MATHEMATICAL AND PHYSICAL BILLIARD IN PYRAMIDS

The paper states there are three possible 4-cycle orders up to reversal and that each order yields either one cycle or none. It then proves that when one dihedral angle is right, the two orders in which the orthogonal faces are adjacent cannot give two distinct cycles (their reflections commute, forcing the same starting vector), so at most one cycle arises from those two orders; the remaining order can contribute at most one more, giving a global maximum of two. This matches the candidate’s conclusion. The model’s proof is more explicit: it classifies the orders into two types (adjacent vs. opposite), uses a half-turn conjugacy to pin the axis in type (A) (v ∥ CD), and invokes uniqueness of the rotation axis in SO(3) for type (B), hence at most one cycle of each type. The paper’s Proposition 4.2 is terse and contains a minor typo in one listed order, but in light of the earlier Proposition 2.1 (uniqueness per order), the argument is substantively correct. The model’s proof is correct and more detailed, using a different approach. See the paper’s enumeration of orders and uniqueness per order, and its Proposition 4.2 argument based on commuting reflections in orthogonal faces, as well as the general linear-operator formulation of 4-cycles in Section 2 .