Operational Mathematics of Elliptic Functions: Extending the Iteration Count of sn(a+z) and cn(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 elliptic sine operation ₙ^ (a, b) and the elliptic cosine operation ₙ^ (a, b), together with their inverses (the inverse elliptic sine and inverse elliptic 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 elliptic iterations is analyzed in depth, revealing a fundamentally novel phenomenon: the simultaneous presence of algebraic branch points (square-root type) at the preimages of \ 1, 1/k\ (for) and \ 1, i k'/k\ (for), and logarithmic branch points arising from the period lattice = 4K 2iK', 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 elliptic 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 elliptic kernels are shown to be special cases of the elliptic operational framework, thereby unifying discrete elliptic hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of elliptic operations is established, yielding a field isomorphism between the elliptic hyperfield and the complex numbers. The connection between elliptic iteration values and the arithmetic of elliptic curves is explored, with all previously announced conjectures either proved as theorems or reduced to precisely formulated statements with supporting evidence. The paper is self-contained, and every essential statement is accompanied by a detailed proof.