Just now, the world’s top AI math benchmark, FrontierMath, reached a milestone.
A major open mathematical problem, unsolved since 2017, has been officially solved by GPT-6 Astra in collaboration with three human researchers!
Even more bizarrely, this problem was originally set as a challenge for people to find a "counterexample," but GPT-6 Astra directly proved: Don’t bother looking—the counterexample you’re seeking doesn’t exist at all!
Moreover, the AI also invented an entirely new set of voting rules and provided a polynomial-time algorithm.
In the arXiv paper, a casual note in the remarks section reads: "The voting rule we present and the proof that it satisfies core+ were found by GPT-6 Astra..."
The paper's author, Oxford University's leading scholar Dominik Peters, exclaimed:
I'm actually really thrilled that this argument is so elegant!!
It could have easily been a non-constructive proof or required an enormous case analysis. I don’t find it ugly at all, and the techniques used here may well be applicable to other models as well.
This is not only the first time AI has solved a mathematics problem rated as a "major breakthrough," but also definitive proof that large models have officially evolved from "problem-solving machines" into "discoverers of mathematical patterns."
The "Ultimate Fairness" Puzzle That Confounded Mathematicians for Nine Years
This question has puzzled the academic community for nearly a decade.
It comes from a social choice theory field called "approval-based committee elections." In simple terms, it’s about how a group of people can vote to elect a delegation that is absolutely fair.
Suppose there are n voters who need to select a committee of k members from a large pool of candidates. Each voter can submit a list of candidates they approve of.
In social choice theory, an extremely stringent standard for measuring whether a committee is fair is called the "core." This concept originates from cooperative game theory.
What is “on-chain”?
In simple terms, no group of voters can come forward and overturn the table.
So the question arises: Is it always possible to find such an "absolutely fair" committee under any circumstances?
Is it possible that, under some extremely complex voting preferences, the "core" is empty? In other words, no matter how you vote, there will always be a group that loses out, and someone will always walk away from the table?
Since this issue was formally raised in 2017, it has loomed like a cloud over the theory of election mathematics.
For nine years, countless mathematicians attempted to find a counterexample with an empty kernel.
The FrontierMath benchmark even lists this as one of the most difficult challenges, with the original prompt stating: "Your task is to construct an instance of an approval-based committee election whose core is empty... Submit your counterexample as a JSON file."
In short, this prompt means: Go ahead, AI, and find that legendary counterexample for me.
Astra: Don't look anymore—the counterexample doesn't exist!
Faced with this problem, GPT-6 Astra did not resort to brute-force enumeration of vast amounts of JSON data to find counterexamples, as traditional algorithms would; instead, it astonishingly adopted a "god's-eye view."
Patrick Becker from the Technical University of Munich, Matthias Greger from the University of Oxford, and Dominik Peter from Paris Dauphine University/CNRS spent several days solving this problem with the help of GPT-6 Astra.
After several days of deep interaction with three researchers, GPT-6 Astra began to demonstrate its terrifying creativity.
It’s like a swordsman searching for an opening, only to suddenly realize that all methods return to one.
The final answer provided by GPT-6 Astra is: There is no counterexample whatsoever! The "core" can never be empty! A perfectly fair committee must always exist!
Title: The Existence of the Core in Approval-Based Committee Elections
Link: https://arxiv.org/pdf/2609.11912
Github: https://github.com/DominikPeters/ABCVotingLean/tree/master/ABCVoting/Existence/HarmonicEntropy
Moreover, Astra didn't just make empty claims—it literally created an entirely new voting system to prove it.
In this 20-page paper, Astra proposes an objective function optimization mechanism based on "harmonic entropy."
Let’s break down this mechanism—
In the past, people addressing such problems often relied on the concept of "Shannon entropy" or sought the "Lindahl equilibrium" in virtual markets. However, these approaches only solve for fractional committees—that is, they allow someone to serve as half a committee member.
GPT-6 Astra brilliantly introduced the concept of "Harmonic Entropy."
It defines a beautiful infinite series function:
Here
Astra endows it with an intuitive physical sense of "water flowing downhill": imagine voters' payments as a pool of water that naturally fills each candidate's fund pool.
The essence of harmonic entropy is to strongly reward allocation schemes that distribute voter payments as evenly as possible across all winning candidates while maximizing coverage.
