THE GROWTH OF A FIXED CONJUGACY CLASS IN NEGATIVE CURVATURE

The paper proves the δ/2 law with an exponential error via an adjusted-counts framework, smoothing of discontinuous test functions, and a Radius Hölder-Continuity (RHC) property for Bowen–Margulis balls; these technical ingredients are established and tied to exponential mixing under the stated curvature assumptions. By contrast, the model’s outline assumes an effective Parkkonen–Paulin orbital counting theorem with an exponential error for convex subsets in variable curvature as a black box. In this generality that step is not available without the extra smoothing/RHC machinery the paper develops. The model therefore relies on an unproven claim and omits necessary hypotheses, even though it reaches the correct final asymptotic.