The Toroidal QDC Knot: A Closure-Stable Geometric Substrate Mode for the Quantized Dimensional Ledger

This Zenodo full-version preprint introduces the Toroidal QDC Knot as a closure-stable geometric substrate mode for the Quantized Dimensional Ledger. The paper extends the QDL substrate interpretation by modeling Planck-scale candidate fluctuations as compact two-cycle recurrence modes rather than as formless microscopic disturbances or a material medium. A toroidal QDC knot is defined as a candidate mode T₍, ₌ with winding data (n, m), sectoral closure data, and effective persistence cell QDCT = VT omega₁ omega₂ ~ L³ F². This gives the QDL Quantized Dimensional Cell a geometric interpretation: compact spatial occupancy multiplied by two-cycle recurrence. Each candidate is assigned a toroidal closure vector GammaT (T), and physical persistence is defined by the closure condition CTQDLGammaT (T) = 0. The resulting closure sequence is T₍, ₌ -> QDCT -> GammaT (T) -> CTQDL = 0 -> RTQDL. Conditional on the toroidal QDC representation, the paper develops a unified closure mechanism for several QDL targets: a minimal three-family recurrence-class structure Ngen = 3, a toroidal interpretation of the charged-lepton relational phase thetaₗ = 2/9, a charged-lepton mass-ratio reconstruction pathway through the Koide occupancy-amplitude cone, a vacuum residual selection rule, a gauge-sector admissibility criterion, a non-SM sector exclusion test, the Compton-gravity recurrence threshold m_*/mP = 1/sqrt (2), and a candidate toroidal QDC Hilbert space. The paper includes a minimal toroidal family-class lemma, a physical justification for primitive parity reduction, a reproducible toroidal projection toy model for the Koide cone, a finite toroidal vacuum-filter illustration, and a worked non-SM exclusion audit for an uncompensated anomalous U (1) ' sector. The proposal is explicitly conditional and falsifiable: if a physically necessary structure requires a non-compensable toroidal closure violation, the toroidal QDC substrate hypothesis fails or must be revised. This record is intended as the canonical QDL geometric substrate-mode paper. It does not claim a completed derivation of the Standard Model, a full quantum theory of gravity, a numerical derivation of the cosmological constant, or a final theory of dark matter, inflation, black-hole microstates, or time. Its purpose is to define the toroidal QDC knot as a compact closure object from which these problems become structurally attackable within QDL.