Boundary–Bulk Scaling Transitions: A Unified Exponent Framework

Abstract Classical examples such as Gabriel's Horn demonstrate that a geometric body may possess finite volume yet infinite surface area. While this mathematical result is well known, its structural origin and generalization remain under-formalized. We present a minimal exponent-based framework for classifying rotational geometries according to asymptotic decay laws of their radial profiles. Analyzing radius functions of the form $r(x) = Cx^{-p}$, we prove that bulk and boundary measures obey distinct convergence thresholds, yielding three universal regimes: bulk-divergent ($p < 1/2$), boundary-dominant ($1/2 \leq p < 1$), and self-closing ($p \geq 1$). Gabriel's Horn ($p=1$) emerges as the canonical threshold case separating finite-boundary from infinite-boundary regimes. The framework generalizes to systems where interior and interface contributions scale with different exponents, providing a compact classification principle. We demonstrate applicability to biological branching networks, confirm the framework recovers Murray's Law scaling, and discuss extensions to transport systems, heat transfer, and network theory. The criterion is falsifiable: systems with equivalent asymptotic scaling must exhibit predicted boundary-bulk transitions or the framework fails. Keywords scaling laws, phase transitions, Gabriel's Horn, interface dominance, Murray's Law