Based on the analytic algebraic finite representation theory, this paper systematically constructs an analytic algebraic classification system for exterior integral equations (i.e., equations satisfied by the inverse operation of exterior differentiation). The main contributions include: (1) Defining differential-algebraic definability of exterior integral equations in the generalized representation framework (Fi, Ωj ), where Fi are formal constant fields (coefficient fields of differential forms generated by algebraic numbers, π, e, etc.), and Ωj are formal operation sets (exterior product, exterior differentiation, exterior integration, Hodge star operator, etc.), and proving that all exterior integral equations induced by algebraic manifolds (such as harmonic form equations, Feynman integral equations, linear problems of integrable systems, etc.) are definable in the (F0, Ω2) framework; (2) Introducing the characteristic invariant spectrum of exterior integral equations: exterior monodromy rank (geometric rank), exterior differential Galois group dimension (algebraic rank), isomonodromic moduli space dimension (moduli rank), exterior arithmetic rank, exterior analytic rank, exterior Tate rank, exterior Beilinson rank, exterior Iwasawa rank, exterior automorphic rank, and exterior Galois rank, and proving that they satisfy the unified rank correspondence law; (3) Generalizing the period number theorem to: for the exterior integral equation satisfied by harmonic k-forms on a smooth compact manifold M, the rank of the exterior period lattice equals the k-th Betti number bk of M (real coefficients) or 2bk (complex coefficients); (4) Establishing the double spectrum theorem for exterior integral equations, precisely relating the problem complexity (topology of the manifold, Hodge numbers) of the equation to the geometric complexity (period lattice rank, moduli space dimension) of the solution forms; (5) Proving the analytic-algebraic spectral theorem for the eigenform problem of exterior integral operators, revealing the intrinsic relationship between π-type and e-type differential forms; (6) Introducing the Analytic-Algebraic Self- onsistency Axiom (AASA) and verifying it on elliptic curves and abelian varieties; (7) Under the AASA framework, providing rigorous proofs of the BSD theorem, Tate rank correspondence, Beilinson rank correspondence, Iwasawa rank correspondence, and automorphic-Galois rank correspondence, establishing the unified rank correspondence law; (8) Providing a complete proof of the definability of solutions to the quantum KZ equation, showing the embedding of multiple zeta values in Qe2; (9) Generalizing the theory to non-K¨ahler manifolds, establishing the period theory for ∂∂-manifolds, and giving the period lattice rank formula determined by Bott-Chern numbers; (10) Establishing the analytic-algebraic theory of Feynman periods, proving that the Feynman period lattice rank equals the grade of the corresponding mixed Tate motive; (11) Exploring applications of the theory in arithmetic geometry, quantum field theory, integrable systems, and the Langlands program, and providing rigorous proofs of key conjectures (verified for elliptic curves and abelian varieties) and future research directions. This paper provides a unified geometric and representation-theoretic framework for the classification and arithmetic theory of differential forms.
shifa liu (Wed,) studied this question.