{
  "schema_version": 1,
  "id": "evt-2026-claude-riemann-lower-bound",
  "slug": "claude-riemann-zeta-lower-bound",
  "title": {
    "en": "Claude improves a longstanding lower bound related to the Riemann hypothesis",
    "zh-CN": "Claude 改进与黎曼猜想相关的长期下界"
  },
  "dates": {
    "event": "2026-08-10",
    "first_public_claim": "2026-08-10",
    "paper_release": null,
    "artifact_release": null,
    "independently_verified": "2026-08-11",
    "added_to_chronicle": "2026-09-10",
    "last_updated": "2026-09-10"
  },
  "event_types": [
    "discovery",
    "proof",
    "formalization",
    "system"
  ],
  "significance": {
    "tier": "H1",
    "rationale": {
      "en": "A concrete AI-generated improvement to a longstanding analytic-number-theory bound is a historical AI4Math milestone because it contributes new mathematics rather than merely demonstrating benchmark competence; H1 does not replace the separate verification state.",
      "zh-CN": "AI 对解析数论中长期下界给出具体改进，属于历史级 AI4Math 里程碑，因为它贡献的是新的数学结果，而非仅展示基准能力；H1 不替代独立的 verification 状态。"
    }
  },
  "mathematical_novelty": {
    "type": "new_result"
  },
  "interfaces": [
    "informal",
    "formal",
    "symbolic",
    "hybrid"
  ],
  "claim": {
    "en": "Anthropic reported that an unreleased Claude research model raised the known lower bound for the fraction of Riemann-zeta zeros on the critical line from 41.6% to 67.2%.",
    "zh-CN": "Anthropic 报告一个未发布 Claude 研究模型把已知的“位于临界线上的黎曼 zeta 零点比例下界”从 41.6% 提升到 67.2%。"
  },
  "summary": {
    "en": "Claude coordinated dozens of subagents, searched literature and numerical evidence, and produced a paper; Anthropic mathematicians reviewed the result and a Lean formalization passed comparator.",
    "zh-CN": "Claude 协调数十个子 Agent、检索文献并进行数值检查，随后生成论文；Anthropic 数学家审核结果，Lean 形式化通过 comparator。"
  },
  "why_it_matters": {
    "en": "It is a notable example of autonomous mathematical exploration yielding an unexpected publishable result while failing at the original, much harder Riemann-hypothesis objective.",
    "zh-CN": "这是自主数学探索产生意外可发表结果的代表案例，同时也清楚显示模型并未解决最初更困难的黎曼猜想。"
  },
  "ai_role": {
    "level": "autonomous_primary",
    "description": {
      "en": "Claude generated and explored thousands of ideas with subagents, found the core argument and produced a formalization workflow.",
      "zh-CN": "Claude 通过多个子 Agent 生成并探索大量想法，找到核心论证并参与形式化流程。"
    }
  },
  "human_contribution": {
    "description": {
      "en": "Anthropic mathematicians reviewed the result, external experts examined it, and staff assisted formalization and publication.",
      "zh-CN": "Anthropic 数学家审核结果，外部专家进行审阅，工作人员协助形式化与发布。"
    }
  },
  "systems": [
    "claude-research",
    "lean"
  ],
  "organizations": [
    "anthropic"
  ],
  "people": [],
  "problems": [
    "riemann-hypothesis-related-zero-density"
  ],
  "methods": [
    "multi-agent",
    "literature-search",
    "analytic-number-theory",
    "formalization"
  ],
  "verification": {
    "evidence_level": "E3",
    "status": "independently_verified",
    "formal_assurance": "independently_replayed",
    "history": [
      {
        "date": "2026-08-10",
        "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": "2026-08-11",
        "status": "partially_verified",
        "evidence_level": "E3",
        "note": {
          "en": "Riemann Labs independently rebuilt the public Lean formalization in its own CI and reported successful statement-equality checks plus replay under the Lean default kernel and nanoda. This supports `independently_replayed` formal assurance, while broader mathematical peer review remains distinct and the Chronicle does not yet assign `independently_verified` status.",
          "zh-CN": "Riemann Labs 在其独立 CI 中重新构建公开 Lean 形式化，并报告 statement-equality 检查以及 Lean 默认内核与 nanoda 重放均通过。这支持将 formal assurance 记为 `independently_replayed`；更广泛的数学同行评审仍是另一层问题，Chronicle 目前不授予 `independently_verified` 状态。"
        }
      },
      {
        "date": "2026-09-10",
        "status": "independently_verified",
        "evidence_level": "E3",
        "note": {
          "en": "Under the Chronicle's event-level verification semantics, Riemann Labs' independent rebuild, statement comparison and dual-kernel replay substantively corroborate the core event description and released formal artifact. This status does not mean the Chronicle independently adjudicates the ultimate correctness, priority or broader mathematical significance of the underlying research claim.",
          "zh-CN": "按 Chronicle 的事件级 verification 语义，Riemann Labs 的独立重建、statement comparison 与双内核重放已经对该事件的核心描述和公开形式化产物形成实质性独立佐证。该状态不表示 Chronicle 独立裁决底层研究主张的最终正确性、优先权或更广泛数学意义。"
        }
      }
    ]
  },
  "sources": [
    {
      "id": "anthropic-riemann-zeta-2026",
      "tier": "S2",
      "type": "official_announcement",
      "title": "Learning more about Claude's mathematical capabilities",
      "url": "https://www.anthropic.com/research/riemann-zeta",
      "primary": true,
      "published_date": "2026-08-10",
      "accessed_at": "2026-09-10"
    },
    {
      "id": "anthropic-zeta-23-lean",
      "tier": "S1",
      "type": "formal_artifact",
      "title": "Zeta23 Lean 4 formalization",
      "url": "https://github.com/anthropics/zeta-23-lean",
      "primary": true,
      "published_date": null,
      "accessed_at": "2026-09-10"
    },
    {
      "id": "riemann-labs-zeta-independent-replay",
      "tier": "S3",
      "type": "independent_analysis",
      "title": "Independent Verification — More Than Two Thirds of the Zeros of the Riemann Zeta Function Lie on the Critical Line",
      "url": "https://github.com/primaryhosting/zeta-two-thirds-verification",
      "primary": false,
      "published_date": "2026-08-11",
      "accessed_at": "2026-09-10"
    }
  ],
  "artifacts": [
    {
      "id": "anthropic-zeta-23-lean-artifact",
      "kind": "proof",
      "title": "Lean 4 formalization of the zeta lower-bound result",
      "url": "https://github.com/anthropics/zeta-23-lean"
    },
    {
      "id": "riemann-labs-zeta-replay",
      "kind": "supplement",
      "title": "Independent rebuild and dual-kernel replay evidence",
      "url": "https://github.com/primaryhosting/zeta-two-thirds-verification"
    }
  ],
  "relationships": {
    "predecessors": [
      "evt-2026-first-proof-openai",
      "evt-2026-frontiermath-open-problems"
    ],
    "successors": [],
    "related": []
  },
  "timeline": {
    "featured": true,
    "significance": "primary"
  },
  "tags": [
    "riemann-hypothesis",
    "number-theory",
    "claude",
    "multi-agent"
  ],
  "editorial": {
    "created_at": "2026-09-10",
    "last_reviewed": "2026-09-11",
    "reviewers": [
      "Charlie-Wang-03"
    ]
  },
  "corrections": []
}