Outer billiards on the manifolds of oriented geodesics of the three dimensional space forms

The paper proves (i) the F-construction and diffeomorphism properties (Theorem 3), (ii) the geodesic characterization F(u,t)=Γ_{J(N(u))}(t) for κ=±1 (Theorem 4), and (iii) symplecticity of the outer billiard with respect to ω_K (Theorem 5), with full Jacobi-field computations and a matrix check for symplecticity. By contrast, the model’s Phase C asserts B is symplectic because it is obtained by following the Hamiltonian flow of JN for a variable time 2ρ(q); this inference is invalid in general (variable-time Hamiltonian flow maps need not be symplectomorphisms). The model also replaces the paper’s explicit nondegeneracy check for dF by informal convexity arguments and omits needed details. Hence the paper’s arguments are correct and complete, while the model’s proof is flawed in the symplectic step and lacks rigor in parts of A.