PHASE RESPONSE CURVES AND THE ROLE OF COORDINATES

The paper explicitly states the delay-coordinate PRC identity as equation (4) in Example 1 and indicates it follows from Remark 2 (diffeomorphism invariance of the PRC), together with the explicit form of the delay map R(x) = (h(x), h∘Φ_{−δ1}(x), …, h∘Φ_{−δm}(x)) and the chain rule for pushforwards; see the statement of Remark 2 and Example 1 with (3)–(4) in the paper . The candidate solution reproduces exactly this argument: Step 1 transports the pairing via the naturality of pullbacks to obtain ρ_Z(x) = ⟨ϕ̃^*dx, Ψ_*Z⟩(Ψ(x)) (the content of Remark 2), and Step 2 computes Ψ_*Z by differentiating Ψ = (h, h∘Φ_{−δ1}, …, h∘Φ_{−δm}) and expanding in the ambient coordinate basis to yield the sum ∑_k (Dh∘DΦ_{−δ_k} Z) ∂_{x_k}, precisely matching (4) . The only minor omission in the model write-up is that it uses Ψ^{-1} without restating the paper’s assumption that R is an embedding (hence Ψ a diffeomorphism onto its image) near Γ; this assumption is made explicitly in the paper before (4) . Overall, the two proofs are the same in substance and both are correct.