Geometric Resolution Thresholds and the Emergence of Distinguishable Structure

This work investigates the conditions under which structure can be meaningfully distinguished from noise under finite resolution. Through geometric analysis and computational experiments across multiple vertex systems, we demonstrate that geometric distinguishability has a universal threshold at ε = d/2, where ε denotes resolution and d the separation between objects. Below this threshold, distinct structures cannot exist geometrically; any apparent structure arises from projection or operator-induced effects rather than intrinsic separation. We further show that previously observed thresholds near ~0.1 originate from statistical or representational criteria (e.g., requiring a fixed fraction of unambiguous samples or using base-10 discretization), and should not be conflated with geometric distinguishability. By explicitly separating geometric, statistical, and dynamical thresholds, the work clarifies how apparent coherence can be induced by alignment between selection and measurement operators, as demonstrated in related analyses of prime gap statistics and hierarchical decompositions. The result provides a resolution-based diagnostic framework for assessing claims of structure across geometry, data analysis, spectral methods, and dynamical systems. It establishes a hard geometric bound for distinguishability, offers falsification criteria for projection-induced patterns, and helps distinguish intrinsic structure from artifacts of representation or measurement.