Exterior Differential–Interior Integral Meta-Operational Mathematics: From Iteration of Exterior Differentiation and Interior Integration to an Axiomatic Framework of Meta-Operations

We develop Exterior Differential--Interior Integral Meta-Operational Mathematics, a systematic axiomatic framework that elevates the exterior differential operation d and the interior integral operation δ to the status of independent mathematical objects and performs operations upon them. We study meta-operations (composition, exponentiation, logarithm, exterior differential meta-operation, interior integral meta-operation, variation, infinite sums, infinite compositions) acting on d and δ. An axiomatic system of twelve axioms incorporating graded structures is established. The category of exterior differential--interior integral meta-operations is shown to carry a differential graded Hopf operad structure. An explicit Hopf algebra morphism to the de Rham cohomology Hopf algebra is constructed. Graded bornological convergence is introduced to handle infinite exterior differential meta-operations and is applied to elliptic operators and spectral sequences. The path integral is reinterpreted as a supertrace on the graded operad, connecting to topological quantum field theory and Morse theory. All classical spectral invariants (heat kernel coefficients, spectral zeta functions, η-invariants, analytic torsion) are proved to belong to the exterior differential--interior integral meta-operational universe, and deep theorems such as the Atiyah-Singer index theorem become identities among meta-operations. Six core conjectures formulated in the original framework are completely resolved and integrated: the classification of degree $+1$ primitives (Theorem {thm:primitive-classification}), the bornological convergence of general elliptic spectral sequences (Theorem {thm:general-ss-conv}), the rigidity of the differential graded Hopf operad (Theorem {thm:rigidity}), the complete identification of the regularized path integral supertrace with the de Rham co-unit for interacting theories (Theorem {thm:Str-Phi-epsilon}), the finite generation of elliptic operators (Theorem {thm:elliptic-generation-ops}), and the construction of the ∞-operad of meta-operations (Theorem {thm:MetOp-infty-operad}). This work provides a unified meta-operational language connecting analysis, geometry, topology, and quantum field theory.