Differential Algebraic Closure Framework for Exterior Delta-Geometry: A Unified Constructive Approach to Geometric Delta-Equations and Exterior Delta-Equations

This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to geometric partial delta-equations and exterior delta-equations on delta-manifolds. We define the exterior delta-geometric closure K∆DiffGeo and linear exterior delta-differential closure K∆LEDE, differentially closed field extensions constructed through recursive adjunction processes that incorporate discrete geometric objects, discrete harmonic forms, discrete curvature tensors,and discrete fundamental solutions.Within these closures, we prove that solutions to broad classes of geometric delta-equations—including the discrete Yamabe equation, discrete Ricci flow, discrete Einstein field equations, and linear exterior delta-equations—admit unified representations that respect the underlying discrete geometric and algebraic structures. The framework rigorously addresses nonlinearity, discrete geometric constraints, and multi-dimensional challenges while preserving graded algebraic structures and discrete geometric compatibility conditions.We provide detailed constructive proofs, derive explicit solution formulas with rigorous error bounds, and establish convergence criteria in appropriate discrete geometric 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 discrete interval arithmetic and discrete exterior calculus demonstrates the practical effectiveness of our approach.The work demonstrates that explicit algebraic solutions exist within appropriately constructed differential algebraic closures for discrete geometric problems,providing new algebraic perspectives on discrete geometric solvability while maintaining consistency with discrete geometric theory. Extensions to discrete spectral theory, discrete curvature flows, discrete topological invariants, geometric deep learning on discrete structures, and structure-preserving numerics establish connections across mathematical disciplines