One-dimensional inelastic collapse of four particles: asymmetric collision sequences and spherical billiard reduction

The paper’s Theorem 4.1 (spherical billiard reduction) proves that, for four inelastic 1D particles with 0<r<1, the sequence of planes P(k)=Span(p(k),q(k)) and, hence, the collision order depend only on the initial plane and the initial colliding pair; the proof is by explicit case computations (ab, ac, bc) and a careful analysis showing P(1) depends only on P(0), not on the velocity’s radial component, together with a rule determining the next collision from the (oriented) great circle in P(k) on S^2 . The candidate model solution proves the same statement via a more abstract linear-algebraic update map F_j(P) on planes, showing its well-definedness and the same “next-collision depends only on P(k)” rule, followed by induction. Thus, both are correct; the proofs are substantively different in style (computational vs. structural) but agree on content and conclusions. The paper’s mapping (ck,P(k))↦(ck+1,P(k+1)) explicitly “keeps track of the order of the collisions,” matching the model’s claim that collision labels are determined by P(k) alone (given the current colliding pair) .