Nevanlinna’s half-plane omitted-values problem

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. Postcomposition 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.

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. This 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.