  It was recently noted that the entanglement entropy for a subsystem of a
chaotic eigenstate exhibits an enhanced correction when the subsystem
approaches a phase transition at half the total system size. This enhanced
correction was derived for general subsystems by Dong and Wang by summing over
noncrossing permutations, which can be thought of as ``saddles'' either in a
sum emerging from averaging over Wick contractions or in an analogous
gravitational calculation. We extend these results to the case of entanglement
negativity, an entanglement measure defined on a bipartite density matrix. We
focus on a particular transition previously studied in a toy model of JT
gravity, one for which the sum over permutations was found to give similar (or
even stronger) enhanced corrections. We derive and resum the relevant
permutations to give a form for the averaged negativity spectrum, reproducing
the gravitational answer for some quantities and finding tension with other
quantities, namely the partially transposed entropy. Along the way, we extend
the results of Dong and Wang to the case of $n < 1$ R\'enyi entropy, showing
that it always receives volume law corrections.
