On the pullback relation on curves induced by a Thurston map

The paper explicitly states the dichotomy for #P=4 as Theorem 3.6 and sketches the correct mechanism via the holomorphic correspondence on moduli space: either σ_f is constant (trivial pullback), or else the induced map to the triply punctured sphere is nonconstant and forces all cusps (curve classes) to be hit, yielding surjectivity on curves . By contrast, the model’s argument contains two substantive flaws: (i) it reverses the Selinger boundary–preimage dictionary, which sends the stratum for β to the stratum for f^{-1}(β), not the other way around ; and (ii) it asserts that any nonconstant continuous self-map of a circle is surjective, which is false (continuous maps S^1→S^1 can have proper arc images). The conclusion matches the paper, but the reasoning is invalid in key steps.