On Planar Shadowing Curves to Closed Escaping Curves

The candidate solution reproduces the paper’s program: reduce to the RSE θ' = F_R(t,θ), construct the 2π-time Poincaré map P_R on the circle, use continuity of the rotation number in R, show ̺(R) = ω0 for small R (defining the critical distance), prove ̺(R) → 0 as R → ∞, and apply circle dynamics. The paper states these ingredients explicitly: the RSE and shadowing domain DR (Eq. (2.9), (2.11) , the rotation number framework and continuity (Theorems 3.1, 3.3) , the small‑R plateau ̺(R) ≡ ω0 (Theorem 3.10) , large‑R decay/expansion and eventual monotonicity (Lemma 3.8, Theorem 3.12) , and the definitions of the critical and turning distances R(E), R(E) under hypothesis (H) with 0 < R(E) ≤ R(E) < ∞ (Defs. 3.13–3.15) . The classification of SCs follows from circle dynamics: for rational ̺ there are periodic orbits and other orbits approach them (Theorem 3.3, Theorem 3.17) , and for irrational ̺ with P C2 all orbits are dense (Denjoy minimality; existence of R with prescribed irrational ̺ is Theorem 3.18) , leading to density in DR (e.g., Eq. (4.33) in the circle case) . The model’s proof sketch differs only in presentation (e.g., it records an explicit C0 bound for P_R(R)–P_S, and it invokes topological conjugacy as a standard C2 refinement of Denjoy). No substantive logical gap or contradiction relative to the paper is found.