{ "schema_version": "1.0.0", "source": "vibemath", "problem_id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem", "title": "Nevanlinna’s half-plane omitted-values problem", "canonical_statement": "We construct a real meromorphic function $F$ on $\\mathbb{C}$ such that $F^{-1}(\\{0,1,\\infty\\})\\subset\\mathbb{R}$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\\in\\widehat{\\mathbb{C}}\\setminus\\{0,1,\\infty\\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction provides an independent negative answer to a question going back to Nevanlinna’s 1925 work that had remained open for over a century.\n\nPostcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.", "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.", "current_status": "resolved", "solution_events": [ { "id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:event", "problem_id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem", "type": "disproved", "status": "resolved", "occurred_at": "2026-08-24T00:00:00.000Z", "title": "arXiv", "summary": "The paper constructs a real meromorphic function $F$ on $\\mathbb{C}$ satisfying $F^{-1}(\\{0,1,\\infty\\})\\subset\\mathbb{R}$, with each of the three fibers $F^{-1}(0)$, $F^{-1}(1)$, and $F^{-1}(\\infty)$ infinite, such that for every $a\\in\\widehat{\\mathbb{C}}\\setminus\\{0,1,\\infty\\}$, the $a$-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, $F$ is not of bounded type in either half-plane.\n\nThis gives a negative answer to Nevanlinna’s century-old question asking whether an entire-plane meromorphic function that omits three distinct values in a half-plane must be of bounded type there. By postcomposition with Möbius transformations, the three exceptional values $\\{0,1,\\infty\\}$ may be replaced by any prescribed triple of distinct values in $\\widehat{\\mathbb{C}}$. By affine change of variables, the construction applies to any Euclidean half-plane.", "ai_contribution": "ai_discovered", "attempt_ids": [ "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:attempt" ], "method_family_ids": [ "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:method" ], "source_assertion_ids": [ "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:assertion" ] } ], "attempts": [ { "id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:attempt", "problem_id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem", "model": "GPT-5.6 Sol Ultra", "provider": "OpenAI", "attempted_at": "2026-08-24T00:00:00.000Z", "human_collaborators": [ "Quanyu Tang", "Bokai Cui", "Wei He", "Tao Hu", "Yanyang Li", "Ke Wang", "Zijun Yu" ], "exposure": "unknown", "prompt_public": null, "outcome": "Disclosed twice, in the abstract and in a dedicated section 1.2 \"Declaration of AI usage\": \"The core mathematical construction and proof underlying this paper were generated during an autonomous run of GPT-5.6 Sol using OpenAI's ultra setting. The run lasted 7 hours, 14 minutes, and 14 seconds. The resulting candidate proof was subsequently subjected to human checking, mathematical auditing, and revision. The initial draft of the manuscript was also generated by AI and was subsequently reviewed and lightly revised by the authors.\" For a disproof the construction is the whole result, and an autonomous run produced it, so this is AI-discovered on the authors' own account.\n\nWorth noting alongside: the independent concurrent counterexample of He and Zhang is also AI-involved, its acknowledgements stating that \"This counterexample was obtained through AI-assisted exploration under the authors' mathematical supervision and guidance\" - though it names no model. Both routes to this century-old problem, within two days of each other, had a model in the loop.", "failed_routes": [], "artifacts": [ { "label": "ChatGPT prompt", "url": "https://chatgpt.com/s/cx_6a8c620c99308191972b12d53e3d314f", "kind": "transcript", "license": null }, { "label": "Independent contemporary result by He and Zhang", "url": "https://arxiv.org/abs/2608.24829", "kind": "paper", "license": null }, { "label": "Eremenko, Kulikov, Sodin - where the problem is explicitly recorded (Question 2)", "url": "https://arxiv.org/abs/2604.06136", "kind": "problem-record", "license": null }, { "label": "Timestamped candidate-proof archive, 24 Aug 2026 (the priority record)", "url": "https://github.com/QuanyuTang/three-omitted-values-bounded-type-candidate-proof/releases/tag/private-archive-2026-08-24-v1", "kind": "other", "license": null } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.26062", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Nevanlinna’s half-plane omitted-values problem", "url": "https://vibemathed.com/problem/nevanlinna-s-half-plane-omitted-values-problem", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "cost_usd": null, "independence": "unknown" } ], "verifications": [ { "id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:verification", "problem_id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem", "solution_event_id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:event", "level": "unreviewed", "mathematical_correctness": "unknown", "statement_fidelity": "unaudited", "peer_review": "none", "verifier": "VibeMathed (source-reported label)", "verified_at": null, "note": "An arXiv preprint (v1, 26 August 2026, math.CV), unrefereed, with no formalization and no computational certificate, so there was nothing mechanical to re-run here and no mathematics was checked. What was verified on 27 August 2026: the paper exists at arXiv:2608.26062 with the title and all seven authors this entry lists; the AI declaration is quoted above verbatim from section 1.2; the problem it answers is real and traceable to Eremenko, Kulikov and Sodin's Question 2 (arXiv:2604.06136, confirmed by title and authors); the concurrent work of He and Zhang exists at arXiv:2608.24829 dated 25 August and does contain its own AI-assistance statement; and the timestamped GitHub release of 24 August that the priority claim rests on resolves and is live. So the chronology the paper asserts is independently checkable, which is more than most priority claims offer.", "sources": [ { "label": "VibeMathed: Nevanlinna’s half-plane omitted-values problem", "url": "https://vibemathed.com/problem/nevanlinna-s-half-plane-omitted-values-problem", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" }, { "label": "arXiv", "url": "https://arxiv.org/abs/2608.26062", "kind": "primary-mathematical-source", "license": null } ] } ], "sources": [ { "label": "arXiv", "url": "https://arxiv.org/abs/2608.26062", "kind": "primary-mathematical-source", "license": null }, { "label": "VibeMathed: Nevanlinna’s half-plane omitted-values problem", "url": "https://vibemathed.com/problem/nevanlinna-s-half-plane-omitted-values-problem", "kind": "source-assertion", "license": "https://vibemathed.com/data-license" } ], "recommended_mode": "verify", "recommended_exposure": "result_only", "opportunity_signals": [ { "id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem:signal:watch_only", "problem_id": "vibemathed:nevanlinna-s-half-plane-omitted-values-problem", "kind": "watch_only", "reason": "Current public evidence does not meet the default replay threshold.", "generated_at": "2026-09-13T16:28:40.178Z" } ], "uncertainties": [ "VibeMath has not independently verified the mathematical claim.", "AI-attempt independence and training-data exposure are unknown unless explicitly documented.", "VibeMath has not independently audited the mathematical statement, proof, or novelty claim." ], "generated_at": "2026-09-13T16:28:40.178Z" }