Meta-Operational Mathematics: From Iteration of Operations to Operations on Operations

This paper develops Meta-Operational Mathematics, a systematic framework that elevates operations themselves to the status of independent mathematical objects. We study meta-operations (composition, translation, exponentiation, logarithm, differentiation, integration, variation, infinite sums, infinite compositions) acting on operations. An axiomatic system of ten axioms is established. The category of meta-operations is shown to carry an endomorphism operad structure,which is further endowed with a Hopf operad structure. A concrete Hopf algebra morphism from the one-ary meta-operations to the Connes–Kreimer renormalization Hopf algebra is constructed, thereby embedding renormalization group theory into the meta-operational framework. Bornological convergence is introduced to handle infinite meta-operations, and is applied to spectral triples in noncommutative geometry. The path integral is reinterpreted as a trace on the operad, connecting to topological quantum field theory. All classical special functions (trigonometric, elliptic, Gamma, Zeta, Theta, hypergeometric, etc.) are shown to belong to the meta-operational universe, and their fundamental identities become equations of meta-operations. Open problems are reformulated as precise conjectures. This work provides a unified language connecting analysis, algebra, geometry, topology,and quantum field theory.