GPT-6 Astra Solves Decade-Old Voting Theory Problem

Researchers Becker, Greger, and Dominik Peters have utilized GPT-6 Astra to resolve a mathematical problem open since 2017. The problem, 'The Core in Approval-Based Committee Elections,' was originally posed by Aziz, Brill, Conitzer, Elkind, Freeman, and Walsh. By proving that the core of an approval-based multiwinner election is always non-empty, the team showed that a requested counterexample cannot exist. Epoch AI subsequently marked this as the first solved problem in the Major Advance category of its FrontierMath: Open Problems benchmark.

The breakthrough represents a new 'human + AI' classification for Epoch AI, reflecting interactive collaboration rather than autonomous, one-shot success. Peters noted that the proof required sustained human guidance alongside Astra's core conceptual contributions. This milestone brings the FrontierMath board status to 1 of 6 Major Advance, 2 of 18 Solid Result, and 5 of 22 Moderately Interesting problems solved, with 0 of 3 Breakthroughs resolved. The active board now contains 49 problems, down from 50 after Epoch removed a question regarding stretched Littlewood-Richardson coefficients.

Astra has demonstrated strong capabilities across other benchmarks, raising the top score on FrontierMath Tier 4 to 98% from a previous high of 5% over the 14 months following its July 2025 launch. On the more difficult FrontierMath Erdős benchmark, Astra solved 2 of 68 problems during its official run and 5 across all attempts. In contrast, rival models like GPT-5.6 Sol and Claude Fable 5.1 failed to solve any. However, these advanced reasoning attempts carry significant resource requirements. While one Astra counterexample took 15 hours and $218 in compute, generating the five Erdős solutions across all attempts cost more than $220,000.

This interactive proof represents a step beyond earlier AI-credited mathematical achievements, which mostly involved lower-tier construction tasks. Previous successes include an Anthropic team using Claude to find a Hadamard construction of order 668, as well as discoveries of rational-point constructions on genus-2 curves and short superpermutations over 8, 9, and 10 symbols.