Measure-Theoretic Time-Delay Embedding

The paper’s main claim (Theorem 4 and Corollary 1) — that if f: M→N is an embedding then the pushforward F=f#: P2(M)→P2(N) is an embedding, and hence generically Ψh,ϕτ is an embedding when Φh,ϕτ is — is essentially correct. However, the proof of the derivative’s injectivity picks ς=φ∘f−1 as a Cc∞(N) test function even though φ∘f−1 is only defined on f(M); an explicit smooth extension to N (e.g., via a tubular neighborhood and retraction) is not provided in the main text, and the justification offered (that ς∈C1(N) because φ and f are C1) is insufficient as written . The model’s solution provides that missing step cleanly. Apart from this fixable gap and some imprecision about “invertible” (meant: invertible onto its image) in Theorem 3, the paper’s structure aligns with standard OT/AGS theory and Takens, and the core result stands .