{ "schema_version": 1, "id": "evt-2026-claude-3sum-apsp", "slug": "claude-3sum-apsp-polynomial-speedup", "title": { "en": "Claude is credited with discovering truly subquadratic 3SUM and truly subcubic APSP algorithms", "zh-CN": "Claude 被归因于发现真正次二次 3SUM 与真正次三次 APSP 算法" }, "dates": { "event": "2026-10-05", "first_public_claim": "2026-10-05", "paper_release": "2026-10-05", "artifact_release": null, "independently_verified": null, "added_to_chronicle": "2026-10-07", "last_updated": "2026-10-07" }, "event_types": [ "discovery", "proof", "formalization" ], "significance": { "tier": "H1", "rationale": { "en": "This result is a Historical Milestone because it reports the first polynomial exponent improvements for 3SUM and APSP, two problems whose conjectured near-quadratic and near-cubic barriers structured fine-grained complexity for decades. The paper attributes the core algorithmic discovery to an Anthropic research model operating in a long session without human input, after which Josh Alman and Virginia Vassilevska Williams independently understood, simplified, strengthened and extended the method. Anthropic subsequently released a substantial Lean formalization of the principal deterministic claims.", "zh-CN": "该结果被列为 Historical Milestone，因为它报告了 3SUM 与 APSP 的首次多项式级 指数改进，而这两个问题长期以来的近二次与近三次复杂度假设塑造了 fine-grained complexity 的研究框架。论文把核心算法发现归因于一个 Anthropic 研究模型在无人类 输入的长程 session 中完成；随后 Josh Alman 与 Virginia Vassilevska Williams 独立理解、简化、强化并扩展该方法。Anthropic 又公开了主要 deterministic claims 的大规模 Lean 形式化。" } }, "mathematical_novelty": { "type": "new_result" }, "interfaces": [ "informal", "formal", "symbolic", "hybrid" ], "claim": { "en": "Alman and Vassilevska Williams report deterministic algorithms solving 3SUM on polynomial-size integer inputs in O(n^1.9992) time and APSP with polynomially bounded integer weights in O(n^2.99942) time in their word-RAM formulation, together with related improvements for Exact Triangle, min-plus product and zero-weight k-Clique. The paper states that these are the first truly subquadratic and truly subcubic algorithms in the corresponding settings and that they refute the standard 3SUM and APSP hypotheses used in fine-grained complexity.", "zh-CN": "Alman 与 Vassilevska Williams 报告 deterministic algorithms：在其 word-RAM 模型中，对多项式大小整数输入的 3SUM 达到 O(n^1.9992)，对多项式有界整数权重的 APSP 达到 O(n^2.99942)，并同时改进 Exact Triangle、min-plus product 与 zero-weight k-Clique。论文称这些是在相应设定下首次真正次二次与真正次三次算法， 并由此推翻 fine-grained complexity 中常用的 3SUM 与 APSP hypotheses。" }, "summary": { "en": "The paper's methodology section says an Anthropic employee initially asked an internal research model to study a cryptographic question based on average-case Zero-k-Clique. Claude unexpectedly developed a new algorithm, first for the average-case setting and then for the worst case, during a session producing about 16 million output tokens with no human input. Anthropic shared the result with Alman and Vassilevska Williams, who reconstructed the argument, simplified it, improved constants and exponents, derived additional consequences and wrote the research paper. After the paper was complete, Anthropic used a research model to formalize major deterministic claims in Lean.", "zh-CN": "论文的方法部分说明，Anthropic 员工最初让内部研究模型研究一个基于 average-case Zero-k-Clique 的密码学问题。Claude 在约 1,600 万 output tokens、无人类输入的 session 中意外发展出新算法，先处理平均情形，再推广到最坏情形。Anthropic 将结果 提供给 Alman 与 Vassilevska Williams；两位作者随后重新理解论证、简化方法、 改进常数与指数、推出更多后果并撰写论文。论文完成后，Anthropic 又使用研究模型 把主要 deterministic claims 形式化到 Lean 中。" }, "why_it_matters": { "en": "This event provides unusually explicit provenance for a frontier mathematical algorithm originating in a long autonomous model run and then surviving expert human reconstruction. The result does not merely improve a benchmark: it changes the conjectural landscape of fine-grained complexity and forces reconsideration of reductions and hardness assumptions built on 3SUM and APSP. The accompanying Lean artifact further makes a subset of the headline claims mechanically inspectable.", "zh-CN": "本事件为“前沿数学算法由长程自主模型运行产生、随后经专家重新理解”的过程提供了罕见的 明确 provenance。