Integrable Birkhoff Billiards inside Cones

The paper proves: (i) a quadratic first integral I=|Q|^2 for conical billiards and spherical caustics (Theorem 1), (ii) finiteness of reflections in C^3 convex cones (Theorem 2), and (iii) integrability in a strong sense via 2n−1 (a.e. smooth) first integrals that separate trajectories, obtained by lifting functions from ψ+ after extending the billiard map by the identity there (Theorem 3) . The construction of integrals uses the final direction components v_i and a tangential “moment” built from projecting Q against a continuous extension X(v) of the normal field from Γ to D, plus the quadratic integral; smoothness holds off a codimension-1 set ∆ and the quadratic integral disambiguates on ∆ (Examples/Lemmas 12–15) . The candidate solution mirrors these exact steps: identifying Ψ⊂TS^{n−1}, extending μ by Id on ψ+, lifting functions via τ(x), using final direction plus a projected “tangential moment,” noting smoothness loss only on a glancing set, and using |Q|^2 to separate trajectories on ∆. Minor omissions in the candidate (e.g., the compatibility condition f(v,Q)=f(σ(v,Q)) on ∆0 for continuity of the lifted integral, and the precise boundary decomposition of Ψ) do not alter the core argument, which is materially the same as the paper’s construction .