Wang Hong has solved the three-dimensional Kakeya conjecture, and OpenAI has released a 175-page four-dimensional proof manuscript.
Wang Hong has conquered the 3D case, and OpenAI has directly laid the 4D proof manuscript on the table!
On October 6, OpenAI publicly released the first batch of 722 mathematical manuscripts on GitHub.
One of these manuscripts is 175 pages long, explicitly targeting the 4D Kakeya conjecture.
The other one is 97 pages long, aiming to make further progress on the 3D case, targeting a stronger maximal function version than the set theorem established by Wang Hong and Zahl.
Group 074 in the OpenAI mathematical manuscript catalogue corresponds to the 3D maximal function version and the 4D set version respectively.
This problem that has occupied mathematicians for more than a century has a surprisingly simple original form.
How can a needle turn around in the smallest possible space?
The Ultimate U-Turn of a Needle
In a crowded carriage, turning a folded long-handled umbrella around while shifting and rotating usually takes less space than rotating it directly sideways.
In 1917, Japanese mathematician Soichi Kakeya proposed a similar problem. For a needle of length 1 with no thickness, what is the minimum area it needs to sweep to turn 180 degrees on a plane?
Besicovitch gave the answer: The area can be arbitrarily small.
If we further relax the requirement, not requiring the needle to rotate continuously, but only requiring that a unit line segment can fit in every direction, the area of this set can even be zero!
The area has been reduced to zero, but the problem is not over yet. Do these line segments pointing in all directions still have to span a two-dimensional structure?
The Kakeya conjecture asserts that: In n-dimensional space, as long as a set contains unit line segments in all directions, its dimension must be n, that is, "full dimension".
The 2D case was proven in 1971, but when it came to 3D, mathematicians got stuck for more than half a century.
What Did Wang Hong Prove
To solve the 3D problem, mathematicians first slightly thicken these line segments into thin tubes, then observe how the overlapping volume changes as the tubes keep getting thinner.
More than 20 years ago, Katz, Łaba and Terence Tao had already discovered some usable arrangement rules.
However, the bundles of tubes that look crowded from a distance may have a large number of gaps when magnified, and cannot be directly calculated as solid thick tubes. The results at different scales may not be connected to each other either.
To solve this problem, Wang Hong and Zahl started with the more regular "sticky" case and completed the proof in 2022.
In February 2025, the two published a 127-page paper, finally advancing the proof to the general case.
The 3D "set version" conjecture has been formally solved at this point.
Wang Hong and Zahl's 3D Kakeya paper, the abstract concludes that both the Hausdorff dimension and Minkowski dimension are 3
The left figure shows the "sticky" case; in the right figure, the thick tubes overlap heavily, but the inner thin tubes are very sparse
OpenAI's New 3D Proof
The Same Problem, A Stronger Version
The 3D case has been conquered, but the tough problem is far from over.
Wang Hong and Zahl solved the dimension problem of the set, while the more demanding "maximal function version" is still waiting for an answer.
This 97-page manuscript from OpenAI is precisely targeting this step.
The title and abstract of OpenAI's 3D manuscript, the main theorem targets the stronger Kakeya maximal function conjecture
The difference lies in the coloring of the thin tubes. Only part of each tube is colored, and the overlapping area is counted only once. What is the minimum total volume occupied by all these colored parts combined?
The estimates from Wang Hong and Zahl are sufficient to deduce that "the dimension is 3". OpenAI goes one step further to precisely control the relationship between the coloring ratio and the total volume. Even if the colored part gets smaller and smaller, this stricter lower bound must still hold.
According to OpenAI's argument, this stricter requirement is satisfied, and it can further deduce the complete 3D maximal function estimate.
The trouble arises after magnification.
With the same 10% coloring, if the colored part is concentrated at one end or scattered everywhere, the density may be completely different after intercepting a small section.
