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

Abstract

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 $a\in \hat{\mathbb{C}}\setminus \{0,1,\infty \}$, 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.

11 pages, 1 figure. Preliminary version. Not intended for journal submission at this stage. Comments are welcome