Maxwell Conjecture Falls: GPT-5.6 Sol Beginner’s Guide

The #1 mistake people are making with the Maxwell conjecture news? Reading the headline, opening ChatGPT, typing “solve an open problem in math,” and expecting GPT-5.6 Sol to pull a rabbit out of a hat. That’s not what happened in the paper everyone is sharing. Sol didn’t sit alone and think for eight hours – three human mathematicians did the framing, the verification, and the writing. Sol suggested one construction.

If you reverse the workflow – you frame, Sol proposes, you verify – the result blowing up on X this week (arXiv:2607.27197) becomes a template you can copy for your own hard problems. This tutorial walks through what actually happened, then shows you how to use the same pattern.

What the paper actually claims

“The Maxwell Conjecture is False” was posted to arXiv on July 29, 2026 by Philip Arathoon, Gavin Ball, and Matthew D. Kvalheim. Short note – four pages – but it settles a question open in one form or another for over a century. The claim: a configuration of five point charges in Euclidean space whose electrostatic potential admits at least 24 critical points, all non-degenerate. Maxwell’s conjecture that the field of n point charges has at most (n-1)² critical points is therefore false.

For five charges, (n-1)² = 16. The counterexample hits 24. Bound broken.

The historical context matters for one reason – it tells you why nobody found this earlier. In J. C. Maxwell’s 1873 treatise on electricity and magnetism (§113), he discusses the number of equilibria of the electric field generated by n point charges. Apparently unaware of this, M. Morse and S. S. Cairns in 1969 posed the problem of finding an upper bound for the number of equilibria (per arXiv:2607.27197). Smart people had been staring at this for over 50 years in its modern form.

The construction Sol proposed (and why it’s a repeatable pattern)

Here’s the setup, straight from the paper: the counterexample starts with three unit charges placed at the vertices of an equilateral triangle. That configuration has four equilibria – one at the centre, three displaced inwards around the triangle’s edges. Then two small charges get added to the centre and moved slightly apart along the orthogonal axis, forming a shallow triangular bipyramid.

The key move: the three edge equilibria survive the addition of the small axial charges, but the central equilibrium bifurcates into a family of 21 equilibria. Three plus 21 equals 24. Done.

Notice the shape of the idea: start with a symmetric configuration whose equilibria you understand, then perturb one point slightly so a degenerate equilibrium unfolds into many. That’s not a Maxwell-specific trick. That’s a general prompting pattern any beginner can steal.

How to use GPT-5.6 Sol for your own hard problems

You don’t need a math PhD to use the same workflow. Here’s the beginner version.

1. Open ChatGPT on a paid plan. Logged-out users and free-tier accounts cannot access GPT-5.6 Sol (OpenAI Help Center).
2. Pick the right reasoning level. Medium = standard reasoning with GPT-5.6 Sol. High = extended reasoning. Extra High = the highest reasoning effort available. Start on Medium. Move up only if Medium’s answer is clearly wrong.
3. Frame the problem for it. Give it the known bounds, the symmetric “easy case,” and ask it to propose perturbations that might break the bound. Don’t ask “solve X.” Ask “suggest a construction that could violate X – I’ll verify.”
4. Verify by hand or by code. The humans in the Maxwell paper did the linear algebra themselves. Sol’s job is proposal; yours is falsification.
5. Iterate small. If the first construction fails, ask Sol what parameter to vary. This is the loop the paper implicitly ran.

Pro tip: When you ask Sol to “propose a construction,” force it to include a symmetry-breaking parameter (angle, distance, weight) as an explicit variable. Vague proposals waste output tokens; parameterised ones let you sweep them yourself in a notebook.

Pitfalls nobody is mentioning

Three specific things burn beginners the first week they use Sol.

The “frozen screen” trap. As of July 2026, Artificial Analysis benchmarks put Sol (max)’s time to first token at 133.04 seconds – compared to a median of 2.80s for other reasoning models in a similar price tier. That’s over two minutes before you see any output. New users assume the request died and kill it. Don’t. This is normal for the max reasoning setting – the model is thinking. Set a timer and walk away.

Silent downgrades in ChatGPT. Per the OpenAI Help Center: “If you reach a GPT-5.6 reasoning limit, ChatGPT may continue with GPT-5.4 Thinking mini.” Your conversation keeps going, but you’re no longer talking to Sol. The next “answer” that looks weirdly shallow? Check the model label.

The 272K token cliff. Standard API pricing (as of July 2026) is $5.00 per 1M input tokens and $30.00 per 1M output tokens. But requests above 272,000 input tokens trigger higher long-context pricing – a threshold not prominently displayed on the main pricing page, per community documentation. Dumping an entire codebase in one shot can quietly double your bill.

Sol vs Sol (non-reasoning) vs Terra: what to actually pick

This is where most beginners overpay. The two “Sol” variants on the API cost the same per token, but their intelligence isn’t the same.

Same price, 18 points of intelligence difference. If you’re paying Sol prices, use the reasoning variant – otherwise you’re buying a Ferrari and driving it in first gear.

For everything that isn’t a genuinely hard problem, drop down to Terra or Luna. The Maxwell-style workflow – one clever construction, verified by you – only needs Sol at the proposal step. Verification prompts (“does this configuration satisfy X?”) can run on cheaper tiers.

Why humans missed this for 50 years

Worth a moment to sit with. The construction isn’t exotic. Five charges on a triangular bipyramid is the kind of thing a physics undergraduate could draw. The reason nobody wrote it down, I think, is that experienced mathematicians have strong priors about which perturbations “look promising,” and this one probably didn’t – small axial charges near a symmetric equilibrium feel too tame to unfold into 21 new critical points.

Sol doesn’t have that prior. It tries things that look boring. Sometimes boring things are exactly what was missing.

FAQ

Did GPT-5.6 Sol actually prove the Maxwell conjecture false on its own?

No. Sol suggested the construction; the three human authors verified it and wrote the paper. From the acknowledgments: “The idea behind this construction was suggested by an LLM (OpenAI’s GPT-5.6 Sol).” That’s the full extent of the credit.

Is this the same thing as the Cycle Double Cover result from earlier this month?

No, different result, different setup. The Cycle Double Cover work was OpenAI’s own demo where GPT-5.6 Sol Ultra reportedly produced a proof in under an hour, using 64 subagents working in parallel on a conjecture that had been unsolved for 50 years (reported by The Decoder and Crypto Briefing – not yet peer-reviewed as of this writing). That’s very different from the Maxwell paper, where three named mathematicians did the human work and used Sol as a suggestion engine. Treat them as separate data points, not one story.

Can I try this with a cheaper model?

Sometimes. If your problem has a clear symmetric base case and you just want brainstorming, Terra will often propose reasonable perturbations. Sol earns its price on problems where the useful idea is counterintuitive – where you specifically need a model that doesn’t get discouraged by the obvious approach failing.

Do this next: pick one problem in your own field that has a known symmetric “easy case” and an unproven bound. Open ChatGPT on a paid plan, switch to Medium reasoning, and ask Sol for three parameterised perturbations of the easy case. Verify one by hand. That’s the whole Maxwell workflow, scaled down.