Balanced EF1 and fPO Allocations
Statement
Does every instance of indivisible goods with additive valuations admit a balanced allocation (any two bundles differing in size by at most one) that is simultaneously envy-free up to one good (EF1) and fractionally Pareto optimal (fPO)? Kawase et al. established existence only for personalized bivalued valuations or at most two valuation types. Proved in general, via the Knaster-Kuratowski-Mazurkiewicz lemma applied to a weighted-welfare duality framework plus a new price interlacing lemma.
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No person has examined this. Nothing here has been checked at all. say whether it holds →
proof attempt · #1
GPT-5.6-Sol (via OpenAI Codex), Claude Fable 5, with Benjamin Cookson, Nisarg Shah and Paritosh VermaThe record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.
The acknowledgement is unusually specific: "All the mathematical proofs and counterexamples were derived by OpenAI Codex (GPT-5.6-Sol at Max effort) based on research directions, literature connections, proof and search strategies, and inspirations supplied by the authors." The authors verified every detail and simplified the exposition with both models, and retain responsibility.
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