  A $t$-emulator of a graph $G$ is a graph $H$ that approximates its pairwise
shortest path distances up to multiplicative $t$ error. We study fault tolerant
$t$-emulators, under the model recently introduced by Bodwin, Dinitz, and
Nazari [ITCS 2022] for vertex failures. In this paper we consider the version
for edge failures, and show that they exhibit surprisingly different behavior.
  In particular, our main result is that, for $(2k-1)$-emulators with $k$ odd,
we can tolerate a polynomial number of edge faults for free. For example: for
any $n$-node input graph, we construct a $5$-emulator ($k=3$) on $O(n^{4/3})$
edges that is robust to $f = O(n^{2/9})$ edge faults. It is well known that
$\Omega(n^{4/3})$ edges are necessary even if the $5$-emulator does not need to
tolerate any faults. Thus we pay no extra cost in the size to gain this fault
tolerance. We leave open the precise range of free fault tolerance for odd $k$,
and whether a similar phenomenon can be proved for even $k$.
