{"data":{"event":{"id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","slug":"gpt6-astra-abc-core-existence","title":"GPT-6 Astra proves the core is nonempty in approval-based committee elections","short_summary":"On 10 September 2026 Becker, Greger, and Dominik Peters posted a proof that every approval-based committee election has a core-stable committee. They attribute the primary idea and proof to GPT-6 Astra in a lengthy interactive session. Epoch marked the FrontierMath Open Problems item Solved (human + AI).","full_description":"Whether the core of an approval-based multi-winner (ABC) election is always nonempty has been open since Aziz, Brill, Conitzer, Elkind, Freeman, and Walsh (2017). It is the standard group-fairness notion of proportional representation for committee elections. Epoch listed a request for an empty-core counterexample as a Major-advance FrontierMath: Open Problems item; Peters had suggested the problem for the benchmark.\n\nThe 10 September 2026 preprint (arXiv:2609.11912) proves there is no such counterexample. The argument introduces a voting rule that maximizes an entropy-like “harmonic entropy” objective over committees and payment systems; every local optimum lies in the core, so a core committee can be found in polynomial time. The authors write that the proof was obtained with GPT-6 Astra. Epoch’s 18 September note records a human + AI solution: the authors attribute the primary idea and proof to Astra in a lengthy interactive session; Peters told Epoch he doubts the team would have found the proof without Astra, and also that Astra does not appear to solve it from a simple prompt. Because the theorem shows the stated counterexample task is impossible, Epoch still marks the problem solved under its policy. This is newly created social-choice knowledge, not a FrontierMath exam score, and it is separate from Astra’s August ten-proofs cluster and from FrontierMath Erdős.\n\nUpdate (2026-09-14): Open-Source AI and Open Models Reading List. Article URL: https://www.interconnects.ai/p/open-source-ai-reading-list\nComments URL: https://news.ycombinator.com/item?id=49690260\nPoints: 18\n# Comments: 1\n\nUpdate (2026-09-19): Laya the open source version of Jev. Article URL: https://laya.convaiinnovations.com/\nComments URL: https://news.ycombinator.com/item?id=49765348\nPoints: 12\n# Comments: 2\n\nUpdate (2026-09-19): Microsoft director: AI scraping 'the largest theft of labor in human history'. Article URL: https://www.tomshardware.com/tech-industry/artificial-intelligence/microsoft-director-called-ai-scraping-the-largest-theft-of-labor-in-human-history-while-openai-head-brands-chatgpt-an-existential-threat-to-publishers-revelations-come-from-legal-briefs-filed-in-nyt-lawsuit\nComments URL: https://news.ycombinator.com/item?id=49768921\nPoints: 79\n# Comments: 21\n\nUpdate (2026-09-22): GPT-6 Sol and Luna. Article URL: https://openai.com/index/introducing-gpt-6-sol-and-luna/\nComments URL: https://news.ycombinator.com/item?id=49805509\nPoints: 370\n# Comments: 176\n\nUpdate (2026-09-22): The current balance of power in open models. Article URL: https://www.interconnects.ai/p/the-current-balance-of-power-in-open\nComments URL: https://news.ycombinator.com/item?id=49808816\nPoints: 16\n# Comments: 3\n\nUpdate (2026-09-23): GPT-6 Astra has gained the ability to drive a car. Article URL: https://drivingbench.com/\nComments URL: https://news.ycombinator.com/item?id=49817404\nPoints: 41\n# Comments: 26\n\nUpdate (2026-09-24): Best LLM for every budget, updated daily. Article URL: https://bestmodelforyourbudget.terrydjony.com/\nComments URL: https://news.ycombinator.com/item?id=49830866\nPoints: 43\n# Comments: 20\n\nUpdate (2026-09-24): Open Source Replication of the Auditing Game Model Organism We release an open source replication of the model organism from our previous auditing game paper.. Open Source Replication of the Auditing Game Model Organism We release an open source replication of the model organism from our previous auditing game paper.\n\nUpdate (2026-09-22): AI Exec: We May Have Pulled Off “The Largest Theft of Labor in Human History” - Mother Jones. AI Exec: We May Have Pulled Off “The Largest Theft of Labor in Human History”  Mother Jones","ledger_type":"benefit","primary_domain_id":"23d935bb-b857-4b2b-9471-d568bb848a66","event_status":"provisional","event_date":"2026-09-10T00:00:00.000Z","discovery_date":"2026-09-10T00:00:00.000Z","first_published_date":"2026-09-10T00:00:00.000Z","last_reviewed_date":"2026-09-22T00:00:00.000Z","geographic_scope":"International","affected_population":"Computational social choice and multi-winner voting research community","base_impact_tier":3,"base_score":"10.00","attribution_multiplier":"0.8000","evidence_multiplier":"0.5000","realization_multiplier":"0.6000","durability_multiplier":"1.0000","current_event_score":"2.400000","confidence_level":"medium","score_explanation":"Settling the main open question of ABC core existence is major social-choice theory (tier 3), not a contest score and not a Historic cluster. Authors attribute the primary idea to GPT-6 Astra in an interactive session (0.80). Evidence is a domain-expert preprint plus Epoch’s independent Solved (human + AI) mark (0.50), not peer review. Realization is a complete public manuscript without a machine-checkable formalization (0.60). A correct existence theorem is permanent (1.00). Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source. Follow-up ingest: added a follow-up source.","methodology_version_id":"8745541b-0f8e-4d5a-9b47-5291d3813a8d","original_methodology_version_id":"8745541b-0f8e-4d5a-9b47-5291d3813a8d","published_at":"2026-09-10T00:00:00.000Z","created_at":"2026-09-18T19:11:40.608Z","updated_at":"2026-09-26T19:01:05.489Z","flags":["Independent reconstruction pending"],"domain_name":"Mathematics","domain_slug":"mathematics","methodology_version":"0.2"},"contributions":[{"id":"f0cb6f6b-30cf-43ed-bdc5-6fdf5477ba2e","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","model_id":"0f0d18fc-d3f1-4551-9eaf-a13f1f93b1a5","role_description":"Named Astra release that, in a lengthy interactive session with Becker, Greger, and Peters, produced the primary idea and proof of ABC core existence via a harmonic-entropy voting rule.","attribution_multiplier":"0.8000","credit_share":"1.0000","contribution_score":"2.400000","attribution_rationale":"The preprint and Epoch both attribute the primary idea and proof to GPT-6 Astra. Humans elicited the result; Epoch classifies it as human + AI, not an autonomous one-shot solve.","attribution_confidence":"medium","first_used_date":"2026-09-10T00:00:00.000Z","model_version_if_known":null,"review_status":"approved","created_at":"2026-09-18T19:11:40.608Z","model_slug":"gpt-6-astra","model_name":"GPT-6 Astra","identity_class":"named_release","is_internal":false,"family_name":"GPT-6","family_slug":"openai-gpt-6","organization_name":"OpenAI","organization_slug":"openai"}],"sources":[{"id":"4e12c409-03a7-4673-abab-b0f4e43bf8d7","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://arxiv.org/abs/2609.11912","canonical_url":"https://arxiv.org/abs/2609.11912","source_type":"preprint","publisher":"arXiv","author":null,"publication_date":"2026-09-10T00:00:00.000Z","retrieved_at":"2026-09-18T19:11:40.608Z","title":"Existence of the Core in Approval-Based Committee Elections","excerpt":null,"content_hash":null,"source_reliability_class":"medium","is_primary_source":true,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-18T19:11:40.608Z"},{"id":"88a1f620-9deb-470c-bddd-542e26fcfca7","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://epoch.ai/frontiermath/open-problems/committee-election","canonical_url":"https://epoch.ai/frontiermath/open-problems/committee-election","source_type":"other","publisher":"Epoch AI","author":null,"publication_date":"2026-09-18T00:00:00.000Z","retrieved_at":"2026-09-18T19:11:40.608Z","title":"The Core in Approval-Based Committee Elections (FrontierMath Open 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1","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-18T20:00:49.770Z"},{"id":"0fd176f9-15cf-4d28-bf7c-ec86de4b7b98","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://x.com/EpochAIResearch/status/2100986494873989227","canonical_url":"https://x.com/EpochAIResearch/status/2100986494873989227","source_type":"other","publisher":"Epoch AI","author":null,"publication_date":"2026-09-18T00:00:00.000Z","retrieved_at":"2026-09-18T19:11:40.608Z","title":"Another problem from FrontierMath: Open Problems has been solved","excerpt":null,"content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-18T19:11:40.608Z"},{"id":"8fafe27a-7156-4d11-92a3-7e01be6ab75b","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://laya.convaiinnovations.com/","canonical_url":"https://laya.convaiinnovations.com/","source_type":"journalism","publisher":"Hacker News","author":null,"publication_date":"2026-09-19T00:00:00.000Z","retrieved_at":"2026-09-19T12:00:59.647Z","title":"Laya the open source version of Jev","excerpt":"Article URL: https://laya.convaiinnovations.com/\nComments URL: https://news.ycombinator.com/item?id=49765348\nPoints: 12\n# Comments: 2","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-19T12:00:59.647Z"},{"id":"77ad3a4d-4772-4baa-ad6b-22d06b1b55eb","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://www.tomshardware.com/tech-industry/artificial-intelligence/microsoft-director-called-ai-scraping-the-largest-theft-of-labor-in-human-history-while-openai-head-brands-chatgpt-an-existential-threat-to-publishers-revelations-come-from-legal-briefs-filed-in-nyt-lawsuit","canonical_url":"https://www.tomshardware.com/tech-industry/artificial-intelligence/microsoft-director-called-ai-scraping-the-largest-theft-of-labor-in-human-history-while-openai-head-brands-chatgpt-an-existential-threat-to-publishers-revelations-come-from-legal-briefs-filed-in-nyt-lawsuit","source_type":"journalism","publisher":"Hacker News","author":null,"publication_date":"2026-09-19T00:00:00.000Z","retrieved_at":"2026-09-19T23:01:01.670Z","title":"Microsoft director: AI scraping 'the largest theft of labor in