{
  "schema_version": 1,
  "id": "evt-2026-unit-distance-disproof",
  "slug": "openai-unit-distance-conjecture-disproof",
  "title": {
    "en": "OpenAI reports an autonomous disproof of Erdős's unit-distance growth conjecture",
    "zh-CN": "OpenAI 报告 AI 自主反驳 Erdős 单位距离增长猜想"
  },
  "dates": {
    "event": "2026-05-20",
    "first_public_claim": "2026-05-20",
    "paper_release": "2026-05-20",
    "artifact_release": null,
    "independently_verified": "2026-05-20",
    "added_to_chronicle": "2026-09-10",
    "last_updated": "2026-09-10"
  },
  "event_types": [
    "discovery",
    "proof",
    "system"
  ],
  "significance": {
    "tier": "H1",
    "rationale": {
      "en": "A substantive AI-generated disproof of a longstanding Erdős conjecture is a historical AI4Math milestone because the central contribution is new mathematics rather than benchmark performance or a general model release; the H1 designation does not itself assert independent verification.",
      "zh-CN": "AI 对长期存在的 Erdős 猜想给出实质性新反例属于历史级 AI4Math 里程碑，因为核心贡献是新的数学结果，而非基准成绩或通用模型发布；H1 定级本身不等同于已完成独立验证。"
    }
  },
  "mathematical_novelty": {
    "type": "new_result"
  },
  "interfaces": [
    "informal",
    "symbolic"
  ],
  "claim": {
    "en": "OpenAI reported that an internal general reasoning model disproved the longstanding conjectured n^(1+o(1)) upper-growth behavior for planar unit distances by constructing infinitely many configurations with at least n^(1+delta) unit distances for fixed delta>0.",
    "zh-CN": "OpenAI 报告内部通用推理模型反驳了平面单位距离问题长期流行的 n^(1+o(1)) 增长猜想，构造出无穷多个具有至少 n^(1+delta) 个单位距离的点集，其中 delta>0 为常数。"
  },
  "summary": {
    "en": "The model found a construction using tools from algebraic number theory; OpenAI released the proof and a companion paper by external mathematicians who checked and contextualized the argument.",
    "zh-CN": "模型使用代数数论工具发现构造；OpenAI 同时发布证明与由外部数学家撰写的 companion paper，后者检查并解释了论证。"
  },
  "why_it_matters": {
    "en": "Unlike benchmark progress, the event concerns a longstanding open problem in active mathematics and an AI-generated argument judged substantive by independent experts.",
    "zh-CN": "与基准进展不同，该事件涉及活跃数学领域的长期开放问题，且 AI 生成论证获得外部专家的实质性认可。"
  },
  "ai_role": {
    "level": "autonomous_primary",
    "description": {
      "en": "OpenAI states that the core construction and proof were generated autonomously by an internal general-purpose reasoning model.",
      "zh-CN": "OpenAI 表示核心构造与证明由内部通用推理模型自主生成。"
    }
  },
  "human_contribution": {
    "description": {
      "en": "External mathematicians checked the proof and wrote companion analysis; OpenAI researchers ran the evaluation and publication process.",
      "zh-CN": "外部数学家检查证明并撰写 companion analysis；OpenAI 研究者组织评测与发布。"
    }
  },
  "systems": [
    "openai-internal-reasoning-model"
  ],
  "organizations": [
    "openai"
  ],
  "people": [],
  "problems": [
    "erdos-unit-distance-growth-conjecture"
  ],
  "methods": [
    "algebraic-number-theory",
    "autonomous-research",
    "proof-generation"
  ],
  "verification": {
    "evidence_level": "E3",
    "status": "independently_verified",
    "formal_assurance": "artifact_available",
    "history": [
      {
        "date": "2026-05-20",
        "status": "partially_verified",
        "evidence_level": "E3",
        "note": {
          "en": "The Chronicle records this event from the cited primary or official source; the status does not imply independent Chronicle verification.",
          "zh-CN": "Chronicle 依据所引 primary / official source 记录该事件；该状态不表示 Chronicle 已完成独立验证。"
        }
      },
      {
        "date": "2026-05-20",
        "status": "partially_verified",
        "evidence_level": "E3",
