Spectral Wall Arithmetic D: The Singularity Dichotomy of the Spectral Curve: Boundary Nodes, Discriminants, and Arithmetic of the Rational-Flux Hofstadter Model

For the Hofstadter operator with coprime flux p/q, we define the spectral curveΣq : y2 = Q(E)(Q(E) +4) by means of the Chambers polynomial Q. We provethe singularity dichotomy structure theorem for the spectral curve: when 4 ∤ q, Σqis a smooth hyperelliptic curve of geometric genus q − 1; when 4 | q, the uniquesingular point of Σq is a single A1 node at E = 0, with local model y2 = 2Q′′(0)E2,and geometric genus q−2. The proof chain is synthesized from three parts: an elementary proof of the identity Q = ∆−2, the degeneration classification of the vanMouche gap nondegeneracy theorem, and an external explicit formula for the nonvanishing of the node coefficient Q′′(0). The identity proof uses only the evaluationof the Chambers identity at the origin phase, the transfer-matrix correspondence,and elementary coefficient expansion; its root count is supplied by the ν = 0 row ofthe van Mouche theorem, and the proof chain closes at this point without reverselydepending on subsequent conclusions at the matrix layer or the curve layer. Thenode coefficient Q′′(0) lies in the cyclotomic real field K = Q(ζq+ζ−1q )and satisfies the Galois covariance vap = σa(vp); for the tabulated instances with q ≤ 28 and4 | q, exact rational arithmetic shows that the conjugate values in the orbit arepairwise distinct, hence Q(Q′′(0)) = K and the minimal polynomial has degreeφ(q)/2. At the node the density of states does not diverge but takes the finiteplateau √︁|Q′′(0)|/(πq√2), which constitutes an observational criterion for 4 | q.All structural assertions are supported by independent double-pipeline recomputation over pairwise inequivalent fluxes with q ≤ 30; machine checks do not enter theload-bearing proof.