Unified Analytic Solution of Nonlinear Inverse Difference Equations in Generalized Difference Algebraic Closure

This paper establishes a rigorous generalized difference algebraic framework for solving nonlinear inverse difference equations, extending the integral algebraic closure approach developed for continuous integral equations to discrete settings. We prove that solutions to general nonlinear inverse difference equations of Hammerstein type can be analytically expressed within an extended difference algebraic closure Kgdiff, which incorporates discrete polynomial approximations, rational function representations, and specialized basis functions for handling general nonlinearities.We provide constructive proofs for general nonlinearities, derive extended combinatorial expressions for correction coefficients Γ(L)m , develop adaptive approximation strategies for various nonlinear types, and present comprehensive numerical algorithms with enhanced parameter selection methodologies. Extensive validation demonstrates machine-precision accuracy across diverse nonlinear inverse difference equation types including exponential, trigonometric, rational, and piecewise nonlinearities.Key Innovations:• First extension of integral algebraic closure to discrete inverse difference equations with general nonlinearities.• Introduction of nonlinear feature spaces and adaptive basis systems for capturing complex nonlinear behavior in discrete settings.• Unified solution representation combining discrete polynomial algebraic structure with spectral decomposition.• Rigorous error bounds and adaptive regularization strategies for ill-posed discrete inverse problems, with explicit condition number control.