A Minimal Discretized Model for Observable Boundary Length

This technical note presents a minimal discretized model for defining and measuring an observable boundary length under finite spatial resolution.The model is intentionally simple and is not designed to introduce new mathematical results or physical laws. Instead, its purpose is to illustrate how a continuous geometric quantity can emerge from an explicit operational definition applied to a discretized observation scheme. A unit disk in two-dimensional Euclidean space is approximated by a collection of square cells placed on a regular grid. Cells are selected according to a local and resolution-dependent rule based solely on whether the geometric center of each cell lies inside the disk. The boundary length is then defined operationally as the total length of cell edges that are externally visible, while all internal shared edges are excluded by construction. As the discretization resolution is increased, the observable boundary length is evaluated numerically. The resulting behavior exhibits convergence toward a stable effective value consistent with the known circumference of the disk. No analytical convergence rate is claimed, and the emphasis is placed on conceptual transparency rather than numerical optimization or rigor. The model highlights the role of observable definitions, coarse-graining, and resolution dependence in numerical measurements of geometric quantities. Owing to its minimal assumptions and straightforward implementation, it may serve as a conceptual illustration for discretization effects in numerical modeling, as well as a pedagogical example suitable for educational and exploratory contexts.