Generalization of Dual Variational Calculus to Partial 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 partial differential equations (PDEs). We define higher-order variation operators in the multi-variable context, prove the PDE versions of the Great Descent Theorem and the Great Ascent Theorem, and introduce spectral manifolds to replace spectral curves. Descent towers are constructed using Hilbert schemes of points on the spectral manifold, while ascent towers are given by the corresponding intermediate Jacobians. We develop a Hierarchical Period Number Theorem and a duality pairing of period lattices. 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 PDEs such as KdV, KP, and 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. Finally, an axiomatic formulation is presented, capturing the universal duality principle underlying all these structures. All theorems are provided with complete, rigorous proofs.