COMPUTER VALIDATION OF OPEN GAPS FOR THE ALMOST MATHIEU OPERATOR WITH CRITICAL COUPLING

The uploaded paper explicitly validates, with rigorous numerics, that at critical coupling b=2 the first eight labelled gaps are open for ω=(√5−1)/2 and twelve are open for ω=e−2, exactly as the candidate states. The abstract and introduction announce these counts, and Tables 1–4 give the certified intervals and labels, including ±1, ±2, ±3, ±4 for the golden mean and ±1, ±2, ±3, ±4, ±5, ±9 for e−2. The methods match the model’s outline: (i) a dynamical, uniform-hyperbolicity validation (Theorem 2.1) and (ii) a spectral/rational-approximant route using quantitative Hausdorff continuity. One sentence in the paper misstates Johnson’s characterization (it says “in the spectrum iff uniformly hyperbolic”), but the authors use the correct implication elsewhere and in Theorem 2.1; the model states the direction correctly. Overall, both the paper and the model are aligned on results and proof strategy. See abstract and roadmap (, ); dynamical method and Theorem 2.1 (, ); spectral continuity bounds and periodic method (, ); tables with validated gaps: Table 1 (ω=(√5−1)/2, dynamical), Table 2 (ω=e−2, dynamical), Table 3 (ω=(√5−1)/2, spectral), Table 4 (ω=e−2, spectral) (, , , ). Note the isolated typo in the text around Johnson’s theorem ().