Differential Algebraic Closure Framework for Discrete Exterior Variational Geometry: A Unified Constructive Approach with Certified Computations
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| Format: | Recurso digital |
| Sprache: | Englisch |
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2026
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| _version_ | 1866901757318135808 |
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| author | liu, shifa |
| author_facet | liu, shifa |
| contents | <p>This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to discrete exterior variational problems on simplicial complexes and discrete manifolds. We define the discrete exterior variational geometric closure KExtVar,h and the discrete quantum exterior variational closure KExtQVar,h, differentially closed field extensions constructed through a well-founded recursive adjunction process that incorporates discrete exterior differential forms, conservation laws, topological invariants, and quantum corrections. Within these closures, we prove that solutions to broad classes of discrete exterior variational problems—including discrete Maxwell equations, discrete Yang-Mills theory, discrete Chern-Simons theory, discrete Einstein-Hilbert action, and quantum effective actions for discrete differential forms—admit unified algebraic representations that respect the underlying geometric, algebraic, and physical structures. The framework rigorously addresses nonlinearity, exterior constraints, topological changes, and quantum effects while preserving graded algebraic structures and compatibility conditions.We provide detailed constructive proofs for all core theorems, including the existence and minimality of the discrete exterior variational closure (Theorem 3.1), the algebraic representation of solutions via discrete Hodge decomposition (Theorem 4.1), the algebraic classification of topological singularities (Theorem 5.1), the algebraic formulation of discrete Noether’s theorem (Theorem 3.3), the conservation of discrete topological charges (Theorem 5.3), the algebraic convergence of quantum effective actions (Theorem 6.2), the algebraic convergence of tensor network variational principles (Theorem 6.3), the algebraic equivariance of geometric graph neural networks (Theorem 7.3), the algebraic structure of discrete variational integrators (Theorem 8.1), the algebraic computability of surgery error bounds (Theorem 5.2), and the algebraic construction of parallel certification error bounds (Theorem 9.3). All conjectures from the original framework are either proved as theorems or reformulated with precise conditions. Comprehensive algorithms with precise complexity analysis are presented, including parallel implementations with optimal speedup (Theorem 8.1), adaptive precision control with certified error bounds (Theorem 9.2), and real-time contact solvers with verified convergence (Theorem 8.3). A rigorous validation framework employing interval arithmetic and discrete exterior calculus demonstrates the practical effectiveness of our approach, with explicit error bounds and complexity estimates.The work demonstrates that explicit analytic solutions exist within appropriately constructed differential algebraic closures, providing new algebraic perspectives on discrete exterior variational solvability while maintaining consistency with classical theory. Extensions to quantum field theory, topological dynamics, geometric machine learning, and real-time physical simulation establish connections across mathematical disciplines.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18672535 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Differential Algebraic Closure Framework for Discrete Exterior Variational Geometry: A Unified Constructive Approach with Certified Computations liu, shifa Discrete exterior variational geometry, differential algebraic closure, Euler-Lagrange equations for discrete differential forms, Noether's theorem, topological changes, quantum exterior variational, geometric machine learning, certified computation, interval arithmetic, adaptive precision <p>This paper establishes a comprehensive differential algebraic framework for constructing explicit solutions to discrete exterior variational problems on simplicial complexes and discrete manifolds. We define the discrete exterior variational geometric closure KExtVar,h and the discrete quantum exterior variational closure KExtQVar,h, differentially closed field extensions constructed through a well-founded recursive adjunction process that incorporates discrete exterior differential forms, conservation laws, topological invariants, and quantum corrections. Within these closures, we prove that solutions to broad classes of discrete exterior variational problems—including discrete Maxwell equations, discrete Yang-Mills theory, discrete Chern-Simons theory, discrete Einstein-Hilbert action, and quantum effective actions for discrete differential forms—admit unified algebraic representations that respect the underlying geometric, algebraic, and physical structures. The framework rigorously addresses nonlinearity, exterior constraints, topological changes, and quantum effects while preserving graded algebraic structures and compatibility conditions.We provide detailed constructive proofs for all core theorems, including the existence and minimality of the discrete exterior variational closure (Theorem 3.1), the algebraic representation of solutions via discrete Hodge decomposition (Theorem 4.1), the algebraic classification of topological singularities (Theorem 5.1), the algebraic formulation of discrete Noether’s theorem (Theorem 3.3), the conservation of discrete topological charges (Theorem 5.3), the algebraic convergence of quantum effective actions (Theorem 6.2), the algebraic convergence of tensor network variational principles (Theorem 6.3), the algebraic equivariance of geometric graph neural networks (Theorem 7.3), the algebraic structure of discrete variational integrators (Theorem 8.1), the algebraic computability of surgery error bounds (Theorem 5.2), and the algebraic construction of parallel certification error bounds (Theorem 9.3). All conjectures from the original framework are either proved as theorems or reformulated with precise conditions. Comprehensive algorithms with precise complexity analysis are presented, including parallel implementations with optimal speedup (Theorem 8.1), adaptive precision control with certified error bounds (Theorem 9.2), and real-time contact solvers with verified convergence (Theorem 8.3). A rigorous validation framework employing interval arithmetic and discrete exterior calculus demonstrates the practical effectiveness of our approach, with explicit error bounds and complexity estimates.The work demonstrates that explicit analytic solutions exist within appropriately constructed differential algebraic closures, providing new algebraic perspectives on discrete exterior variational solvability while maintaining consistency with classical theory. Extensions to quantum field theory, topological dynamics, geometric machine learning, and real-time physical simulation establish connections across mathematical disciplines.</p> |
| title | Differential Algebraic Closure Framework for Discrete Exterior Variational Geometry: A Unified Constructive Approach with Certified Computations |
| topic | Discrete exterior variational geometry, differential algebraic closure, Euler-Lagrange equations for discrete differential forms, Noether's theorem, topological changes, quantum exterior variational, geometric machine learning, certified computation, interval arithmetic, adaptive precision |
| url | https://doi.org/10.5281/zenodo.18672535 |