Krauth–Mézard capacity conjecture for the Ising perceptron

Prove that, as the dimension N tends to infinity, the capacity M_N/N of the Ising perceptron concentrates around the explicit constant α⋆(κ) predicted by the replica method (approximately 0.833 when κ = 0), where M_N is the largest number of random constraints for which the feasible set S_N^M is nonempty.

Background

The Ising perceptron, defined as the set of spin configurations in the hypercube satisfying M random Gaussian half-space constraints with margin κ, has a capacity (maximum constraints per dimension) that has been a central question in statistical physics and probability.

Using the non-rigorous replica method, Krauth and Mézard predicted a precise limiting capacity α⋆(κ). This work provides a conditional upper bound matching a previously established conditional lower bound, yielding a conditional proof under numerical assumptions, but a fully unconditional proof remains open.