This summer, the mathematics world was shaken by continuous shocks—
OpenAI's Astra model solved 10 long-standing unresolved mathematical problems at once (including the existence of non-sofic groups, new results on high-dimensional sphere packing, etc.); Claude Fable 5 found a counterexample to the nearly century-old Jacobian conjecture and later significantly advanced the Riemann Hypothesis!
This left Sidharth Hariharan, a mathematics graduate student at Carnegie Mellon University, even more confused.
Previously, after receiving an email from 2022 Fields Medalist Maryna Viazovska, he learned that his research had been preemptively published by the AI agent 'Gauss', and he wept.
A year ago, you could still dismiss such progress as 'curiosity or hype, not very useful,' but the situation has changed. Even Fields Medalist Terence Tao admitted: Now, that argument no longer holds water.
Just as the ripples in the mathematics community hadn't subsided, physicist Gavin E. Crooks tossed an open problem to Claude, and within days the entire problem was completely solved. Even a physics graduate student with solid mathematical skills might have taken months on this problem.
In physics, humans are also being dimensionally reduced by AI!
Against this backdrop, he, one of the founders of non-equilibrium statistical mechanics, asserted:
Physics will follow in mathematics's footsteps.
He acknowledged that academia is about to undergo a major transformation.
Academia has two roles: advancing knowledge and teaching/training/mentoring the next generation.
Science may soon develop at a pace humans cannot keep up with. And the other role also faces impact:
So, when any question I pose might be answered faster by Claude, how do we teach and train students?
What Exactly Was the Problem?
To understand the weight of this matter, one must return to Crooks himself. Gavin E. Crooks is highly esteemed in non-equilibrium thermodynamics and statistical mechanics.
In 1998–1999, while still a graduate student at Berkeley, he proposed the famous Crooks fluctuation theorem, which precisely relates non-equilibrium work to equilibrium free energy differences, becoming one of the cornerstones of stochastic thermodynamics.
A single 1999 paper alone has garnered thousands of citations.
His long-term research traverses the intersection of thermodynamics, information theory, and computational science, profoundly influencing nanoscale thermodynamics, free energy estimation methods, and even the later formation of thermodynamic computing paradigms.
He received the Presidential Early Career Award for Scientists and Engineers (PECASE, one of the highest honors bestowed by the U.S. government on early-career researchers) and was elected a Fellow of the American Physical Society (APS Fellow) in 2019, enjoying high prestige in relevant academic circles.
This time, Claude solved a stochastic thermodynamics problem given by Crooks.
In the microscopic world, a system can occasionally briefly 'go against the second law of thermodynamics,' but the probability ratio of逆行事件 to normal events is not arbitrary; it is precisely controlled by entropy production and decreases exponentially rapidly.
This is described by the beautiful, symmetric Detailed Fluctuation Theorem (DFT)—
The problem: Under this functional constraint, what constraints exist for the statistics of entropy production?
This had already spawned a series of results.
Timpanaro et al.'s 2019 exchange TUR (exchange thermodynamic uncertainty relation) started from the exchange fluctuation theorem, giving a saturable bound in matrix form. Salazar's work continued to excavate tight bounds on skewness, tail probabilities, and information content from the DFT.
The 2023 TUT upgraded TUR from an inequality to a 'theorem,' clarifying the precise current achieving minimal scaled variance, and emphasized the influence of higher moments of entropy production.
They were unearthed one after another, yet a unified geometric picture was always missing.
Claude, however, identified all these local results as different projections of the same convex body and provided a complete hierarchical moment characterization.
The entire paper, from the abstract onward, was written by Claude.
Claude's Unified Answer: A Convex Region, Only a Lower Bound
Claude's core insight is remarkably clean:
Every distribution satisfying the DFT uniquely corresponds to a 'gap distribution' ν (i.e., the distribution of |σ|).
For each fixed gap 'a', the distribution P_a has only two outcomes ±a, with weights strictly fixed by e^σ.
Therefore, any DFT distribution is a unique mixture of these two basic outcome distributions.
The joint reachable region of statistics is thus a convex body (moment body). More crucially, this convex body has an exact characterization at each order:
Given the first n−1 moments, the n-th moment can only take values in [some sharp lower bound, +∞), with no upper bound. The lower bound is achieved by a unique 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 nearly saturate all these bounds simultaneously.
This theory can also be generalized to the general two-distribution Crooks fluctuation theorem.
The mechanism stems from an overlooked structural fact: the mean function of DFT, a·tanh(a/2), can be written as a sum of simple relaxation terms with positive weights, with poles at odd squares.
This directly maps the entire problem to the classical moment problem. Using precise identities involving Wronskian and Hankel determinants, all optimization falls onto a few atomic distributions.
The result: Nearly all published DFT bounds become low-dimensional projections (shadows) of this unified convex body. The same two-outcome distribution often saturates multiple different bounds simultaneously—because at a fixed mean, it is a vertex of the convex body.
For the general two-distribution Crooks fluctuation theorem, the symmetric channel (sum of forward and reverse moments) completely inherits the same hierarchy, while the asymmetric channel is almost unconstrained. This precisely explains why one-sided free energy estimators can be arbitrarily poor, while two-sided estimation is inherently necessary.
New Physics
The Thermodynamic Uncertainty Theorem itself was not a completely new discovery by Claude. The foundational version of this theorem was published in 2023 by Kyle J. Ray, Alexander B. Boyd, Giacomo Guarnieri, and James P. Crutchfield.
Claude placed all these 'projections' back into the same geometric object and proved that this object can be completely described.
The novel and fascinating aspect lies in the AI-driven research process.
Claude was not merely asked to explain an existing physics concept.
It was used to explore a difficult theoretical problem, connect existing ideas, and search for new mathematical structures.
This may become a story more significant than a single thermodynamic result:
AI is shifting from solving problems humans already know how to solve, to helping scientists explore problems we don't yet know how to solve.
Physics is entering its own AI-accelerated era.
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 public account "新智元" (New Wisdom), author: ASI Revelation, editor: David
