PFH spectral invariants and C∞ closing lemmas

The paper’s Theorem 1.3 states precisely that if a Hamiltonian isotopy class Φ is rational and some iterate Φ_n has the U-cycle property, then Φ has the C∞ closing property, and the proof in §7 implements a quantitative spectral-gap argument directly for φ^n using PFH spectral invariants, the U-map, and a ball-packing inequality in a graph cobordism, yielding c_{Uσ}(φ^n,γ) ≤ c_σ(φ^n,γ) − δε and a contradiction when d(γ) is large; see the setup and inequality in §7 (including the ball embedding into M_δ and the inequality (7.5)), together with the definition of U-cycles and the U-cycle property (Defs. 2.17 and 2.20), and the spectrality/continuity properties in the rational case (Prop. 4.5) . This proves Theorem 1.3 as stated in the introduction . By contrast, the model’s Step 3 asserts a functorial “transfer” of U-cyclic classes along the n-fold S^1-cover Y_{φ^n}→Y_φ that preserves U and q and thereby shifts degrees; such a covering naturality is neither constructed nor used in the paper (the invariance maps in §3 concern Hamiltonian isotopies, not coverings) . The paper avoids this by running the spectral-gap argument for φ^n and then inferring a periodic orbit for φ, which suffices for the closing property. The model also mis-cites some results (e.g., a non-existent “Theorem 7.4” used for the forcing argument, whereas the key quantitative step is derived via Lemma 6.1 and the §7 setup). Hence the paper’s argument is correct and complete for the stated theorem, while the model’s proof contains an unsubstantiated functorial step and incorrect citations.