Structural Probability Theory: Pattern Partitioning on Closed Surfaces

This work redefines probability as a structural ratio between a defined closed surface (the total event space) and the patterns distributed upon it (possible outcomes). It argues that probability is not an inherent randomness, but a reflection of how a surface is partitioned. The denominator of probability—the total structure—contains physical meaning, while the numerator (individual outcomes) may be as small as a point. Using metaphors like grains of sand on Earth's surface, the paper illustrates that no collection of points can reconstruct the underlying geometry. Thus, probability must be understood as dependent on—and subordinate to—structure.