INVARIANT FORMS IN HYBRID AND IMPACT SYSTEMS

The paper’s Theorem 4.4 states that a differential form α is hybrid-invariant if and only if L_X α = 0 together with the two reset-side equalities Δ* ι_{S̃}^* i_X α = ι_S^* i_X α and Δ* ι_{S̃}^* α = ι_S^* α, where S̃ = Δ(S) and ι denotes inclusion; the proof proceeds via the augmented differential and a decomposition into the X-direction and the tangent space to S (energy and specular conditions) . The candidate solution proves the same equivalence using a flow-box argument around the impact and s-independence from L_X α = 0, then matching traces at s = 0 to recover the same two conditions. Both arguments require the paper’s ‘smooth hybrid system’ regularity (notably transversality X ⟂ S), formalized as (A.1)–(A.2) in Definition 2.5 . The candidate is essentially complete for times with finitely many impacts and, like the paper’s proof of Theorem 4.4, does not engage Zeno completions; the paper treats Zeno separately later. Net: same statement, logically consistent, different proof styles.