Planck-Scale Closure Stiffness as a Selection Principle for Toroidal Recurrence

This preprint develops a conditional selection principle for toroidal recurrence in Planck-normalized theoretical models. The starting point is the persistence problem: finite-energy projected modes must avoid unbounded uncompensated residuals under recurrence. The Planck force, F_P = c^4/G = E_P/l_P, is interpreted not as a new interaction but as the natural Planck-scale energy-per-length normalization for closure-gradient cost. The paper defines a closure-residual framework in which projected configurations are evaluated by whether their residuals are exact, compensable, or noncompensable. Under a monotone residual-cost model, open noncompensable residuals fail finite persistence when residual cost accumulates without bound. A single recurrence cycle S^1 can support return or phase closure, but cannot internally test both return closure and nontrivial projection closure. Projection-stable persistent modes therefore require recurrence rank at least two: one cycle for return or phase closure, and one independent cycle for projection, winding, orientation, or sector closure. Under this recurrence-rank condition, the minimal closed product topology is T^2 = S^1 x S^1. The paper’s contribution is not the elementary topological identity S^1 x S^1 = T^2, but the recurrence-rank selection argument connecting Planck-normalized residual cost, finite persistence, and independent projection closure. The result is interpreted within the Quantized Dimensional Ledger as a substrate-selection gate supporting later work on toroidal QDC closure, vacuum projection, chirality, sector mapping, and operator admissibility. The conclusion is conditional: toroidal recurrence is the minimal admissible two-cycle topology under Planck-normalized residual minimization and projection-stability assumptions. The paper does not claim that the physical vacuum is a literal macroscopic torus, nor does it replace quantum field theory, general relativity, Kaluza-Klein theory, string compactification, or standard gauge theory.