Levent Alpöge, a mathematician at Anthropic and a former Harvard Society of Fellows junior fellow, has done it again. A paper he co-authored with John-Paul Smith, built with Claude, contains an explicit counterexample that refutes the Carathéodory conjecture — a problem in differential geometry that has stood for more than a hundred years — and, in the same stroke, takes down the parallel Loewner index conjecture. One construction, two conjectures.
The PDF had been circulating quietly in math circles for a week before it was made public. Neither conjecture was obscure. The Carathéodory conjecture is one of the classic open problems in the geometry of surfaces; generations of mathematicians tried and failed to resolve it. A single explicit example did what a century of arguments could not.
This is also the second time Alpöge has used Claude to kill a famous conjecture. In July, during the World Cup final, he posted a three-line counterexample to the 87-year-old Jacobian conjecture in algebraic geometry. Mathematicians verified it within hours. Now he is back with a construction that removes two conjectures at once.
One counterexample, two century-old conjectures
The result is worth spelling out because the two conjectures it kills have braced each other for a century. An umbilic point on a surface is a point where the surface curves the same way in every direction — locally spherical. At such a point both principal curvatures are equal, and every tangent direction is a principal direction.
A sphere is all umbilic: no matter which way you run your hand over it, the curvature is the same. An ellipsoid has exactly two. The Carathéodory conjecture, proposed in 1924, asserted that any closed, convex surface in 3D that is smooth enough must have at least two umbilic points. You can deform, pinch, and smooth a surface however you like, the conjecture said — you can never get the count below two.
It felt obviously true. The sphere has infinitely many, the ellipsoid has two, and for a hundred years nobody could construct a smooth convex surface with just one. Alongside it sat the Loewner conjecture: the index of an isolated umbilic point cannot exceed 1. The two conjectures braced each other — a pair of locks that had kept an entire area of differential geometry sealed.
The umbilic point: why the conjecture felt obvious
Alpöge and Smith break the lock with an extremely specific construction. They define a family of functions g_k, taking k=2. The function g₂ serves as the support function of a sphere-like convex body, and the corresponding body produces an umbilic point of index 2 at the origin.
The index of 2 already violates Loewner's upper bound of 1. More subtly, the exponential-decay term in g₂ guarantees the function is C∞ — infinitely differentiable. That means the convex body satisfies every smoothness condition the Carathéodory conjecture demanded. And it has only one umbilic point.
Carathéodory requires at least two umbilic points; the example has one, so the conjecture falls. Loewner requires isolated umbilic indices no greater than 1; the example delivers an index of 2, so it falls too. Two conjectures, one explicit, verifiable counterexample.
In mathematics, a single counterexample is enough to kill a conjecture — but finding that example is usually harder than proving the conjecture itself. That is exactly what makes this result representative rather than freakish.
The construction that broke the lock
Look at the last few months and the pattern is unmistakable:
- Claude set a new Riemann zeta record, pushing the lower bound on the proportion of zeros from 41.6% to 67.2% — a span humans had advanced by just 0.8% in 37 years.
- OpenAI published a batch of ten advances in mathematics and theoretical computer science.
- Alpöge has now used Claude to refute two classic conjectures in a single summer.
- A newly minted Fields medalist has joined OpenAI.
The trend is no longer that AI is a faster calculator. It is becoming an intuition supplement for mathematicians — the thing that keeps searching and constructing where human intuition has been stuck for decades. A century-old conjecture was not shattered by a smarter human. It was shattered by a different division of labor: the human picks the target and interprets, the model exhausts the search space and produces constructions that would take a human lifetime to stumble onto.
This is the same conclusion the mathematics community is slowly absorbing: the value is shifting from generating arguments to digesting them. When AI can produce proofs and counterexamples at scale, the scarce skill becomes understanding, verifying, and organizing what the machine returns. For a closer look at why AI math breakthroughs keep arriving as counterexamples, see Gowers' analysis of why AI math breakthroughs keep arriving as counterexamples.
A paradigm shift, not a fluke
For research mathematics, the practical change is immediate. Conjectures can now be stress-tested against a search tool that will happily hunt for a counterexample before a mathematician commits years to a proof. A disproof is cheap and decisive; you would rather find it in a week with a model than spend a career on a false statement.
The harder change is cultural. Terence Tao has spent days digesting an AI-assisted proof end to end, as he did with the Sendov conjecture, and both he and Fields medalist Wang Hong now argue that the community must learn to digest AI output rather than ignore it. A correct proof is only the first gate; a result is not fully usable until professional mathematicians understand, reproduce, and reorganize it. That human layer becomes more valuable, not less, as generation gets cheaper.
There is also a warning buried in the result. If AI can construct a C∞ surface that violates a century of intuition, then intuition — even the shared intuition of the best mathematicians alive — is not a reliable guide to what is true. The frontier has moved from can we prove it to did we guess correctly in the first place.
What changes for mathematics
If you are a researcher: put a good model in the loop before you start proving. State the conjecture, ask the model to hunt for a counterexample, and treat a negative search as a confidence check. You are not looking for the model to replace your thinking; you are looking for it to find the thing your intuition filters out.
If you are a mathematician: invest in the skills of verification and digestion. Knowing how to check a machine-produced construction, formalize it, and reorganize it into something the community can use is becoming a core competency rather than a curiosity.
If you are just watching: stop counting AI solved a hard problem as an event. It is now the normal operating mode. The interesting question is no longer whether a model can break a conjecture — it is which intuition, which field, is next.
What you can do about it
Turn the argument into three concrete moves. First, put a model in the counterexample-hunting loop: before you commit years to a conjecture, spend a week letting a model stress-test it — if a counterexample appears, the bet collapses from years to a week. Second, make digestion a formal skill: join or build a group that checks, formalizes, and reorganizes machine-produced constructions, because the cheaper generation gets, the more this human layer is worth. Third, reset the default on intuition: treat what once felt obviously true as provisional until it has been stress-tested. Carathéodory is not the end of the story, just the first domino; the next intuition to fall is already on its way.
