QUASI-INTERMEDIATE VALUE THEOREM AND OUTFLANKING ARC THEOREM FOR PLANE MAPS

The Outflanking Arc Theorem (Theorem 4.4) is carefully proved in the paper via a case reduction and a boundary-degree/contradiction argument using an auxiliary map G; the proof is complete and self-contained in the provided text. In contrast, the candidate solution contains critical gaps: it misidentifies an endpoint when excluding boundary fixed points (checks f(y)≠y instead of addressing the endpoint v on [v,u]A), makes an unjustified inference about f(Γ2)∩Γ1=∅ by “applying f,” and leaves the key interior/exterior “sidedness” claims and corner-rotation accounting insufficiently justified for the degree computation. Hence, the paper’s argument stands, but the model’s proof outline does not yet constitute a valid proof. See the statement and proof of Theorem 4.4, including the construction leading to the degree −1 boundary map G, and the case reductions via topological conjugacies.