human history'","excerpt":"Article URL: https://www.tomshardware.com/tech-industry/artificial-intelligence/microsoft-director-called-ai-scraping-the-largest-theft-of-labor-in-human-history-while-openai-head-brands-chatgpt-an-existential-threat-to-publishers-revelations-come-from-legal-briefs-filed-in-nyt-lawsuit\nComments URL: https://news.ycombinator.com/item?id=49768921\nPoints: 79\n# Comments: 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news","author":null,"publication_date":"2026-09-22T00:00:00.000Z","retrieved_at":"2026-09-26T19:01:05.465Z","title":"AI Exec: We May Have Pulled Off “The Largest Theft of Labor in Human History” - Mother Jones","excerpt":"AI Exec: We May Have Pulled Off “The Largest Theft of Labor in Human History”  Mother Jones","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-26T19:01:05.465Z"},{"id":"dbbfc9cf-061e-4b05-8446-ede8330fae0c","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://openai.com/index/introducing-gpt-6-sol-and-luna/","canonical_url":"https://openai.com/index/introducing-gpt-6-sol-and-luna/","source_type":"journalism","publisher":"Hacker News","author":null,"publication_date":"2026-09-22T00:00:00.000Z","retrieved_at":"2026-09-22T19:01:14.334Z","title":"GPT-6 Sol and Luna","excerpt":"Article URL: https://openai.com/index/introducing-gpt-6-sol-and-luna/\nComments URL: https://news.ycombinator.com/item?id=49805509\nPoints: 370\n# Comments: 176","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-22T19:01:14.334Z"},{"id":"f916b6aa-7668-4452-94a3-89b1f54b97dc","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://www.interconnects.ai/p/the-current-balance-of-power-in-open","canonical_url":"https://www.interconnects.ai/p/the-current-balance-of-power-in-open","source_type":"journalism","publisher":"Hacker News","author":null,"publication_date":"2026-09-22T00:00:00.000Z","retrieved_at":"2026-09-23T00:01:06.090Z","title":"The current balance of power in open models","excerpt":"Article URL: https://www.interconnects.ai/p/the-current-balance-of-power-in-open\nComments URL: https://news.ycombinator.com/item?id=49808816\nPoints: 16\n# Comments: 3","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-23T00:01:06.090Z"},{"id":"69308c0f-8b3c-4c56-806e-b297f0ed871d","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://drivingbench.com/","canonical_url":"https://drivingbench.com/","source_type":"journalism","publisher":"Hacker News","author":null,"publication_date":"2026-09-23T00:00:00.000Z","retrieved_at":"2026-09-23T16:01:05.179Z","title":"GPT-6 Astra has gained the ability to drive a car","excerpt":"Article URL: https://drivingbench.com/\nComments URL: https://news.ycombinator.com/item?id=49817404\nPoints: 41\n# Comments: 26","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-23T16:01:05.179Z"},{"id":"4929bb54-78cc-4eed-bfab-a26aba662121","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://bestmodelforyourbudget.terrydjony.com/","canonical_url":"https://bestmodelforyourbudget.terrydjony.com/","source_type":"journalism","publisher":"Hacker News","author":null,"publication_date":"2026-09-24T00:00:00.000Z","retrieved_at":"2026-09-24T15:01:01.248Z","title":"Best LLM for every budget, updated daily","excerpt":"Article URL: https://bestmodelforyourbudget.terrydjony.com/\nComments URL: https://news.ycombinator.com/item?id=49830866\nPoints: 43\n# Comments: 20","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-24T15:01:01.248Z"},{"id":"b69a7f0f-82b7-40eb-83bd-82ed7a8e8ffc","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","url":"https://alignment.anthropic.com/2025/auditing-mo-replication/","canonical_url":"https://alignment.anthropic.com/2025/auditing-mo-replication/","source_type":"developer_announcement","publisher":"Anthropic Alignment Science","author":null,"publication_date":"2026-09-24T00:00:00.000Z","retrieved_at":"2026-09-24T23:00:53.811Z","title":"Open Source Replication of the Auditing Game Model Organism We release an open source replication of the model organism from our previous auditing game paper.","excerpt":"Open Source Replication of the Auditing Game Model Organism We release an open source replication of the model organism from our previous auditing game paper.","content_hash":null,"source_reliability_class":"medium","is_primary_source":false,"is_independent":true,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-09-24T23:00:53.811Z"}],"claims":[{"id":"34d4fcd5-b0a0-4b1c-aef8-f3bee052bac3","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","claim_text":"Every approval-based committee election has a nonempty core; a harmonic-entropy voting rule finds a core committee in polynomial time.","claim_type":"outcome","claim_status":"supported","confidence_score":null,"created_at":"2026-09-18T19:11:40.608Z","updated_at":"2026-09-18T19:11:40.608Z"},{"id":"e906dc86-3907-4be2-b715-687582442e20","event_id":"0c339522-fcc5-469d-a4e6-82d5cc121dd0","claim_text":"GPT-6 Astra produced the primary idea and proof in a lengthy interactive session with Becker, Greger, and 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