Listening to the rain, from Aofeisi Temple
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In July this year, Wang Hong and Deng Yu won the Fields Medal at the Philadelphia International Congress of Mathematicians.
During those days when the messages flooded the screen, Anthropic employee Alek Dimitriev posted a message on X:
This year’s Fields Award will be the last time humans receive this award.
Past winner Timothy Gowers commented that he had similar thoughts.
However, this process has a delay, so I think they can probably hold out until 2030.
Who would have thought… Gowers, you might be optimistic too.
On October 7th, OpenAI released 722 mathematical manuscripts at once, and all the papers along with some Lean proofs were uploaded to Github.
Among them, only two articles are in set 074, which is the same direction as that of Wang Hong’s award: Han Gu Conjecture.
A paper claims to prove the three-dimensional Hatake-Matsumoto function conjecture, which is stronger than the version proven by Wang Hong and Zahl.
Another paper claims to prove the four-dimensional Hau-Ki conjecture.
Two manuscripts total 272 pages, and the author line contains only one word: OpenAI.
Wang Hong and Joshua Zahl just solved the three-dimensional Hagu Valley dimension conjecture, and AI has now advanced the related results to four dimensions.
Is it really the last Fellis Prize in human mathematics this year?
What did OpenAI’s two Haguya papers specifically do?
In 1917, Japanese mathematician Katai Sōji posed a question: What is the minimum area required on a plane so that a needle of length 1 will rotate one full circle?
△ Schematic diagram of the Kugura problem
Besicovitch gave an counterintuitive answer: the area can be arbitrarily small. Even more absurdly, a set can hold one needle of unit length in every direction, yet the area is zero.
The area isn’t accurate, so mathematicians replaced the measuring tool with “dimension” to measure it.
The “dimension” here is not just the everyday concepts of length, width, and height. For such a highly irregular shape, mathematicians use tools such as Hausdorff dimension and Minkowski dimension to measure how complex it is at different scales.
The volume of a set can be 0, but it may still have a full spatial dimension.
Hagakure conjecture refers to the idea that: in an n-dimensional space, if a set can accommodate a unit line segment in every direction, then its dimension must be n.
The challenge is how these seemingly thin line segments can interlock and overlap, and whether the set can be further reduced after they overlap.
△ Schematic diagram of the Kugura problem
The planar case was proven by Davies in 1971; the three-dimensional one took half a century, until it was taken by Wang Hong and Joshua Zahl in February 2025.
Four-dimensional and above; yet no proof has been presented so far.
The two manuscripts publicly released by OpenAI address two versions of the Haga problem respectively.
Article 1: Conjecture on the Three-Dimensional Kagura Giant Function
Still 3D, but the problem has been changed to a more difficult version.
It can be thought of as a “tube overlap control” problem: many thin, long tubes are placed in three-dimensional space, and each tube represents one direction of the needle.
△ OpenAI’s First Page of the 3D Hatake Great Function Conjecture Paper
What mathematicians want to study is how much these tubes can overlap when they are oriented in different directions, and whether a unified estimate can be used to control this overlap.
Wang Hong and Zahl proved the “set version”, and studied how much space these piles of needles occupy at least.
And the “maximal function version” proven by OpenAI is studied in greater detail:
Make each needle thick into a thin tube. Assume that only a part of each tube is “solid”, and let the ratio be denoted by λ. If all solid parts are combined together, can the volume ensure that it is not less than the order of λ³?
The smaller λ is, the emptier the tube is, and the harder it is to control. The collection version only requires a relatively loose power of λ, while the maximum function version requires this power to be exactly 3.
This is also clearly stated in the introduction of the OpenAI paper: Wang Hong and Zahl’s theorem gives the λ raised to the power of K(ε), and the “need to replace it with λ³” was already clearly stated after their theorem.
That is, this is the next step left by Wang Hong and Zahl. And OpenAI’s 97-page paper claims to have solved this step.
Why is this important? Because the Hanaga-Goto large function turns “how thin tubes in various directions are packed together” into a calculable analytical problem.
It is related to some key conjectures in Fourier analysis, and has long been considered an important tool for understanding waves, frequencies, and spatial concentration phenomena.
In 1971, Fefferman used the Hauzelandt set to construct a counterexample and proved that the higher-dimensional ball products are not bounded except in L². Since then, the Hauzelandt problem has been closely linked to the Fourier limit conjecture and the Bochner-Riesz conjecture.
Article 2: The Hausdorff dimension of Siwei Guaguji
In the four-dimensional realm, the progress of humans so far is as follows:
In 1995, Wolff proved that the degree of vertex connectivity was at least 3 using the “hairbrush” method.
Later, Guth and Zahl used a polynomial method to approximate 3 + 1/40, which is approximately 3.025;
In 2019, Katz and Zahl used a “plane brush” to reach 3.059;
The subsequent progress mainly involved shifts in the second and third places after the decimal point.
And OpenAI’s 175-page paper claims to directly prove 4.
△ OpenAI’s Public Paper on the Four-Dimensional Hausdorff Dimension Conjecture
What is difficult about four-dimensional space? Guth pointed out in his review that the key theorem in Wang Hong and Zahl’s three-dimensional proofs no longer holds when moved to four-dimensional space.
Because in high-dimensional spaces, tubes can gather together near low-degree algebraic surfaces, forming a structure that has no counter-example in three dimensions.
And the core tool of OpenAI’s four-dimensional paper targets this problem: using quadratic polynomials at different scales for local fitting, to track those lines that continue to accumulate across scales.
