Spectral Wall Arithmetic C: Genus, Complex Multiplication, and Chowla–Selberg Periods of the Spectral Curve

We take the Chambers polynomial Q for rational flux p/q as the starting point,construct the spectral curve Σq : y2 = Q(E)(Q(E)+4), identify the wall set as thebranch divisor, and identify spectral bands and gaps as real ovals and real gaps.In the weighted projective model P(1,q,1), the curve is smooth at infinity and hasarithmetic genus q−1; the finite branch points are given by the roots of Q = 0 andQ=−4, counted with multiplicity 2q in total. The spectral curve Σ2 for flux 1/2gives the first rigorous example in this series in which a spectral period equals anexplicit arithmetic constant: the Legendre parameter is λ = (√2−1)4 = 17−12√2,j = 663 = 287496; the period lattice is Ω(Z ⊕ (i/2)Z), with half-oval period Ω/4,full-oval period Ω/2, and gap period Ω, where Ω = √2K(1/√2) = Γ(1/4)2/(2√2π).The CM order of Σ2 is the non-maximal order Z+2Z[i] of discriminant −16, andit differs from the Legendre curve of the maximal order Z[i] by a 2-isogeny ofindex 2; the rationality of the period ratios 1 and 1/2 is explained by the scalingconversion for conductor 2, while the numerical identities are verified in layersby Beta integrals and an independent AGM path. On the 23 pairwise inequivalentfluxes with q ≤ 12, 4 divides q if and only if a unique A1 node appears at E = 0, thegeometric genus drops to q−2, and the arithmetic genus remains q−1; the generalassertion of this equivalence is registered as a conjecture based on full certificationand is not upgraded to a theorem by finite-range machine checks. For q = 3,exact integer elimination gives Res(f,f′) = −221 · 36 and disc(f) = +221 · 36. Forq ≥3, the Igusa invariants and CM criterion, high-genus isogeny correspondences,and period arithmetic after node resolution are all open items, whose commonobstruction is a compatibility formula between a canonical homology basis for theperiod matrix and node resolution. The numerical conclusions of this paper do notdepend on public benchmark data sets; all are generated by deterministic symbolicarithmetic and arbitrary-precision floating-point arithmetic. The isolation of thetwo-route implementation layers, the prohibition on writing preset constants innormalization, and the equal treatment of incident registration and theorem proofconstitute mandatory constraints of the methodology.