{ "id": "AIBD-0075", "title": "GPT-5.5 Pro solves Erdős problem #38; solution human-checked, Lean-formalized and recorded as solved on erdosproblems.com", "summary": "GPT-5.5 Pro solved Erdős problem #38; human-checked 26 April 2026, Lean-formalized by Aristotle 1 May, recorded as solved on erdosproblems.com", "date_observed": "2026-04-25", "date_basis": "unassessed", "date_precision": "unassessed", "benefit_status": null, "source_collection": null, "source_entry": null, "account": "Erdős problem #38 is an open question in additive number theory. It asks for a set B that is not an additive basis yet raises the Schnirelmann density of every set A. OpenAI's GPT-5.5 Pro solved it. On 25 April 2026 a user (gebyjaff) posted the model's solution on the erdosproblems.com forum, with a PDF cleanup by Liam Price. On 26 April forum commenter Nat Sothanaphan reported that a standard check found no issues in the write-up. On 1 May the Aristotle system autoformalized the argument in Lean, which Sothanaphan confirmed matches the paper. Site maintainer Thomas Bloom added an alternative sketch on 2 May 2026. The problem page now records the positive solution by GPT 5.5 Pro. Terence Tao's wiki lists it as an AI-standalone full solution.", "category": "benefit created", "unexpected": false, "purpose": "General-purpose reasoning model prompted to solve an open research problem", "system": "GPT-5.5 Pro, with Aristotle used for Lean autoformalisation", "model_version": "GPT-5.5 Pro", "developer": "OpenAI (GPT-5.5 Pro); Aristotle (developer not named in the cited sources)", "deployer": "Individual user (gebyjaff) posting on erdosproblems.com", "location": null, "domain": "Mathematics", "segment": "Research mathematics (additive number theory)", "benefit_type": "Improved Knowledge Discovery", "taxonomy_note": null, "registry_ids": {}, "ratings": [ { "group": "organizational", "depth": 1, "reason": "Small: one more entry resolved in the erdosproblems.com project, accepted by its maintainer" }, { "group": "societal", "depth": 1, "reason": "Small: a single open problem settled. The problem page records the solution, and Tao's wiki catalogs it among many AI contributions" } ], "ai_role": { "status": "assessed", "statement": "The model produced the proof after a human prompt. Humans cleaned up the write-up and checked it, and the maintainer accepted it. A separate AI system (Aristotle) produced the Lean formalization that a human confirmed matches the paper. Tao's wiki classifies the case as AI standalone with non-significant human involvement. The attribution of GPT-5.5 Pro to OpenAI rests on the chatgpt.com share link in gebyjaff's forum post. No cited source names OpenAI in prose. The cited sources do not name Aristotle's developer." }, "evidence": [ { "kind": "other", "citation": "Bloom, T. (ed.) (2026). Erdős Problem #38. erdosproblems.com.", "title": "Erdős Problem #38", "url": "https://www.erdosproblems.com/38", "date": "2026-05-02", "written_by": "independent", "quote": "A positive solution was given by GPT 5.5 Pro (prompted by gebyjaff) in the comments; in fact a sparse random set B has this property" }, { "kind": "other", "citation": "erdosproblems.com forum (2026). Thread for problem 38. erdosproblems.com.", "title": "Forum thread: Problem 38", "url": "https://www.erdosproblems.com/forum/thread/38", "date": "2026-04-26", "written_by": "independent", "quote": "Standard check found no issues in the writeup" }, { "kind": "other", "citation": "Tao, T. et al. (2026). AI contributions to Erdős problems. GitHub wiki, teorth/erdosproblems.", "title": "AI contributions to Erdős problems", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "date": null, "written_by": "independent", "quote": "This AI Wiki collects contributions of AI systems to the understanding of Erdős problems." } ], "replication": { "produced_by": [ "A user prompting GPT-5.5 Pro with the problem statement", "Human cleanup of the written proof", "Independent human check and Lean autoformalization by Aristotle", "Acceptance by the site maintainer" ], "must_hold": [ "The problem has a precise, formalized statement", "The proof is checkable by humans or in Lean", "A community venue reviews and records AI-produced solutions" ], "echoes": [ { "date": "2026-05-01", "setting": "Erdős problem #694, GPT-5.5 Pro, full solution listed in Tao's wiki section 1(a)", "model_version": "GPT-5.5 Pro", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" }, { "date": "2026-05-26", "setting": "Erdős problem #90, Claude Mythos, full solution listed in Tao's wiki section 1(a); the wiki also lists a full solution of #90 by an OpenAI internal model on 20 May 2026", "model_version": "Claude Mythos", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" } ] }, "paired_incident": { "database": "AI Incident Database", "identifier": "1044", "title": "Reported Emergence of 'Vegetative Electron Microscopy' in Scientific Papers Traced to Purported AI Training Data Contamination", "url": "https://incidentdatabase.ai/cite/1044", "severity": null, "group": "societal", "basis": "Same developer, OpenAI, and same stakeholder, the research record: the nonsense term “vegetative electron microscopy” entered published papers via AI training contamination that GPT-3 and GPT-4 reproduce, harming researchers, publishers and the scientific