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We prove almost Lipshitz continuity of spectra of singular quasiperiodicJacobi matrices and obtain a representation of the critical almost Mathieufamily that has a singularity. This allows us to prove that the Hausdorffdimension of its spectrum is not larger than 1/2 for all irrationalfrequencies, solving a long-standing problem. Other corollaries include twovery elementary proofs of zero measure of the spectrum (Problem 5 in [41]) anda similar Hausdorff dimension result for the quantum graph graphene.

Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum