A Minimal Finite-Update Field-Equation Model for Stability, Nonzero-Mode Residuals, and Continuum-Limit Consistency

This work presents a numerical investigation of a minimal finite-update field-equation model in which field evolution is defined by an explicit discrete-time update rule. The study examines one-dimensional stability diagnostics, two-dimensional nonzero-mode residual structures, stability phase maps, continuum-limit consistency, and intrinsic finite-update residuals. The purpose of the work is not to claim a proof of physical spacetime discreteness, nor to replace continuum field theory. Instead, the paper evaluates whether a conservative finite-update framework exhibits internally consistent numerical behavior under controlled validation tests. The results show stable relaxation below a CFL-like boundary, reproducible finite-amplitude nonzero-mode residuals, approximately second-order convergence, and separately diagnosable finite-update residuals. This manuscript is released as a research preprint.