record." }, "headline": "GPT-5.5 Pro solves Erdős problem #38; solution Lean-formalized", "metric": null, "comparator": null, "later": { "status": "persisted", "date": "2026-05-01", "note": "Six days after the solution was posted, the Aristotle system autoformalized the argument in Lean and a forum checker confirmed that the formalization matches the write-up; Thomas Bloom added an alternative sketch the next day, and Tao's wiki (data to 30 June 2026) lists the problem as a full AI solution.", "quote": "I have confirmed the Lean matches the paper and that both use the asymptotic additive basis definition", "url": "https://www.erdosproblems.com/forum/thread/38" }, "value": null, "developer_country": "United States", "deployer_country": null, "unrated_because": null, "brief_notes": null, "brief": null, "revisions": [ { "date": "2026-10-20", "change": "Record drafted from the cited sources.", "source": null, "editor": "AIBD" }, { "date": "2026-10-20", "change": "Paired with AI Incident Database incident 1044 (medium confidence); severity not given by the source.", "source": "https://incidentdatabase.ai/cite/1044", "editor": "AIBD" }, { "date": "2026-10-20", "change": "Dropped Harmonic as Aristotle's developer because no cited source names it, noted that the OpenAI attribution rests on the ChatGPT share link, called Nat Sothanaphan a forum commenter rather than a reviewer, and removed the unsupported “human-verified” and “Lean-verified” labels from the two wiki echoes while adding the 20 May 2026 OpenAI internal model entry for #90.", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Added what happened since from the cited sources.", "source": "https://www.erdosproblems.com/forum/thread/38", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Country of the developer added from the organization's own site or a registry.", "source": "https://en.wikipedia.org/wiki/OpenAI", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Prose pass: a rating reason, account, the replication conditions, the system's part rewritten in shorter sentences with abbreviations expanded; facts, figures and quotations unchanged.", "source": null, "editor": "AIBD" } ], "confidence": "claimed", "shade": "low", "claim_status": "Listed, not verified", "claim_basis": "Sources: three other independent sources. The cited evidence includes no independent study or report and no finding by a regulator or court.", "profile": { "case_id": "AIBD-0075", "catalog_version": "1.0.0", "source_fingerprint": "a937c036c24fcfe7fb4a28e5da6f1ecb04e3d8447e3e51b0ba66b44be46e3ed4", "collection_refs": [ { "collection_id": "aibd", "version": "snapshot-8bf7cfec9cb185aa" }, { "collection_id": "aibd", "version": "2026-10-25" } ], "source_ids": [ "AIBD-0075-source-0001", "AIBD-0075-source-0002", "AIBD-0075-source-0003" ], "families": { "strategic assessments": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "record_ids": [] }, "evaluations": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "record_ids": [] }, "monetary observations": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "record_ids": [] }, "institutional decisions": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "record_ids": [] } }, "fields": { "actors[]": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "developer": "OpenAI (GPT-5.5 Pro); Aristotle (developer not named in the cited sources)", "deployer": "Individual user (gebyjaff) posting on erdosproblems.com" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "developer", "deployer" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "actors[].ownership_control": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "ai_contribution": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "ai_role": { "status": "assessed", "statement": "The model produced the proof after a human prompt. Humans cleaned up the write-up and checked it, and the maintainer accepted it. A separate AI system (Aristotle) produced the Lean formalization that a human confirmed matches the paper. Tao's wiki classifies the case as AI standalone with non-significant human involvement. The attribution of GPT-5.5 Pro to OpenAI rests on the chatgpt.com share link in gebyjaff's forum post. No cited source names OpenAI in prose. The cited sources do not name Aristotle's developer." } } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "ai_role" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "ai_relationship": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "category": "benefit created", "ai_role": { "status": "assessed", "statement": "The model produced the proof after a human prompt. Humans cleaned up the write-up and checked it, and the maintainer accepted it. A separate AI system (Aristotle) produced the Lean formalization that a human confirmed matches the paper. Tao's wiki classifies the case as AI standalone with non-significant human involvement. The attribution of GPT-5.5 Pro to OpenAI rests on the chatgpt.com share link in gebyjaff's forum post. No cited source names OpenAI in prose. The cited sources do not name Aristotle's developer." } } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "ai_role", "category" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "assessments[].alternatives": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "assessments[].assessor_date": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "assessments[].depth": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "ratings": [ { "group": "organizational", "depth": 1, "reason": "Small: one more entry resolved in the erdosproblems.com project, accepted by its maintainer" }, { "group": "societal", "depth": 1, "reason": "Small: a single open problem settled. The problem page records the solution, and Tao's wiki catalogs it among many AI contributions" } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "ratings" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "assessments[].display_confidence": { "state": "value_present", "value": { "kind": "structured", "data": { "confidence": "low", "evidence_word": "claimed", "claim_status": "Listed, not verified", "basis": "Sources: three other independent sources. The cited evidence includes no independent study or report and no finding by a regulator or court.", "interpretation": "Existing display rule preserved; the richer outcome-specific confidence question still needs review." } }, "provenance": { "kind": "curatorial", "basis": "Derived from the existing case display properties; not a new evidentiary judgment.", "record_fields": [], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "assessments[].rationale": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "ratings": [ { "group": "organizational", "depth": 1, "reason": "Small: one more entry resolved in the erdosproblems.com project, accepted by its maintainer" }, { "group": "societal", "depth": 1, "reason": "Small: a single open problem settled. The problem page records the solution, and Tao's wiki catalogs it among many AI contributions" } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "ratings" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "assessments[].rubric_version": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "assessments[].status_basis": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "assessments[].target": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "ratings": [ { "group": "organizational", "depth": 1, "reason": "Small: one more entry resolved in the erdosproblems.com project, accepted by its maintainer" }, { "group": "societal", "depth": 1, "reason": "Small: a single open problem settled. The problem page records the solution, and Tao's wiki catalogs it among many AI contributions" } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "ratings" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "availability": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "brief": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "case_kind": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "case_links[]": { "state": "value_present", "value": { "kind": "structured", "data": { "answer": "Same developer, OpenAI, and same stakeholder, the research record: the nonsense term “vegetative electron microscopy” entered published papers via AI training contamination that GPT-3 and GPT-4 reproduce, harming researchers, publishers and the scientific record.", "links": [ { "database": "AI Incident Database", "identifier": "1044", "title": "Reported Emergence of 'Vegetative Electron Microscopy' in Scientific Papers Traced to Purported AI Training Data Contamination", "url": "https://incidentdatabase.ai/cite/1044", "group": "societal", "basis": "Same developer, OpenAI, and same stakeholder, the research record: the nonsense term “vegetative electron microscopy” entered published papers via AI training contamination that GPT-3 and GPT-4 reproduce, harming researchers, publishers and the scientific record.", "relation": "comparable_harm" } ] } }, "provenance": { "kind": "legacy_record", "basis": "Existing paired_incident classification; no new source review.", "record_fields": [ "paired_incident" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "claims[].evidence_links": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "claims[].evidence_strength": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "claims[].missingness": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "claims[].proposition": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "cross_border_flows[]": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "decisions[].action_outcome": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "institutional decisions" }, "decisions[].grounds": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "institutional decisions" }, "decisions[].institution_task": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "institutional decisions" }, "decisions[].success_criterion": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "institutional decisions" }, "deployment_stage": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "economics[].amount": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].costs[]": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].counterfactual": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].currency_price_basis": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].gross_net": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].measure_type": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].period_unit": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].realization": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].recipient_payer": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].return_payback": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "economics[].valuation_method": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "monetary