Abstract Tumor growth has traditionally been described through empirical laws—exponential, logistic, or Gompertz curves that capture observed dynamics but do not explain their origin 4–5. In this work, we instead derive tumor growth from first principles by treating it as a boundary-mediated transport process. We propose a geometric–transport framework in which growth is governed by flux across a reactive boundary, yielding the scaling law dMdt = C M^, = d-1+d where the exponent α is not assumed but emerges from two coupled physical factors: geometric boundary scaling and transport amplification 6–8. The parameter β encodes the efficiency with which biological systems overcome geometric constraints on resource delivery. Within this framework, we show that biologically sustained tumor growth is confined to a restricted exponent spectrum 23 1 with the upper bound representing a critical transition. When β exceeds unity, the system enters a superlinear regime characterized by finite-time divergence, signaling instability and breakdown of regulated growth. This formulation unifies diffusion-limited growth, vascularized tumor expansion, exponential growth, and pathological runaway dynamics within a single governing equation 9–14, 22–26. Importantly, it establishes a falsifiable structure: measurable biological variables map directly to the growth exponent, allowing the theory to be tested against empirical data. Tumor growth thus emerges not as a collection of empirical laws, but as a manifestation of a deeper physical principle—structure formation through boundary-mediated flux under geometric constraint.
Oleg Sirotnikov (Tue,) studied this question.