Inelastic Particle Clusters from Cumulative Momenta

The paper (Chien–Mager–Safranek–Zariski, 2021) defines the cumulative momentum diagram Γ(f) via points P_k=(∑_{i=1}^k m_i, ∑_{i=1}^k m_i v_i), its convex envelope r, and the associated polygons T_j, then proves the main theorem: clusters A_j are in bijection with these polygons T_j and each cluster’s terminal velocity equals the slope of T_j (Theorem in §2.3 and §3) . The proof relies on a ‘cutting’ lemma using convex-envelope slope monotonicity to show no inter-polygon collisions (Lemma 3.1), followed by an induction to match polygons to clusters and identify velocities with slopes . The candidate model presents a different, self-contained argument: it proves contiguity of clusters, conservation-of-momentum averaging (2.1), monotonic ordering of terminal cluster velocities, and crucial one-sided momentum inequalities within a cluster; from these it shows each cluster chord lies below Γ(f) and that the concatenation of such chords is the convex minorant r, hence identifies polygons with clusters and slopes with velocities. The building blocks used by the model (stickiness, momentum conservation, and cumulative-sum geometry) are consistent with the paper’s preliminaries (especially formula (2.1)) . Both arguments are sound; they reach the same conclusion by different routes (paper: cutting + induction; model: convex-minorant maximality).