A Discrete Weighted Geometric Framework with Spectral Convergence of Weighted Graph Laplacians

We introduce a discrete weighted geometric framework defined on admissible lattice structures endowed with a metric, differential operators, and a spectral theory. The construction is fully explicit and dimensionally consistent. We prove that, under suitable regularity conditions, the associated discrete Laplacian converges to the classical Laplace operator in the continuum limit. The framework is designed to be directly implementable in numerical settings and to provide a stable spectral structure suitable for modeling wave and diffusion phenomena. Unlike purely combinatorial approaches, the present construction preserves dimensional coherence and admits a rigorous convergence analysis toward smooth Riemannian geometry.