Many natural and artificial systems exhibit a fundamental relation between boundary and bulk structure. In geometry, this appears in surface-to-volume ratios; in physics, through surface tension and energy minimization; in fractals, through scaling dimensions; and in machine learning, through the geometry of latent spaces. Despite their differences, these systems share three basic quantities: a boundary measure A, a bulk measure V, an effective dimensionality functional N. This motivates the introduction of a universal normalized boundary density, first formalized in the boundary--bulk functional of D. Pivec (2026). In this article we extend that framework by deriving exact spherical proportionality laws, proving isoperimetric optimality, and providing physical interpretations.
Davorin Pivec (Tue,) studied this question.