In the entire text, the word “polynomial” appears 213 times.
There is another detail: one of the key inputs for a four-dimensional paper is a lemma from that OpenAI three-dimensional paper. It’s like two papers forming a chain; the stronger version of three dimensions has paved the way for four dimensions.
Therefore, it can also be said that it pushed Wang Hong’s achievements a big step forward. In three dimensions, her theorem was enhanced to a version of a large function; in four dimensions, a problem stuck at around 3 points was directly advanced to 4.
But this paper still has limitations. The four-dimensional case only proves the Hausdorff dimension version, and it does not resolve the maximal function conjecture in four dimensions; the five-dimensional case and above have not been proven either. The paper only provides a lower bound using projections.
Also, it is worth noting that the results of group 074 have not been formalized in Lean format, nor have they been peer-reviewed.
OpenAI wrote in the README: Some unformalized results “may be problematic”.
What did Wang Hong do?
Some people say that Wang Hong won the Fields Prize before AI did is a stroke of luck.
Is this really the case?
When reading the full texts of these two papers by OpenAI, one of the most frequently mentioned names is Wang Hong.
In just “Three-Dimensional”, “Wang” appears 23 times.
The three-dimensional Hanggu Mountain peak has been climbed by human mathematicians for many years.
Katz, Łaba, and Tao early on discovered that nearly extreme tube configurations exhibit three structures: adhesiveness, planarity, and granularity.
But over several decades, no one was able to combine them into a complete proof.
Wang Hong and Zahl divided things into three steps:
In 2022, a case of a “sticky” set of Hakuya was demonstrated, and it was later published in the American Mathematical Society Journal.
In 2024, the Assouad version with V-count was achieved.
In February 2025, a 127-page full proof was released, solving the Hausdorff and Minkowski dimensions version of the three-dimensional Hatao set conjecture.
△ Wang Hong and Joshua Zahl’s paper on the three-dimensional Hagu Valley conjecture
Nets Katz said it is “a once-in-a-century event”.
During the same period, Wang Hong also resolved the Furstenberg set conjecture in topology with Ren Kang. This is also an old problem in harmonic analysis.
These works have now become the foundation of OpenAI papers.
The three-dimensional paper directly uses the simplified proofs of Guth, Wang Hong, and Zahl. Guth was Wang Hong’s supervisor during her PhD at MIT; it also employs the planar Furstenberg estimates of Ren Kang and Wang Hong.
The four-dimensional paper uses a dot product theorem by Wang Hong and Zahl.
If the two papers are ultimately confirmed to be valid, they also add two more layers on top of the framework that Wang Hong established.
Moreover, Wang Hong’s attitude towards AI is not rejection.
In an interview after receiving the award, she described AI as a “proactive booster” for mathematical research, believing that asking questions, creating concepts, and building theories remain the core work of mathematicians.
In the field of human mathematics, it’s already exploded...
Now, it is probably the “critical moment of survival” in human mathematics.
The lifecycle of an important mathematical result in the past is roughly: a person or a small team spends several years on it, posts it on arXiv, peers spend one to two years reviewing it, and the academic community gradually absorbs it.
When the conclusions are made clear, and after a few more years, the awards will be finally finalized.
Wang Hong and Zahl took nearly four years to walk this path, from the sticky situation in 2022 to the Fields Prize in 2026.
And this time, OpenAI released 722 mathematical manuscripts overnight. On average, each result took about three hours of ChatGPT Pro thinking power, and only about 42% of the results were formalized in Lean format.
It’s just like a provocation…
Tao Zhexiu has led the Association of Human Mathematical Societies (AHM) to issue a major joint statement, fully opposing OpenAI.
The wording is quite serious:
The mathematicians did not demand these tasks. The simultaneous release of over 700 files is not about academia, but about power.
The statement finally strongly urges all mathematicians to stop cooperating with OpenAI and return to the scientific vision centered on human understanding.
Tao Zhexuan has always been an active user of AI-assisted research. What he opposes is presenting the mass resolution of famous problems as a product.
In his view, once a problem is solved by AI, it can no longer return to an unsolved state. The new methods and new understandings that human mathematicians might have developed for this problem are also lost.
And now, the review process has become the biggest bottleneck.
A 175-page four-dimensional proof of Haguya, there are only a few people in the world who can truly understand it seriously. Who will read it, how long it will take, and whose workload it will be—all these answers remain unknown.
The signatures and attribution also begin to become chaotic. Scholars studying the same issue might wake up one day to find that a problem they have been working on for several years has been “claimed” solved by a model.
It is still unknown whether these AI-generated proofs are correct, but it has become extremely difficult to continue with the topic.
Of course, some people are also excited.
For example, Daniel Litt, a mathematics assistant professor at the University of Toronto, believes that mathematicians will have a lot of exciting work to do in the future.
Yann LeCun also believes that on the contrary, mathematics is entering a new era.
Formal proofs will be greatly automated, with focus shifting to developing new concepts, new abstractions, new definitions, and new conjectures.
The invention of ships reduced the importance of swimming, but it led us to discover the New World.
The field of mathematics is at a critical point where old and new systems collide violently. The old research methods are being challenged, and the new path has not yet taken shape.
Where we will go in the future, there is no answer yet. But we prefer to believe: thousands of sails pass by the sunken boat, and ten thousand trees bloom in front of the sick tree.
Gowers said that humans can probably last until 2030, and the next Fields Prize is scheduled to be awarded in 2030.
By then, will there still be humans on the podium?