The Statistical Emergence of Extremal Behaviour in Physical and Computational Systems

Extremal principles -- least action in mechanics, free-energy minimization in thermodynamics, and minimal description length in computation -- recur across disparate scientific domains with striking regularity. Their global and endpoint-constrained form has often invited teleological interpretations, as if systems somehow "seek" optimal states. We develop a statistical account in which such behavior emerges naturally without foresight or intrinsic optimization. Let a configuration space Ω be equipped with a scalar resistance functional Rx and probability measure μx = Z⁻¹ exp(-Rx/κ). In the limits of high dimensionality or small fluctuation scale κ, concentration-of-measure phenomena imply that probability mass localizes sharply near configurations minimizing, or rendering stationary, R. Deviations are exponentially suppressed. Observable behaviour therefore becomes effectively indistinguishable from the satisfaction of a variational principle. This framework places classical path selection, Boltzmann statistics, algorithmic complexity distributions, and stochastic learning dynamics within a common mathematical structure. As a concrete illustration, we show a structural correspondence between Fermat's principle in optics and probabilistic shortest-path search on weighted graphs. Extremal laws are thus reinterpreted not as fundamental prescriptions, but as emergent macroscopic descriptions arising from probability geometry in large spaces. This preserves their predictive role while dissolving the need for teleological language, and leaves open the possibility that deeper principles determine the landscapes on which concentration occurs.