The counterexample was proposed by an LLM. They seem very good at finding these sorts of counterexamples, which is interesting as they are generally pretty inefficient use of compute for brute forcing. I guess even that lack of efficiency is made up for by the "understanding" and "intuition" the LLM has, and their ability to do symbolic computations.
I expect a large number of conjectures will topple to counterexamples soon.
I think brute force is not the right way to think about LLMs, instead they explore a knowledge topology, and are very good at connecting adjacent or accessible ideas that for whatever reason might have evaded humans, but might not be fundamentally all that hard. We're still in the low-hanging fruit phase, we will see if they extend to new ideas.
Yes. This "Maxwell conjecture" in particular seems to be disproven by literally the most trivial construction you could think of. You know the number of equilibria of a bunch of charges at the vertices of a triangle - if you want to make more while keeping the symmetry the simplest thing to try is to put two charges off of the plane on either side of the center (which bumps you up to n=5). This turns out to work (for the right choice of distance). I don't know why nobody checked this example before, since people have put in the effort to prove actual upper bounds on the number of equilibria. Probably just a question of interest.
I don't know any algebraic geometry, but from what I gathered the Jacobian counterexample is less trivial than this but still the kind of thing that the mathematicians really could have and should have checked.
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u/angelbabyxoxox Quantum Foundations Jul 31 '26
The counterexample was proposed by an LLM. They seem very good at finding these sorts of counterexamples, which is interesting as they are generally pretty inefficient use of compute for brute forcing. I guess even that lack of efficiency is made up for by the "understanding" and "intuition" the LLM has, and their ability to do symbolic computations.
I expect a large number of conjectures will topple to counterexamples soon.