Emergent Spectral Dimension in the Universal Correlational Lattice

The spectral dimension provides an operational notion of dimensionality based on diffusion processes rather than geometric embedding. In this work we study the scale-dependent spectral dimension $d_s(s)$ of simple graph geometries, focusing on the path graph as a controlled and fully analyzable example. Using the heat kernel trace and a robust logarithmic-derivative estimator, we compute $d_s(s)$ across ultraviolet, intermediate, and infrared diffusion scales. We observe the emergence of a well-defined plateau in an intermediate range of diffusion times, corresponding to an effective dimensionality that is stable under variations of graph size, smoothing procedure, and numerical differentiation scheme. At large diffusion times, the spectral dimension exhibits a characteristic drop driven by finite-size effects and saturation of the diffusion process. These results demonstrate how effective dimensionality can emerge dynamically from purely combinatorial structures, providing a clean benchmark for studies of dimensional flow in more complex networks and emergent-geometry frameworks.