A note on the capacity of the binary perceptron

Determining the capacity αc of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of αc, approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that αc < .996, while Krauth and Mezard outlined an argument which can be used to show that αc < .847. The purpose of this expository note is to record a complete proof of the bound αc < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron