This paper systematically develops a complete theory of higher-order variations and their inverse problems for summation equations (discrete analogues of integral equations), based on the fundamental insight that the k-th variation descends to the first variation through successive applications of the variation operation. We define higher-order variational operators in the context of discrete summation operators, prove the summation equation versions of the Great Descent Theorem and the Great Ascent Theorem, and introduce spectral manifolds (now associated with discrete Lax pairs). 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 Discrete Hierarchical Period Number Theorem and a duality pairing of period lattices. A Discrete Hierarchical Unified Rank Correspondence is established, linking geometric, algebraic, moduli, arithmetic, and analytic ranks. We formulate a Discrete Hierarchical Birch--Swinnerton--Dyer Conjecture and prove it in the function field case. The theory is applied to classify integrable summation equations such as discrete Painlev\'e equations by their descent length. Furthermore, we develop a quantized version of the dual calculus, relating discrete Schwinger--Dyson equations to the effective action in the presence of summation operators. 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, with particular attention paid to the essential characteristics of summation operators throughout the development.
Ye Liu (Wed,) studied this question.