Coexistence of zero Lyapunov exponent and positive Lyapunov exponent for new quasi-periodic Schrödinger operator

The paper proves an exact Lyapunov exponent formula on the spectrum, L(E) = max{ 0, log(|αE + 2λ ± sqrt((αE + 2λ)^2 − 4α^2)| / (2(1 + sqrt(1 − α^2)))) }, via Herman-type bounds plus Avila’s global theory, and then derives the mobility-edge line αE = 2 sgn(λ)(1 − |λ|). The candidate solution asserts a different exact formula, L(E) = max{0, log(|λ| + sgn(λ)·αE/2)}, which is incompatible with the paper’s Theorem 1.2 except for reproducing the correct zero-set boundary that yields the same mobility-edge line. Thus the model’s exact-value claim is wrong even though the phase-boundary conclusion matches the paper.