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

Based on the theory of analytic algebraic finite representations, this paper systematically constructs an analytic algebraic classification system for inverse differential equations, 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 inverse differential equations. The core contributions include: (1) Defining differential-algebraic definability of inverse differential equations in the representation framework (Ci, Oj ), and proving that all inverse differential equations induced by algebraic curves (such as inverse elliptic function equations, inverse KdV equations, inverse Painlev´e equations, etc.) are definable in the framework (C0, O2); (2) Introducing a spectrum of characteristic invariants for inverse differential equations: inverse monodromy rank (geometric rank), inverse differential Galois group dimension (algebraic rank), inverse isomonodromic moduli space dimension (moduli rank), inverse rational solution rank (arithmetic rank), and order of vanishing of inverse 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 inverse integrable systems on an algebraic curve of genus g is 2g, and equals the inverse monodromy rank; (4) Establishing a double spectrum theorem for inverse differential equations, precisely correlating the problem complexity of the equation (order, singularity structure, spectral curve genus) 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 inverse differential operators, elucidating the spectral symmetry of π-type and e-type inverse transcendental functions; (6) Exploring applications of the theory in arithmetic inverse differential equations, inverse integrable systems, and inverse mathematical physics, and indicating deep connections with the inverse BSD conjecture and the inverse Langlands program. This paper provides a unified geometric and representation-theoretic framework for the classification and arithmetic theory of inverse differential equations.