Generalization of Analytic Algebraic Finite Representation Theory to Exterior Differential Equations Unified Rank Correspondence Law and Geometric Classification of Exterior Differential Systems

Based on the theory of analytic algebraic finite representations, this paper systematically constructs an analytic algebraic classification system for exterior differential equations (exterior differential systems), fully generalizing the period number theorem, the double spectrum theorem, and the unified rank correspondence law established in the case of algebraic equations to the field of exterior differential systems. The core contributions include: (1) Defining differential-algebraic definability of exterior differential systems in the representation framework (Ci, Oj ), and proving that all exterior differential systems induced by algebraic curves (such as elliptic function equations, periodic solutions of KdV equations, Euler–Lagrange equations, etc.) are definable in the framework (C0, O2); (2) Introducing a spectrum of characteristic invariants for exterior differential systems: monodromy rank (geometric rank), differential Galois group dimension (algebraic rank), isomonodromic moduli space dimension (moduli rank), rational solution rank (arithmetic rank), and order of vanishing of L-functions (analytic rank), and proving that they satisfy a unified rank correspondence law; (3) Generalizing the period number theorem to: the period lattice rank of solutions of integrable exterior differential systems on an algebraic curve of genus g is 2g, and equals the monodromy rank; (4) Establishing a double spectrum theorem for exterior differential systems, precisely correlating the problem complexity of the system (order, singularity structure, spectral manifold dimension) with the geometric complexity of the solution functions (period number, moduli rank); (5) Proving a form of the analytic algebraic spectral theorem for eigenvalue problems of exterior differential operators, elucidating the spectral symmetry of π-type and e-type transcendental functions; (6) Exploring applications of the theory in arithmetic exterior differential equations, integrable systems, and mathematical physics, and indicating deep connections with the BSD conjecture and the Langlands program. This paper provides a unified geometric and representation-theoretic framework for the classification and arithmetic theory of exterior differential systems.