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Unlike its cousin 3SAT, the NAE-3SAT (not-all-equal-3SAT) problem has theproperty that spectral/SDP algorithms can efficiently refute random instanceswhen the constraint density is a large constant (with high probability). But dothese methods work immediately above the "satisfiability threshold", or isthere still a range of constraint densities for which random NAE-3SAT instancesare unsatisfiable but hard to refute?  We show that the latter situation prevails, at least in the context of randomregular instances and SDP-based refutation. More precisely, whereas a random$d$-regular instance of NAE-3SAT is easily shown to be unsatisfiable (whp) once$d \geq 8$, we establish the following sharp threshold result regardingefficient refutation: If $d < 13.5$ then the basic SDP, even augmented withtriangle inequalities, fails to refute satisfiability (whp), if $d > 13.5$ theneven the most basic spectral algorithm refutes satisfiability~(whp).

The threshold for SDP-refutation of random regular NAE-3SAT