SUB-EXPONENTIAL STABILITY FOR THE BEAM EQUATION

The paper studies the one‑dimensional beam equation with frequencies ωj=√(j^4+m) and proves sub‑exponential stability in log‑analytic spaces H_{s,p}; its main sub‑exponential result (Theorem 1.4, including the thresholds δsE and the lifespan T0) is internally derived via a detailed Birkhoff normal form scheme with precise small‑divisor bounds and width bookkeeping, and is coherent across Sections 2–7 . By contrast, the candidate solution sets up the Klein–Gordon equation (ωj=√(j^2+m)) rather than the beam equation, and thereby works with the wrong linear frequencies and Hamiltonian. Although the high‑level normal‑form outline and the final formulas for δsE and T0 mimic the paper’s statements, the solution omits the paper’s beam‑specific Diophantine set and small‑divisor architecture (Section 3 and Proposition 4.1), and does not reconcile the finite‑order nonresonance structure used in the paper. Hence, relative to the paper’s question, the model solves the wrong PDE and misses essential hypotheses; the paper’s argument is correct.