{ "schema_version": 1, "id": "evt-2026-astra-liouville-goldbach", "slug": "astra-liouville-goldbach", "title": { "en": "Captain Sude credits GPT-6 Astra with an unconditional proof of the Liouville–Goldbach problem", "zh-CN": "Captain Sude 将 Liouville–Goldbach 问题的无条件证明归因于 GPT-6 Astra" }, "dates": { "event": "2026-09-16", "first_public_claim": "2026-09-16", "paper_release": "2026-09-16", "artifact_release": "2026-09-16", "independently_verified": null, "added_to_chronicle": "2026-09-21", "last_updated": "2026-09-27" }, "event_types": [ "discovery", "proof", "formalization" ], "significance": { "tier": "H2", "rationale": { "en": "The release addresses a named open Liouville–Goldbach problem by giving an unconditional all-even theorem where the closest direct predecessor required GRH and sufficiently large even inputs. The result is accompanied by a complete Lean formalization and an independent technical replay, making it a substantive AI-associated research milestone while remaining distinct from the classical binary Goldbach conjecture and below the Chronicle's H1 historical threshold.", "zh-CN": "该成果针对一个已有明确记录的 Liouville–Goldbach 开放问题，给出覆盖全部偶数的 无条件定理，而最直接的前驱结果仍依赖 GRH 且只适用于充分大的偶数。公开成果同时 提供完整 Lean 形式化，并已有独立技术重放，因此足以构成实质性的 AI 参与研究里程碑； 但它不同于经典二元哥德巴赫猜想，且目前尚不足以达到 Chronicle 的 H1 历史级门槛。" } }, "mathematical_novelty": { "type": "new_result" }, "interfaces": [ "informal", "formal", "symbolic", "hybrid" ], "claim": { "en": "Captain Sude publicly attributed to GPT-6 Astra an unconditional proof that every even integer N > 2 can be written as N = a + b with positive integers a and b satisfying lambda(a) = lambda(b) = -1. The released paper and Lean theorem state this all-even result without GRH or a sufficiently-large restriction. This is a Liouville analogue of binary Goldbach, not a proof of the classical Goldbach conjecture.", "zh-CN": "Captain Sude 公开将一项无条件证明归因于 GPT-6 Astra：对每个大于 2 的偶数 N， 都存在正整数 a、b，使 N = a + b 且 lambda(a) = lambda(b) = -1。公开论文与 Lean 定理均陈述这一覆盖全部偶数的结论，不依赖 GRH，也没有“充分大”限制。 这是二元哥德巴赫问题的 Liouville 类比，并非经典哥德巴赫猜想的证明。" }, "summary": { "en": "The public project supplies an authorless mathematical paper, a complete Lean 4 formalization against mathlib's Liouville function, pinned toolchain metadata, build and axiom audits, and proof-route checks. A separate audit repository independently rebuilt the frozen v1.0.0 artifact on Linux, matched the source hashes, repeated the axiom and route audits, and reported no sorry, custom theorem axiom, or native_decide. The public paper and repository do not preserve a detailed model–human derivation trace, so the precise division of intellectual labor remains source-attributed rather than independently reconstructed by the Chronicle.", "zh-CN": "公开项目提供一篇未署名数学论文、基于 mathlib Liouville 函数的完整 Lean 4 形式化、固定工具链元数据、构建与公理审计以及证明路线检查。另一个独立审计仓库 在 Linux 上重新构建冻结的 v1.0.0 产物，对照源码哈希并重复公理与路线审计， 报告未发现 sorry、自定义定理公理或 native_decide。由于公开论文和仓库没有保存 详细的模型—人类推导轨迹，Chronicle 对具体智力分工只做来源归因，不作独立重建。" }, "why_it_matters": { "en": "Mangerel's 2024 Shusterman paper obtained the same (-1,-1) sign-pattern conclusion only under GRH and for sufficiently large even N. The new release claims to remove both restrictions and packages the argument as a reproducible formal artifact. If the mathematical novelty and reported Astra provenance continue to withstand specialist scrutiny, it is evidence of AI participation in resolving a concrete open research problem rather than merely improving a benchmark.", "zh-CN": "Mangerel 2024 年关于 Shusterman 问题的论文，只在 GRH 假设下并对充分大的偶数 N 得到相同的 (-1,-1) 符号模式结论。此次发布声称同时去掉这两个限制，并把论证 封装为可复现的形式化产物。如果其数学新颖性与 Astra 来源归因继续经受专业审查， 该事件将体现 AI 参与解决具体开放研究问题，而不是仅仅提升基准成绩。" }, "ai_role": { "level": "unclear", "description": { "en": "Captain Sude publicly attributes the proof discovery to GPT-6 Astra, including a first multiples-of-four argument followed by a new route covering every even integer greater than two. The released paper and repository do not document enough interaction provenance to independently separate model-generated ideas from human problem selection, prompting, editing, proof repair, or formalization work.", "zh-CN": "Captain Sude 公开将证明发现归因于 GPT-6 Astra，包括先得到 4 的倍数情形， 随后找到覆盖全部大于 2 偶数的新路线。但公开论文与仓库没有记录足够详细的 交互 provenance，无法独立区分模型生成思想与人类的问题选择、提示、编辑、 证明修复或形式化工作。" } }, "human_contribution": { "description": { "en": "Captain Sude maintained and released the public project materials and publicly described the Astra attribution. The released record does not document the exact human contribution to target selection, prompting, manuscript editing, Lean proof repair, or verification, so those roles are not inferred more specifically here.", "zh-CN": "Captain Sude 维护并发布公开项目材料，同时公开说明 Astra 的归因。