Toroidal QDC Closure Cell: Planck-Normalized L 3 F 2 Geometry, QDL String Modes, and the Sector-Complete Particle Assignment Problem

This paper introduces the Toroidal Quantized Dimensional Cell, or Toroidal QDC, as a compact closure-space representation of the Quantized Dimensional Ledger cell L3F2L^3F^2L3F2. The construction represents the three length powers and two frequency powers of the QDC as five compact closure cycles, TQDC5=(SL1)3×(SF1)2T^5_{\mathrm{QDC}}=(S_L^1)^3 \times (S_F^1)^2TQDC5=(SL1)3×(SF1)2. A QDL string mode is defined as an admissible harmonic winding path on this compact ledger-space geometry, with integer winding vector in Z5\mathbb{Z}^5Z5. The main formal result is the derivation of the QDC closure functional CQDC(w)=2(wL1+wL2+wL3)−3(wF1+wF2)C_{\mathrm{QDC}}(w)=2(w_{L1}+w_{L2}+w_{L3})-3(w_{F1}+w_{F2})CQDC(w)=2(wL1+wL2+wL3)−3(wF1+wF2) as the unique primitive integer homomorphism on the five-cycle QDC winding lattice satisfying additivity, permutation symmetry among equivalent length and frequency cycles, nontriviality, primitiveness, and neutrality on the primitive L3F2L^3F^2L3F2 winding. Its kernel defines the first-pass QDC lattice, and closed modes admit a quantized closure level kQDCk_{\mathrm{QDC}}kQDC. The Toroidal QDC is not proposed as a literal Planck-scale torus in physical space. It is a compact closure geometry in dimensional-ledger space. Its purpose is to provide an auditable admissibility framework for classifying winding modes, operator transitions, Regge-zero kernel events, ultraviolet nonleakage, Planck-scale gravitational closure, mass-frequency occupancy, Koide-type flavor projection, Schwarzschild threshold behavior, and the particle-assignment problem. The paper proves the Planck-QDC identity GMP=LP3FP2GM_P=L_P^3F_P^2GMP=LP3FP2, derives a Schwarzschild closure-level ladder RS(k)=2kQDCLPR_S(k)=2k_{\mathrm{QDC}}L_PRS(k)=2kQDCLP, reconstructs minimal Regge-zero residue factors as QDC winding-kernel events, develops a Planck-normalized mass-occupancy relation, and derives the Koide charged-lepton relation from a balanced QDC flavor-projection postulate. It also proves that QDC closure alone cannot uniquely assign physical particle identities. A complete particle assignment requires a sector-complete projection map that supplies spin, gauge, charge, chirality, flavor, and mass-sector information. The claim is structural and bounded: the Toroidal QDC does not by itself derive the Standard Model, prove string theory, or solve quantum gravity. It provides a compact, derived, and reproducible admissibility lattice on which more complete physical assignments and operator/amplitude audits can be built.