A Worked Example of Finite Refinement Closure: Wave-Like Propagation

This revised manuscript develops a finite relational refinement model in which wave-like dynamics arise from a local admissibility constraint imposed across discrete refinement layers and adjacency relations. The paper defines a propagation generator on finite measurement structures, derives the corresponding centered finite-difference update rule, examines residuals as diagnostics of refinement-closure failure, and shows how the familiar one-dimensional wave equation appears as a coarse-grained representation under additional regularity and scaling assumptions. This version clarifies the status of the chosen propagation generator. The generator is not presented as uniquely forced by finite measurement…