Examples of Non-Busemann Horoballs

The paper proves that for any finite symmetric projection-symmetric generating set on the generalized discrete Heisenberg group there exists an S-horoball that is not a limit of Busemann horoballs, by combining a half-space projection property of Busemann limits with a symmetric limit of balls around central elements. The candidate solution proves the same statement via a different route: quantitative isoperimetric control of vertical displacement, constructing a bounded-projection cylinder horoball, and a stronger structural property for Busemann horoballs. The model’s argument is sound at a high level; one step (uniformity in limits of Busemann horoballs) needs a small clarification, which can be patched using the paper’s half-space lemma. Overall, both are correct and essentially independent.