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

This manuscript presents a worked example of how a finite measurement and refinement framework may be applied to a concrete model. The example considered is wave-like propagation on a one-dimensional refinement chain. Starting from finite contexts, admissible assignments, formal refinements, and local generator constraints, the manuscript shows how a candidate closure rule selects an explicit second-order update. The resulting rule has the same algebraic form as the centered finite-difference scheme associated with the one-dimensional wave equation, but it is interpreted here as an admissibility condition under refinement rather than as a discretization of a continuum law. The manuscript also illustrates residual analysis as a refinement test: failures of the candidate generator are treated not as arbitrary curve-fitting errors, but as diagnostics of incomplete context, missing relational structure, or failure of refinement stability. The work is exploratory and methodological in scope. It does not claim a unique derivation of wave dynamics or a complete empirical theory of wave phenomena. Its purpose is to provide a concrete, checkable example of how finite records, generator constraints, residual tests, and coarse-grained continuum representations may be organized within the proposed framework.