Traversal-Induced Geometry: A Discrete Path-Based Measurement Framework for Circular Metrics

This work introduces Traversal-Induced Geometry, a discrete framework in which geometric quantities are defined through operational traversal rules rather than continuous limits. By modeling measurement as path-dependent movement on a 2D integer lattice, the framework examines how alternative traversal constraints induce distinct geometric invariants that deviate from classical Euclidean expectations. The study investigates several traversal protocols—including axis-priority rules, alternating-step rules, and noise-perturbed paths—and evaluates their effects on length estimation, scaling behavior, and invariant stability across resolutions. Simulation groups systematically explore: (1) baseline discrete scaling, (2) traversal sensitivity, (3) resolution robustness, and (4) rule-dependent metric divergence. The results demonstrate that measurement, when grounded in discrete operational rules, can yield geometric outcomes that differ fundamentally from smooth-limit assumptions. The framework is positioned as a starting point for a broader exploration of operational geometry, discrete topology, and path-induced metrics, with potential relevance for computational geometry and algorithmic modeling.