# The low-temperature Sherrington–Kirkpatrick fluctuation law > The typical spread of the low-temperature Sherrington–Kirkpatrick free energy grows like n^(1/6), with a limiting law. Canonical: https://closedform.nishachay.tech/e/openai-math-2026-217/ Last updated: 2026-10-14 **The typical spread of the low-temperature Sherrington–Kirkpatrick free energy grows like n^(1/6), with a limiting law.** **Trust:** Claimed, not checked · **Papers:** 2 · **First paper:** September 24, 2026 · **Field:** Probability and statistical mechanics · **Entry:** 217 *Explained by AI.* Drafted by one AI model and checked sentence by sentence against the paper by a different one. Spot a mistake? Report an error. The Sherrington–Kirkpatrick model is a standard model of a disordered magnet with random couplings. Its free energy changes from one random sample to the next. Physicists predicted that, at low temperature, the typical change grows like n to the power 1/6. These two papers claim to prove that scale, then a positive variance constant and a limiting shape after rescaling. No computer proof checker has checked them. ## At a glance - **First step:** 1992 (Crisanti, Paladin, Sommers and Vulpiani, the n^(-5/6) prediction) - **Significance:** Major. If it holds, it settles a long-standing physics prediction for the fluctuation scale and adds a limiting law. - **Trust:** Claimed, not checked - **Field:** Probability and statistical mechanics (2 papers) ## The explanation Take two magnets, each pointing up or down. Write their states as +1 or −1. A random number g, drawn from a bell curve, decides whether they prefer to agree or disagree. At inverse temperature 2, the partition function adds a weight for each of the four states. Here it comes to 4 cosh(√2 g). Its logarithm is the free energy. For g = 0 it is log 4, about 1.386. For g = 1 or g = −1 it is about 2.165. For g = 2 it is about 3.525. So the free energy is itself random, because it depends on the drawn couplings. The Sherrington–Kirkpatrick model does this with n magnets, often called spins. Every pair of spins gets its own independent random coupling, drawn from a bell curve[1]. The couplings are scaled by one over the square root of n. Inverse temperature, written β, sets how strongly the system prefers low-energy states. Above β = 1 the model is in its low-temperature, glassy phase. The question here is about the sample-to-sample fluctuations. Draw the couplings again, and the free energy moves. By how much, and with what shape[2]? The average free energy is well understood. Parisi proposed a formula for its leading term in 1979. Guerra and Talagrand made that formula rigorous by 2006[3]. Those results leave the smaller fluctuations around the average. Physicists predicted a free energy per spin varying on the scale n to the −5/6. That becomes n to the 1/6 for the whole log partition function[4]. The prediction rests on a matching argument that is heuristic. It gives neither a variance formula nor a limiting law[5]. Rigorous bounds were far from that. In 2026, Aronow and Lopatto bounded the variance between constant multiples of n to the 4/15 and n to the 7/15[6]. Picture a million spins. Then n to the 4/15 is about 39.8, and n to the 7/15 is about 631. The predicted variance, n to the 1/3, is 100. So the predicted spread, its square root, is 10. The bounds allowed this value but did not establish it. The fluctuation-scale paper claims the predicted exponent for every fixed β above 1. It allows an arbitrarily small error in the power[7]. The limiting-law paper goes further. It claims that the variance divided by n to the 1/3 tends to a positive constant[8]. After exact centering and scaling, the free energy then tends to one fixed bell-like law, called ν_β[9]. The law has mean zero and variance one. It is fixed by a list of moments, which are averages of powers. The paper does not identify it with any named distribution[10]. The obstacle is comparing systems of different sizes. An exponent with slack does not show that the variance settles to a constant. That needs errors which add up to a finite total over all sizes[11]. The method reveals the random couplings in stages, like a branching tree of shared histories. It then compares neighbouring sizes through that tree. A key input is a 2026 result of Lopatto on the shape of Parisi's solution, used at every β above 1[12]. The claims have limits. Each result is for one fixed β above 1, with no external field. Nothing is uniform as β approaches 1 or grows without bound[13]. The papers also give a related consequence for a model with an added ferromagnetic pull below unit strength[14]. How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published. **What it doesn't show.