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    ChatGPT 5.5 Pro Helps Advance Fair Division Proof

    Analyst reports on AI tool advancing a fair division proof for four people and nine goods.

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    1 Source, 27d ago, first seen 27d ago

    TLDR

    Kim Isenberg posted about a call with Eyad Alkassar. They examined how ChatGPT 5.5 Pro contributed to an open problem in computer science. The example involves four people dividing nine objects that each may value differently. The post included a screenshot of an academic paper abstract. The paper states it proves every fair-division instance with four agents, additive valuations, and at most nine indivisible goods admits a complete allocation that is EFX.

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    1 Source, first seen 27d ago

    Combined views

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    1 Source, first seen 27d ago

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    1 Source

    @kimmonismusI just had a fascinating call with Eyad Alkassar (@AlkassarEyad) about how ChatGPT 5.5 Pro helped move the frontier of an open problem in computer science. The problem sounds abstract, but it is surprisingly intuitive: Imagine four people dividing nine objects. Each person may value them differently. The goal is to find a division so fair that, after any single object is removed from someone else’s pile, nobody would prefer what remains to their own pile. This criterion is called envy-freeness up to any good, or EFX. Whether complete EFX allocations always exist for four or more people remains open. For four people, the previous general guarantee stopped at seven objects. A new preprint by Eyad Alkassar, Mahmoud Fouz and renowned computer scientist Kurt Mehlhorn now presents a proof for up to nine. The story behind it is remarkable. Over dinner, Eyad challenged Mehlhorn to give AI one of the harder open problems in his field. Eyad and Mahmoud are both computer science PhDs who had spent years building startups and were newcomers to fair division. They built a research workflow around several AI systems. At one point, Fable had questions about one of Mehlhorn’s papers and repeatedly urged Eyad to contact him. After Eyad refused twice, the AI pointed out that they were only around 30 kilometers apart and suggested that a short drive would be worth it. According to Eyad, the situation became even stranger once the proof was finished. Fable then advised him against sending the result to Mehlhorn, warning that he might be a competitor and that an incorrect proof could be embarrassing. Eyad contacted him anyway. Mehlhorn reviewed and verified the proof architecture, refined the arguments and joined the paper as a co-author. According to the paper, Claude Fable performed the analysis and lemma proofs, authored the trusted verification layer and audited its soundness. ChatGPT 5.5 Pro proposed most of the candidate attacks and optimizations. Eyad and Mahmoud selected the systems, assigned their roles, coordinated their communication and provided the computing infrastructure. The proof divided the space into 36,152 canonical cases after symmetry reduction. The closing run verified all 122,553 certificates using Z3 and completed 141,878,161 per-clause soundness checks with zero reported failures. The general theorem for an unlimited number of goods remains open. This result moves the known frontier from seven goods to nine. The authors describe the process as “AI led, human assisted and verified.” For me, this is one of the clearest examples yet of current AI systems contributing substantial work to a new mathematical proof. I found the story too remarkable not to share. Paper: https://arxiv.org/abs/2608.08590

    1 Source

    @kimmonismusI just had a fascinating call with Eyad Alkassar (@AlkassarEyad) about how ChatGPT 5.5 Pro helped move the frontier of an open problem in computer science. The problem sounds abstract, but it is surprisingly intuitive: Imagine four people dividing nine objects. Each person may value them differently. The goal is to find a division so fair that, after any single object is removed from someone else’s pile, nobody would prefer what remains to their own pile. This criterion is called envy-freeness up to any good, or EFX. Whether complete EFX allocations always exist for four or more people remains open. For four people, the previous general guarantee stopped at seven objects. A new preprint by Eyad Alkassar, Mahmoud Fouz and renowned computer scientist Kurt Mehlhorn now presents a proof for up to nine. The story behind it is remarkable. Over dinner, Eyad challenged Mehlhorn to give AI one of the harder open problems in his field. Eyad and Mahmoud are both computer science PhDs who had spent years building startups and were newcomers to fair division. They built a research workflow around several AI systems. At one point, Fable had questions about one of Mehlhorn’s papers and repeatedly urged Eyad to contact him. After Eyad refused twice, the AI pointed out that they were only around 30 kilometers apart and suggested that a short drive would be worth it. According to Eyad, the situation became even stranger once the proof was finished. Fable then advised him against sending the result to Mehlhorn, warning that he might be a competitor and that an incorrect proof could be embarrassing. Eyad contacted him anyway. Mehlhorn reviewed and verified the proof architecture, refined the arguments and joined the paper as a co-author. According to the paper, Claude Fable performed the analysis and lemma proofs, authored the trusted verification layer and audited its soundness. ChatGPT 5.5 Pro proposed most of the candidate attacks and optimizations. Eyad and Mahmoud selected the systems, assigned their roles, coordinated their communication and provided the computing infrastructure. The proof divided the space into 36,152 canonical cases after symmetry reduction. The closing run verified all 122,553 certificates using Z3 and completed 141,878,161 per-clause soundness checks with zero reported failures. The general theorem for an unlimited number of goods remains open. This result moves the known frontier from seven goods to nine. The authors describe the process as “AI led, human assisted and verified.” For me, this is one of the clearest examples yet of current AI systems contributing substantial work to a new mathematical proof. I found the story too remarkable not to share. Paper: https://arxiv.org/abs/2608.08590