No-overlap gap conjecture for the Parisi minimizer in SK

Prove that the minimizer γ* of the Parisi functional for the SK model (ξ(t)=t^2/2) is strictly increasing on [0,1), establishing the "no-overlap gap" condition that implies equality of the algorithmic value _L^*(ξ) and the Parisi optimum _U^*(ξ) and thereby enabling (1−ε)-approximation via convex-optimization-based methods.

Background

The paper’s algorithmic discussion notes that recent convex-optimization-based methods can match message-passing guarantees for SK if a widely believed structural property of the Parisi minimizer holds—namely, that the minimizer γ* is strictly increasing on [0,1). This condition, referred to as the "no overlap gap," ensures that the algorithmic value equals the Parisi ground-state value.

While widely believed in statistical physics, this property has not been proved for SK; its resolution would have direct algorithmic implications by closing the gap between achievable values and the Parisi optimum.