L1-FLAT POLYNOMIALS AND SIMPLE LEBESGUE SPECTRUM FOR CONSERVATIVE MAPS EXIST: A SIMPLE PROOF.

The paper proves L^α-flatness (0<α<2) for 0–1 analytic polynomials built from Singer sets via a careful Marcinkiewicz–Zygmund interpolation argument that uses only the values of the auxiliary polynomial Q_q on q-th roots of unity, where Q_q(1)=|S|-1 and Q_q(z_r)=-1/|S| for r=1,...,q-1, then controls the L^α norms and passes to α≤1 by monotonicity (see (2.5), (2.12)–(2.19) in the paper) . The model’s core step asserts the global identity |P_q(z)|^2 = 1 + r^{-1}∑_{x=1}^{q-1} z^x for all z on the circle; that identity holds only at q-th roots of unity and is false for general z. Without the paper’s interpolation step (or a correct inequality bound), the model’s reduction to Dirichlet kernel norms is invalid. The real-line windowed flatness and the spectral constructions (rank-one map/flow, generalized Riesz products) in the paper are standard and correctly executed (Fejér kernel step and rank-one spectral formula) . Hence: paper correct; model’s argument has a crucial flaw.