Working with the S-matrix formulation of thermal physics, I resum a series of diagrams that contribute to the free energy of $\lambda\phi^4$ theory in 3+1 dimensions. The diagrams correspond to tree level amplitudes that have, at fixed order in $\lambda$, the maximum number of singular propagators in the forward limit. It was recently argued that this set of diagrams saturates the free energy of a certain integrable 1+1 dimensional theory, and it might play a special role in thermal physics in general.