Local Lipschitz LID Bounds and Full-Domain Sphere Average in the Poincaré Ball

We refine the finite-scale LID bridge of [1] and extend the sphere-averaged monotonicity result of [2]. First, we replace the global Lipschitz constant L in the multiplicative envelope between G_ε(r) and the population finite-scale LID by a local constant L_z(δ) defined as the supremum of ∥∇ log p∥H over the closed hyperbolic ball of radius δ centred at z. Under a C¹ regularity assumption on the density p, we prove the local envelope e^{−2L_z (δ)δ} G_ε(r) ≤ LID_δ(z) ≤ e^{2L_z (δ)δ} G_ε(r), which refines the global bound. As an asymptotic consequence, if ∇log p is continuous at z and δ → 0⁺ at fixed z, then the relative error satisfies |LID_δ(z)/G_ε(r) − 1| ≤ 2∥∇ log p(z)∥_Hδ + o(δ), replacing the global constant L by the pointwise gradient norm. Second, we prove that the sphere-averaged hyperbolic step \( \bar{\delta} \)(r, ε) introduced in [2] remains strictly increasing in r on the full domain a + ε < 1, extending the result from the restricted domain ε < a < 1 − ε. The proof uses a symmetrization argument that reduces the integral over the sphere to a pair of pointwise inequalities. Both results preserve the multiplicative envelope structure of [1] and transfer to all five directional steps δ₊, δ₋, δ_⊥, \( \bar{\delta} \), δ_vol on their corresponding domains.