# GPT 5.6 Sol Helps Mathematicians Prove That 150-Year-Old Maxwell Conjecture Is False > Source: > Published: 2026-07-31 10:31:16+00:00 AI is helping mathematicians come up with more and more results. The latest one comes from a trio of researchers at Babson College, the University of Missouri, and the University of Maryland, Baltimore County, who have used an idea suggested by OpenAI’s GPT 5.6 Sol to disprove a 150-year-old conjecture in electrostatics that had stumped mathematicians since James Clerk Maxwell first wrote about it in 1873. The [paper](https://arxiv.org/pdf/2607.27197), titled “The Maxwell Conjecture is False,” was posted to arXiv on July 29 by Philip Arathoon, Gavin Ball, and Matthew D. Kvalheim. It’s a short, four-page note, but it settles a question that had been open in one form or another for over a century, and more recently had been formalized into a specific conjecture that mathematicians had been trying to crack since 2007. ## What The Maxwell Conjecture Actually Says To understand the result, it helps to picture a handful of electric charges scattered in space. Each charge pushes and pulls on the space around it, creating a field. At most points, that field points somewhere, pulling a test particle in some direction. But at a few special points, the pushes and pulls from all the charges cancel out perfectly, and the field is zero. Physicists call these points equilibria, or critical points. Maxwell noticed that the number of these balance points seems to be capped by how many charges you place. In his 1873 treatise on electricity and magnetism, he made an offhand remark about this. Nearly a century later, mathematicians Marston Morse and Stewart Cairns picked up the thread and asked, more formally, what the actual limit was. It took until 2007 for Alexandre Gabrielov, Dmitry Novikov, and Boris Shapiro to read Maxwell’s original passage and turn it into a precise mathematical statement, now known as the Maxwell conjecture: for n charges, if all the equilibrium points are “non-degenerate” (a technical way of saying they’re clean, isolated balance points rather than points where the math gets ambiguous), then there can be at most (n-1)² of them. For two charges, that bound is trivially satisfied. For three charges, mathematicians still don’t know for certain whether four is really the maximum, except in special cases where all three charges are equal in strength. The bound had been chipped away at over the years, most recently by researchers who tightened the upper estimates in 2023 and again earlier this year. But nobody had actually broken it. Until now. ## Five Charges, More Balance Points Than Allowed The authors built a configuration of five point charges that produces at least 24 non-degenerate critical points. Under Maxwell’s conjecture, five charges should top out at (5-1)² = 16. Twenty-four blows past that. Here’s the construction, in plain terms. Start with three equal charges sitting at the corners of an equilateral triangle. This setup has exactly four equilibrium points: one sitting at the centre of the triangle, and three more pushed slightly inward along each edge. That’s a clean, well-behaved system that already respects Maxwell’s bound. Then the trick: place two much smaller charges just above and below the centre of the triangle, along the axis running perpendicular to it, forming a shallow triangular bipyramid. The three edge equilibria stick around largely undisturbed. But the single equilibrium point that used to sit at the centre doesn’t just shift, it splinters. Under the right choice of how small and how far apart those two extra charges are, that one point breaks apart into 21 separate equilibria. Add those 21 to the three surviving edge points, and you get 24. The mathematical heavy lifting involves zooming into the centre of the configuration at just the right rate as the added charges shrink toward zero, using what the authors call a deformed potential. Rescale space and the charge strength together in a particular way, and the messy, shifting central region resolves into a fixed polynomial function whose critical points can be counted exactly. That fixed function turns out to have 21 non-degenerate critical points, all of which can be traced back, via the implicit function theorem, to genuine equilibria of the real five-charge system for small enough values of the shrinking parameter. The result doesn’t stop at disproving the conjecture once. The authors show the same trick can be repeated: keep adding pairs of small charges along carefully chosen axes, and each round adds another 20 non-degenerate critical points at the cost of two more charges. That gives a family of configurations with 3 + 2m charges producing at least 4 + 20m critical points, which is a far better ratio of equilibria to charges than what had been reported by other researchers earlier this year. ## Where GPT 5.6 Sol Comes In The paper carries a short disclosure section that’s arguably as interesting as the mathematics itself. The authors write plainly that the idea behind the construction was suggested by GPT 5.6 Sol. They’re careful to note that they verified the mathematical details themselves and wrote up the argument in their own words, with Mathematica and Maple used to check computations and generate the figures. That’s a fairly narrow, specific role for the model to have played, suggesting a particular geometric trick, the triangular bipyramid with an axial pair of shrinking charges, that the human authors then had to formalize, prove rigorously, and push through the technicalities involving the implicit function theorem and parametric transversality. It fits a pattern that’s become familiar over the past year, where models are increasingly used to seed ideas or spot constructions that mathematicians then verify and write up. [Terence Tao](https://officechai.com/ai/ai-is-allowing-me-to-experiment-and-try-crazier-things-mathematician-terrance-tao/) has talked about using models this way to try out ideas he might not have otherwise attempted, and there’s been a broader run of AI-assisted results chipping away at long-standing open problems, including several tied to Erdős conjectures. OpenAI has also been on something of a hiring spree in this space, most notably bringing on this year’s [Fields Medal winner](https://officechai.com/ai/fields-medal-winner-jacob-tsimerman-joins-openai-says-ai-will-soon-do-everything-mathematicians-do-better-and-faster/) to work on AI safety. What makes the Maxwell conjecture case notable is how targeted the contribution was. Rather than the model attempting an end-to-end proof, it appears to have supplied the geometric seed, add small charges off-axis to a symmetric base configuration and watch a degenerate point bifurcate, which the authors then had to turn into hard analysis involving Taylor expansions of harmonic polynomials, Hessian classifications, and a transversality argument to rule out any lingering degenerate points elsewhere in the configuration. The three authors also note they haven’t fully closed the book on the problem. Whether three charges can ever exceed four equilibria remains open, and the general question of exactly how many critical points n charges can produce is still unresolved beyond this counterexample. But the specific bound proposed in 2007, the one that had been carrying Maxwell’s name and resisting every attempt to break it, no longer holds.