A SYMPLECTIC HILBERT-SMITH CONJECTURE

The paper proves that there is no non-trivial continuous Z_p-action by Hamiltonian homeomorphisms on closed symplectically aspherical manifolds (Theorem A), via a refined Floer/barcode argument that uses a quantitative almost-homogeneity inequality for the spectral norm (Theorem F) and coefficient-field comparisons (Lemma 14) to contradict the existence of arbitrarily large roots produced by the Z_p action . In contrast, the candidate solution hinges on an incorrect claim that the Oh–Schwarz spectral norm satisfies exact homogeneity under iteration γ(φ^k) = |k| γ(φ) on Ham(M, ω) (and that this equality extends by C0-continuity). The paper explicitly relies on a weaker inequality, q·γ(ψ; F_q) − γ(ψ^q; F_q) ≤ 2∑β_i(ψ^q; F_q), not on equality, and develops additional barcode bounds to close the argument . The spectral norm’s C0-continuity and non-degeneracy on Ham(M, ω) do hold , but exact homogeneity is not available; indeed the paper’s Theorem F quantifies the deviation from linear scaling under iteration. Therefore the model’s short proof is invalid, while the paper’s proof is coherent and complete.