Boundary Fingerprints as Observables

This technical note reinterprets truncation errors and Gibbs phenomena arising in finite Fourier series as observable boundary fingerprints, rather than numerical artifacts to be eliminated.By treating the Fourier band limit, the value of π, and spatial sampling as explicitly finite, boundary oscillations are shown to encode structural information about underlying update rules in discrete space-division models. The note defines a minimal set of boundary observables (overshoot amplitude, ringing energy, and effective boundary shift) and proposes an inversion workflow that classifies update-rule families without invoking continuum limits or infinite sums.Residuals, non-convergent behavior, and finite-precision effects are preserved as first-class scientific records. This document is intended as a methodological and archival technical note, compatible with discrete physics, cellular automaton models, numerical simulation diagnostics, and space-division interpretations of physical laws.