A new mathematics paper co-authored by Shing-Tung Yau thanks GPT 6 Astra and Claude Pro for helping explore proof ideas and calculations. The paper addresses a classic conjecture about whether the 27 exotic spheres in seven dimensions can admit strictly positive sectional curvature, a problem Yau listed in 1982. If accepted, the result would resolve a long-standing open question.
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A paper involving Yau Shing-Tung, which thanks GPT 6 Astra and Claude Pro, claims that all 28 smooth spheres in seven-dimensional space (including 27 exotic spheres) can be equipped with metrics of strictly positive sectional curvature, potentially completing the final step from "non-negative" to "positive" for a classical problem that has been suspended for about 70 years. However, the paper explicitly states that the responsibility for manual verification and writing rests with the authors, and whether the conclusion can pass peer review remains unknown.
In 1956, American mathematician Milnor constructed an exotic sphere that is topologically identical to the standard seven-dimensional sphere but has a different smooth structure. Seven years later, Kervaire and Milnor proved that there are 28 smooth spheres in seven-dimensional space, forming Θ₇ ≅ Z/28Z, of which 27 are exotic spheres. The standard sphere has strictly positive sectional curvature everywhere, so mathematicians naturally asked: can the other 27 exotic spheres also achieve this?
In 1974, Gromoll and Meyer constructed a non-negatively curved metric on one exotic sphere; in 2020, Goette, Kerin, and Shankar generalized this result to all 28 seven-dimensional spheres. But from "non-negative" to "positive"—from allowing some places not to bend to requiring every place to bend—there is still a final step in between. In 2008, Petersen and Wilhelm released a preprint claiming to achieve positive curvature on one exotic sphere, but their proof never passed peer review.
In 1982, Yau Shing-Tung placed the question of whether exotic spheres can have positive sectional curvature on his personal problem list, ranked second. Now, a latest paper involving him claims to have completed this step from non-negative to positive, proving that each of the 28 seven-dimensional spheres can be equipped with a metric of strictly positive sectional curvature. If this result is ultimately accepted by the mathematical community, this problem suspended for about 70 years will be completely solved. But the paper itself only claims to complete this step, and whether the conclusion can pass peer review and be ultimately accepted by the mathematical community remains unknown.
The differentiable sphere theorem of Brendle and Schoen states that if the ratio of the maximum to minimum curvature at any point in any direction on a manifold does not exceed 4, then the manifold must be smoothly equivalent to the standard sphere. This means that positive curvature on an exotic sphere must be positive, but not too uniformly positive, otherwise it would no longer be an exotic sphere, leaving extremely narrow room for construction.
The core idea of the paper is to cut each exotic sphere into two halves. The authors adopt the two-disk model proposed by Durán, Putmann, and Rigas, viewing each seven-dimensional sphere as two seven-dimensional disks glued along their boundaries, with the 28 spheres corresponding to 28 gluing methods. The problem then becomes: can metrics that bend inward everywhere be constructed on each disk separately, and can the two disks fit perfectly at the seam.
The authors do not directly construct metrics on the seven-dimensional disks. They first construct a principal bundle with the three-dimensional sphere S³ as fiber, moving the problem to ten-dimensional space, then project the metric back to the seven-dimensional disks via Riemannian submersion and O'Neill's formulas. O'Neill's formulas guarantee that after projecting from ten dimensions to seven dimensions, the curvature can only increase or stay the same, not decrease, so it suffices to make the curvature positive in ten-dimensional space.
The core technique for achieving positive curvature in ten-dimensional space is to make the radius of the three-dimensional fiber extremely small. The thinner the fiber, the greater its own bending, and this high curvature is sufficient to suppress negative curvature contributions that may appear in other directions.
The constructions of the two disks are not symmetric: the south disk uses a curved base plus connection, while the north disk uses a flat product connection plus a warped radial profile. They must satisfy two conditions at the boundary: the induced metrics on the boundary are completely identical, and the sum of the second fundamental forms from both sides is strictly positive in every direction. The paper first establishes a common coordinate system on the shared boundary of the two disks, proving that the derivatives of the fiber radii on both sides with respect to the normal direction are exactly opposite, so that the fiber part's contribution to the second fundamental form cancels out exactly at the gluing, while the angular part's contributions add up to strictly positive. Once these two conditions are satisfied, the gluing theorem of Reiser and Wraith guarantees the existence of a smooth positive curvature metric on the entire glued sphere.