        "note": {
          "en": "OpenAI released the AI-generated proof together with a companion paper by nine external mathematicians describing a human-verified, digested version of the argument. Will Sawin independently published a stronger explicit lower bound building on the new construction. These sources materially strengthen the mathematical evidence while the Chronicle retains the human editorial gate for `independently_verified`.",
          "zh-CN": "OpenAI 同日公开 AI 生成证明及由九位外部数学家撰写的 companion paper，后者给出经人工核查和整理的论证版本；Will Sawin 还独立发表了建立在该新构造之上的更强显式下界。这些来源显著加强了数学证据，但 Chronicle 仍保留 `independently_verified` 的人工编辑门槛。"
        }
      },
      {
        "date": "2026-09-10",
        "status": "independently_verified",
        "evidence_level": "E3",
        "note": {
          "en": "Under the Chronicle's event-level verification semantics, the nine-mathematician human-verified companion paper and Will Sawin's independent follow-up work substantively corroborate the core event description. This label records independent corroboration of the event and its public evidence trail rather than a Chronicle adjudication of ultimate mathematical correctness.",
          "zh-CN": "按 Chronicle 的事件级 verification 语义，九位数学家的 human-verified companion paper 与 Will Sawin 的独立后续工作已经对该事件核心描述形成实质性独立佐证。该标签记录的是事件与公开证据链获得独立确认，而不是 Chronicle 对底层数学结论最终正确性的裁决。"
        }
      }
    ]
  },
  "sources": [
    {
      "id": "openai-unit-distance-2026",
      "tier": "S2",
      "type": "official_announcement",
      "title": "An OpenAI model has disproved a central conjecture in discrete geometry",
      "url": "https://openai.com/index/model-disproves-discrete-geometry-conjecture/",
      "primary": true,
      "published_date": "2026-05-20",
      "accessed_at": "2026-09-10"
    },
    {
      "id": "openai-unit-distance-proof-2026",
      "tier": "S1",
      "type": "paper",
      "title": "Planar Point Sets with Many Unit Distances",
      "url": "https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf",
      "primary": true,
      "published_date": "2026-05-20",
      "accessed_at": "2026-09-10"
    },
    {
      "id": "unit-distance-remarks-2026",
      "tier": "S3",
      "type": "paper",
      "title": "Remarks on the disproof of the unit distance conjecture",
      "url": "https://arxiv.org/abs/2605.20695",
      "primary": false,
      "published_date": "2026-05-20",
      "accessed_at": "2026-09-10"
    },
    {
      "id": "sawin-unit-distance-explicit-bound-2026",
      "tier": "S3",
      "type": "paper",
      "title": "An explicit lower bound for the unit distance problem",
      "url": "https://arxiv.org/abs/2605.20579",
      "primary": false,
      "published_date": "2026-05-20",
      "accessed_at": "2026-09-10"
    }
  ],
  "artifacts": [
    {
      "id": "openai-unit-distance-proof-pdf",
      "kind": "proof",
      "title": "AI-generated proof manuscript",
      "url": "https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf"
    },
    {
      "id": "unit-distance-human-verified-remarks",
      "kind": "supplement",
      "title": "Human-verified companion remarks",
      "url": "https://arxiv.org/abs/2605.20695"
    }
  ],
  "relationships": {
    "predecessors": [
      "evt-2026-first-proof-openai",
      "evt-2025-alphaevolve"
    ],
    "successors": [],
    "related": []
  },
  "timeline": {
    "featured": true,
    "significance": "primary"
  },
  "tags": [
    "open-problem",
    "discrete-geometry",
    "erdos",
    "autonomous-research"
  ],
  "editorial": {
    "created_at": "2026-09-10",
    "last_reviewed": "2026-09-11",
    "reviewers": [
      "Charlie-Wang-03"
    ]
  },
  "corrections": []
}