Unified Algebraic Framework for Discrete Inverse Exterior Variation Geometry: Constructive Methods and Certified Computation

This paper establishes a complete differential-algebraic framework for discrete inverse exterior variation geometry, providing unified constructive solutions to discrete inverse variational problems. We define the discrete exterior variational geometry closure KDiscExtVar, the quantum discrete exterior variational closure KDiscExtQVar, and the central discrete inverse exterior variational geometry closure KDiscInvExtVar. These closures are constructed as differential field extensions through recursive adjunction processes, integrating discrete exterior differential forms, conservation laws, topological invariants, as well as discrete Helmholtz integrability conditions for inverse variational problems, action reconstruction, and the discrete inverse Noether theorem. Within these closures, we prove that solutions to a large class of discrete exterior variational problems (including discrete Maxwell equations, discrete Yang-Mills theory, discrete Chern-Simons theory) and discrete inverse exterior variational problems admit unified representations that respect underlying discrete geometric, algebraic, and physical structures. The framework rigorously handles discrete nonlinearity, exterior constraints, topological changes, and variational invertibility problems, while preserving discrete graded algebra structures and compatibility conditions. We provide detailed constructive proofs, derive explicit solution formulas with rigorous error bounds, and establish convergence criteria in appropriate norms for discrete differential forms. Complete algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with certified error bounds. The practical effectiveness of the method is demonstrated through a rigorous verification framework using interval arithmetic and discrete exterior calculus. This work shows that within appropriately constructed differential-algebraic closures, explicit analytic solutions to discrete exterior variational and inverse exterior variational problems exist, thereby providing a new algebraic perspective on discrete variational solvability while maintaining consistency with continuous theory. Connections to quantum field theory, topological dynamics, geometric machine learning, and real-time physical simulation establish cross-disciplinary mathematical links.