Operational Mathematics of the Riemann Zeta Function: Extending the Iteration Count 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 transcendental binary operations: the Riemann zeta function operation ₍^ (s, z) and its inverse ₍^^{-1} (s, z). A complete set of seven 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 Schr\"oder's equation, Abel's equation, and a suitably adapted Kneser construction. Uniqueness theorems under natural regularity conditions are provided. The singularity structure of complex-order zeta iterations is analyzed in depth, revealing a fundamentally novel phenomenon: the branch points are of mixed algebraic (square-root type, from the critical values of) and logarithmic type (from the unique simple pole of, from its trivial and non-trivial zeros, and from the essential singularity). Three distinct families of logarithmic branch points are identified and proved distinct. The union of these branch points accumulates densely on the negative real axis, forming a natural boundary. Conditionally on the Riemann Hypothesis, the branch points from non-trivial zeros also accumulate on the critical line (w) =-1/2 - (s), forming a secondary natural boundary. The local monodromy group contains both Z₂ and Z factors. A fundamental structural discovery is rigorously proved: the zeta 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 zeta kernels are shown to be special cases of the zeta operational framework, thereby unifying discrete zeta hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of zeta operations is established, yielding a field isomorphism between the zeta hyperfield and the complex numbers. A functorial relationship between the zeta and gamma hyperfields, reflecting the functional equation (s) = (s) (1-s), is constructed. The connection between zeta iteration values and the arithmetic of the zeta function is explored, with particular emphasis on transcendence of special values and the Zeta Riemann Hypothesis, which is proved unconditionally via a Hilbert--P\'olya self-adjoint operator construction applied to the corrected zeta function (defined using backward iterates). The corrected zeta function satisfies an exact functional equation and an Euler product over prime periodic orbits. A conditional reduction of the classical Riemann Hypothesis to the compactification of the zeta iteration generator is established. The paper is self-contained, and every essential statement is accompanied by a detailed proof.