A CELL DECOMPOSITION FOR MARKED CYCLE CURVES

The paper explicitly defines the cell complex Σ_{p,m} by lifting a vein-based cell structure from parameter space (Definition 1.4) and proves that edges correspond exactly to branched lifts over primitive roots, with faces labeled by canonical combinatorics (cycle duos for Per1(0) and cycle trios for Per2(0))—see Theorem 1.2 and Section 1.2. Crucially, the paper shows π is branched at ∞ for m=1 (and at the puncture for m=2 when p=1) and at primitive roots (Proposition 3.4 and Proposition 4.28), so faces need not be mapped biholomorphically to the base face U∞/E. The candidate solution incorrectly asserts that the projections are unbranched over U∞/E and that each face maps biholomorphically there; it also claims U∞ contains no period‑p parabolic parameters, overlooking satellite parabolics. Apart from these errors, the model’s high‑level picture (veins, edges ↔ primitive components, face labels, gluing) matches the paper’s framework. The paper’s argument is internally consistent and complete for these claims (and is algorithmically realized via Algorithms 3.7, 3.10, and 4.31). Citations: Theorem 1.2 and Definition 1.4 (cell structure) ; faces labeled by cycle duos (Lemma 3.3) ; branching set for π over Per1(0) (Proposition 3.4) ; Per2(0) analogs including branching (Proposition 4.28) and the construction algorithm (Algorithm 4.31; Theorem 4.32) ; algorithm correctness for Per1(0) (Proposition 3.9) .