Dual Variational Calculus for Exterior Integral Equations: Spectral Hierarchy, Moment Duality, and Arithmetic Correspondences

This paper develops a systematic theory of higher-order variations and their inverses for exterior integral equations, fully respecting the nonlocal nature of integral operators. We define a class of admissible exterior integral operators characterized by their spectral decomposition, smooth kernels, and compactness properties. The k-th variation ᵏ S is shown to be expressible as a multilinear integral form involving k copies of the integral operator I. The Great Descent Theorem is proved: for any k 2, ᵏ S = ¹ S₊-₁ for some functional S₊-₁ depending parametrically on k-1 test forms, with an explicit construction using the resolvent kernel. The inverse problem is solved under the k-th order Helmholtz conditions, leading to the Great Ascent Theorem via a multi-parameter Vainberg integral. The descent length _ (P) is defined as the maximal k for which P admits a k-th order variational representation, and is shown to satisfy _ (P) (₀ I - I) where ₀ is the principal eigenvalue of I. The dual ascent length satisfies ^ = _ + n - 1, where n is the multiplicity of the eigenvalue. The spectral manifold of an exterior integral equation is defined as the set = \ₙ\ \\ of eigenvalues of I, compactified by adding a point at infinity. The descent tower is realized by the symmetric products Sᵏ, whose topology is studied via Dold-Thom theory. The ascent tower is realized by the determinant line bundles ᵏ (Sᵏ). A Fourier-Mukai-type transform is established between Sᵏ and ᵏ (Sᵏ). Periods are replaced by spectral moments ₘ = ₙ ₙᵐ where ₙ are the eigenvalues. The Hierarchical Moment Theorem states that the Hankel matrix Hₖ = (₈+₉) ₈, ₉=₀^k-1 has rank equal to the descent length _. A duality pairing between descent and ascent moment lattices is constructed. The Hierarchical Unified Rank Correspondence is proved for integral equations over function fields: the geometric rank (dimension of spectral eigenspace), algebraic rank (dimension of abelianized differential Galois group), arithmetic rank (rank of K-theory group K₁ () ), and analytic rank (order of vanishing of the spectral zeta function at s=0) all coincide and equal _ + 1. The Hierarchical Spectral BSD Conjecture is formulated: for an integral equation over a number field, the order of vanishing of the spectral L-function L (, s) at s=1 equals the rank of the regulator map on K₁ (). This is proved in the function field case using the Artin-Tate theorem applied to the determinant line bundle. Applications include the classification of Fredholm-type and Volterra-type integral equations by their descent length, and the analysis of integral forms of Painlev\'e equations. A quantum version is developed, replacing the integral operator by its quantized counterpart and relating Schwinger-Dyson equations to the effective action via a quantum Legendre transform. Higher-dimensional generalizations are considered, where the spectral manifold is replaced by a spectral scheme. An axiomatic framework is presented, capturing the universal duality principle underlying all these structures. All theorems are provided with complete, rigorous proofs, paying particular attention to the challenges arising from the nonlocality of exterior integral operators. The proofs utilize spectral decomposition, kernel representations, and careful convergence arguments to establish the results in full generality.