Based on the analytic continuation of the fundamental theorem of exterior calculus—where the complex-order exterior derivative dz and the complex-order exterior integral (or co-derivative) δz are mutual inverses modulo exact and co-exact terms—this paper establishes a complete dual hierarchy theory for complex-order differential forms. We prove the Complex-Order Exterior Descent Theorem: for any z ∈ C with Re(z) ≥ 1, the operator dz descends to the classical firstorder exterior derivative d1, i.e., dzω = d1ωz−1 for some form ωz−1. Dually, the Complex-Order Exterior Ascent Theorem states that δ1 ascends to δz under appropriate integrability conditions. We introduce the invariants cohomological descent depth ℓ↓(M) and homological ascent depth ℓ†(M) for a Riemannian manifold M, proving the fundamental duality relation ℓ†(M) = ℓ↓(M) + dim M−1. Geometrically, descent corresponds to complex-order jet bundles and, in the context of integrable systems, to complex-order Hilbert schemes Xz of points on a spectral variety X, forming a natural descent tower X1, X2, . . . , Xz, . . .. The Complex-Order Period Number Theorem establishes that the rank of the level-z period lattice, generated by integrals of level-z harmonic forms over cycles, remains invariant and equal to the Betti numbers bk(M) for all z. We establish a Unified Hodge-Rank Correspondence, relating the geometric rank (Betti numbers), the algebraic rank (dimension of the abelianized differential Galois group), the moduli rank, the arithmetic rank of higher Chow groups, and the analytic order of vanishing of motivic L-functions. We formulate the Complex-Order Beilinson-Bloch Conjecture, relating the rank of the complex-order higher Chow group CHz+1(M, 1)hom to the order of vanishing of L(H2⌈Re(z)⌉+1(M), s) at the shifted central point s = z + 1. The Painlevé-type equations are classified by their descent length; via a continuous family P(z)V I we obtain ℓ↓(P(z)V I ) = Re(z) ∈ 0, 3. Furthermore, we develop a complete duality theory—Complex-Order Exterior Dual Calculus—showing that the descent (covariant) direction is dual to an ascent (contravariant) direction, with dual lengths satisfying ℓ† = ℓ↓ + dim M − 1. This duality extends to Hodge theory, arithmetic, and physics. The theory is unified in an axiomatic framework and extended to interdisciplinary applications, revealing a universal cohomological duality principle.
shifa liu (Wed,) studied this question.