Vacuum Projection and Sector Mapping from a Typed Toroidal Closure Lattice: A Quantized Dimensional Ledger Construction for Standard-Model Admissibility

This paper develops the mathematical completion mechanism required by the Quantized Dimensional Ledger (QDL) after the formal definition of the Planck-scale toroidal closure substrate. Starting from the typed toroidal Quantized Dimensional Cell TQDC5=(SL1)3×(SF1)2T^{5}_{QDC}=(S^{1}_{L})^{3}\times(S^{1}_{F})^{2}TQDC5=(SL1)3×(SF1)2, the paper defines integer winding vectors, the QDC closure functional CQDCC_{QDC}CQDC, and the closure lattice LQDC=ker⁡CQDC\mathcal{L}_{QDC}=\ker C_{QDC}LQDC=kerCQDC. It then shows why closure alone is insufficient for physical-sector derivation and introduces the required completion structures: the vacuum projection Πvac\Pi_{\mathrm{vac}}Πvac, the sector map Σ\SigmaΣ, and the closure-vector operator audit Γ(O)\Gamma(O)Γ(O). The paper provides explicit algebraic properties of the QDC closure lattice, including subgroup structure, rank, basis construction, typed permutation symmetry, projection examples, admissibility gates, compensator logic, and a toy Standard-Model-oriented sector partition. It does not claim to complete a derivation of the Standard Model. Instead, it defines the mathematical gate through which any future QDL derivation of Standard-Model sectors must pass. The work is intended as the formal sequel to the QDL substrate paper and as the foundation for subsequent papers on anomaly cancellation, operator-governance audits, hypercharge closure, and sector admissibility.