{"data":{"event":{"id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","slug":"alphaproof-nexus-open-mathematics-2026","title":"AlphaProof Nexus produces Lean proofs for nine Erdős instances and 44 OEIS conjectures","short_summary":"A 21 May 2026 DeepMind preprint reports formal proofs for nine of 353 Erdős problem instances and 44 of 492 OEIS conjectures, plus research collaborations. Its Gemini-led agents publish Lean artifacts; independent records distinguish full, partial, and variant results.","full_description":"Google DeepMind’s AlphaProof Nexus combines Gemini 3.1 Pro proof search, Lean feedback, optional AlphaProof subgoal solving, and Gemini 3.0 Flash sketch rating. The paper reports nine solved Erdős instances and 44 OEIS conjectures, with public Lean proofs and selected prose proofs. Human researchers provided formal statements and reviewed their correspondence to the intended questions. Additional collaborations cover optimization, algebraic geometry, graph theory, additive combinatorics, and quantum optics.\n\nThe nine Erdős instances are #12(i), #12(ii), #125, a #138 variant, #152, #741(i), #741(ii), #846, and a strengthened #26 variant; they are not nine distinct catalogue numbers. The independent AI-contributions wiki records #125 and #741 as full resolutions, #138 and #12 as partial progress, and prior comparable literature for #152. Its data stops on 30 June 2026 and it explicitly disclaims a definitive assessment. The general graph reconstruction conjecture remains open.\n\nThis single aggregate event remains provisional pending broader novelty and statement review. Public formal artifacts support mathematical realization, but this ingestion does not independently rebuild them. The simpler Gemini-plus-Lean agent reproduced the nine Erdős successes in the authors’ ablation; benchmark reruns receive no additional discovery credit. This corpus is separate from the February Aletheia event already recorded.","ledger_type":"benefit","primary_domain_id":"23d935bb-b857-4b2b-9471-d568bb848a66","event_status":"provisional","event_date":"2026-05-21T00:00:00.000Z","discovery_date":"2026-05-21T00:00:00.000Z","first_published_date":"2026-05-21T00:00:00.000Z","last_reviewed_date":"2026-10-28T00:00:00.000Z","geographic_scope":"International mathematics research","affected_population":"Researchers in number theory, combinatorics, optimization, algebraic geometry, and mathematical quantum optics","base_impact_tier":3,"base_score":"10.00","attribution_multiplier":"0.7950","evidence_multiplier":"0.6000","realization_multiplier":"0.9000","durability_multiplier":"1.0000","current_event_score":"4.293000","confidence_level":"medium","score_explanation":"This is a substantial cross-domain corpus of formal research proofs, including open Erdős instances and dozens of sequence conjectures, but the aggregate does not establish a set of landmark theorems: the nine Erdős results include variants and partial progress, one has comparable prior literature, and broader novelty review remains incomplete. Score the body of work as Major (tier 3, base 10), not Historic. Evidence 0.60 reflects a detailed preprint, public checkable Lean artifacts, and limited independent catalogue corroboration; it does not establish peer review or universal novelty. Realization 0.90 reflects published machine-verifiable proofs; correct mathematical knowledge is permanent (durability 1.00). Editorial AI credit shares are 0.85 Gemini 3.1 Pro, 0.10 AlphaProof, and 0.05 Gemini 3.0 Flash, with causal attribution 0.90, 0.25, and 0.10 respectively. These shares are estimates, not measured author-supplied percentages. Weighted attribution is 0.795; contributions sum to 10 × 0.60 × 0.90 × (0.90 × 0.85 + 0.25 × 0.10 + 0.10 × 0.05) = 4.2930 points. Human formalization, validation, and existing literature limit AI attribution. No additional credit is awarded for reproducing the same proofs in ablations.","methodology_version_id":"8745541b-0f8e-4d5a-9b47-5291d3813a8d","original_methodology_version_id":"8745541b-0f8e-4d5a-9b47-5291d3813a8d","published_at":"2026-05-21T00:00:00.000Z","created_at":"2026-10-10T23:53:23.452Z","updated_at":"2026-10-28T20:22:47.621Z","flags":["Aggregate corpus; nine instances are not nine distinct Erdős numbers","Full, partial, and variant results; novelty review remains incomplete","Public Lean artifacts; independent rebuild not performed at ingestion","Credit split is an editorial estimate","Independent wiki frozen on 2026-06-30"],"domain_name":"Mathematics","domain_slug":"mathematics","methodology_version":"0.2"},"contributions":[{"id":"ce94dec6-cddf-4be8-a057-3ae4f47d02a4","event_id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","model_id":"68b40d82-d353-4a9d-b6d9-33a6f91311c2","role_description":"Existing reinforcement-learning prover queried as an optional tool to discharge proof subgoals in the full-featured AlphaProof Nexus agent.","attribution_multiplier":"0.2500","credit_share":"0.1000","contribution_score":"0.135000","attribution_rationale":"The reported full-featured discovery pipeline used AlphaProof for