Meta-Operational Mathematics of Exponentiation and Logarithm: From Iteration of Operations to Operations on Operations

This paper systematically extends the framework of Meta-Operational Mathematics to exponentiation E and logarithm L, establishing a self-contained theory centered on the exponential-logarithmic duality. We study meta-operations acting on exponential and logarithmic operations—composition, translation, exponentiation, logarithm, differentiation, integration, variation, infinite sums, and infinite compositions. A specialized axiomatic system of eight axioms (E1, L1, EL1–EL8) is established, characterizing the essential properties of exponentiation and logarithm as meta-operations. We prove that the category of exponential-logarithmic meta-operations carries an endomorphism operad structure PEL, which is further endowed with a Hopf operad structure wherein E is group-like, L is primitive, and the antipode satisfies S(E) = L. A concrete Hopf algebra morphism from the unary exponential-logarithmic meta-operations to the Connes–Kreimer renormalization Hopf algebra is constructed, thereby embedding the renormalization group flow into the exponential-logarithmic meta-operational framework. Bornological convergence is introduced to handle infinite exponential and logarithmic series, and is applied to spectral triples in noncommutative geometry. The path integral is reinterpreted as a trace on the operad, expressed as TrE(S) = ϵCK(ΦEL(S)), establishing a direct connection to topological quantum field theory. All classical special functions—trigonometric, hyperbolic, exponential, logarithmic, Gamma, Zeta, Theta, hypergeometric, Bessel, elliptic—are shown to be generated by the minimal set {E,L,D,⊕,⊙,◦,1,0} through finite compositions of meta-operations. Their fundamental identities become elegant equations of exponential-logarithmic meta-operations. The 2-category 2MetOpEL is constructed, and the adjunction E ⊣ L is proved. The ∞-operad ∞MetOpEL is constructed and shown to be complete in the sense of Rezk. Applications span quantum field theory, information geometry, noncommutative geometry, homological algebra, topology, combinatorics, probability, number theory, dynamical systems, Riemannian geometry, provability logic, and machine learning. We resolve ten major conjectures from the literature, including the minimality of the generating set, the structure of primitives in the Hopf operad, the global convergence of the RG flow under Borel summability, the completeness of TQFT invariants, the ∞-categorical extension, the classification of abelian exponential-logarithmic operads, the generation of modular forms, the abelian Langlands correspondence as exponential-logarithmic duality, the fractional exponential-logarithmic duality, and the Koszul duality of the exponential-logarithmic operad. We prove the relative consistency of the entire framework with respect to ZFC plus an inaccessible cardinal. This work provides a unified language connecting analysis, algebra, geometry, topology, and quantum field theory through the fundamental duality of exponentiation and logarithm.