Complex-Order Dual Calculus: Descent, Ascent, and Unified Correspondences

Based on the analytic continuation of the fundamental theorem of calculus to complex order (fractional) – where complex-order differentiation and complex-order integration are mutual inverses modulo initial conditions and branch cuts – this paper establishes a complete dual hierarchy theory for complex-order differential operators Dz and integral operators Iz for z ∈ C with ℜ(z) > 0. We prove the Complex-Order Great Descent Theorem: every z-th order differential operator descends to a first-order differential operator (modulo a compact operator and a branch adjustment), i.e., Dzf = D1fz−1; and the Complex-Order Great Ascent Theorem: every first-order integral operator ascends to a z-th order integral operator, i.e., Izg = I1gz. We introduce the invariants differential depth ℓdiff(P) and integral depth ℓint(P), proving the fundamental duality ℓint(P) = ℓdiff(P) +n−1, where n is the number of marked points on the spectral curve, now extended by analytic continuation in the complex order parameter. Geometrically, descent corresponds to complex-order Hilbert schemes Cz of points on the spectral curve C, forming a natural descent tower C1 = C, C2, C3, . . .with analytic continuation to complex z. The Complex-Order Period Number Theorem gives the rank of the level-z period lattice as ρz = 2g (where g is the genus of C), proving invariance of the period rank throughout the complex tower. The Complex-Order Hierarchical Unified Rank Correspondence establishes, under an analytic-algebraic self-consistency condition, that the geometric rank, algebraic rank, and twice the arithmetic and analytic ranks satisfy ρz = dz =2rarithz = 2ranalz = 2g. We formulate the Complex-Order Birch–SwinnertonDyer Conjecture, relating the rank of the higher Chow group CHz+1(C, 1)hom (interpreted via analytic continuation in the motivic index) to the order of vanishing of L(H2ℜ(z)+1(C), s) at s = z + 1, connecting the theory to motivic cohomology and Beilinson’s conjectures. The Painlevé equations are classified by their descent length; via a continuous family P(z)V I we obtain ℓdiff(P(z)V I ) = ℜ(z) ∈ 0, 3, revealing its universal nature as the “master equation” of the descent hierarchy. Furthermore, we develop a complete duality theory – Complex-Order Dual Calculus – showing that the descent (covariant) direction is dual to an ascent (contravariant) direction, with dual lengths satisfying ℓ† = ℓ↓ + n − 1. This duality extends to geometry (complex-order Hilbert schemes vs. complex-order intermediate Jacobians), arithmetic (period lattices vs. dual period lattices), and analysis (single vs. multiple integral representations). The theory is unified in an axiomatic framework and extended to interdisciplinary applications in physics, computer science, biology, economics, information theory, and engineering, revealing a universal duality principle underlying all natural systems.