observations" }, "evaluations[].ai_uplift": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].comparator": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].design": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].effect": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].exposure": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].id": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].measurement_method": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].metric": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].reproduction_materials": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].setting_sample": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "evaluations[].uncertainty_limitations": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "evaluations" }, "external_references[]": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "paired_incident": { "database": "AI Incident Database", "identifier": "1044", "title": "Reported Emergence of 'Vegetative Electron Microscopy' in Scientific Papers Traced to Purported AI Training Data Contamination", "url": "https://incidentdatabase.ai/cite/1044", "group": "societal", "basis": "Same developer, OpenAI, and same stakeholder, the research record: the nonsense term “vegetative electron microscopy” entered published papers via AI training contamination that GPT-3 and GPT-4 reproduce, harming researchers, publishers and the scientific record." } } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "paired_incident" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "follow_up[]": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "later": { "status": "persisted", "date": "2026-05-01", "note": "Six days after the solution was posted, the Aristotle system autoformalized the argument in Lean and a forum checker confirmed that the formalization matches the write-up; Thomas Bloom added an alternative sketch the next day, and Tao's wiki (data to 30 June 2026) lists the problem as a full AI solution.", "quote": "I have confirmed the Lean matches the paper and that both use the asymptotic additive basis definition", "url": "https://www.erdosproblems.com/forum/thread/38" } } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "later" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "human_oversight": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "id": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "id": "AIBD-0075" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "id" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "infrastructure_dependencies[]": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].beneficiaries": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].beneficiary_count": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].benefit_type": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "benefit_type": "Improved Knowledge Discovery" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "benefit_type" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "outcomes[].claim": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "account": "Erdős problem #38 is an open question in additive number theory. It asks for a set B that is not an additive basis yet raises the Schnirelmann density of every set A. OpenAI's GPT-5.5 Pro solved it. On 25 April 2026 a user (gebyjaff) posted the model's solution on the erdosproblems.com forum, with a PDF cleanup by Liam Price. On 26 April forum commenter Nat Sothanaphan reported that a standard check found no issues in the write-up. On 1 May the Aristotle system autoformalized the argument in Lean, which Sothanaphan confirmed matches the paper. Site maintainer Thomas Bloom added an alternative sketch on 2 May 2026. The problem page now records the positive solution by GPT 5.5 Pro. Terence Tao's wiki lists it as an AI-standalone full solution.", "benefit_type": "Improved Knowledge Discovery" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "account", "benefit_type" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "outcomes[].distribution": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].duration": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].id": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].stakeholder_group": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "ratings": [ { "group": "organizational", "depth": 1, "reason": "Small: one more entry resolved in the erdosproblems.com project, accepted by its maintainer" }, { "group": "societal", "depth": 1, "reason": "Small: a single open problem settled. The problem page records the solution, and Tao's wiki catalogs it among many AI contributions" } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "ratings" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "outcomes[].status": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].tradeoffs": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "outcomes[].unexpected": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "unexpected": false, "purpose": "General-purpose reasoning model prompted to solve an open research