This paper develops a systematic extension of Meta-Operational Mathematics Liu2026a, Liu2026b, Liu2024 that elevates the variational operation and its inverse, the involutional integration, to the status of mutually inverse meta-operations. Building upon the original ten axioms, we establish an extended system of twelve axioms that fully incorporate the - duality. We construct the variational operad - and the involutional operad -, proving that they are Koszul dual sub-operads of () and carry a bordered Hopf operad structure in which and correspond via the antipode (modulo boundary terms). The variational Poincar\'e lemma is formulated in meta-operational language, and a complete theory of bornological convergence for variational sequences is developed, establishing compactness and regularization properties of the involutional integration. The Euler--Lagrange equations are recast as the meta-operational equality ₀ S = 0, while the reconstruction of the action from field equations is given by ₁. The path integral measure is realized as the continuous bornological limit of finite-dimensional involutional integrations, with the stationary phase approximation corresponding to the - duality expansion. A variational version of the Connes--Kreimer renormalization Hopf algebra correspondence is established. In noncommutative geometry, we define the spectral variational complex and prove stability of the Dirac operator variation under bornological convergence, linking the spectral flow to involutional integration and variational cohomology to K-theory. All classical special functions are shown to arise as critical points of variational principles or as images of involutional integrations; their functional equations become meta-operational equalities. Higher-order self-variations and iterated involutional integrations are studied, leading to a variational dynamical systems theory. The framework is categorified to a strict 2-category -2Cat in which ₀ ₁ is an adjunction, and the extension to an -operad is outlined. Numerical algorithms for automatic variational differentiation and involutional quadrature are provided with rigorous error bounds. Open problems are formulated as precise conjectures and many are resolved as theorems within the paper. This work provides a unified operadic language for the calculus of variations, quantum field theory, noncommutative geometry, and special function theory, centered on the fundamental inverse pair (, ).
Liu S (Wed,) studied this question.