LARGE HYPERBOLIC CIRCLES

The paper’s Theorem 1.8 explicitly bounds the coefficients and the remainder with a 1/θ factor, not θ: ∑_{μ>0}(||D^+_{θ,μ}f||_∞ + ||D^-_{θ,μ}f||_∞) ≤ C′_Spec C_{1,s−3} (1/θ) ||f||_{W^s} and |R_θ f(p,t)| ≤ C_Spec C_{1,s−3} (1/θ) ||f||_{W^s} (t+1)e^{-t} (eqs. (1.16)–(1.18)). This is repeatedly substantiated in the proofs (e.g., (4.11), (5.18)–(5.19), (5.22)–(5.25)). Hence the model’s purported contradiction (which relies on assuming a factor θ on the right-hand side) does not arise. The right–K–invariant test function is consistent with the paper’s results; the 1/θ scaling is natural since k_{f,θ} is an average with a prefactor 1/θ. Therefore, no correction is needed; the model misread the θ-dependence. See Theorem 1.8 and its bounds (1.16)–(1.18) and their derivation in Sections 4–5 of the paper , and the definition of the circle-arc average (1.6) .