{"paper":{"title":"Complete EFX Allocations Exist for Four Additive Agents and Up to Nine Goods","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.GT","authors_text":"Eyad Alkassar, Kurt Mehlhorn, Mahmoud Fouz","submitted_at":"2026-08-09T09:08:27Z","abstract_excerpt":"We prove that every fair-division instance with four agents, additive valuations over the non-negative reals, and at most nine indivisible goods admits a \\emph{complete} allocation that is envy-free up to any good in the strong, zero-tolerant sense ($\\EFXo$). The case $m=9=n+5$ lies beyond the previously known frontier for complete EFX with four agents ($m\\le n+3$). The proof combines a small set of hand-proven reduction lemmas with a machine-verified certificate corpus. The valuation polytope is covered by a collection of smaller polytopes. For each smaller polytope $P$, a family $F$ of alloc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08590","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.08590/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}