problem" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "unexpected", "purpose" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "places[]": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "developer_country": "United States" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "developer_country" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "purpose": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "purpose": "General-purpose reasoning model prompted to solve an open research problem" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "purpose" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "record_dates": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "date_observed": "2026-04-25", "revisions": [ { "date": "2026-10-20", "change": "Record drafted from the cited sources.", "editor": "AIBD" }, { "date": "2026-10-20", "change": "Paired with AI Incident Database incident 1044 (medium confidence); severity not given by the source.", "source": "https://incidentdatabase.ai/cite/1044", "editor": "AIBD" }, { "date": "2026-10-20", "change": "Dropped Harmonic as Aristotle's developer because no cited source names it, noted that the OpenAI attribution rests on the ChatGPT share link, called Nat Sothanaphan a forum commenter rather than a reviewer, and removed the unsupported “human-verified” and “Lean-verified” labels from the two wiki echoes while adding the 20 May 2026 OpenAI internal model entry for #90.", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Added what happened since from the cited sources.", "source": "https://www.erdosproblems.com/forum/thread/38", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Country of the developer added from the organization's own site or a registry.", "source": "https://en.wikipedia.org/wiki/OpenAI", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Prose pass: a rating reason, account, the replication conditions, the system's part rewritten in shorter sentences with abbreviations expanded; facts, figures and quotations unchanged.", "editor": "AIBD" } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "date_observed", "revisions" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "replication.attempts[]": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "replication": { "produced_by": [ "A user prompting GPT-5.5 Pro with the problem statement", "Human cleanup of the written proof", "Independent human check and Lean autoformalization by Aristotle", "Acceptance by the site maintainer" ], "must_hold": [ "The problem has a precise, formalized statement", "The proof is checkable by humans or in Lean", "A community venue reviews and records AI-produced solutions" ], "echoes": [ { "date": "2026-05-01", "setting": "Erdős problem #694, GPT-5.5 Pro, full solution listed in Tao's wiki section 1(a)", "model_version": "GPT-5.5 Pro", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" }, { "date": "2026-05-26", "setting": "Erdős problem #90, Claude Mythos, full solution listed in Tao's wiki section 1(a); the wiki also lists a full solution of #90 by an OpenAI internal model on 20 May 2026", "model_version": "Claude Mythos", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" } ] } } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "replication" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "replication.conditions": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "replication": { "produced_by": [ "A user prompting GPT-5.5 Pro with the problem statement", "Human cleanup of the written proof", "Independent human check and Lean autoformalization by Aristotle", "Acceptance by the site maintainer" ], "must_hold": [ "The problem has a precise, formalized statement", "The proof is checkable by humans or in Lean", "A community venue reviews and records AI-produced solutions" ], "echoes": [ { "date": "2026-05-01", "setting": "Erdős problem #694, GPT-5.5 Pro, full solution listed in Tao's wiki section 1(a)", "model_version": "GPT-5.5 Pro", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" }, { "date": "2026-05-26", "setting": "Erdős problem #90, Claude Mythos, full solution listed in Tao's wiki section 1(a); the wiki also lists a full solution of #90 by an OpenAI internal model on 20 May 2026", "model_version": "Claude Mythos", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" } ] } } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "replication" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "revision_history[]": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "revisions": [ { "date": "2026-10-20", "change": "Record drafted from the cited sources.", "editor": "AIBD" }, { "date": "2026-10-20", "change": "Paired with AI Incident Database incident 1044 (medium confidence); severity not given by the source.", "source": "https://incidentdatabase.ai/cite/1044", "editor": "AIBD" }, { "date": "2026-10-20", "change": "Dropped Harmonic as Aristotle's developer because no cited source names it, noted that the OpenAI attribution rests on the ChatGPT share link, called Nat Sothanaphan a forum commenter rather than a reviewer, and removed the unsupported “human-verified” and “Lean-verified” labels from the two wiki echoes while adding the 20 May 2026 OpenAI internal model entry for #90.", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Added what happened since from the cited sources.", "source": "https://www.erdosproblems.com/forum/thread/38", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Country