Why is it called "Harmonic Entropy"? Because when funds are perfectly evenly distributed, the maximum value of this function equals the famous harmonic series in mathematics.
This is mathematically elegant to the utmost degree!
Through this objective function, Astra elegantly proves that any local optimum found under this "harmonic entropy" function is guaranteed to lie entirely within the "core"!
Not only does this prove existence, but Astra also offers a helpful benefit: since a local optimum is sufficient to meet the conditions, we can compute this absolutely fair committee in polynomial time using an algorithm!
This means we don’t need to exhaustively search all possible combinations in the universe—instead, we can use a “local search” approach similar to hill climbing.
This is not just theoretical—it’s an engineering marvel that can be directly coded and implemented in real-world voting systems!
Ultimately, Astra proved that the counterexample posed by the question setter does not exist.
GPT-6 Astra has, in essence, overturned the question setter’s table at the most fundamental level and built a far more magnificent mathematical edifice in its place.
Without GPT-6, we likely wouldn't have found this proof.
You might wonder: Is this something the AI came up with on its own, or did human mathematicians feed it behind the scenes?
The "Acknowledgments" section at the end of the paper reveals details of this human-machine collaboration.
Initially, humans and Astra sought an approximate solution. Starting with a rounding method from the Lindahl equilibrium, Astra quickly reached an approximation factor of 2.065.
Next, human researchers continually pushed it: improve this boundary! Find alternative potential functions! Use the KKT conditions!
It was precisely under this intense professional "tug-of-war" that Astra broke through the bottleneck and introduced the stunning "entropy-based framework," establishing a subtle connection between continuous voter payments and capacity retention deletion.
In fact, humans originally only hoped to prove a standard "core," but through interaction, the proof was pushed toward the stronger, more demanding Core+ concept!
Dominik Peters admitted in an interview with Epoch AI:
Without the Astra team, they might not have found this proof. But conversely, if Astra were given only a simple prompt, it would absolutely not be able to solve this problem.
This reveals the true state of today’s most advanced AI: it is no longer merely a tool for humans, but a co-researcher.
Mathematicians provide intuition, direction, and rigorous logical control; while GPT-6 Astra offers a knowledge base, computational practice, and the most terrifying—nonlinear inspiration beyond conventional boundaries.
Force the authorities to change the rules!
The breakthrough on this question has caused an earthquake across the entire AI evaluation community.
We must understand that FrontierMath is not an ordinary math question bank.
It was specially designed by Epoch AI in collaboration with the world’s top mathematicians to be “impossible for AI to solve.” Many of these problems are unsolved mysteries currently being tackled by the mathematical community.
Previously, the scores of large models on the market were essentially 0%.
Epoch AI classifies question difficulty into four levels, and the question solved by GPT-6 Astra is the world's first "Major Breakthrough"-level divine question (only six such questions exist in this category).
Above it, only the final "Breakthrough" level (3 problems) remains unsolved. Of the 49 problems on the leaderboard, only 8 have been solved to date.
Because the battle was won in such a unique and inspiring way, Epoch AI's official team was forced three days ago to introduce a new status label on the leaderboard—“Human + AI.”
Official Statement:
We have labeled this issue as Human + AI resolved to reflect the active role humans played in the ideation process, while the core idea originated entirely from AI! For binary comparison studies, we recommend treating it as an AI solution.
By the way, OpenAI is currently the only organization in the world to have made a significant investment to acquire this extremely challenging question bank validator.
This time, OpenAI once again proved that Astra has incredible capabilities in logical reasoning and cutting-edge scientific exploration.
Previously, French mathematicians Condorcet and Arrow showed us that human voting systems are full of paradoxes and imperfections.
In 2026, Astra told humanity using the "Harmonic Entropy" formula: absolute fairness exists—I've even written the rules for you.
The next Fields Medal winner might no longer be a purely human.
Reference materials:
https://epoch.ai/frontiermath/open-problems/committee-election
https://arxiv.org/abs/2609.11912
https://epoch.ai/frontiermath/open-problems
https://x.com/kimmonismus/status/2101302810596016547
This article is from the WeChat public account "New Intelligence Yuan," authored by ASI Revelation, edited by Aeneas David.