Exact frustration cancellation and uniqueness at all positive temperatures for diluted ±J spin glasses slightly above the percolation threshold

We consider the Edwards–Anderson spin glass on ℤ^d, d ≥ 2, with independent couplings equal to 0, +1 and −1 with probabilities 1−p, p/2 and p/2. When p exceeds the bond-percolation threshold p_c(d), the nonzero couplings percolate, the ferromagnet with couplings |J_e| has distinct plus and minus states at low temperature, and frustrated plaquettes occupy a positive density of the infinite cluster. We prove that nevertheless, for p < p_c(d) + g_d/2 with an explicit g_d > 0, almost surely the infinite-volume Gibbs measure is unique simultaneously at every positive temperature. Moreover, the disorder average of the squared two-point function decays exponentially, uniformly in the temperature, and the quenched two-replica pressure has zero derivative at zero replica coupling in the overlap channel and, more generally, in every channel formed by the product of the overlaps at an odd number of sites. In dimension two this answers the "for some μ" form of Research Problem 4.2 in Newman's Topics in Disordered Systems. The mechanism is the exact frustration cancellation behind physical predictions of a spin-glass threshold above p_c(d), used rigorously on hierarchical lattices by Okuyama and Ohzeki: a frustrated plaquette whose two opposite corners have degree two transmits zero effective coupling at every temperature. Deleting such plaquettes is a non-monotone local diminishment of Bernoulli percolation, and an Aizenman–Grimmett-type argument, with a local surgery constructed for this rule, shows that it remains subcritical slightly above p_c(d). In d = 2 the result holds for every p < 1/2 + 1.1692·10⁻⁵; here the surgery is replaced by an exact, computer-verified comparison of local pivotal probabilities. For every d ≥ 3 a sparse surgery with exact accounting of preimages gives a window in closed form, proved by hand, and in d = 3 a computer-assisted certificate extends the result to every p < p_c(3) + 7.4055·10⁻⁹. The argument also applies to diluted symmetric laws whose magnitude distribution has an atom.