of the developer added from the organization's own site or a registry.", "source": "https://en.wikipedia.org/wiki/OpenAI", "editor": "AIBD" }, { "date": "2026-10-21", "change": "Prose pass: a rating reason, account, the replication conditions, the system's part rewritten in shorter sentences with abbreviations expanded; facts, figures and quotations unchanged.", "editor": "AIBD" } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "revisions" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "sector": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "domain": "Mathematics", "segment": "Research mathematics (additive number theory)" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "domain", "segment" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "sources[].access_version": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "sources[].citation_locator": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "evidence": [ { "kind": "other", "citation": "Bloom, T. (ed.) (2026). Erdős Problem #38. erdosproblems.com.", "title": "Erdős Problem #38", "url": "https://www.erdosproblems.com/38", "date": "2026-05-02", "written_by": "independent", "quote": "A positive solution was given by GPT 5.5 Pro (prompted by gebyjaff) in the comments; in fact a sparse random set B has this property" }, { "kind": "other", "citation": "erdosproblems.com forum (2026). Thread for problem 38. erdosproblems.com.", "title": "Forum thread: Problem 38", "url": "https://www.erdosproblems.com/forum/thread/38", "date": "2026-04-26", "written_by": "independent", "quote": "Standard check found no issues in the writeup" }, { "kind": "other", "citation": "Tao, T. et al. (2026). AI contributions to Erdős problems. GitHub wiki, teorth/erdosproblems.", "title": "AI contributions to Erdős problems", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "written_by": "independent", "quote": "This AI Wiki collects contributions of AI systems to the understanding of Erdős problems." } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "evidence" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "sources[].id": { "state": "value_present", "value": { "kind": "identifiers", "identifiers": [ "AIBD-0075-source-0001", "AIBD-0075-source-0002", "AIBD-0075-source-0003" ], "role": "Existing evidence items, in their original order" }, "provenance": { "kind": "curatorial", "basis": "Stable within the fingerprinted record version; no source content or source review has been added.", "record_fields": [], "source_refs": [] }, "review_state": "record_mapped", "coverage": "complete" }, "sources[].language": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "sources[].relationship": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "evidence": [ { "kind": "other", "citation": "Bloom, T. (ed.) (2026). Erdős Problem #38. erdosproblems.com.", "title": "Erdős Problem #38", "url": "https://www.erdosproblems.com/38", "date": "2026-05-02", "written_by": "independent", "quote": "A positive solution was given by GPT 5.5 Pro (prompted by gebyjaff) in the comments; in fact a sparse random set B has this property" }, { "kind": "other", "citation": "erdosproblems.com forum (2026). Thread for problem 38. erdosproblems.com.", "title": "Forum thread: Problem 38", "url": "https://www.erdosproblems.com/forum/thread/38", "date": "2026-04-26", "written_by": "independent", "quote": "Standard check found no issues in the writeup" }, { "kind": "other", "citation": "Tao, T. et al. (2026). AI contributions to Erdős problems. GitHub wiki, teorth/erdosproblems.", "title": "AI contributions to Erdős problems", "url": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems", "written_by": "independent", "quote": "This AI Wiki collects contributions of AI systems to the understanding of Erdős problems." } ] } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "evidence" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "sources[].upstream_sources": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "strategic_assessments[]": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial", "family": "strategic assessments" }, "summary": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "summary": "GPT-5.5 Pro solved Erdős problem #38; human-checked 26 April 2026, Lean-formalized by Aristotle 1 May, recorded as solved on erdosproblems.com", "account": "Erdős problem #38 is an open question in additive number theory. It asks for a set B that is not an additive basis yet raises the Schnirelmann density of every set A. OpenAI's GPT-5.5 Pro solved it. On 25 April 2026 a user (gebyjaff) posted the model's solution on the erdosproblems.com forum, with a PDF cleanup by Liam Price. On 26 April forum commenter Nat Sothanaphan reported that a standard check found no issues in the write-up. On 1 May the Aristotle system autoformalized the argument in Lean, which Sothanaphan confirmed matches the paper. Site maintainer Thomas Bloom added an alternative sketch on 2 May 2026. The problem page now records the positive solution by GPT 5.5 Pro. Terence Tao's wiki lists it as an AI-standalone full solution." } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "summary", "account" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "systems[].autonomy_permissions": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "systems[].component_role": { "state": "value_present", "value": { "kind": "text", "text": "GPT-5.5 