Birkhoff Normal Forms for Hamiltonian PDEs in their Energy Space

The candidate solution reproduces the paper’s partial Birkhoff/Lie normal form relative to low super-actions, using the generalized strong non-resonance (Definition 2.3) to solve the homological equations with small-divisor bounds, builds a near-identity canonical map, and derives almost-conservation of low super-actions over times |t| ≤ (ε1/η)^{-(r-p)} with drift O(η^{-(p-2)}⟨n⟩^b ε1^p). This matches the paper’s Theorem 4.1 construction and Theorem 5.1 dynamical corollary, including the L/U decomposition by κω(σ,n), the commutation {Jn, Q≤N_res}=0, the remainder gradient estimate, and the push-forward of forcing F̃, e.g., Theorem 4.1 and its Lie-series control and commuting property, and the final bound (54) in Theorem 5.1. No substantive logical gaps or missing hypotheses beyond those already stated in the paper are introduced by the model; the proof strategy and estimates are essentially the same as in the paper. See the paper’s decomposition (46)–(47), κω definition (8), the generalized strong non-resonance (Def. 2.3), the flow/near-identity bounds, and the dynamical corollary (53)–(54) for alignment.