Bell Factorization, Trace-Class Conditions, and Functional Fine-Tuning: A Dictionary between Quantum Field Theory and Classical Causal Models

This work develops a functional-analytic dictionary between Bell-type factorizations and correlation kernels in relativistic quantum field theory (QFT). We show that uniformly bounded Bell factorizations correspond to finite local trace conditions on quadratic forms, while divergence of the trace forces any approximating factorization to exhibit unbounded L² norms. This provides a precise functional interpretation of fine-tuning in classical causal models, identifying it with instability under approximation. The results establish a bridge between three frameworks: (i) hidden-variable factorizations, (ii) quadratic forms on test functions, and (iii) structural properties of QFT correlation functions. In particular, typical short-distance singularities suggest that infinite trace — and therefore functional fine-tuning — is generic in QFT. This paper serves as a conceptual dictionary linking previous results on trace-class obstructions with causal-model interpretations of quantum correlations.