Spectral Obstruction to Stable Bell Factorization in Relativistic Quantum Field Theory

We establish a structural obstruction to bounded Bell-type factorizations of two-point functions in relativistic quantum field theory. Working at the level of bilinear forms on Schwartz space, we show that any Bell-factorizable distribution with uniformly bounded L² response functions satisfies a universal quadratic bound. We then prove that Wightman two-point functions with unbounded spectral density (a standard ultraviolet property in interacting QFT) violate this bound. As a consequence, no such function admits a stable Bell factorization: any approximating sequence of Bell-factorizable distributions necessarily involves response functions whose norms diverge. This provides a functional-analytic formulation of the fine-tuning phenomenon identified by Cavalcanti in causal models, linking spectral properties of quantum field theory to the instability of classical causal representations. The results avoid assumptions of spacetime localization and instead rely solely on spectral structure and positivity, offering a new perspective on the incompatibility between relativistic QFT and classical causal factorizations.