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Three omitted values and non-Blaschke point divisors in half-planes

Quanyu Tang, Bokai Cui, Wei He, Tao Hu, Yanyang Li, Ke Wang, Zijun Yu

Tang et al.’s Theorem 1.2 constructs a real meromorphic F on C with all 0-, 1- and ∞-points on the real axis, each infinite, and for every other spherical value a, the a-point divisor in each half-plane fails the Blaschke condition. Consequently F is not of bounded type in either half-plane despite omitting three prescribed values there; Möbius postcomposition and affine precomposition extend the example to any triple and Euclidean half-plane.

Sources for this record

What was already known

Nevanlinna proved bounded type with a finite-order assumption; Ostrovskii weakened that growth condition. Eremenko, Kulikov and Sodin noted half-plane-only counterexamples but left the entire-plane continuation question open. He and Zhang independently submitted a negative answer on 25 August 2026. Tang et al. cite a timestamped 24 August candidate archive and distinguish their explicit Gaussian/Farey orbit non-Blaschke mechanism and stronger fiberwise conclusion from He–Zhang’s finite Farey-refinement and Green-area method.

AI involvement AI-autonomous

A 7-hour-14-minute autonomous GPT-5.6 Sol Ultra run supplied the core construction and proof. Quanyu Tang, Bokai Cui, Wei He, Tao Hu, Yanyang Li, Ke Wang and Zijun Yu checked, revised and reviewed the candidate and final manuscript. No separate human explanation of the central Gaussian/Farey mechanism is documented.

The authors explicitly identify an autonomous model run as the source of the core construction and proof, with human checking and revision afterward.

Importance Notable

The construction gives a negative answer with a stronger conclusion for every other value in both half-planes.

Limitations

The reported human audit does not specify complete coverage of the final proof. He and Zhang independently answered the basic question by a different method, so priority for that yes-or-no answer is shared.

Technical verification Partial

The authors report human checking and mathematical auditing of the candidate plus review of the final manuscript, but do not state the exact scope or completion of a rigorous check of every final step. The separate He–Zhang proof does not verify this construction.

Human understanding Few

The human authors report audit and revision, but no named human account explains the central Gaussian/Farey orbit mechanism; Few is the evidence default, not a finding of inability to understand.

Correction and publication history

2026-10-18 · publication

The construction gives a negative answer with a stronger conclusion for every other value in both half-planes.

Read this published revision