebraic Framework

This paper extends the theory of exterior anti-difference to nonlinear discrete differential forms,establishing a comprehensive constructive framework based on difference algebra.Classical discrete exterior calculus,while well-developed for linear forms, lacks systematic methods to handle nonlinear constraints and their integration along discrete fibers. We introduce the nonlinear exterior difference algebraic closure KNLEAD,a minimal difference field extension closed under solutions of nonlinear difference equations, radical extensions, roots of unity, and special nonlinear forms.Within this closure,we define the nonlinear exterior anti-difference operator Π∗ :Ωk+mnl (E)→Ωknl(B) for discrete fiber bundles π:E→B,which generalizes discrete fiber integration by incorporating combinatorial correction terms arising from nonlinear constraints.We establish fundamental properties of Π∗: nonlinearity,degree reduction, interaction with the discrete exterior derivative (nonlinear discrete Stokes theorem),projection formula, naturality, and a Fubini-type theorem.Explicit combinatorial coefficients Γ(n,d)m,k,expressed via discrete multivariate Beta functions,characterize the nonlinear corrections.We develop a complete computational algorithm with complexity analysis and provide arigorous numerical validation protocol.Applications include nonlinear discrete dimensional reduction,nonlinear discrete characteristic classes (Euler,Chern), and connections with nonlinear discrete Yang-Mills theory,Kaluza-Klein reduction, and index theory. The framework bridges discrete differential geometry,difference algebra, and nonlinear field theory,providing new tools for analyzing nonlinear discrete structure singeometry and mathematical physics.