它不是 benchmark 提升，而是改变 fine-grained complexity 的 猜想格局，并迫使研究者重新审视建立在 3SUM 与 APSP 假设上的归约和困难性框架。 配套 Lean artifact 还使部分 headline claims 可以接受机械检查。" }, "ai_role": { "level": "ai_primary_human_verified", "description": { "en": "The paper attributes the core algorithmic discovery to an Anthropic research model in a 16-million-output-token session with no human input. Human authors then independently understood the method, simplified and strengthened it, derived new consequences and took responsibility for the final paper. This classification applies to the reported discovery workflow, not to every theorem and exposition choice in the final manuscript.", "zh-CN": "论文把核心算法发现归因于 Anthropic 研究模型在约 1,600 万 output tokens 且 无人类输入的 session 中完成。人类作者随后独立理解该方法、简化并强化论证、 推出新的后果，并对最终论文负责。该分类描述的是公开披露的发现流程，并不表示 最终论文中的每个定理和阐释选择都由模型完成。" } }, "human_contribution": { "description": { "en": "Josh Alman and Virginia Vassilevska Williams reconstructed the model's algorithm, simplified and strengthened it, improved quantitative bounds, developed further consequences and wrote the paper. Anthropic researchers supplied the model run and later formalization infrastructure.", "zh-CN": "Josh Alman 与 Virginia Vassilevska Williams 重新构建模型算法、简化并强化方法、 改进定量界、发展进一步后果并撰写论文；Anthropic 研究人员提供模型运行以及后续 形式化基础设施。" } }, "systems": [ "claude", "lean", "comparator", "nanoda" ], "organizations": [ "anthropic" ], "people": [ "josh-alman", "virginia-vassilevska-williams" ], "problems": [ "three-sum", "all-pairs-shortest-paths" ], "methods": [ "fine-grained-algorithms", "sparse-lopsided-triangles", "word-ram", "formal-verification" ], "verification": { "evidence_level": "E3", "status": "under_verification", "formal_assurance": "machine_checked", "history": [ { "date": "2026-10-05", "status": "paper_released", "evidence_level": "E2", "note": { "en": "The research paper publicly states the algorithmic results and gives an explicit account of the AI discovery workflow and subsequent human reconstruction.", "zh-CN": "研究论文公开陈述算法结果，并明确记录 AI 发现流程及随后的人类重建过程。" } }, { "date": "2026-10-07", "status": "under_verification", "evidence_level": "E3", "note": { "en": "Anthropic's public Lean project proves five headline deterministic claims, including the 3SUM and APSP word-RAM bounds, and documents Comparator/Nanoda checking procedures. The repository also explicitly lists paper claims that are not formalized in the same machine model. Chronicle has not located an independent external replay or specialist review sufficient for independently_verified status.", "zh-CN": "Anthropic 的公开 Lean 项目证明五项 headline deterministic claims，其中包括 3SUM 与 APSP 的 word-RAM 界，并记录 Comparator/Nanoda 检查流程。仓库也明确 列出未在同一机器模型中形式化的论文主张。Chronicle 尚未找到足以支持 independently_verified 状态的外部独立重放或专业审查。" } } ] }, "sources": [ { "id": "alman-vw-3sum-apsp-2026", "tier": "S1", "type": "paper", "title": "Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs", "url": "https://arxiv.org/abs/2610.06783", "primary": true, "published_date": "2026-10-05", "accessed_at": "2026-10-07" }, { "id": "anthropic-3sum-apsp-formalization", "tier": "S1", "type": "formal_artifact", "title": "Truly Subquadratic 3SUM and Truly Subcubic APSP — Lean 4 formalization", "url": "https://github.com/anthropics/formal-math/tree/main/3sum-apsp", "primary": true, "published_date": null, "accessed_at": "2026-10-07" } ], "artifacts": [ { "id": "anthropic-3sum-apsp-lean", "kind": "proof", "title": "Lean formalization of the principal deterministic 3SUM and APSP claims", "url": "https://github.com/anthropics/formal-math/tree/main/3sum-apsp" } ], "relationships": { "predecessors": [], "successors": [], "related": [] }, "timeline": { "featured": true, "significance": "primary" }, "tags": [ "claude", "3sum", "apsp", "fine-grained-complexity", "algorithms", "lean", "ai-generated-mathematics" ], "editorial": { "created_at": "2026-10-07", "last_reviewed": "2026-10-07", "reviewers": [ "Charlie-Wang-03" ] }, "corrections": [] }