Pro generated a proof, and Aristotle converted it into Lean for verification." }, "provenance": { "kind": "legacy_record", "basis": "Restated the mechanical task already described in the existing system and AI-role fields. No new source review is claimed.", "record_fields": [ "system", "ai_role" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "systems[].deployment_configuration": { "state": "value_present", "value": { "kind": "text", "text": "The reported configuration was GPT-5.5 Pro." }, "provenance": { "kind": "legacy_record", "basis": "Separated the model, software, hardware, or operation already described in the existing record. No new source review is claimed.", "record_fields": [ "model_version", "system", "ai_role" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "systems[].inputs_outputs": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "systems[].model_name": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "systems[].model_version": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "model_version": "GPT-5.5 Pro" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "model_version" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial", "display": { "answer": "Exact release not established.", "editor_note": "" } }, "systems[].product_name": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "system": "GPT-5.5 Pro, with Aristotle used for Lean autoformalisation" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "system" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "systems[].provider": { "state": "value_present", "value": { "kind": "structured", "data": { "application_developer": "OpenAI (GPT-5.5 Pro); Aristotle (developer not named in the cited sources)" } }, "provenance": { "kind": "legacy_record", "basis": "Mapped from the existing case description without a new external source review.", "record_fields": [ "developer" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "taxonomy_crosswalks[]": { "state": "not_assessed", "review_state": "not_assessed", "coverage": "partial" }, "title": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "title": "GPT-5.5 Pro solves Erdős problem #38; solution human-checked, Lean-formalized and recorded as solved on erdosproblems.com", "headline": "GPT-5.5 Pro solves Erdős problem #38; solution Lean-formalized" } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "title", "headline" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" }, "version": { "state": "value_present", "value": { "kind": "identifiers", "identifiers": [ "a937c036c24fcfe7fb4a28e5da6f1ecb04e3d8447e3e51b0ba66b44be46e3ed4" ], "role": "SHA-256 of normalized original case data" }, "provenance": { "kind": "curatorial", "basis": "Deterministic content version assigned by the local migration.", "record_fields": [], "source_refs": [] }, "review_state": "record_mapped", "coverage": "complete" }, "workflow": { "state": "value_present", "value": { "kind": "record_fields", "fields": { "account": "Erdős problem #38 is an open question in additive number theory. It asks for a set B that is not an additive basis yet raises the Schnirelmann density of every set A. OpenAI's GPT-5.5 Pro solved it. On 25 April 2026 a user (gebyjaff) posted the model's solution on the erdosproblems.com forum, with a PDF cleanup by Liam Price. On 26 April forum commenter Nat Sothanaphan reported that a standard check found no issues in the write-up. On 1 May the Aristotle system autoformalized the argument in Lean, which Sothanaphan confirmed matches the paper. Site maintainer Thomas Bloom added an alternative sketch on 2 May 2026. The problem page now records the positive solution by GPT 5.5 Pro. Terence Tao's wiki lists it as an AI-standalone full solution.", "replication": { "produced_by": [ "A user prompting GPT-5.5 Pro with the problem statement", "Human cleanup of the written proof", "Independent human check and Lean autoformalization by Aristotle", "Acceptance by the site maintainer" ], "must_hold": [ "The problem has a precise, formalized statement", "The proof is checkable by humans or in Lean", "A community venue reviews and records AI-produced solutions" ], "echoes": [ { "date": "2026-05-01", "setting": "Erdős problem #694, GPT-5.5 Pro, full solution listed in Tao's wiki section 1(a)", "model_version": "GPT-5.5 Pro", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" }, { "date": "2026-05-26", "setting": "Erdős problem #90, Claude Mythos, full solution listed in Tao's wiki section 1(a); the wiki also lists a full solution of #90 by an OpenAI internal model on 20 May 2026", "model_version": "Claude Mythos", "outcome": "recurred", "source": "https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems" } ] } } }, "provenance": { "kind": "legacy_record", "basis": "Preserved from the existing AIBD record; external sources were not rechecked.", "record_fields": [ "account", "replication" ], "source_refs": [] }, "review_state": "record_mapped", "coverage": "partial" } }, "summary": { "total": 91, "addressed": 91, "known": 34, "reviewed": 0, "record_mapped": 34, "not_assessed": 57, "complete": false, "by_state": { "value_present": 34, "not_reported": 0, "unresolved": 0, "not_assessed": 57, "not_applicable": 0, "withheld": 0, "source_unavailable": 0 } } } }