From Weighted Microscopic Jumps to Operational Geometry in RZS

This paper develops a conservative route from weighted microscopic jumps to an operational notion of geometry within the Relational Zero State (RZS) framework. The central claim is deliberately narrow. A scalar stability law may remain useful at the macroscopic level, but it does not carry enough structure to recover distance, anisotropy, effective dimension, or curvature. The relevant object is instead the weighted microscopic transition operator. I formulate the dynamics as a normalized continuous-time random walk on a weighted graph, derive the associated random-walk Laplacian and heat semigroup, and use standard results from graph-Laplacian convergence, diffusion maps, and short-time heat-kernel asymptotics to motivate an operational RZS distance. No new theorem is claimed. The proposed bridge is then tested on two public benchmark networks with observed edge weights: Zachary’s Karate Club and the Les Misérables coappearance graph. The empirical protocol uses exact matrix exponentials on the observed graphs, descriptive bootstrap intervals, and Monte Carlo permutation tests. On Karate Club, the weighted operator yields a slightly larger faction-separation score than the unweighted baseline, but the gain is not decisive at the main diffusion scale. On Les Misérables, the weighted operator aligns much more strongly with observed coappearance intensities than the unweighted baseline. The conclusion is limited but useful: weighted microscopic jumps provide a rigorous first bridge from transition structure to emergent spatial geometry in RZS, but they do not by themselves derive the full Lorentzian space-time of physics.