Krauth–Mézard conjecture for the capacity of the binary perceptron

Let $A\in\mathbb{R}^{\alpha N\times N}$ have independent standard Gaussian entries, and consider the set of $\sigma\in\{-1,+1\}^N$ satisfying $A\sigma>0$ entrywise. The largest solvable value of $\alpha$ is the capacity, and denote by $\alpha_c$ the constant around which this capacity concentrates. Krauth–Mézard conjecture. The capacity of the binary perceptron concentrates around an explicit constant $\alpha_c\approx.833$. This is a long-standing open problem; a matching rigorous lower bound is known, while the corresponding upper bound remains difficult.