Small-Gain Theorem for Safety Verification under High-Relative-Degree Constraints

The paper’s Theorem 2 (items i–iii) is carefully proved using a comparison lemma, the auxiliary ZBF transform for high relative degree, and a σ-based gain construction; the proof details and intermediate bounds are explicit in the appendices, including the delicate equality I(x0,u)=J(x0,u), the robust forward invariance of C, and the set-ISS estimate via a comparison system and time-slicing argument . The candidate solution reproduces the same high-level structure and even the same constructions for φ̂i,k, γ̂i,k, and di,k−1, but it relies on an unproven “σ-decomposition (Young-type) lemma,” treats the interval equality I=J as “trivial,” and compresses the set-ISS derivation into a single scalar Dini-derivative inequality without establishing the required bounds. These gaps are substantive: the paper’s Appendix IV shows why I=J needs a careful argument, and how the σ-chain and min-of-two bounds are obtained via Lemma 2 and monotonic compositions, not via the candidate’s algebraic lemma. Hence, while the candidate’s outline is close in spirit, it is not a complete or correct proof.