Strongly commuting interval maps

The paper’s Theorem 5.21 gives a precise three-case decomposition for strongly commuting, piecewise monotone interval maps, with the “moreover” openness/monotonicity implications, and it is proved via a careful machinery of hats/endpoints, primary critical values, and exacting points. The candidate solution reaches the same statement but its proof relies on incorrect set-equality manipulations and unproved invariance claims (e.g., f(Cg) ⊂ Cg and f(P) ⊂ P), and it allows discontinuities not permitted by the paper’s setting. As a result, key steps (the finite invariant partition and the path-automorphism argument) are unsupported. The paper’s argument appears coherent and correct; the model’s proof is flawed/incomplete.