OpenAI therefore records the distribution of coloring at different scales, tracks the changes of each segmentation and magnification, and combines geometric estimation and entropy analysis to try to exclude arrangements that have excessive overlap and violate the target lower bound.
This set of arguments still uses the set estimates of Guth, Wang Hong and Zahl, as well as the planar Furstenberg theorem of Wang Hong and Ren Kang.
If the main theorem holds, the gains will be far more than that.
One category is Nikodym maximal function estimation, which shifts from looking for thin tubes by direction to studying thin tubes passing through each point.
The other category uses existing transformation theorems to extend the conclusion from straight thin tubes to curved thin tubes that meet specific conditions, and obtain corresponding local results.
OpenAI's New 4D Proof
One More Dimension, The Difficulty Is More Than Doubled
The previous lower bound of dimension stayed at 3.059; the full dimension breakthrough by Wang Hong and Zakharov in September this year only covered the special "sticky" case.
From 3.059 straight to 4! This 175-page manuscript from OpenAI is precisely challenging this step.
The title and abstract of OpenAI's 4D Kakeya manuscript. The main conclusion targets the Hausdorff dimension of general 4D Kakeya sets
But the 3D proof cannot be directly moved to the 4D case.
Guth pointed out that thin tubes in 4D space can gather in large numbers along quadric surfaces, forming overlaps that cannot be controlled by 3D methods.
OpenAI thus takes this aggregation itself as a clue. It uses quadratic polynomials to describe the local line segment relationships, and then tracks how far these relationships can extend along the line segments.
But it is not enough to explain only a few line segments.
OpenAI uses a weighted estimate from the 97-page 3D manuscript to make the selected local area contain enough line segment intersections.
And the source of this lemma is still the set estimates of Guth, Wang Hong and Zahl.
Page 20 of OpenAI's 3D manuscript, Lemma 3.3 and formulas (3.7), (3.8); cited as "Lemma 2.3" in the 4D manuscript, and restated in its own Lemma 2.15
When dealing with planar structures, OpenAI's 4D argument also uses the dot product theorem of Wang Hong and Zahl.
This is a bit like unlocking a tech tree. When OpenAI continues to move forward, it still needs to use the pre-unlocked skills that Wang Hong participated in developing.
The last step is to let the hypothetical counterexample reveal its own flaws.
OpenAI first assumes that there exists a Kakeya set with dimension less than 4, then handles different cases according to the aggregation mode of line segments. Some cases allow local relationships to extend further along line segments, while others can restrict line segments to thinner regions.
Next, these improvements need to be connected, and it must be confirmed that they are still valid after returning to the original scale. According to OpenAI's argument, this will conflict with the conditions that the hypothetical counterexample must satisfy, excluding the possibility that the dimension is less than 4.
Once these 175 pages can stand the test, the full-dimension conclusion for 4D will advance from the special case to the general case. As long as the unit line segments in all directions are complete, the Hausdorff dimension of the set must be 4.
However, this is still the "set version" for 4D. The stronger 4D maximal function version has not been solved by this. For 5D and above, we only get the conclusion that "the dimension is at least 4", and cannot deduce their respective full dimensions.
The Last Fields Medal Belonging to Humans
So the question arises: Will this year's Fields Medal be the last one belonging to humans?
Just a few months after Wang Hong won the medal, OpenAI submitted 272 pages of new manuscripts in the same research direction. The two arguments still need independent verification. But if they eventually hold, this part of mathematical history will not be able to bypass the name OpenAI when written later.
Wang Hong's name has not faded away because of this. When OpenAI moves forward, it still uses the estimates and methods established by her and her collaborators.
The real thing hanging in the balance is the next breakthrough. If AI starts to continuously propose new methods and open up new directions, can humans always take that step ahead?
When the next Fields Medal is awarded in 2030, what is worth asking may not only be who is standing on the podium.
But also who came up with the most critical step in that award-winning achievement.
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