  The Euclidean fermionic determinant in four-dimensional quantum
electrodynamics is considered as a function of the fermionic mass for a class
of $O(2)\times O(3)$ symmetric background gauge fields. These fields result in
a determinant free of all cutoffs. Consider the one-loop effective action, the
logarithm of the determinant, and subtract off the renormalization dependent
second-order term. Suppose the small-mass behavior of this remainder is fully
determined by the chiral anomaly. Then either the remainder vanishes at least
once as the fermionic mass is varied in the interval $0 < m < \infty$ or it
reduces to its fourth-order value in which case the new remainder, obtained
after subtracting the fourth-order term, vanishes at least once. Which
possibility is chosen depends on the sign of simple integrals involving the
field strength tensor and its dual.
