RAYS TO RENORMALIZATIONS

The paper’s Theorem 1 states, under (p1)–(p2), the existence of a finite-to-one correspondence λ from P-external rays having accumulation points in K_f onto the set of polynomial-like rays {ℓ_t}, with uniqueness up to K_f-equivalence, non-tangential convergence to the same prime end, surjectivity, and the described “almost injective” alternatives (i)–(ii); it also excludes (i) when K_P is connected (all rays are smooth). This is explicitly formulated and proved in the paper (see the statement of Theorem 1 and its proof outline and ingredients, including Lemma 2.1 and Lemma 2.2 establishing non-tangential access and the Ψ–construction) . The candidate solution reproduces this result with the same structural ideas: it fixes a straightening h and pulls back the external geometry of the straightened polynomial G to define polynomial-like rays ℓ_t , invokes (p2) to control passages through a puzzle/collar representative (W*, W_1*), uses prime ends of Ω = C \ K_f to assert existence/uniqueness and non-tangential convergence, and argues surjectivity and the almost-injectivity dichotomy with an iteration/Markov-collar argument (aligned with the paper’s g-plane and Ŷ(K) strategy). Small differences are expository (e.g., the candidate references a “puzzle collar” and locates Y on ∂W_1*, whereas the paper defines Y via the unit circle S and pulls back under ψ), but they do not affect correctness. Minor imprecision (“K_f full ⇒ Ω simply connected”) is harmless here because K_f is connected, and the paper itself also defines prime ends for Ω = C \ K_f . Overall, the paper’s argument is sound and the model’s proof is substantially the same in content and logic.