Dual Variational Calculus for Exterior Differential Equations: The Descent Hierarchy and Its Geometric, Arithmetic, and Analytic Correspondences

This paper systematically generalizes the theory of higher-order variations, duality, descent hierarchies, geometric realizations, and arithmetic correspondences, as established by Liu for ordinary differential equations, to the realm of exterior differential equations (EDEs). We define higher-order variation operators in the context of differential forms, respecting the graded structure of the exterior algebra. We prove the EDE versions of the Great Descent Theorem and the Great Ascent Theorem, with careful attention to the signs arising from the graded Leibniz rule and integration by parts for differential forms. Spectral manifolds, characterized by de Rham cohomology \ (H¹₃ₑ\) and Hodge numbers \ (h^p, q\), replace spectral curves. Descent towers are constructed using Hilbert schemes \ (X^k\) of points on the spectral manifold, while ascent towers are given by the corresponding intermediate Jacobians \ (Jᵏ (X^k) \). We develop a Hierarchical Period Number Theorem and a duality pairing of period lattices, consistent with the Hodge structure. A Hierarchical Unified Rank Correspondence is established, linking geometric, algebraic, moduli, arithmetic, and analytic ranks. We formulate a Hierarchical Birch-Swinnerton-Dyer Conjecture and prove it in the function field case. The theory is applied to classify integrable EDEs such as self-dual Yang-Mills equations by their descent length. Furthermore, we develop a quantized version of the dual calculus, relating Schwinger-Dyson equations to the effective action. The entire framework is extended to higher-dimensional spectral manifolds (e. g. , twistor spaces). Finally, an axiomatic formulation is presented, capturing the universal duality principle underlying all these structures. All theorems are provided with complete, rigorous proofs that consistently incorporate the graded structure of differential forms.