The QDC Completion Theorem: Matter-Basis Minimality, Three-Family Automorphism, Charged-Lepton Closure, and Gravitational Recurrence in the Quantized Dimensional Ledger

This work presents the QDC Completion Theorem as a comprehensive substrate-level completion architecture within the Quantized Dimensional Ledger. Starting from the toroidal Quantized Dimensional Cell, the closure functional C_QDC(w)=2Σ_L−3Σ_F, and the gravitational closure cell [GM]=L^3F^2, the paper develops a finite closure framework for Standard-Model admissibility, matter-basis minimality, anomaly/Yukawa hypercharge closure, scalar Higgs compensation, three-family residual automorphism structure, charged-lepton mass-ratio access, gravitational recurrence, dark-sector residuals, and cosmological background closure. The paper distinguishes exact anchors, conditional theorems, restricted minimality results, constrained branches, candidate derivations, and open numerical/dynamical gates. Its central results are the Matter-Basis Projection Theorem, the Primitive Three-Family Automorphism Computation, and the Charged-Lepton Mass Closure Theorem. The paper does not claim to be a completed final theory. It identifies the finite mathematical and physical gates that must close for QDL to become a stronger candidate substrate-level completion theory. The graphical abstract included with this record is a schematic representation of the QDC recurrence substrate and completion architecture. The blue cycles L_1, L_2, L_3 denote the three length-type recurrence channels, and the orange cycles F_1, F_2 denote the two frequency-type recurrence channels of T^5_QDC=(S^1_L)^3×(S^1_F)^2. The toroidal rendering is a visual schematic of typed closure recurrence, not an assertion that an ordinary two-dimensional torus has five independent topological cycles.