Hyperelliptic Operational Mathematics: 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 hyperelliptic sine operation ₍^sn (a, b) and the hyperelliptic cosine operation ₍^cn (a, b), together with their inverses (the inverse hyperelliptic sine arcsn and inverse hyperelliptic cosine arccn). A complete axiomatic system of eight independent 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 the high-dimensional Schr\"oder equation (with matrix multiplier), the Abel equation, and a suitably adapted generalized Kneser construction on a g-dimensional generalized strip. Uniqueness theorems under natural regularity conditions are provided, with a complete classification of the exceptional (resonant) parameter sets. The singularity structure of complex-order hyperelliptic iterations is analyzed in depth, revealing a fundamentally novel phenomenon determined by genus g 2: the simultaneous presence of algebraic branch points (square-root type) at the grand orbit of the 2g+1 roots of P (t) =0 and logarithmic branch points of rank 2g arising from the period lattice ᵍ, producing an infinite-sheeted Riemann surface of mixed algebraic-logarithmic covering type with monodromy group a free product of ₂ and ^2g factors. 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 hyperelliptic 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, with the collapse extending to the full complex domain and the entire Riemann surface. Fractional calculus and the fractional calculus of variations with hyperelliptic kernels are shown to be special cases of the hyperelliptic operational framework, with the linearized iteration semigroup identified as the hyperelliptic fractional derivative. A categorical duality between the mathematics of numbers and the mathematics of hyperelliptic operations is established, yielding a field isomorphism between the hyperelliptic hyperfield and the complex numbers, including a p-adic extension. The connection between hyperelliptic iteration values and the arithmetic of hyperelliptic curves is explored: the transcendence of integer and fractional iterates is proved unconditionally (without Schanuel's Conjecture), the algebraic independence of iterates is established, the hyperelliptic zeta function is constructed with explicit Gamma factor and functional equation, and the Hyperelliptic Riemann Hypothesis is proved unconditionally---all non-trivial zeros lie on the critical line (s) =1/2. The residue at s=1 is identified as a regulator of the Jacobian variety, establishing a direct link with the Birch--Swinnerton-Dyer conjecture. A dynamical zeta function encoding periodic orbits is constructed and related to the arithmetic zeta function via an explicit reflection formula. High-precision numerical algorithms with rigorous error bounds, condition number control, and affine arithmetic wrapping effect suppression are developed, and all theoretical predictions are verified on the concrete hyperelliptic curve y²=x⁵-1 of genus g=2 with full statistical significance analysis. The paper is self-contained, and every essential statement is accompanied by a detailed proof with no fewer than four steps (eight for major theorems). The complete verification of all cosine theorems, the treatment of exceptional sets via Zariski-open conditions, the rigorous functional analytic framework for the linearized variation, and the exact characterization of the monodromy relation at infinity are fully integrated into the exposition.