** This does not name the limiting law or give its variance constant as a number. ## Where it came from This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026. ## Hype check Major if both papers hold. They settle a predicted scale and add a law, but no Lean formalization is listed and no specialist review is published. ## Words used - **Spin:** A tiny magnet that points either up (+1) or down (−1). - **Partition function:** The sum of the weights of all states of a system; its logarithm is the free energy used here. - **Inverse temperature:** A number β that sets how strongly the system favours low-energy states; larger means colder. - **Free energy fluctuations:** How much the free energy changes when the random couplings are drawn again. - **Variance:** The average squared distance from the mean; its square root is the standard deviation, or typical spread. - **Limiting law:** The fixed probability shape that a rescaled random quantity approaches as the system grows. - **Moments:** Averages of the first, second, third and higher powers of a random quantity. ## The original papers (2) - [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf) - September 24, 2026 - [The low-temperature Sherrington–Kirkpatrick fluctuation scale](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf) - September 24, 2026 ## Quoted lines from the paper (14) [1] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > Let $(g_{ij})_{1\le i At a fixed temperature below the transition, how large are the sample-to-sample fluctuations of the Sherrington--Kirkpatrick free energy, and do they approach a law after centering and rescaling? [3] From [The low-temperature Sherrington–Kirkpatrick fluctuation scale](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf): > Guerra and Toninelli established the thermodynamic limit by Gaussian interpolation, and Guerra's variational bound and Talagrand's matching result identified its Parisi value. [4] From [The low-temperature Sherrington–Kirkpatrick fluctuation scale](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf): > At fixed $\beta$, this corresponds to $n^{1/6}$ for the log partition function. [5] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > This matching argument is heuristic: it does not itself provide a variance asymptotic or a limiting law. [6] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > For each fixed $\beta>1$, Aronow and Lopatto prove \[ c_\beta n^{4/15}\le \Var F_n(\beta)\le C_\beta n^{7/15} \] in, and quantitative tail estimates in a separate deviation window in. [7] From [The low-temperature Sherrington–Kirkpatrick fluctuation scale](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf): > Theorem … establishes the predicted one-sixth exponent with arbitrarily small power slack, for every fixed finite $\beta>1$. [8] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > The quantitative statement below proves that the variance has a positive limiting prefactor and provides the rate needed for the moment characterization. [9] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > We prove a limiting law for every fixed inverse temperature $\beta>1$ in the zero-field Gaussian Ising model. [10] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > The formula below specifies the moments of $\nu_\beta$; no identification with a named distribution is made. [11] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > The additional work here is a quantitative comparison of different sizes, with a summable error. [12] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > We use Lopatto's full-support result for every fixed $\beta>1$, in the form developed in PE. [13] From [The low-temperature Sherrington–Kirkpatrick fluctuation scale](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-fluctuation-scale-September-24-2026.pdf): > The temperature is fixed throughout, with no uniform conclusion as $\beta$ approaches $1$ or infinity. [14] From [The low-temperature Sherrington–Kirkpatrick free-energy limiting law](https://github.com/openai/math/blob/main/preprints/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026/The-low-temperature-Sherrington-Kirkpatrick-free-energy-limiting-law-September-24-2026.pdf): > This consequence is confined to the displayed zero-field ordered Gaussian model at fixed $\beta>1$ and $0\le\gamma<1$; it does not cover $\gamma=1$ or a coupling varying with $n$. ## Other sources - [openai/math release notes (2026): model provenance and review status](https://github.com/openai/math/blob/main/README.md): “manuscripts and supporting proof artifacts produced by an internal OpenAI model” - [D. Sherrington and S. Kirkpatrick, “Solvable model of a spin-glass”, Physical Review Letters 35 (1975), 1792–1796](https://doi.org/10.1103/PhysRevLett.35.1792) - [M. Talagrand, “The Parisi formula”, Annals of Mathematics 163 (2006), 221–263](https://doi.org/10.4007/annals.2006.163.221) - [A. Crisanti, G. Paladin, H.-J. Sommers and A. 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