Viscosity subsoltions of Hamilton-Jacobi equations and Invariant sets of contact Hamilton systems

The paper’s Theorem A states strict statements (T^-_t φ > φ and open-epigraph invariance) but its proofs only establish non-strict monotonicity and (via Lemma 2.5) an inclusion that yields the strictness only if one already assumes open-epigraph invariance; moreover, the Grönwall argument in Section 2.2 rules out T^-_t φ < φ yet does not preclude equality, which contradicts the theorem as stated and also conflicts with the existence of stationary viscosity solutions where T^-_t φ ≡ φ. See Theorem A and its proof snippets and Lemma 2.5 in the paper . The candidate model goes beyond the paper by asserting graph-transform equalities Γ_{T^-_t ψ} = Φ^t_H(Γ_ψ) and Γ_{T^+_t ψ} = Φ^{-t}_H(Γ_ψ), which are not proved in the paper and are unlikely under the stated assumptions; it also implicitly uses an ODE comparison principle in the u-variable that is not available from the mere Lipschitz bound |∂_u H| ≤ λ (no monotonicity in u is assumed), so those steps are not justified. Therefore, both the paper and the model require corrections: the paper should change > to ≥ in Theorem A and clarify Γ to be the closed epigraph to match what the proofs actually show; the model should drop the unproven graph-transform equalities and the comparison claim.