现有记录没有 具体说明人类在目标选择、提示、论文编辑、Lean 证明修复或验证中的精确贡献， 因此本记录不对这些角色作进一步推断。" } }, "systems": [ "gpt-6-astra", "lean" ], "organizations": [], "people": [], "problems": [ "liouville-goldbach-problem" ], "methods": [ "multiplicative-functions", "finite-field-symmetry", "descent", "quadratic-reciprocity", "formalization" ], "verification": { "evidence_level": "E4", "status": "partially_verified", "formal_assurance": "independently_replayed", "history": [ { "date": "2026-09-16", "status": "paper_released", "evidence_level": "E3", "note": { "en": "The public repository released the mathematical paper and a complete Lean formalization whose final theorem matches the stated all-even Liouville–Goldbach claim. The project records successful local build, axiom, and proof-route audits.", "zh-CN": "公开仓库发布数学论文与完整 Lean 形式化，最终定理与所述覆盖全部偶数的 Liouville–Goldbach 主张一致；项目同时记录了成功的本地构建、公理审计和 证明路线审计。" } }, { "date": "2026-09-17", "status": "partially_verified", "evidence_level": "E4", "note": { "en": "An independent audit repository reports a Linux rebuild of the frozen v1.0.0 release, zero source-hash mismatches, a fresh axiom audit, a successful route audit, and no sorry, custom theorem axiom, or native_decide. This supports independently_replayed formal assurance but is explicitly a technical reproduction rather than independent mathematical peer review or priority adjudication.", "zh-CN": "一个独立审计仓库报告对冻结的 v1.0.0 版本完成 Linux 重建，源码哈希零差异， 重新执行公理审计和证明路线审计，并未发现 sorry、自定义定理公理或 native_decide。 这些证据支持 independently_replayed 的形式保障，但该审计明确属于技术复现， 并非独立数学同行评议或历史优先权裁决。" } } ] }, "sources": [ { "id": "captain-sude-liouville-goldbach-repository", "tier": "S1", "type": "repository", "title": "A Goldbach theorem for the Liouville function", "url": "https://github.com/CaptainSude/Liouville-Goldbach", "primary": true, "published_date": "2026-09-16", "accessed_at": "2026-09-21" }, { "id": "captain-sude-liouville-goldbach-paper", "tier": "S1", "type": "paper", "title": "A Goldbach theorem for the Liouville function", "url": "https://github.com/CaptainSude/Liouville-Goldbach/blob/main/paper/liouville-goldbach.pdf", "primary": true, "published_date": "2026-09-16", "accessed_at": "2026-09-21" }, { "id": "captain-sude-astra-liouville-goldbach-announcement", "tier": "S2", "type": "official_announcement", "title": "Captain Sude announcement attributing the Liouville–Goldbach proof to GPT-6 Astra", "url": "https://x.com/captain_sude/status/2100277626120163664", "primary": true, "published_date": "2026-09-16", "accessed_at": "2026-09-21" }, { "id": "sunnyspot-liouville-goldbach-independent-audit", "tier": "S3", "type": "independent_analysis", "title": "Liouville–Goldbach Independent Audit Bundle", "url": "https://github.com/sunnyspot114514/Liouville-Goldbach-audit", "primary": false, "published_date": "2026-09-17", "accessed_at": "2026-09-21" }, { "id": "mangerel-shusterman-liouville-goldbach", "tier": "S1", "type": "paper", "title": "On Shusterman's Goldbach-type problem for sign patterns of the Liouville function", "url": "https://arxiv.org/abs/2412.17199", "primary": false, "published_date": "2024-12-23", "accessed_at": "2026-09-21" }, { "id": "mangerel-goldbach-liouville-correlation", "tier": "S1", "type": "paper", "title": "On a Goldbach-type problem for the Liouville function", "url": "https://arxiv.org/abs/2404.12117", "primary": false, "published_date": "2024-04-18", "accessed_at": "2026-09-21" }, { "id": "mathoverflow-liouville-goldbach-2018", "tier": "S5", "type": "community", "title": "Goldbach's conjecture for the Liouville function", "url": "https://mathoverflow.net/questions/307479/goldbachs-conjecture-for-the-liouville-function", "primary": false, "published_date": "2018-08-03", "accessed_at": "2026-09-21" } ], "artifacts": [ { "id": "liouville-goldbach-lean-proof", "kind": "proof", "title": "Complete Lean 4 formalization of the Liouville–Goldbach theorem", "url": "https://github.com/CaptainSude/Liouville-Goldbach/tree/main/lean" }, { "id": "liouville-goldbach-final-theorem", "kind": "proof", "title": "LiouvilleGoldbach.liouville_goldbach", "url": "https://github.com/CaptainSude/Liouville-Goldbach/blob/main/lean/LiouvilleGoldbach/Main.lean" }, { "id": "liouville-goldbach-independent-replay", "kind": "supplement", "title": "Independent Linux replay and audit bundle", "url": "https://github.com/sunnyspot114514/Liouville-Goldbach-audit" } ], "relationships": { "predecessors": [], "successors": [], "related": [ "evt-2026-openai-ten-advances", "evt-2026-openai-prime-gaps-186" ] }, "timeline": { "featured": true, "significance": "primary" }, "tags": [ "liouville-function", "goldbach-type-problem", "number-theory", "gpt-6-astra", "lean", "ai-generated-mathematics" ], "editorial": { "created_at": "2026-09-21", "last_reviewed": "2026-09-27", "reviewers": [ "Charlie-Wang-03" ] }, "corrections": [] }