Generalization of Dual Variational Calculus to Exterior Summation Equations: The Descent Hierarchy and Its Geometric, Arithmetic, and Analytic Correspondences on Discrete Structures

This paper systematically generalizes the theory of higher-order variations, duality, descent hierarchies, geometric realizations, and arithmetic correspondences, as established by Liu for ordinary and partial differential equations and subsequently extended to exterior integral equations, to the realm of exterior summation equations defined on discrete manifolds (simplicial complexes or graphs). Unlike integral equations which involve continuous integration kernels, exterior summation equations are characterized by discrete summation operators with finite-range kernels k_ (x, y), reflecting the intrinsic discreteness of the underlying structure. We define higher-order variation operators in the context of discrete exterior calculus and summation operators, prove the discrete versions of the Great Descent Theorem and the Great Ascent Theorem, and introduce spectral graphs to replace spectral curves. Descent towers are constructed using configuration spaces of points on the spectral graph, while ascent towers are given by the corresponding discrete intermediate Jacobians (Picard varieties of configuration spaces). 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 in the discrete setting. We formulate a Hierarchical Birch–Swinnerton–Dyer Conjecture for arithmetic graphs and prove it in the function field case (graphs over finite fields). The theory is applied to classify integrable exterior summation equations such as discrete Fredholm-type, discrete Volterra-type, and discrete Painlev\'e-type equations by their descent length. Furthermore, we develop a quantized version of the dual calculus on lattices, relating Schwinger–Dyson equations to the effective action in the presence of exterior summation operators. The entire framework is extended to higher-dimensional spectral complexes. Finally, an axiomatic formulation is presented, capturing the universal duality principle underlying all these structures. All theorems are provided with complete, rigorous proofs, and particular attention is paid to the challenges arising from the discrete nature of summation operators.