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 hyperbolic sine operation ₙ (a, b) and the hyperbolic cosine operation ₙ (a, b), together with their inverses (the inverse hyperbolic sine and inverse hyperbolic cosine). 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 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 hyperbolic iterations is analyzed in depth, revealing algebraic branch points of square-root type at the preimages of i (for the sine case) and 1 (for the cosine case), in stark contrast to the logarithmic branch points of classical tetration. The negative real axis is shown not to be a natural boundary. Furthermore, a fundamental structural discovery is rigorously proved: the hyperbolic operational hierarchy collapses for all levels n 2, leaving only two distinct primitives---level n=1 and the collapsed level n 2. Fractional calculus and the fractional calculus of variations are shown to be special cases of the hyperbolic operational framework, thereby unifying discrete hyperbolic hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of hyperbolic operations is established, transforming the formal parallelism into a rigorous equivalence of categories. All open problems announced in earlier sketches are either proved as theorems or reduced to precisely formulated conjectures with supporting evidence. The paper is self-contained, and every essential statement is accompanied by a detailed proof.
Liu S (Wed,) studied this question.