A preprint posted to arXiv on September 10, 2026, settles a nine-year open question in computational social choice: every approval-based committee election has a non-empty core. Researchers Patrick Becker, Matthias Greger, and Dominik Peters prove that a fair committee can always be found, and introduce a polynomial-time algorithm based on "harmonic entropy" to compute one.

The problem was listed as a "major breakthrough" level challenge on Epoch AI's FrontierMath benchmark. Rather than construct a counterexample as the benchmark asked, GPT-6 Astra proved the opposite—that the core is never empty. The collaboration led Epoch AI to introduce a new "Human + AI" classification, recognizing that the model contributed the core ideas during an extended interactive session with the three mathematicians.

Confirmed

  • The paper "Existence of the Core in Approval-Based Committee Elections" (arXiv:2609.11912v1) proves that for any approval-based multi-winner election with committee size k, a committee in the core always exists under both the Hare quota (n/k) and the Droop quota (n/(k+1)).
  • The proof constructs a voting rule that maximizes harmonic entropy — a new entropy-like objective over committees and payment systems — and shows all local optima satisfy core+, a strengthening of the core that admits a payment-based certificate.
  • A core committee can be found in polynomial time via local search on the harmonic entropy objective, combined with a linear-program simplification.
  • The result has been formally verified in Lean as part of the ABCVotingLean project.
  • The problem is marked "Solved (human + AI)" on Epoch AI's FrontierMath page, with GPT-6 Astra credited as the source of the primary idea and proof structure.

Unknown

  • Whether the harmonic entropy voting rule satisfies other proportionality axioms beyond core+ (e.g., FJR+, priceability) in all cases.
  • Practical performance of the local-search algorithm on large-scale real-world election instances; the paper establishes polynomial-time computability but does not report empirical runtime or implementation benchmarks.
  • Whether the human-AI interaction pattern observed here generalizes to other open problems in social choice or mathematics more broadly.

Our take

The breakthrough is significant both for closing a decade-old gap and for how it happened. During multi-day collaboration, the AI model proposed the central mathematical object—harmonic entropy—and sketched the proof structure, while human researchers supplied direction, logical checks, and refinement. This marks a shift in AI-assisted mathematics from verification toward concept invention. The polynomial-time algorithm also converts the core from existential guarantee to a computable rule, potentially influencing real-world committee selection if performance scales.

Sources