This work presents a structural theorem showing that any local diffusion-driven hierarchical amplification chain in a physical or abstract dynamical system must terminate after finitely many levels if each amplification step requires strictly positive resource expenditure and the total available resource is finite. The result is independent of the specific PDE, geometry, or physical interpretation of the “resource,” and applies broadly to all recursive local amplification mechanisms. The theorem directly excludes all blow-up constructions that rely on an infinite cascade of increasingly fine-scale structures, providing a structural constraint complementary to classical PDE-based regularity approaches. Implications for 3D Navier–Stokes, turbulence cascades, and other multiscale nonlinear systems are discussed.