THURSTON OBSTRUCTIONS AND TROPICAL GEOMETRY

The paper explicitly constructs ν^trop_φ from weighted multicurves (Definition–Lemma 7.4) and proves the commutative equalities ρ_trop = π^trop_1 ∘ ν^trop_φ and ρ_trop ∘ TP^trop_φ = π^trop_2 ∘ ν^trop_φ (Proposition 7.6) . It then shows that any local branch MSC^trop_{φ,Λ} has the same matrix as the Thurston linear transformation TLT_{φ,Γ} (Proposition 7.8) and identifies weakly fixed cones/rays with φ′-stable multicurves, with scaling equal to the Thurston eigenvalue, and λ ≥ 1 characterizing Thurston obstructions up to Hurwitz equivalence (Propositions 7.9, 7.12; Corollary 7.13) . The candidate solution reconstructs the same structure and conclusions via the standard “covering-of-annuli” rule (lengths divide by local degree), which matches the paper’s length formulas for tropical admissible covers (Section 6.4) and its proofs of the three claims, differing mainly in exposition rather than substance .