The paper includes a set of verification code written in SageMath. The entire construction involves a large number of precise estimates of curvature inequalities and parameter selections, which would be nearly impossible to check manually one by one; computer verification is an important support for the credibility of the proof here.
However, whether the SageMath verification code has been independently reviewed, and what its coverage and reliability are, the source does not state. This constitutes an uncertainty factor in the acceptance of the conclusion.
The paper thanks GPT 6 Astra and Claude Pro in the acknowledgments for their help in exploring some proof strategies and calculations. The paper immediately notes that the responsibility for manual verification and paper writing rests with the authors themselves.
Exactly which proof ideas and calculation steps GPT 6 Astra and Claude Pro provided help with, and to what extent, the source does not state. AI is only acknowledged as providing help in exploring some proof strategies and calculations, while manual verification and writing responsibility rest with the authors; this boundary is key to understanding the nature of this achievement.
Yau Shing-Tung's attitude toward AI gradually shifted from saying in April 2023 that AI could not affect the most cutting-edge mathematicians, to acknowledging in 2024 that AI has many benefits for mathematics, to advocating in 2025 that students learn to use AI tools as early as possible.
In April 2023, on the eve of attending the Pujiang Science Masters Forum at Fudan University, he said that AI could not have any impact on the most cutting-edge mathematicians, that it could only integrate existing materials and could hardly form breakthroughs in scientific concepts. In April 2024, his statement at the Zhongguancun Forum softened, changing from "will not have any impact" to "has many benefits for mathematics, can help the development of mathematics," but he still insisted that AI could not change mathematics. In January 2025 at the World Congress of Chinese Mathematicians, he said that what AI changes is research efficiency, but it is not yet a paradigm revolution; in June, he also said that AI can solve many "nice, cute questions," but is still far from a breakthrough that changes all of mathematics; in July at the International String Theory Conference, he already advocated that students should learn to use AI tools as early as possible.
The causal relationship between Yau Shing-Tung's attitude shift toward AI and the AI assistance in this paper is only presented as a timeline by the source, with no direct evidence given. Whether this achievement constitutes the mathematical paradigm revolution that Yau Shing-Tung expects, the source shows that he himself still holds reservations.
The source is a report from the QbitAI WeChat public account, a secondhand account, whose unique numbers and conclusions lack first-party source support. The paper was released around October 2026, with arXiv number 2609.29426.
Whether the paper's conclusion can pass peer review and be ultimately accepted by the mathematical community remains unknown. The specific reasons why the 2008 Petersen and Wilhelm preprint failed review, the source does not state. Exactly which proof ideas and calculation steps GPT 6 Astra and Claude Pro provided help with, and to what extent, the source does not state. Whether the SageMath verification code has been independently reviewed, and what its coverage and reliability are, the source does not state.
The concerns of Yau Shing-Tung and Terence Tao about AI are similar: as AI becomes increasingly capable of solving problems, will mathematicians still be willing to continue asking the questions that AI cannot solve. This question itself is more worthy of sustained attention than the conclusion of a single paper.
This article is based on a report from the QbitAI WeChat public account (source_tier 2, secondhand account). The paper's claimed conclusion, the specific extent of AI assistance, the reliability of the SageMath verification code, etc., all lack first-party source support and should be regarded as source_claim or unresolved uncertainties. Whether the paper's conclusion can pass peer review and be ultimately accepted by the mathematical community remains unknown.
This paper claims to have completed the final step from "non-negative" to "positive" for the positive curvature problem of seven-dimensional exotic spheres, and thanks GPT 6 Astra and Claude Pro for their help in exploring some proof strategies and calculations, but manual verification and writing responsibility rest with the authors. Whether the conclusion can be accepted by the mathematical community remains unknown; the source is a secondhand account, and its unique numbers and conclusions lack first-party source support.
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