Sharp Localization on the First Supercritical Stratum for Liouville Frequencies

The paper’s Theorem 1.1 states Anderson localization for even analytic potentials on the first supercritical stratum {E: L(ω,E) > β(ω) ≥ 0, κ(ω,E) = 1} for all non-resonant phases θ in Θ, precisely matching the claim the model aims to establish . The paper’s proof hinges on new large deviation estimates and sharp complexity bounds for an analytic proxy g_{m,E}, a zero-count mechanism on an annulus, and a rational-approximation analysis of traces at strong Liouville scales, together with the evenness of v and the non-resonance condition on 2θ+nω to eliminate double resonances . The model’s solution follows a different route: it invokes Avila’s acceleration with κ=1 and a small imaginary phase shift, postulates sharp LDT off the real axis, and then uses the Avalanche Principle to push estimates back to the real line before a Bourgain–Goldstein–Schlag elimination step. While Avila’s acceleration law and regular/dominated inputs are indeed recorded in the paper , the specific off-axis LDT-and-AP transfer is not the paper’s method; instead, the paper develops direct real-axis LDT for g_{m,E} and a trace-based strong-scale analysis. Hence, both arrive at the same conclusion under the same assumptions, but by substantially different proofs.