Remark on the existence of phi44

The bound |S_ε⁽²⁾(fg)| 0, is sufficient to yield construction of a φ⁴ field theory. In perturbation theory, this bound holds in spacetime dimensions d< 4. The bounds on the n-point Schwinger functions follow from the given bound by the use of Lebowitz's inequalities. They yield the growth |S⁽ⁿ⁾|< O(n!), consistent with the Osterwalder-Schrader axioms, which allow reconstruction of a real-time theory.