Extended Topological Quantum Field Theory (Part I): Formal Foundations and Categorical Operator Dynamics

Topological Quantum Field Theory (TQFT) traditionally provides a strict mathematical framework in which physical systems and observables remain invariant under continuous geometric deformation. However, in its classical formulation defined by the Atiyah-Segal axioms, TQFT is structurally restricted to finite-dimensional vector spaces and finite geometric manifolds, rendering it inadequate for modeling infinite computational networks, recursive architectures, or large-scale distributed aggregation. This paper formalizes the theoretical extension of TQFT from physical manifolds to abstract, recursive, and infinite computational systems, introducing the Extended Topological Quantum Field Theory (DFT-TQFT) framework. We define the transition from the classical axiomatic framework to an enriched category of system states, and explicitly formulate the four core mathematical operators that constrain infinite networks: the Fractal decomposition operator, the Zeta aggregation operator, the Well-foundedness constraint, and the Invariant mapping operator. By rigorously delineating proven topological theory from proposed systemic extensions, we establish the formal mathematical foundations required to define transformation-invariant computational structures.