This work investigates structural constraints imposed by analyticity in relativistic quantum field theory on classical causal explanations of quantum correlations. Building on previous results on analytic rigidity, we establish a conditional obstruction to the realization of classical causal factorizations through spacetime-localized modifications of two-point correlation functions. The argument derives analyticity directly from the spectral condition and identifies clustering as the key mechanism linking probabilistic factorization with analytic constraints. We then extend the analysis to three-point functions. A marginal spectral condition is introduced as a natural generalization compatible with dimensional reduction. Under this framework, we prove that any admissible modification must vanish upon reduction to all two-point marginals. The irreducible three-point sector is analyzed via connected correlation functions. A conditional obstruction is obtained under an extended analytic rigidity hypothesis in higher-dimensional tube domains, which is formulated explicitly as an open problem. The work provides a unified perspective connecting analytic properties of quantum field theory with causal modeling approaches to quantum correlations, and outlines a program for extending these results to higher-point functions.
Eduardo Gonzalez-Granda Fernandez (Sun,) studied this question.