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We construct, using fermionic functional integrals, thermodynamic Green'sfunctions for a weakly coupled fermion gas whose Fermi energy lies in a gap.Estimates on the Green's functions are obtained that are characteristic of thesize of the gap. This prepares the way for the analysis of single scalerenormalization group maps for a system of fermions at temperature zero withouta gap.Comment: 42 page

SINGLE SCALE ANALYSIS OF MANY FERMION SYSTEMS PART 4: SECTOR COUNTING