Operational Mathematics of Abelian Functions: Extending the Iteration Count of A(a+z) and A−1(a +z) to the Complex Domain

This paper 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 a new class of binary operations: the Abelian operation ₙ^A (a, b) and its inverse ₙ^A^{-1} (a, b). Based on the multi-periodic structure of Abelian functions and their intrinsic connection with compact Riemann surfaces of genus g 1, a complete axiomatic system 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 the generalized Schr\"oder equation, Abel equation, and a suitably adapted Kneser construction for higher-dimensional tori. Uniqueness theorems under natural regularity conditions are provided. The singularity structure of complex-order Abelian iterations is analyzed in depth, revealing a fundamentally novel phenomenon: the simultaneous presence of algebraic branch points of arbitrary order (originating from the local monodromy of the Abelian integral) and logarithmic branch points arising from the 2g independent generators of the period lattice, producing an infinite-sheeted Riemann surface of mixed algebraic-logarithmic covering type. The negative real axis is shown to be a natural boundary for the analytically continued iteration. Furthermore, a fundamental structural discovery is rigorously proved: the Abelian 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 Abelian kernels are shown to be special cases of the Abelian operational framework, thereby unifying discrete Abelian hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of Abelian operations is established, yielding a field isomorphism between the Abelian hyperfield and the complex numbers. The connection between Abelian iteration values and the arithmetic of Abelian varieties is explored, with all previously announced conjectures now proved as theorems or reduced to precisely formulated statements with rigorous proofs. In particular, the unconditional transcendence of fractional iterates, the complete hierarchy collapse, the classification of singularities including the natural boundary on (-, -1], the Abelian Riemann Hypothesis for the Abelian zeta function, and the uniqueness of real-order iterations under a convexity condition are all established as theorems within the present work. The paper is self-contained, and every essential statement is accompanied by a detailed proof.