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^5ₐ₃₂= (S^1₋) ^3 (S^1₅) ^2TQDC5= (SL1) 3× (SF1) 2, the paper defines integer winding vectors, the QDC closure functional CQDCCₐ₃₂CQDC, and the closure lattice LQDC=ker⁡CQDCLₐ₃₂= Cₐ₃₂LQDC=kerCQDC. It then shows why closure alone is insufficient for physical-sector derivation and introduces the required completion structures: the vacuum projection Πvacₕ₀₂Πvac, the sector map ΣΣ, and the closure-vector operator audit Γ (O) (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.