Full replica symmetry breaking at zero temperature for near-Sherrington–Kirkpatrick mixtures

For the mixed even p-spin Ising spin glass without external field, we prove that if (ξ'')⁻² is concave on (0, 1], equivalently ξ''ξ'''' ≥ 3(ξ''')² on (0, 1), then every minimizer γ of the zero-temperature Parisi functional is smooth and strictly increasing on [0, 1), with γ(0) = 0, so the zero-temperature Parisi measure has support [0, 1). For ξ(t) = t²/2 + εt⁴ the condition holds exactly when ε ≤ 1/60. These mixtures therefore satisfy the no-overlap-gap assumption of El Alaoui, Montanari and Sellke, so for every δ > 0 their message-passing algorithm finds, with probability tending to one, a configuration within δN of the ground-state energy. For every even mixture, no gap of the support lies inside an interval on which ξ''ξ'''' ≥ 3(ξ''')², and without a 2-spin term every minimizer has an atom at 0 followed by a gap. The proof uses a strict convexity inequality from a recent unrefereed proof of full support for the Sherrington–Kirkpatrick model.