A MODEL OF THE CUBIC CONNECTEDNESS LOCUS

The paper’s Main Theorem states that the alliances AP form an upper semicontinuous partition of CrP and that P ↦ AP is continuous into the quotient CrP/{AP} (see the Main Theorem and setup defining SP, CP, S^l_P in the introduction and Section 2 ). The paper proves upper semicontinuity of P ↦ CP (Proposition 2.2) using a robust landing lemma and S^l_P, then defines alliances, including the prime alliance, and establishes continuity via Theorem 4.4 with a delicate case split that treats invisible parameters by proving any limit portrait must be prime (Lemmas and proof around Section 4 ). By contrast, the candidate solution asserts several unproven or incorrect steps: (i) it claims S^l_P is itself a σ3-invariant geodesic lamination and that P ↦ S^l_P is upper semicontinuous, which the paper does not assert or require; (ii) it requires uniqueness of the “oldest ancestor” of S^l_P, not established in the paper (the paper instead proves a non-sharing property for regular oldest ancestors, sufficient for the partition); and crucially (iii) its continuity proof omits the invisible case handled in Theorem 4.4, where the paper shows any limit portrait must be prime to maintain closedness of the graph (the candidate’s closed-graph argument implicitly assumes a nonempty oldest-ancestor limit and so fails when SP = ∅) . Friendship closedness and prime-closedness are carefully proven in the paper (e.g., Lemma 3.6 and Lemma 3.11), while the model only sketches these points . Therefore, the paper’s result and proof are correct, whereas the model’s proof is incomplete and, in parts, incorrect.