  We consider refrigeration and heat engine circuits based on the nonlinear
thermoelectric response of point-contacts at pinch-off, allowing for
electrostatic interaction effects. We show that a refrigerator can cool to much
lower temperatures than predicted by the thermoelectric figure-of-merit ZT
(which is based on linear-response arguments). The lowest achievable
temperature has a discontinuity, called a fold catastrophe in mathematics, at a
critical driving current I=I_c. For I >I_c one can in principle cool to
absolute zero, when for I<I_c the lowest temperature is about half the ambient
temperature. Heat back-flow due to phonons and photons stop cooling at a
temperature above absolute zero, and above a certain threshold turns the
discontinuity into a sharp cusp. We also give a heuristic condition for when an
arbitrary system's nonlinear response means that its ZT ceases to indicate
(even qualitatively) the lowest temperature to which the system can
refrigerate.
