This paper establishes a complete discrete analogue of the theory of higher-order variations and their inverse problems, based on the fundamental insight that the k-th discrete variation descends to the first variation through successive applications of the variation operation. We prove the Discrete Great Descent Theorem, which shows that every k-th order discrete variation is the first variation of some other discrete functional, providing the foundation for the entire hierarchy. The Discrete Fundamental Equivalence Theorem demonstrates that the k-th order discrete inverse variational problem is equivalent to the classical problem for all k, establishing that no new equations arise from higher-order discrete variations. However, we introduce a new invariant---the discrete descent length---that stratifies discrete variational equations into a strict hierarchy, with explicit constructions showing the hierarchy is infinite. Geometrically, discrete descent representations correspond to symmetric products of the spectral curve, forming a natural descent tower C^ (1) = C, C^ (2), C^ (3),. The Discrete Hierarchical Period Number Theorem gives the rank of the k-th level period lattice as ₖ = 2g, where g is the genus of C. The Discrete Hierarchical Unified Rank Correspondence establishes relations between the geometric rank, algebraic rank, moduli rank, arithmetic rank, and analytic rank at each level, with explicit recurrence relations across levels. We formulate the Discrete Hierarchical BSD Conjecture, predicting that the rank of the k-th Chow group ᵏ (C) ₀ is g+k-1kgk, connecting discrete variational theory to motivic cohomology and Beilinson's conjectures. The discrete Painlev\'e equations are classified by their descent length, with dPₕ₈ having maximal length 3, revealing its universal nature as the ``master equation'' of the discrete descent hierarchy. This framework creates a new research direction---discrete descent geometry---uniting the discrete calculus of variations, algebraic geometry, combinatorics, number theory, integrable systems, and motivic theory.
S. B. Liu (Wed,) studied this question.