The low-temperature Sherrington–Kirkpatrick fluctuation law

What OpenAI claims

For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to c_β n^1/3, with c_β > 0, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.

Papers (2)

1. The low-temperature Sherrington–Kirkpatrick free-energy limiting law

   24 September 2026 · Preprint, not peer reviewed · PDF

   For every fixed inverse temperature β > 1, we prove that the zero-field Gaussian Ising Sherrington–Kirkpatrick free energy, centered by its expectation and divided by its standard deviation, converges in distribution to a nondegenerate law as the system size tends to infinity through all integers. We also prove that its variance divided by n1/3 converges to a finite positive constant.

2. The low-temperature Sherrington–Kirkpatrick fluctuation scale

   24 September 2026 · Preprint, not peer reviewed · PDF

   For the zero-field Gaussian Sherrington–Kirkpatrick model at every fixed inverse temperature β > 1, we prove that the standard deviation of the log partition function is n^1/6+o(1). The same exponent describes its typical centered absolute fluctuations, establishing the predicted one-sixth exponent in this regime.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.