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 structure alone. Rather, it is treated as a candidate sufficient generator within the finite domain considered: a local, symmetric, lowest-order closure condition whose adequacy depends on admissibility, refinement compatibility, residual stability, and survival under additional finite measurement data. The revision therefore distinguishes finite-domain sufficiency from uniqueness and frames the result as an existence and sufficiency demonstration rather than a uniqueness theorem. The manuscript is part of a broader research program on finite measurement structure, relational constraints, refinement closure, and emergent physical dynamics.