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 hyperbolic and inverse hyperbolic operations. 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 (operations as mappings 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 hyperbolic functions, , , , , and their inverse counterparts, , , , , 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. We provide complete and rigorous proofs for every assertion. The paper is organized as follows. Section~1 establishes the foundational hierarchy of operations and meta-operations. Section~2 rigorously constructs the space Hyp (F) of hyperbolic operations and verifies all ten axioms of Meta-Operational Mathematics. Section~3 proves that Hyp forms a sub-operad of End (C) and computes its Lie algebraic structure. Section~4 endows Hyp with a Hopf operad structure with complete verification of co-associativity, the counit axiom, and the antipode axiom. Section~5 constructs an explicit Hopf algebra morphism to the Connes--Kreimer Hopf algebra for the sinh-Gordon model, including explicit Feynman diagram computations. Section~6 establishes the bornological convergence theory with complete error bounds. Section~7 applies bornological convergence to spectral triples on noncommutative hyperboloids. Section~8 interprets the path integral on hyperbolic actions as a trace on the hyperbolic operad. Section~9 provides a detailed meta-operational analysis of all six inverse hyperbolic functions. Section~10 presents series and continued fraction representations as infinite meta-operations with rigorous convergence proofs. Section~11 analyzes self-action of hyperbolic operations including Julia set characterizations. Section~12 rewrites all classical hyperbolic identities as meta-operation equalities. Section~13 categorifies hyperbolic operations to a strict 2-category 2Hyp and sketches the (, 1) -operad extension. Section~14 provides numerical algorithms with rigorous error bounds. Section~15 formulates and resolves open problems as theorems. Appendices~A--L provide supplementary material, including the complete operational dependency graph and theorem optimality counterexamples.
Liu S (Wed,) studied this question.