This work presents a completely rigorous and self-contained extension of the full apparatus of Meta-Operational Mathematics to the specific and important class of hyperelliptic functions and their inverses. The central philosophical principle—that operations upon operations constitute meta-operations—is established with complete mathematical precision through a hierarchical framework: Level 0 (elements of a base space), Level 1 (hyperelliptic functions and their inverses as operations on the base space), Level 2 (meta-operations as mappings on operations), and Level 3 (meta-meta-operations acting on meta-operations). Within this framework, the hyperelliptic \-functions \₈₉ (z), the Riemann Theta function \ (z;\), their derivatives, and the inverse hyperelliptic functions ₈₉ are shown to admit canonical lifts to meta-operations via composition, and these meta-operations interact with one another through composition, pointwise addition, pointwise multiplication, differentiation, exponentiation, and logarithm in arbitrarily many iterations—integer, fractional, real, and complex. A fundamental distinction from the elliptic case is established: the multiple periodicity of hyperelliptic functions, arising from the Jacobian variety ᵍ / \ of dimension g \ 2, leads to the \ Hyperelliptic Duality Axiom (Axiom 2. 20), replacing the translation group (, +) with the quotient group ᵍ / \ where \ \ ^2g is a full-rank lattice. This crucial modification permeates the entire theory, affecting the operadic algebra, Hopf operad structure, bornological convergence, noncommutative geometry, and topological quantum field theory. We systematically construct the hyperelliptic operation space (F), verify all ten axioms with complete rigor, establish the hyperelliptic operad and its Hopf operad structure, and prove its correspondence with the Connes-Kreimer renormalization Hopf algebra for hyperelliptic quantum field theories. The path integral is reinterpreted as a trace on the hyperelliptic operad, connecting to topological quantum field theory on hyperelliptic surfaces. All classical hyperelliptic identities are reformulated as meta-operation equalities, and a complete theory of bornological convergence for hyperelliptic series and continued fractions is developed. Furthermore, the theory is extended to arbitrary principally polarized abelian varieties of genus g \ 1, to non-Archimedean p-adic settings, and to the construction of L-functions within the meta-operational framework. The hyperelliptic operad is shown to be the universal operad classifying hyperelliptic cohomology theories, with its spectrification rationally equivalent to the spectrum of topological modular forms _.
Liu S (Wed,) studied this question.