  The number partitioning problem is a classic problem of combinatorial
optimization in which a set of $n$ numbers is partitioned into two subsets such
that the sum of the numbers in one subset is as close as possible to the sum of
the numbers in the other set. When the $n$ numbers are i.i.d. variables drawn
from some distribution, the partitioning problem turns out to be equivalent to
a mean-field antiferromagnetic Ising spin glass. In the spin glass
representation, it is natural to define energies -- corresponding to the costs
of the partitions, and overlaps -- corresponding to the correlations between
partitions. Although the energy levels of this model are {\em a priori} highly
correlated, a surprising recent conjecture asserts that the energy spectrum of
number partitioning is locally that of a random energy model (REM): the
spacings between nearby energy levels are uncorrelated. In other words, the
properly scaled energies converge to a Poisson process. The conjecture also
asserts that the corresponding spin configurations are uncorrelated, indicating
vanishing overlaps in the spin glass representation. In this paper, we prove
these two claims, collectively known as the local REM conjecture.
