Nonlinear Anderson Localized States at Arbitrary Disorder

The paper proves existence of quasi-periodic, Anderson-localized solutions at arbitrary disorder via a Lyapunov–Schmidt/Newton scheme that crucially uses a new small-scale ‘clustering of diagonals’ property to overcome the O(δ) size of available parameters. The candidate solution reproduces the high-level LS/KAM-Newton outline but omits this essential small-scale ingredient and relies on generic diagonal dominance and coarea-type exclusions without justifying invertibility at the initial scales; this is precisely where the paper’s new clustering input is needed. Hence, while the overall structure resembles the paper, the candidate solution is missing a key hypothesis and is not a correct proof for the arbitrary-disorder regime.