QDL Spectrum Selection as a Conditional Standard Model Theorem: Gauge Seed Minimality, Hypercharge Closure, Anomaly Cancellation, and One-Generation Matter Completion

The Quantized Dimensional Ledger (QDL) program formulates physical theory construction as a problem of structural admissibility, closure completion, transformation stability, and residual survival. This paper presents a conditional QDL spectrum-selection theorem for the Standard Model. It does not claim to derive the Standard Model from no assumptions. Instead, it shows that under explicit Standard Model-facing QDL closure requirements, the Standard Model gauge seed and one-generation chiral matter package arise as a minimal closure-complete admissible package. Starting from the gauge seed SU (3) c x SU (2) L x U (1) Y and primitive field seed Q, L, H, where Q is a quark doublet, L is a lepton doublet, and H is a Higgs doublet, QDL closure completion requires color-compatible Yukawa closure, electroweak mass-generation closure, hypercharge neutrality of interaction vertices, anomaly cancellation, and minimality. Under these assumptions, closure completion adds the conjugate singlet fields uc, dc, and ec. The combined Yukawa-closure and anomaly-closure equations determine the one-generation Standard Model hypercharge pattern up to normalization. With Y (H) = 1/2, the resulting hypercharges are Y (Q) = 1/6, Y (uc) = -2/3, Y (dc) = 1/3, Y (L) = -1/2, and Y (ec) = 1. The anomaly sums SU (3) ² U (1), SU (2) ² U (1), U (1) ³, and grav² U (1) all vanish. An optional neutrino singlet Nc is closure-neutral and appears only if Dirac neutrino closure is imposed. The paper also addresses the origin of the Standard Model gauge seed by declaring explicit QDL gauge-seed closure axioms: non-Abelian color closure for triplet matter, chiral weak doublet closure for electroweak mass generation, a single Abelian hypercharge ledger capable of supporting Yukawa neutrality and anomaly cancellation, Higgs-doublet mass-generation closure, and minimality. Under these declared requirements, the minimal compact gauge seed is SU (3) c x SU (2) L x U (1) Y with primitive field seed Q, L, H. The paper proves an important negative result: gauge representation closure, Yukawa closure, and anomaly cancellation are generation-neutral. One complete generation closes, and any integer number of identical complete generations also closes. Therefore the observed value Ngen = 3 cannot be derived from gauge and anomaly closure alone. It requires an additional QDL flavor, modular, mass, CP, or neutrino-sector principle. The final result is: QDL gauge closure axioms -> GSM -> Q, L, H -> SSM^ (1). This is a conditional Standard Model spectrum theorem: the Standard Model gauge seed and one-generation chiral matter package are selected as the minimal closure-complete solution under explicit QDL closure axioms.