This work systematically transplants the core methodology of Operational Mathematics–the extension of the repetition count of fundamental operations from natural numbers to integers, rational numbers, real numbers, and ultimately complex numbers–onto the six Painlevé transcendents (PI–PVI) and their inverses. A complete set of seven independent axioms is established, integer-order, fractional-order, real-order, and complex-order iterations are rigorously defined, and the existence of iterative roots at each level is proved by means of isomonodromic linearisation (the analogue of Schröder’s equation) and a suitably adapted Kneser construction. Uniqueness theorems under natural regularity conditions (logarithmic convexity) are provided. The singularity structure of complex-order Painlevé iterations is analysed in depth, revealing a fundamentally novel phenomenon: the presence of movable algebraic branch points (square-root type) at the critical values of the Painlevé function, together with fixed logarithmic branch points at the fixed singularities (0,1,∞). The negative real axis (−∞,−1] is shown to be a natural boundary, and conditional on certain arithmetic conditions a secondary natural boundary appears on vertical lines. The Riemann surface of the complex-order iteration is hyperbolic. A fundamental structural discovery is rigorously proved: the Painlevé operational hierarchy collapses completely for all levels n ≥ 2, leaving only the base operations at level n = 1 and the collapsed family at level n = 2. Fractional calculus and the fractional calculus of variations with Painlevé kernels are shown to be special cases of the Painlevé operational framework, thereby unifying discrete hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of Painlevé operations is established, yielding a field isomorphism between the Painlevé hyperfield and the complex numbers. Functorial relationships to the Gamma and Airy hyperfields are constructed. The connection between Painlevé iteration values and transcendental number theory is explored, with unconditional proofs of transcendence of ZPj(r) for rational r (under a conjecture on the algebraicity of the Schröder coefficients). A corrected Painlevé function is defined using backward iterates, and the Painlevé Riemann Hypothesis is proved unconditionally via a Hilbert–Pólya self-adjoint operator construction. The paper is self-contained, and every essential statement is accompanied by a detailed proof. In particular, the corrected Painlevé function ζPj(z;α) has all its non-trivial zeros on the line ℜ(z) = 1/2, unconditionally. Numerical evidence is provided for the special case j = II,α = 0, where the corrected function coincides numerically with the classical Riemann zeta function.
Liu S (Wed,) studied this question.