Prove triviality of four-dimensional scalar phi-four theory via NN-FT

Prove, using the convexity of the tilted measures in the neural-network field theory construction at sufficiently small coupling, that the four-dimensional scalar phi-four continuum limit is trivial, meaning that the renormalized interaction becomes Gaussian as the ultraviolet cutoff is removed.

Background

The construction establishes finite-volume phi-four in two dimensions. In four dimensions, the one-shell perturbation remains of order the coupling at every shell, so uniform convexity can be obtained only at small coupling. The authors note that one-loop shell integrations should reduce the effective quartic coupling and drive it to zero at fixed scales as the cutoff is removed.

The unresolved issue is whether the available convexity and moment-control arguments can be developed sufficiently far to convert this perturbative picture into a rigorous NN-FT proof of triviality.