AI is accelerating everything. The defense line of mathematics has just been breached, and physics is already within the striking range of AI.

This summer, the mathematics community has been rocked by successive breakthroughs —

OpenAI's Astra model has solved 10 long-standing unsolved mathematical problems in one go (including the existence of non-sofic groups, new results for high-dimensional sphere packing, etc.); Claude Fable 5 found a counterexample to the nearly century-old Jacobian conjecture, and later made tremendous progress toward advancing the Riemann Hypothesis!

This left Sidharth Hariharan, a graduate student majoring in mathematics at Carnegie Mellon University, even more confused.

Previously, after receiving an email from Maryna Viazovska, the 2022 Fields Medal winner, he learned that his research had been pre-emptively published by the AI agent "Guass", and burst into tears.

A year ago, you could dismiss such advances as "curiosity-driven, overhyped and useless", but the situation has changed. Terence Tao, a Fields Medalist, even admitted frankly: this argument no longer holds water.

While the aftershocks in the mathematics community have not subsided, physicist Gavin E. Crooks threw an open problem to Claude, and the entire problem was completely solved in just a few days. Even a physics graduate student with a solid mathematical foundation might spend months working on this problem.

In the field of physics, humans have also been hit by AI with a crushing dimensionality reduction strike!

Against this backdrop, one of the founders of nonequilibrium statistical mechanics asserted:

Physics will follow in the footsteps of mathematics.

He admitted frankly that the academic world is about to undergo a major transformation.

There are two roles in academia: advancing knowledge and teaching / training / mentoring the next generation.

Science may soon develop at a speed that humans cannot keep up with. The other role is also facing impact:

So how do we teach and train students when any question I raise could be answered faster by Claude?

What is the problem itself?

To understand the significance of this event, we must go back to Crooks himself. Gavin E. Crooks is a renowned scholar in the field of nonequilibrium thermodynamics and statistical mechanics.

From 1998 to 1999, when he was still a graduate student at UC Berkeley, he proposed the famous Crooks Fluctuation Theorem, which accurately links nonequilibrium work and equilibrium free energy difference, and has become one of the cornerstones of stochastic thermodynamics.

His single paper published in 1999 alone has thousands of citations.

His research has long spanned the intersection of thermodynamics, information theory and computational science, profoundly influencing nanoscale thermodynamics, free energy estimation methods, and even the later formation of the thermodynamic computing paradigm.

He has received the Presidential Early Career Award for Scientists and Engineers (PECASE, one of the highest honors awarded by the US government to early-career researchers), and was elected an APS Fellow of the American Physical Society in 2019, enjoying a high reputation in the relevant academic community.

This time, Claude solved a stochastic thermodynamics problem proposed by Crooks.

In the microscopic world, systems can occasionally briefly "go against the second law of thermodynamics", but the ratio of the probability of reverse events to normal events is not arbitrary. It is precisely controlled by entropy production and decreases exponentially rapidly.

This is described as the elegant and symmetric Detailed Fluctuation Theorem (DFT) —

The question is: under the constraint of this function, what are the constraints on the statistical quantities of entropy production?

A host of results have been derived from this question.

The exchange TUR (Exchange Thermodynamic Uncertainty Relation) proposed by Timpanaro et al. in 2019 starts from the exchange fluctuation theorem and gives a saturable matrix-form bound. Salazar's work continues to explore the tight bounds of DFT on skewness, tail probability and information content.

The TUT proposed in 2023 has upgraded TUR from an inequality to a "theorem", clarified the exact current that achieves the minimum scaled variance, and emphasized the influence of high-order moments of entropy production.

These results have been unearthed one after another, but a unified geometric picture has always been missing.

Claude identified all these local results as different projections of the same convex body, and gave a complete characterization of the moment hierarchy.

The entire paper, starting from the abstract, was written entirely by Claude.

The unified answer given by Claude: a convex region with only lower bounds

Claude's core insight is extremely concise:

Every distribution that satisfies the DFT uniquely corresponds to a "gap distribution" ν (i.e. the distribution of |σ|).

Each distribution P_a with a fixed gap a has only two outcomes ±a, and their weights are strictly fixed by e^σ.

Therefore, any DFT distribution is the unique mixture of these two basic outcome distributions.

The joint reachable region of the statistics is therefore a convex body (moment body). More critically, this convex body has an exact characterization at every order:

Given the first n−1 moments, the nth moment can only take values in [a sharp lower bound, +∞), with no upper bound. The lower bound is achieved by the only distribution with a finite number of symmetric outcome levels.

Every previously published DFT bound can be recovered as a low-dimensional projection of this unified convex region.

This also explains why the same distribution can almost saturate all these bounds at the same time.

This theory can also be extended to the general two-distribution Crooks Fluctuation Theorem.

The mechanism comes from a neglected structural fact: the mean function a·tanh(a/2) of DFT can be written as the sum of simple relaxation terms with positive weights, with poles located at odd squares.

This maps the entire problem directly to the classical moment problem. With the help of the exact identities of Wronskian and Hankel determinants, all optimizations fall on a small number of atomic distributions.

The result is: almost all published DFT bounds have become low-dimensional projections (shadows) of this unified convex body. The same two-outcome distribution often saturates multiple different bounds at the same time — because at a fixed mean value, it is exactly the corner point of the convex body.

For the general two-distribution Crooks Fluctuation Theorem, the symmetric channel (sum of forward and backward moments) completely inherits the same hierarchy, while the asymmetric channel is almost unconstrained. This accurately explains why unidirectional free energy estimators can be arbitrarily poor, and bidirectional estimation is essentially necessary.

New Physics

The Thermodynamic Uncertainty Theorem itself is not a completely new discovery by Claude. The basic version of this theorem was published in 2023 by Kyle J. Ray, Alexander B. Boyd, Giacomo Guarnieri and James P. Crutchfield.

Claude put all these "projections" back into the same geometric object, and proved that this object can be completely described.

The novel and fascinating part lies in the AI-driven research process.

Claude was not merely asked to explain an existing physical concept.

It was used to explore a difficult theoretical problem, connect existing ideas, and search for new mathematical structures.

This could become a far more significant story than a single thermodynamic result:

AI is shifting from solving problems that humans already know how to solve, to helping scientists explore problems that we do not yet know how to solve.

Physics is entering its own era of AI-accelerated development.

References:

https://www.nytimes.com/2026/06/08/science/ai-scoop-young-mathematicians.html

https://x.com/SciTechera/status/2088872862254084449https://x.com/gavincrooks/status/2088643200038883830

https://x.com/gavincrooks/status/2088643200038883830

https://x.com/gavincrooks/status/2088590113463013582?s=20

This article is from the WeChat Official Account "Xinzhiyuan", Author: ASI Revelation, Editor: David, published with authorization from 36Kr.