Counterexample to Nevanlinna's half-plane omitted-values question
An explicit meromorphic function omits three values in a half-plane without being of bounded type there, answering in the negative a question of Nevanlinna open for more than a century.
- Lab
- Independent
- Model
- GPT-5.6 Sol Ultra
- Field
- Mathematics
- Date
- 2026-08-26
- Human collaborators
- Quanyu Tang, Bokai Cui, Wei He, Tao Hu, Yanyang Li, Ke Wang, Zijun Yu
What was found
A meromorphic function on a domain is of bounded type if it is a quotient of two bounded holomorphic functions. Nevanlinna asked whether omitting three values on a half-plane forces bounded type there. The paper constructs a function omitting 0, 1 and infinity whose a-point divisor fails the Blaschke condition in each half-plane, which is exactly the negation, and notes that postcomposition gives the same conclusion for any prescribed triple of distinct values on the Riemann sphere. The construction runs through a modular covering, a comparison domain and a Farey-edge boundary with Schwarz reflection.
Novelty check
The question is a named one in classical function theory, stated by Nevanlinna and described in the paper's own abstract as open for over a century, with the Blaschke condition and bounded type both standard notions carrying their own literature. The paper devotes a subsection to chronology and priority, which fixes what it believes was known when. No prior counterexample appears. The result is a new construction rather than a retrieval.
Caveats and known objections
A preprint days old at entry, unrefereed and not formalized, and no independent check is recorded. Autonomy graded ai-led on an unusually specific disclosure, which the paper puts in a numbered subsection headed Declaration of AI usage rather than in a footnote: the core mathematical construction and proof were generated during an autonomous run of GPT-5.6 Sol on OpenAI's ultra setting, the run lasted 7 hours 14 minutes and 14 seconds, the candidate proof was then subjected to human checking, mathematical auditing and revision, and the initial draft of the manuscript was also machine-generated and lightly revised. Graded ai-led rather than autonomous because the problem was posed and the output audited by the seven authors; the run itself is described as autonomous, and that distinction is the whole of the difference.
Also recorded at
vibemathednevanlinna-s-half-plane-omitted-values-problem
Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.
Nobody outside the lab has checked this yet.
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Entry history (1 event)
- AddedEntered the registry graded Author verified and AI-led.
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Graded author verified for verification and ai-led for autonomy. What these mean.
Cite this entry
whataifound.org. (2026). Counterexample to Nevanlinna's half-plane omitted-values question. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-08-26-nevanlinna-half-planeBibTeX
@misc{whataifound-independent-2026-plane,
title = {Counterexample to Nevanlinna's half-plane omitted-values question},
author = {{whataifound.org}},
year = {2026},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
url = {https://whataifound.org/finding/2026-08-26-nevanlinna-half-plane}
}