Differential Algebraic Closure Framework for Discrete Inverse Variational Geometry: A Unified Constructive Approach

This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to inverse variational problems on discrete geometric structures, including graphs, simplicial complexes, and combinatorial manifolds. We define the discrete inverse variational geometric closure Kdisc-inv-var and quantum discrete inverse variational closure Kqdisc-inv-var, which are differentially closed field extensions constructed through recursive adjunction processes that incorporate discrete geometric objects, conservation laws, topological invariants, and quantum corrections. Within these closures, we prove that solutions to broad classes of discrete inverse variational problems—including the reconstruction of discrete Lagrangians from discrete equations of motion, discrete Noether current recovery, discrete symplectic structure determination, and quantum inverse problems—admit unified representations that respect the underlying combinatorial, algebraic, and physical structures. The framework rigorously addresses nonlinearity, geometric constraints, topological changes, and quantum effects while preserving graded algebraic structures and compatibility conditions. We provide detailed constructive proofs, derive explicit solution formulas with rigorous error bounds, and establish convergence criteria in appropriate discrete function spaces. Comprehensive algorithms with precise complexity analysis are presented, including stability guarantees and adaptive precision control with certified error bounds. A rigorous validation framework employing interval arithmetic and discrete inverse variational calculus demonstrates the practical effectiveness of our approach.