Resummation of φ4 free energy up to an arbitrary order

The consistency condition, which guarantees a well organized small-coupling asymptotic series for the thermodynamics of massless φ4-theory, is generalized to any desired order of the perturbative treatment. Based on a strong conjecture about forbidden two-particle reducible diagrams, this condition is derived in terms of functions of four-momentum in place of the common toy mass in previous treatments. It has the form of a set of gap equations and marks the position in the space of these functions at which the free energy is extremal.