{"id":"openai-math/The-free-energy-of-the-spherical-random-perceptron","pdf":"https://github.com/openai/math/blob/main/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026/The-free-energy-of-the-spherical-random-perceptron-September-24-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026","tags":["statistical-physics","random-perceptron","spin-glass","probability"],"score":0,"title":"The free energy of the spherical random perceptron","domain":"Probability and statistical mechanics","family":"222","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove an exact variational formula for the limiting pressure of the spherical random perceptron with an arbitrary bounded continuous single-pattern potential. The formula holds at every fixed positive density and inverse temperature, with convergence in expectation and in probability.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"Perceptron free energies and microscopic jamming exponents"}