Optimal Entropic Dimensionality: A Continuous Variational Principle for Geometric Equilibrium

抄録

Systems that encode information into high-dimensional degrees of freedom confront a geometric trade-off between dilution (expansion) and crowding (compression). We introduce Optimal Entropic Dimensionality (OED), a variational framework in which effective dimensionality is treated as a continuous state variable and an equilibrium is selected by minimizing geometric size. Under isotropic Lp geometry, this yields the scaling law N * = pH, which balances geometric overhead against entropic density, with p determined by the underlying norm. To ground this law in a canonical setting, we analyze the Euclidean case (p = 2) via information geometry. We model distinguishable configurations as small geodesic balls with respect to the Fisher information metric under Jeffreys measure. Using Gamma-function and Stirling-type volume asymptotics, we obtain a logarithmic expansion cost and derive N * = 2H, up to constants independent of N. We then extend the variational structure to anisotropic systems. Correlations consume a structural log-volume Hstruct, leaving a free payload H_free = H - H_struct. This leads to the structure corrected optimum N * approx p(H - H_struct), supported by both a geometric bulk hyper-rectangle model and an independent whitening-based entropy transformation. OED thus furnishes a geometric coordinate system [N, H, p] for complexity, organizing regimes from sparse (p approx 1), through a log-base calibration point (p = 1/ ln 2), to robustness-oriented limits (p -> infinity) beyond the Euclidean benchmark.