Bulk-Boundary Correspondences in Nuclear AdS-like Geometries and Emergent Large $N$ Symmetries

A nuclear spectral triple framework is developed to classify topological phases and 4--manifolds via a discrete universal invariant $\mathcal{U}(X)$ that remains stable under $\Sigma$--preserving deformations. Key results include a bulk--boundary equivalence, a holographic partition function correspondence, and a natural $1/N$ expansion arising from the matrix approximation $\mathcal{A}_\Sigma\cong\lim_{N\to\infty}M_N(\mathbb{C})$. These findings provide a robust, topologically based approach that yields precise classification and computational advantages over classical continuum methods.