NaviSpace: A Navilogy Framework for Connecting Manifolds and Functional Spaces

This manuscript introduces \textbf{Navilogy}, a new framework for studying Hilbert manifolds. A Navilogy comprises a smooth immersion, its weak accumulation set, and a continuous retraction. We establish that accumulation sets are weakly compact and form weakly geodesic spaces, where geodesics in the submanifold weakly converge to those in the accumulation set. The weak Laplace operator is defined, and its eigenvalues are shown to be weak limits of the submanifold Laplacian, ensuring spectral stability. Weak homotopy equivalence is demonstrated, preserving topological properties. Numerical examples highlight applications to function spaces and weak curvature flows. This framework enhances the understanding of weak topology in infinite-dimensional geometry and spectral analysis.