Algorithmic Resolution and Geometric Distinguishability in SAT Solution Spaces

This work develops a geometric framework for understanding distinguishability under finite resolution and applies it to computational solution spaces, with particular focus on Boolean satisfiability (SAT). We establish that geometric distinguishability between two points in a metric space has a universal threshold at ε = d/2, independent of dimension, coordinate system, numerical base, or sampling procedure. Apparent thresholds near ∼0.1 are shown to arise from statistical targets, representational artifacts, or projection effects rather than intrinsic geometry. Building on this foundation, we introduce the concept of algorithmic resolution: a measurable quantity describing the scale at which a search algorithm can reliably detect structure in a solution space. Computational experiments demonstrate that different algorithms observing the same SAT instances can perceive radically different solution-space structure, including apparent clustering and fragmentation that are not intrinsic to the instance but induced by algorithm-specific blind zones. These effects are quantified and shown to correlate strongly with operator alignment between selection and measurement procedures. The results do not claim implications for complexity class separations such as P vs NP. Instead, they provide a new geometric and observer-aware language for analyzing algorithm–instance interaction, empirical hardness, and projection-induced structure. The framework offers testable hypotheses linking algorithmic resolution to solver performance and suggests a research direction for studying computational hardness through geometric distinguishability rather than runtime alone.