subgoals, but standalone AlphaProof failed on the tested Erdős set and the basic Gemini-plus-Lean agent reproduced the successes. Supporting attribution and a 10% editorial credit share avoid treating AlphaProof as the sole discovery system.","attribution_confidence":"medium","first_used_date":"2026-05-21T00:00:00.000Z","model_version_if_known":null,"review_status":"approved","created_at":"2026-10-10T23:53:23.452Z","model_slug":"alphaproof","model_name":"AlphaProof","identity_class":"named_release","is_internal":false,"family_name":"AlphaProof","family_slug":"google-deepmind-alphaproof","organization_name":"Google DeepMind","organization_slug":"google-deepmind"},{"id":"6f1847e9-db05-4a10-9d19-f73da4f2f48a","event_id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","model_id":"ee7f3366-c558-45dd-a883-5063da1d094b","role_description":"Named model used for sketch comparisons, Elo-based evolutionary rating, and evaluation synthesis in the full-featured agent.","attribution_multiplier":"0.1000","credit_share":"0.0500","contribution_score":"0.027000","attribution_rationale":"The implementation explicitly identifies Gemini 3.0 Flash as the rating model. Its role supports search rather than generating the principal proofs, and the basic agent reproduced the Erdős successes without the evolutionary rating machinery. Minor assistance and a 5% editorial credit share reflect this limited role.","attribution_confidence":"medium","first_used_date":"2026-05-21T00:00:00.000Z","model_version_if_known":null,"review_status":"approved","created_at":"2026-10-10T23:53:23.452Z","model_slug":"gemini-3-0-flash","model_name":"Gemini 3.0 Flash","identity_class":"named_release","is_internal":false,"family_name":"Gemini","family_slug":"google-deepmind-gemini","organization_name":"Google DeepMind","organization_slug":"google-deepmind"},{"id":"edb39c1c-96c4-4bc0-837a-4515f60392fa","event_id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","model_id":"6c7794c5-9a2f-45e2-b7cd-1afd064b46fb","role_description":"Named model used for multi-turn proof generation and refinement in AlphaProof Nexus, grounded by Lean compiler feedback.","attribution_multiplier":"0.9000","credit_share":"0.8500","contribution_score":"4.131000","attribution_rationale":"The paper explicitly names Gemini 3.1 Pro as the proving model. Its basic agent also reproduced all nine Erdős successes without AlphaProof or evolutionary rating. Humans selected and formalized statements, checked intended meaning, and prepared prose. The 85% AI credit share is an editorial estimate assigning the main proof-search role to this release.","attribution_confidence":"medium","first_used_date":"2026-05-21T00:00:00.000Z","model_version_if_known":null,"review_status":"approved","created_at":"2026-10-10T23:53:23.452Z","model_slug":"gemini-3-1-pro","model_name":"Gemini 3.1 Pro","identity_class":"named_release","is_internal":false,"family_name":"Gemini","family_slug":"google-deepmind-gemini","organization_name":"Google DeepMind","organization_slug":"google-deepmind"}],"sources":[{"id":"c8eef069-eab1-4606-a892-d608b331ebd3","event_id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","url":"https://arxiv.org/abs/2605.22763","canonical_url":"https://arxiv.org/abs/2605.22763","source_type":"preprint","publisher":"arXiv / Google DeepMind researchers","author":null,"publication_date":"2026-05-21T00:00:00.000Z","retrieved_at":"2026-10-10T23:53:23.452Z","title":"Advancing Mathematics Research with AI-Driven Formal Proof Search (v2, revised 8 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prose","excerpt":null,"content_hash":null,"source_reliability_class":"low","is_primary_source":true,"is_independent":false,"is_peer_reviewed":false,"archived_url":null,"created_at":"2026-10-10T23:53:23.452Z"},{"id":"7531ece2-bb66-440e-987c-ec3a38e59657","event_id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","url":"https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems","canonical_url":"https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems","source_type":"other","publisher":"Terence Tao and Erdős-problems community / GitHub","author":null,"publication_date":"2026-06-30T00:00:00.000Z","retrieved_at":"2026-10-10T23:53:23.452Z","title":"AI contributions to Erdős problems (last updated 30 June 2026)","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":"2026-10-10T23:53:23.452Z"}],"claims":[{"id":"c275d8ea-67f2-4f9a-be58-6c4a8b03312b","event_id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","claim_text":"The paper reports successful formal proofs for nine of 353 Erdős problem instances and 44 of 492 OEIS conjectures.","claim_type":"outcome","claim_status":"supported","confidence_score":null,"created_at":"2026-10-28T20:22:47.621Z","updated_at":"2026-10-28T20:22:47.621Z"},{"id":"444cda8e-f891-41a0-ad0c-456032826064","event_id":"b36341d8-c06f-4378-a4c6-a6a3e380e732","claim_text":"Gemini 3.1 Pro generates proofs, AlphaProof supplies optional subgoal proofs, and Gemini 3.0 Flash rates sketches in the full-featured AlphaProof Nexus 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