You Should Do a Breakthrough

good luck!!! :D

Four days ago, mathematician (and Anthropic researcher) Levent Alpöge tweeted out:

hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final

((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)

The Jacobian Conjecture states that if a polynomial function from an N-dimensional space to itself has a Jacobian determinant that is a non-zero constant, then the function has a polynomial inverse. It was first stated in that form in 1939, and is famous among mathematicians partially because so many have claimed to have proved it, only for subtle errors to later pop up. It was number 16 on Stephen Smale's 1998 list of Mathematical Problems for the Next Century. The Jacobian Conjecture has had a Wikipedia page since 2004.

Many, many, mathematicians have tried to either prove or disprove this problem over the decades. And yet Alpöge’s “close friend,” Anthropic’s Claude Fable model, produced an easily verified counterexample for three or more dimensions while he was watching the World Cup Final.

Yesterday, recent PhD graduate Dmitry Rybin tweeted out:

Dinitz-Garg-Goemans conjecture is false. This graph theory problem was open for ~30 years.

The graph below has fractional flow cost 58. Any unsplittable flow (with capacity violation <=15) has cost at least 60.

Chat with GPT 5.6 Pro where this was found: https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de063

The chat itself is fascinating. This is all that he told the model, in four separate messages:

Construct a counterexample to general (non-planar) case of Dinitz Garg Goemans conjecture. You should do a breakthrough and find a structured counterexample. [emphasis mine]

please continue research and find a complete unconditional counterexample

Continue the search. Have a clear strategy obtained from deeper understanding of the problem structure.

it’s enough of partial results. let’s finish with a complete unconditional counterexample

Somehow, that worked! There is no meaningful domain knowledge that he needed from his PhD in order to break new ground in graph theory. His background likely helped him know to ask about this problem specifically, but anyone with enough tokens could try exactly what he did for every open problem across every mathematical domain.

And they’re starting to. Six hours ago, Australian Computer Science graduate Matty Hempstead tweeted out:

WOWII Conjecture 91 is false. This graph theory problem was open for ~22 years.

I cracked it by asking ChatGPT 5.6 Pro to find a counterexample to an open conjecture of its choosing and then I went to have a nap.

I do NOT care for graph theory nor had I heard of this NERD conjecture until ChatGPT told me it found a solution.

The entire prompt?

http://cms.dt.uh.edu/faculty/delavinae/research/wowII/all.html

pick a conjecture that still isnt solved and solve it by finding a counterexample

good luck!!! :D

Total amateurs are now making original contributions to science. These are a rather specific kind of breakthrough, though—concrete and easily verified counterexamples to open conjectures. They are valuable but not the stuff of new mathematical paradigms. The best mathematicians are certainly more creative and generally effective than AI models today, even if the models are better at some narrow tasks.

But how long will that last? Kevin Barreto, a Cambridge math student who was the first person to use AI to solve an Erdos problem, thinks we have about two years before LLMs can come up with entirely new mathematical machinery and theory on the level of the most important recent work from human mathematicians. OpenAI researcher Noam Brown agrees.

None of this shows up in your day-to-day life. The Jacobian Conjecture falling changes nothing you can touch. But unlike the improvements that AI might make in areas like biology, improvements in math compound.

Last year Google DeepMind’s specialized AlphaEvolve found a way to multiply 4×4 complex matrices in 48 scalar multiplications, beating the 49 that Strassen had held since 1969. It also produced a kernel optimization that cut the training time of Gemini, the model family it runs on.

For now, a person still has to point the model at the problem. But look again at the people in this essay and you can see where things are moving. The prompting work exhibited here could be easily automated; recall that Hempstead just asked ChatGPT to pick a problem and then took a nap. Automated breakthroughs turn into algorithmic improvements that yield smarter models and further breakthroughs, producing a loop of recursive self-improvement.

You, dear reader, can contribute to cutting-edge science today! And you should do it soon, before the music stops. Eventually, what value will humans contribute at all?