Boundedness criteria for real quivers of rank 3

The paper’s main theorem states that a rank‑3 real‑weighted quiver has a bounded mutation class iff it is either mutation‑finite or mutation‑acyclic with Markov constant C(Q) ≤ 4; this is proved via a combinatorial/analytic argument built on Proposition 2.2, Lemmas 2.3–2.4, and an explicit growth inequality (Eq. (4)) . The candidate solution proves the same criterion using Felikson–Tumarkin’s geometric realizations (reflections or π‑rotations), showing boundedness in spherical/Euclidean cases and unboundedness via hyperbolic translations. Both arrive at the same characterization; the proofs are substantively different (paper: mutation inequalities and an explicit mutation sequence; model: geometric dynamics). One minor point to flag is that the model invokes a claim about mutation‑cyclic classes necessarily having C(Q) ≤ 4—this condition is used in Felikson–Tumarkin’s geometric framework but is not stated in the audited paper; however, it does not affect the stated equivalence nor the correctness of the model’s geometric proof idea for the boundedness criterion. The paper also records that cyclic classes with C(Q) ≤ 4 need not be bounded (e.g., (3,3,3)), underscoring the necessity of the mutation‑acyclic hypothesis .