Logical Meta-Operational Mathematics: A Complete and Rigorous Extension of Meta-Operational Mathematics to Logical Operations and Their Inverses

This paper systematically extends the framework of Meta-Operational Mathematics to the domain of logical operations, constructing a complete theory of Logical Meta-Operational Mathematics. Boolean logical functions are regarded as operations (Level~1), while the maps acting on them---composition, pointwise addition, Boolean difference, iteration, inversion, and duality---form the meta-operations (Level~2). An axiomatic system comprising ten axioms is established, and their relative consistency and independence are proved. The endomorphism operad on the space of logical operations is shown to be generated by the NAND operation, and is endowed with a Lie algebra structure. By mapping logical circuit diagrams into a Connes--Kreimer-type Hopf algebra, we reveal a deep correspondence between logic circuit simplification and renormalization. Discrete bornological convergence is introduced to handle infinite meta-operations, and the path integral is reinterpreted as a trace on the logical operad, linking it to discrete topological quantum field theory. The compositional inverse of logical functions (logic recovery), self-action and iteration, functional equations as meta-operational identities, and the exact correspondence between meta-operational depth and circuit complexity classes are all studied in detail. Finally, the framework is extended to quantum logic and continuous logic, demonstrating that Meta-Operational Mathematics serves as a unifying language across discrete and continuous, algebraic and logical, computational and physical domains.