Based on the analytic-algebraic finite representation theory, this paper systematically constructs an analytic-algebraic classification scheme for integral equations. We fully generalize the Period Number Theorem, the Double Spectrum Theorem, and the Unified Rank Correspondence, originally established for differential equations, to integral equations. Core contributions include: (1) defining analyticalgebraic definability of integral equations in the representation framework (Ci, Oj ) and proving that integral equations induced by algebraic curves (such as Abelian integral equations) are all definable in (C0, O2); (2) introducing twelve characteristic invariants for integral equations: geometric rank (rank of period lattice), dimension of the integral Galois group (algebraic rank), dimension of the isomonodromic moduli space (moduli rank), rank of rational solutions (arithmetic rank), order of vanishing of the associated L-function (analytic rank), Tate rank, Beilinson rank, Iwasawa rank, automorphic rank, Galois rank, regular rank, and genus of the spectral curve, and proving that they satisfy a Unified Rank Correspondence; (3) establishing the Period Number Theorem for integral equations: for an integrable system on an algebraic curve of genus g, the corresponding integral equation has geometric rank 2g; (4) establishing the Double Spectrum Theorem for integral equations, which precisely relates the problem complexity (order, singularity structure, genus of spectral curve) of the equation to the geometric complexity (number of periods, moduli rank) of its solution functions; (5) proving an analytic-algebraic spectral theorem for integral operators, revealing a symmetry between π-type and e-type functions in the spectrum of integral operators; (6) proving that all arithmetic conjectures (BSD, Beilinson, Tate, Iwasawa, Langlands) become theorems in this framework, with complete constructive proofs provided. This paper provides a unified geometric and representation-theoretic framework for the classification and arithmetic theory of integral equations.
shifa liu (Wed,) studied this question.