Here's a headline I did not expect to write on a Sunday morning: an AI model has just been credited with the central idea behind solving a mathematics problem that has been open since 2017. Epoch AI, the benchmark shop that tracks frontier AI capabilities, has marked "The Core in Approval-Based Committee Elections" as solved on its FrontierMath: Open Problems board. The model behind the breakthrough? OpenAI's GPT-6 Astra. If you're still catching up on what this model even is, we covered GPT-6 Astra in detail when it launched — and honestly, it keeps outdoing itself.
01.What the problem actually asked
The question sounds innocent. Imagine voters approving candidates, and a committee of k winners being chosen. Political scientists have long searched for a notion of a "fair" committee — one where no large enough group of voters could all prefer some other committee. Game theorists call this the core. In 2017, a group of researchers including Haris Aziz and Toby Walsh asked: could there be elections where the core is empty — where no perfectly fair committee exists at all?
Nine years, no answer. The problem was considered hard enough that Epoch put it in the second-highest difficulty tier of its Open Problems board — "Major Advance" — alongside just five others. The way these problems work, the original prompt asked solvers to construct a counterexample: an election whose core is empty, submitted as a JSON file.
The answer that came back? There is no counterexample. The core is never empty. A perfectly fair committee always exists, and the proof — which runs to about 20 pages — came with something better: a new set of voting rules and an algorithm that finds such a committee in polynomial time. You know, as a bonus.
02.Humans and Astra, working as a team
The proof came from three researchers: Patrick Becker (Technical University of Munich), Matthias Greger (University of Oxford), and Dominik Peters (Université Paris Dauphine-PSL / CNRS). Crucially, they did not just ask Astra the question and copy the answer. It was a days-long, intense collaboration — the humans kept pushing: improve this bound, find an alternative potential function, try the KKT conditions. Somewhere in that back-and-forth, Astra proposed a framework built around what the researchers call "harmonic entropy" — a way of measuring how evenly voter satisfaction is spread — and showed that any local optimum under it lands inside the core.
Epoch's response to all this is what I find most interesting. They created a brand-new status label — "Human + AI" — because the old categories stopped fitting. Their note says humans played an active role in prompting, but the core ideas came from the model, and for binary comparisons you should count it as an AI solution. Peters himself put it plainly: without the model, they probably wouldn't have found the proof — but hand Astra a simple prompt, and it definitely won't solve this problem. That distinction matters enormously for anyone wondering whether AI is about to replace mathematicians. It's not. It's changing what a mathematician with a good prompt can do.
03.98% on Tier 4 — but the Erdős wall stands
Zoom out and the trajectory gets wilder. When Epoch launched Tier 4 of FrontierMath in July 2025, the best models on the planet managed roughly 5%. Astra is now reported at between 97.6% and 98% across the 43 problems in that tier. Fourteen months, from 5% to 98% — that's the kind of curve that makes benchmark designers nervous, and Epoch is already building a harder successor called the FrontierMath Erdős set, named after the famously prolific mathematician Paul Erdős.
And here's the reality check: on that Erdős set of 68 historically unsolved problems, Astra solved just 2 during the official benchmark run (about 3%) — and 5 across all its attempts, which together burned more than $220,000 of compute. Every other model tested, including GPT-5.6 Sol and Claude Fable 5.1, solved none. We've written before about how Astra tops other benchmarks while absolute scores stay humbling — same story here, at a much higher altitude.
04.What this means for AI reasoning
Here's the thing nobody's talking about: the result says as much about how the problem was solved as about AI capability. A bare-prompt, autonomous solve? Didn't happen. What happened is that a frontier model, guided by experts who knew exactly which levers to pull, produced genuinely novel mathematics — the "harmonic entropy" idea wasn't in any textbook. That's a new kind of tool: not a calculator, not a search engine, but a research partner that can originate ideas under direction.
For students and researchers in India, this is a quiet signal that the skill of the future is knowing how to direct these models — the prompting craft, the iteration, the judgement of when the model is going in circles. Peters' comment that a simple prompt would never crack this problem is the single best argument I've seen for treating prompting as a serious skill. It's exactly why we put prompting first in the GPT-6 Astra — Zero to Hero Guide, before anything else — the difference between a bare prompt and a well-directed session is now literally the difference between a stuck problem and a published proof.
Key Takeaways
- Epoch AI marked "The Core in Approval-Based Committee Elections" as solved — the first Major Advance problem ever cracked on FrontierMath
- The 2017 open question is resolved: a perfectly fair committee always exists, with a polynomial-time algorithm to find it
- Epoch introduced a new "Human + AI" label — core ideas from GPT-6 Astra, sustained direction from Becker, Greger and Peters
- Astra scores ~98% on Tier 4 (up from ~5% for all models in July 2025) but only 2 of 68 on the Erdős set
- Autonomous math genius? No. A research partner that originates ideas under expert direction? Clearly yes
Frequently Asked Questions
What math problem did GPT-6 Astra help solve?
Epoch AI marked “The Core in Approval-Based Committee Elections” as solved on its FrontierMath Open Problems board — the first-ever “Major Advance” problem to be cracked. The voting-theory question, open since 2017, asked whether an election can exist where no perfectly fair committee exists at all. The answer: no — a fair committee always exists, and the roughly 20-page proof includes a polynomial-time algorithm for finding one.
Did GPT-6 Astra solve the FrontierMath problem on its own?
No. Three mathematicians — Patrick Becker, Matthias Greger and Dominik Peters — worked with Astra over days of iterative collaboration. Epoch AI created a new “Human + AI” label for the result: humans actively prompted and steered, but the core ideas, including the “harmonic entropy” framework, came from the model. Researcher Dominik Peters noted a simple one-shot prompt would never crack this problem.
How good is GPT-6 Astra on FrontierMath benchmarks overall?
Astra scores between 97.6% and 98% on FrontierMath Tier 4 — up from roughly 5% for the best models when that tier launched in July 2025. On the harder Erdős set of 68 historically unsolved problems, Astra solved only 2 during the official benchmark run (about 3%), with attempts burning over $220,000 of compute; every other model tested solved none.
What This Means for You
If you're a student, teacher or researcher in India, this is the moment to stop treating AI as a homework shortcut and start treating it as a collaborator you can push. The mathematicians who got this result spent their time challenging the model, demanding better bounds, and rejecting weak arguments — the model did the heavy lifting, but humans steered. That division of labour is going to define knowledge work for the next decade, from research labs to Indian IT companies. If you want to build that skill properly instead of fumbling with random prompts, the GPT-6 Astra — Zero to Hero Guide walks through prompting, reasoning control and cost management step by step — the same craft the researchers used, minus the PhD.