A Constructive Framework for Discrete Exterior Inverse Variational Topology With Certified Computations

This paper establishes a comprehensive constructive framework for Discrete Exterior Inverse Variational Topology (DEIVT), which unifies discrete geometric structures, exterior calculus, and inverse variational principles in a single algebraic-topological setting. We define the discrete exterior inverse variational topological algebraic closure KDEIVT, a differentially closed structure constructed through recursive adjunction of solutions to discrete exterior inverse variational problems, constructively defined topological invariants of discrete differential forms, and quantum field theoretical objects with certified error bounds. Within this closure, we prove that solutions to fundamental problems in discrete exterior inverse variational topology—including the reconstruction of discrete action functionals from discrete field equations, topological charges of discrete differential forms from given geometric structures, and geometric structures from discrete exterior constraints—admit unified constructive representations with explicit convergence rates and error estimates. The framework rigorously addresses the hallenges of combining discrete exterior calculus with inverse variational principles while preserving both geometric and topological structures inherent in discrete exterior inverse variational topology. We provide detailed constructive proofs with complete error analysis, derive explicit expressions for discrete geometric objects with rigorous bounds, and establish convergence criteria in appropriate discrete Sobolev spaces. Detailed algorithms with precise complexity analysis and stability guarantees are presented, including adaptive precision control with certified error bounds derived from a posteriori error estimation. A comprehensive validation framework is established, employing discrete exterior calculus and numerical verification of topological invariants with mathematically rigorous error certification.