Boundary-bulk geometry

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.