Exact Renormalisation, Stochastic Refinement, and Carrier-Derived Graph Distance in a Relational Event–Field Model

Abstract

We study a discrete relational event–field model generated from a K5K_5 seed by local degree-four K4K_4 vertex replacement, with conserved positive vertex charge, positive lapse, a clock-weighted stochastic rewrite law, quadratic graph energy, and a local massless carrier. For the fully synchronous hierarchy, we derive exact vertex and edge counts, graph diameter, a complete Laplacian spectral recursion, an exact constrained coarse operator, and an exact recursive four-port Dirichlet–to–Neumann representation. The stochastic event law is exactly closed under genealogy aggregation, while a conserved-charge tagged lineage admits an explicit pure-birth representation. For the frozen, positively and locally weighted carrier, a localized velocity impulse at graph distance dd first produces a nonzero response at Taylor order 2d+12d+1, recovering combinatorial graph distance as a threshold-free propagation-order metric. Exact min-plus closures compute the same distance hierarchically. Finite asynchronous ensembles are reported explicitly as finite-size numerical evidence. Physical length per graph hop, asynchronous metric-ball dimension, complete finite coarse closure, and a calibrated continuum force law remain open.