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This paper is a contribution to a program to see symmetry breaking in aweakly interacting many Boson system on a three dimensional lattice at lowtemperature. It is part of an analysis of the "small field" approximation tothe "parabolic flow" which exhibits the formation of a "Mexican hat" potentialwell. Here we prove the existence of and bounds on the background and criticalfields that arise from the steepest descent attack that is at the core of ourrenormalization group step anaylsis of these models.

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 4 - Background and Critical Field Estimates