We systematically develop a theory of extending the repetition count of basic quantum group operations — the coproduct ∆, the R-matrix action R, and the antipode S — and their inverses from natural numbers successively to integers, rational numbers, real numbers, and finally complex numbers. This theory, called Operational Mathematics of Quantum Group Operations, generalises the methodology previously applied to classical arithmetic, quantum channels, and set operations. Acomplete axiomatic system is established. Integer-order, fractional-order, real order and complex-order iterations are rigorously defined. Using spectral decomposition of the R-matrix, we construct fractional iterative roots via a quantum version of Schr¨oder’s equation, and prove uniqueness under regularity conditions. We analyse in depth the singularity structure of complex-order iterations: with a fixed principal branch, RzV,V is entire in z; no finite branch points occur. The quantum group zeta function is defined from the R-matrix eigenvalues; its non trivial zeros are shown to lie on the critical line ℜ(s) = 1/2, proving the Quantum Riemann Hypothesis unconditionally.A fundamental phenomenon is the non collapse of the hierarchy: the second level operation (R-matrix twisted representation) is independent of the first-level operation (tensor product). Higher-level operations are not needed; the hierarchy collapses by design (the only levels used are 0,1,2). We establish a categorical duality between the representation category Rep(G) and the complex numbers, constructing a quantum group hyperfield isomorphic to C. All open problems from earlier operational mathematics are resolved within the quantum group framework. Explicit numerical algorithms based on spectral decomposition are provided, with rigorous error bounds.
Liu S (Wed,) studied this question.
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