This paper establishes a comprehensive constructive algebraic framework for exterior inverse variational topology, extending the previously developed methodology for exterior variational topology and differential topology. We define the exterior inverse variational topological algebraic closure 𝐾EIVT, a differentially closed structure constructed through a recursive adjunction process that incorporates solutions to exterior inverse variational problems, constructively defined topological invariants of differential forms, and quantum field theoretical objects with certified error bounds. Within this closure, we prove that solutions to fundamental problems in exterior inverse variational topology—including the reconstruction of action functionals from critical point equations, topological charges of differential forms from given geometric structures, and geometric structures from exterior constraints—admit unified constructive representations with explicit convergence rates and error estimates. The framework rigorously addresses the challenges of combining exterior calculus with inverse variational principles while preserving both geometric and topological structures inherent in exterior inverse variational topology. We provide detailed constructive proofs with complete error analysis, derive explicit expressions for geometric objects with rigorous bounds, and establish convergence criteria in appropriate Sobolev spaces of differential forms. 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. This work demonstrates that explicit constructive representations of fundamental objects in exterior inverse variational topology exist within the appropriately extended and constructively defined exterior inverse variational topological algebraic closure 𝐾EIVT. The framework is shown to be consistent with classical exterior inverse variational topology while extending the constructive power to include harmonic forms reconstructed from variational constraints, characteristic forms derived from curvature data with certified computation, and special geometric structures that respect both exterior differential and inverse variational properties. Extensive theoretical developments and algorithmic specifications validate the proposed approach, including applications to inverse Chern-Simons theory, topological quantum field theory, and high-dimensional geometric analysis.
shifa liu (Wed,) studied this question.