CORRESPONDENCES ON RIEMANN SURFACES AND NON-UNIFORM HYPERBOLICITY

The paper proves that every M-bounded X-ray for an admissible correspondence has a finite attractor (Theorem A) via a non-uniform contraction scheme built on thick–thin decomposition and cusp analysis, culminating in Theorem C and Corollary 6.2 . In contrast, the candidate solution asserts a per-step affine contraction ℓ_S(g') ≤ λℓ_S(g)+K(M) by claiming a uniform inequality ℓ_S(ψ(h)) ≤ λℓ_S(h)+B. That claim fails in the cusp regime: for peripheral elements, lengths grow like 2 log n and are not uniformly contracted by a fixed λ<1; the paper’s Lemma 5.5 explicitly shows the nuanced behavior near cusps (Puiseux exponent t=c/d) inconsistent with uniform per-step contraction . The model’s “bounded cusp contribution” B independent of h is incorrect. The paper’s proof avoids this pitfall by proving an additive decrease over bounded windows (Proposition 7.1 and the ‘tight sequence’ argument) rather than stepwise uniform contraction .