  It has been argued that many non-perturbative phenomena in quantum mechanics
(QM) and quantum field theory (QFT) are determined by complex field
configurations, and that these contributions should be understood in terms of
of Picard-Lefschetz theory. In this work we compute the contribution from
non-BPS multi-instanton configurations, such as instanton-anti-instanton
$[{I}\bar{I}]$ pairs, and argue that these contributions should be interpreted
as exact critical points at infinity. The Lefschetz thimbles associated with
such critical points have a specific structure arising from the presence of
non-Gaussian, quasi-zero mode (QZM), directions. When fermion degrees of
freedom are present, as in supersymmetric theories, the effective bosonic
potential can be written as the sum of a classical and a quantum potential. We
show that in this case the semi-classical contribution of the critical point at
infinity vanishes, but there is a non-trivial contribution that arises from its
associated non-Gaussian QZM-thimble. This approach resolves several puzzles in
the literature concerning the semi-classical contribution of correlated
$[{I}\bar{I}]$ pairs. It has the surprising consequence that the configurations
that dominate the expansion of observables, and the critical points that define
the Lefschetz thimble decomposition need not be the same, a feature not present
in the traditional Picard-Lefschetz approach.
