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…