Analytic Algebraic Representation Theory for Inverse Variational Equations: Unified Rank Correspondence and Geometric Classification

Based on the theory of analytic algebraic finite representations systematically developed for variational equations in 1, 2, 3, 4, this paper extends its core concepts, methods, and conclusions to the inverse variational problem. We define the definability of inverse variational equations in the representation framework (Ci, Oj ), introduce the spectra of problem complexity and geometric complexity, and establish the inverse variational versions of fundamental theorems. This revised version contains 71 core theorems with complete rigorous proofs, including:• Rigorous differential-algebraic formulation of the Helmholtz conditions with closure properties and decision algorithms. • Explicit constructive formulas for Lagrangians of inverse variational equations, including non-homogeneous, parameter-dependent, constrained, and time-varying parameter cases. • Intrinsic nature and constructive realization of the inverse variational period lattice, with complete homological proof of the Period Number Theorem. • Hodge structure, real structure, p-adic properties, and transcendence degree of period lattices. • Algebraic construction of the isomonodromic moduli space with universal property, local coordinates, period mapping, Gauss-Manin connection, and Deligne-Mumford compactification. • Strict stratification of problem complexity and geometric complexity spectra. • Unified Rank Correspondence Law with motivic, categorical, Galois-theoretic, real, p-adic, tropical, and singularity theoretic interpretations. • Complete geometric classification of inverse variational equations with algorithmic decidability, real, p-adic, tropical, and singularity-theoretic characterizations. • Applications to physics including constrained systems, time- arying parameters, critical phenomena, and p-adic physical phenomena. All theorems are proved within the representation framework (C0, OA), with algorithmic implementability and numerical verification. All previously conjectural statements are transformed into rigorously proven theorems.