Constraint-Induced Concentration in High-Dimensional Function Spaces: Geometric Regularities and Entropy Maximization

We establish that minimal global constraints — zero-mean and unit L² norm — in the space of functions on the circle S¹ induce a profound dimensional hierarchy in the geometric partition ratio. Large-scale Gaussian and Haar ensembles reveal exact degeneracy at low dimension, symmetric multimodality with maximal variance near K≈10–15, and exponential concentration at P=1 as K→∞, rigorously proved via Lévy concentration of measure. This drives binary partition entropy to its maximum ln 2 in high dimensions. The phenomenon — emergent scale selection transitioning to universal equilibration — arises purely from manifold geometry.