Spectral Wall Arithmetic B: Density of States, Rotation Number, and Structural Boundaries of the Rational-Flux Hofstadter Operator

This paper studies the spectral structure and explicit expressions for the densityof states and rotation number of the rational-flux Harper–Hofstadter operator, anddelineates the applicability boundaries of this structure in the irrational-flux, exactalgebraic verification, and many-body directions. Taking the Chambers polynomialQasthe sole algebraic input, the density of states inside a spectral band is written as|Q′(E)| divided by πq times √︁−Q(Q+4); nondegenerate band edges exhibit vanHove singularities, and intra-band touch points vanish linearly. At true spectralband endpoints, the multiplicity of Q = 0 or Q = −4 must be odd; when themultiplicity m ≥ 3, the density of states vanishes as |E − Eedge|m/2−1. The mass1/q of each band holds when Q is monotone along the band or its total variationis exactly π; when Q folds back inside a band, a correction by total variationis required. The rotation number is given by a uniform angle function on thetwisted real curve ˆ︁ XQ: ν2 =−Q(λ)(Q(λ)+4); τ is single-valued and continuous onˆ︁ XQ(R), and after projection to the line u = Q(λ) it becomes a q-sheeted multivaluedfunction. The band-bottom angle φj ∈ {0,π} is uniquely determined by the lowerendpoint of the oval; within the same flux, different φj can be mixed, and theuniform formula reads φj oval by oval without switching according to the bandbottom type. The source of elementarity is the rational angle parametrizationu = −2+2cosθ of the conic ν2 = −u(u +4) (genus 0); the complex genus q −1 of the spectral curve XQ belongs to a different level. Under irrational flux,the wall-point counting measure converges weakly∗ to a measure supported on thecritical almost Mathieu spectrum σ(α), and σ(α) has Lebesgue measure zero, so thelimiting wall set has measure zero; this paper does not package the trivial Hausdorffdimension upper bound into a nontrivial conclusion. The cost of exact algebraicverification is controlled by the discriminant height exp(O(q logq)) of the coefficientfield Q(2cos(2πp/q)); in the many-body case, the discriminant of an effective 2×2reduction replaces the full-matrix discriminant, and the spectral-gap premise andpath-interception failure mode of this criterion are explicitly registered. All closedformulas are verified coefficient by coefficient via the three-path input-layer checksfor q = 3 and q = 5 and compared with forward-layer fiber counts and spectralhistograms; the acceptance specifications for the systematic q ≤ 30 checks arelisted separately