A century-old mystery that remained unsolved for 78 years was cracked in just three days!
The question of whether a complex structure exists on the six-dimensional sphere has been asked since 1948, and now the answer is yes.
The答卷 was submitted by Harvard mathematicians Levent Alpöge and Claude.
Even more aggressive is the approach to breaking the deadlock.
They bypassed the decades-old path everyone else had taken and directly built this thing, then pointed to it and declared, “This is the complex structure on S⁶.”
When Alpöge announced it on X, the first sentence he typed sounded more like the announcement of a new life being born—
Welcome this beautiful new geometric object into the world.
For over seventy years, it has been hanging on this single spherical surface.
Among all spheres, only S² and S⁶ are qualified to discuss complex structures; all other dimensions have long been eliminated.
S² is unquestionably the most fundamental building block in the foundation of complex geometry.
For over seventy years, everyone’s attention was fixated solely on S⁶.
Some people claimed it exists, others swore it absolutely doesn’t—but both sides were wrong.
For example, in 2016, one of the greatest contemporary mathematicians and Fields Medalist Michael Francis Atiyah claimed to have solved it, but his argument was later shown to contain flaws.
The renowned Chinese mathematician Shiing-Shen Chern also studied this problem in his later years.
This time, Alpöge released a 108-page argument document.
He wrote out every matrix, every coordinate, and every bonding method used in the construction, in black and white.
Mathematician Qiaochu Yuan specifically used GPT-5.6 Sol to find flaws, but after staring at it for six minutes, he found no weaknesses whatsoever.
Still unsatisfied, I reviewed it for another 15 minutes but couldn’t find a single flaw—in fact, I gained an even deeper understanding of the reasoning.
Sol concluded that if these 108 pages ultimately hold up, they could be considered the most important AI mathematics achievement to date.
If this work had been completed entirely by humans, it would likely have won the Fields Medal, not necessarily because of its difficulty or fame, but at least because of its impact.
Three integers sealed the fate of this problem.
So the question arises: how exactly is this new object constructed?
First, lay a solid foundation.
Take the (3,4,∞) triangle group and use it to fold the upper half-plane; the resulting shape is intuitively a sphere.
However, this sphere has three special points marked on it: a 3rd-order point, a 4th-order point, and a cusp, located at t = 0, t = 1, and t = ∞, respectively.
Alpöge openly admits his preference, stating that the triangular group and the family of tori hanging from it are his most proud creation.
Step two: Attach the torus to the base.
Except for those three special points, each point on the base is adorned with a complex 2-torus, a structure that is two-dimensional in the complex plane and four-dimensional in real space.
The thing hanging above a certain point is mathematically called the fiber of that point, and the entire space X is constructed by arranging these fibers one by one.
After completing this step, the tops of those three special points remain empty, as if three holes have been forcibly punched through the entire sphere.
Step three: Fill in the three holes.
所谓“filling the holes” means inserting a fiber into each of these three empty points and sewing them together to form a complete, compact manifold.
The most clever part is that the three holes were not sealed using the same method—each opening perfectly fell within the scope of a classic filling technique.
The point at t = ∞, using Mumford’s toroidal degeneration, contains a fiber W obtained by gluing opposite pairs of sides of a hexagonal boundary of a sixth-degree del Pezzo surface.
The two remaining points at t = 0 and t = 1, under Kodaira's logarithmic transformation, have multiplicities of 3 and 4, respectively, precisely matching the third- and fourth-order points on the base.
At the moment the three holes were filled, a compact, simply connected three-dimensional manifold named X came into existence.
It’s been created, but is it really S⁶?
At this point, X is a completely legitimate complex manifold, but the questioning is not yet complete.
Section 7 of the paper explicitly computed the fundamental group of X, resulting in π₁(X) ≅ ℤ / |12ℓ₀ − 4ℓ₁ − 3ℓ₂|.
The fundamental group can be roughly understood as whether there are holes in the space that cannot be avoided. On a sphere, any loop can be shrunk to a point, so the fundamental group of the sphere is trivial.
The three integers (ℓ₀, ℓ₁, ℓ₂) in the formula record the degree of twist in the fibers when repairing those three holes.
Substituting (0, 1, −1) gives 12×0 − 4×1 − 3×(−1) = −1, whose absolute value is firmly fixed at 1.
And ℤ modulo 1 is the trivial group, so the fundamental group vanishes completely here, matching X with the sphere on the first point.
Since X is simply connected and its integral homology perfectly matches that of S⁶, applying the Hurewicz and Whitehead theorems confirms it is homotopy equivalent to a 6-sphere; further, invoking Smale’s 1961 generalized Poincaré conjecture, it is homeomorphic to S⁶.
Finally, only the smooth structure remains.
In topology, a homeomorphism does not equal a diffeomorphism. Two objects may appear identical, but if the way calculus is performed on them doesn’t match, such impostors have a special name: exotic spheres.
Fortunately, as early as 1963, Kervaire and Milnor had worked out the details: in six dimensions, everything is clean, with not a single exotic sphere to be found; but in seven dimensions, there are exactly 28 of them.
Thus, the homeomorphism is upgraded to a diffeomorphism, and the true nature of X is S⁶.
The person who ended it was not originally a specialist in complex geometry.
Alpöge is a Junior Fellow at the Harvard Society of Fellows and a postdoctoral researcher at Anthropic. His primary expertise lies in number theory and arithmetic geometry, and he does not frequently work in complex geometry.
Yuan revealed that just three days before solving this puzzle, he had discussed this deadlock with him.
From seeking answers to creating answers
Solving a problem that had stumped people for over seventy years in just three days is already absurd enough.
But it's actually only the third time in 35 days.
On July 20, Alpöge used Claude Fable 5 to produce a counterexample that disproved the Jacobian Conjecture, a problem posed in 1939 that had remained unsolved for 87 years.
Just three weeks later, on August 10, an unreleased research version of Claude, whose identity remains undisclosed, raised the proven proportion of Riemann zeta function zeros lying on the critical line from 41.6% to 67.2%.
During that computational battle, it orchestrated approximately 60 sub-agents, executed over 2,400 shell commands, and consumed 31 million output tokens.
Next is August 24, which is this S⁶.
Previously, you could still勉强 explain it away—AI is merely a highly powerful search engine, a tool that gropes within a known solution space to find counterexamples, or one that forcibly stitches together two papers already sitting in the database.
This time, the nature has changed completely.
This geometric object never existed originally—it was artificially created by the model.
Justin Curry, Associate Professor of Mathematics and Statistics at the University at Albany, State University of New York, stated directly: "If proven true, this is undoubtedly the most remarkable AI achievement in recent times."
For the past 78 years, everyone has been asking the same question: Is there a复structure on S⁶?
And from this moment on, the question may already be a different one.
How many more are still hidden there?
Reference material: https://alpo.ge/s6.pdf
This article is from the WeChat public account "New Intelligence Yuan," authored by ASI Revelation; edited by Moses David.