Embedded constant mean curvature tori in hemispheres of Berger spheres

We prove that for all real numbers $\kappa ,\tau$ with $\kappa >8{\tau }^2$ and $\tau \mathrm{\ne }0$, the Berger sphere ${\mathbb{E}}^3(\kappa ,\tau )$ contains smooth closed embedded rotational surfaces of genus one with constant mean curvature which are contained in an open hemisphere. This disproves a conjecture of Fernández and Mira. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system, and then verified by the authors.