Meta-perational Mathematics:Differential and Integral Dualities

This paper extends Meta-Operational Mathematics---a systematic framework that elevates operations to independent mathematical objects---by fully incorporating the dual meta-operations of differentiation $\fD$ and integration $\fI$. While the original framework established an axiomatic system of ten axioms, an endomorphism operad, a Hopf operad structure, a correspondence with the Connes-Kreimer renormalization Hopf algebra, bornological convergence for infinite meta-operations, and a path integral trace, the present work embeds the Fundamental Theorem of Calculus as a core duality principle. We introduce two additional axioms governing $\fI$ and its invertibility with $\fD$ on a distinguished subspace $C_0$, and we systematically extend all structural results---operadic derivations and coderivations, Hopf pairing, bornological continuity, spectral triple invertibility, Ward identities for path integrals, special function representations, and $\infty$-adjunctions---to the differential-integral pair. All classical special functions are shown to belong to the meta-operational universe generated by $\fD$, $\fI$, and basic operations. Open problems are reformulated as precise conjectures, and several are proved as theorems within the expanded framework. This work provides a unified language for analysis, algebra, geometry, topology, and quantum field theory, with differentiation and integration as fundamental adjoint meta-operations.