A century-old conjecture has fallen yet again.
Anthropic/Harvard mathematician Levent Alpöge, in collaboration with Claude, has directly disproven the Carathéodory conjecture, which had perplexed the mathematical world for over 100 years, and in doing so, also shattered the Loewner index conjecture.
A PDF circulated within mathematical circles for a full week. Now, it has been officially made public.
This is not the first time Alpöge has teamed up with Claude to shake things up. During the World Cup this year, he proposed a counterexample to the 3-dimensional Jacobian conjecture, sending shockwaves through the mathematical community.
What Exactly is This Century-Old Conjecture About?
First, let's talk about the Carathéodory conjecture itself.
In 1924, German mathematician Hans Ludwig Hamburger proposed the Carathéodory conjecture in differential geometry, naming it after his doctoral advisor, the Greek mathematician Constantin Carathéodory.
The conjecture states that any convex, closed, and sufficiently smooth surface in three-dimensional Euclidean space must have at least two umbilic points.
Constantin Carathéodory
The conjecture is extremely concise but notoriously difficult to prove: any sufficiently smooth closed convex surface must have at least two 'umbilic points'.
Umbilic points can be understood, in simple terms, as points on a surface where the 'curvature is exactly the same in all directions'.
In the differential geometry of surfaces in three-dimensional space, an umbilic point is a point on a surface where it locally resembles a sphere; all normal curvatures are equal at that point, meaning the two principal curvatures are equal, and every tangent vector is a principal direction.
Every point on a sphere is an umbilic point—no matter which direction you move, the curvature feels the same.
Carathéodory asserted: no matter how you shape or deform it, as long as this convex surface is sufficiently smooth, you cannot reduce the umbilic points to just one. At least two must remain.
This conjecture seems intuitively 'correct'.
A sphere has infinitely many umbilic points, an ellipsoid has two.
For a hundred years, countless mathematicians tried, but no one could construct a smooth convex body with only one umbilic point.
Parallel to this was the Loewner conjecture—that the index of an isolated umbilic point does not exceed 1. The two conjectures supported each other, for a century locking away an entire domain of differential geometry like a pair of iron locks.
Now, the lock has been pried open. This discovery shows that if a convex surface is too smooth (mathematically called C∞, infinitely differentiable), Levent Alpöge found a convex surface with only one umbilic point.
One Counterexample, Two Conjectures Fall Simultaneously
Alpöge and John-Paul Smith provided an extremely specific construction.
They defined a family of functions g_k, where for k=2, the function g2 serves as the support function for a sphere-like surface, producing a convex body with an umbilic point of index 2 at the origin.
Note that an index of 2 already directly violates the Loewner conjecture's upper bound of 1.
More crucially, due to the presence of an exponential decay term, g2 is mathematically guaranteed to be C∞ smooth—meaning this convex body satisfies all the smoothness conditions required by the Carathéodory conjecture.
But it has only one umbilic point.
The Carathéodory conjecture requires at least two umbilic points, but the counterexample has one, thus disproving the conjecture.
The Loewner index conjecture requires the index of an umbilic point to be 1, but this one is 2, disproving it as well.
An explicit, verifiable counterexample has taken down two conjectures that survived for over a century.
In mathematics, disproving a conjecture requires only one counterexample. But finding that counterexample can be harder than proving the conjecture itself.
The New Normal in Mathematics
Looking back over the past few months:
Claude's new record on the Riemann Hypothesis—raising the lower bound for the proportion of zeros from 41.6% to 67.2%, whereas humans had only advanced it by 0.8% in 37 years.
OpenAI's concentrated release of 10 advancements in mathematics and theoretical computer science.
Alpöge continuously using Claude to disprove several classic conjectures.
A newly crowned Fields Medalist announcing they are joining OpenAI.
A new trend is emerging.
AI is no longer just a mathematician's calculator. It is becoming a mathematician's 'intuition amplifier'—opening new pathways with its exhaustive search and construction capabilities in areas where human intuition fails.
A century-old conjecture was not shattered by superior human intellect. It was shattered by a brand new paradigm of human-machine collaboration.
This paradigm will continue to shatter more conjectures. The only question is: which one will fall next?
References:
https://x.com/haider1/status/2090034966717677902
https://x.com/alpoge/status/2089971359921156203
https://en.wikipedia.org/wiki/Carath%C3%A9odory_conjecture
This article is from the WeChat public account "New Zhiyuan", author: ASI Revelation
