Axiomatic Reconstruction of Functional Integration via Extended Endomorphic Closure: Resolving the Navier-Stokes Existence and Smoothness Problem

The construction of a mathematically rigorous foundation for continuous functional integration over unconstrained function spaces is constrained by the failure of localized differential operations over unbounded configuration spaces. Pointwise operations over these domains encounter local pointwise singularities and gradient explosions, necessitating exterior regularization infrastructures. This paper demonstrates that these analytical failures are artifacts of an inadequate topological domain rather than inherent to the theory. An axiomatic reconstruction of functional integration is established by projecting the infinite-dimensional abstract configuration field space onto the constituent unit intervals of the Hilbert cube ($\mathcal{M}^\infty$) via a global generalized inverse operator and a topological embedding pullback. Under the condition of Extended Endomorphic Closure, a closed endomorphism is sustained within the normalized internal topology of the Hilbert cube across all functional operations and spectral states. At the differential layer, the Jacobian-dependent divergence of fields is intrinsically eliminated via an exact algebraic cancellation of infinite-dimensional Jacobian matrices with their corresponding invariant metric inverse relations ($\mathbf{J} \cdot \mathbf{J}^{-1} = \mathbf{I}$). Through the algebraic cancellation of Radon–Nikodym density derivatives across the closed boundaries of the continuous infinite unit hypercube, the underlying functional invariants remain strictly preserved, establishing the absolute convergence of continuous functional operations natively within the bounded topological Radon measure space of the Hilbert cube. Consequently, universal configuration stabilization and a strictly bounded functional operator norm ($\mathcal{E} \le \mathcal{E}_0 < \infty$) emerge as necessary geometric consequences of the functional and inverse function isometry, projecting a topological contraction that locks the system within this compact topological domain. This non-perturbative, fully unified framework satisfies the structural requirements of the Millennium Prize criteria, shifting the foundational paradigm from external regularizations on unconstrained function spaces to the intrinsic topological stabilization of compact topological domains.