  The Lopsided Lovasz Local Lemma (LLLL) is a cornerstone probabilistic tool
for showing that it is possible to avoid a collection of "bad" events as long
as their probabilities and interdependencies are sufficiently small. The
strongest possible criterion in these terms is due to Shearer (1985), although
it is technically difficult to apply to constructions in combinatorics.
  The original formulation of the LLLL was non-constructive; a seminal
algorithm of Moser & Tardos (2010) gave an efficient algorithm for nearly all
its applications, including to $k$-SAT instances where each variable appears in
a bounded number of clauses. Harris (2015) later gave an alternate criterion
for this algorithm to converge; unlike the LLL criterion or its variants, this
criterion depends in a fundamental way on the decomposition of bad-events into
variables.
  In this note, we show that the criterion given by Harris can be stronger in
some cases even than Shearer's criterion. We construct $k$-SAT formulas with
bounded variable occurrence, and show that the criterion of Harris is satisfied
while the criterion of Shearer is violated. In fact, there is an exponentially
growing gap between the bounds provable from any form of the LLLL and from the
bound shown by Harris.
