Minimal Radii

The Minimal Radii are the smallest major and minor radii a Zinf torus can have, each just large enough for its circumference to hold N⥥ Zinf cells. For a Zinf-limited toroidal universe to sustain coherent dynamics, its geometry must meet minimum resolution thresholds that act like Nyquist bounds, letting standing modes form without collapsing into trivial or aliased states. Setting each circumference equal to N⥥ spatial quanta gives the two minimal radii; from them follow the Minimal Surface Area, A⥥ = 4π² × a⥥ × b⥥ = (N⥥)² × (ℓ_z)², and the count of boundary cells, (N⥥)². Together these define the smallest closed torus that can sustain nontrivial oscillations. Written with ℓ_z = κ_z × 𝒞→ × ①⥂⧖, the radii scale as a⥥ = b⥥ = (N⥥/(2π)) × κ_z × 𝒞→ × ①⥂⧖, with ①⥂⧖ the Pulse Rhythm. The formal statements are a⥥ = (N⥥/(2π)) × ℓ_z and b⥥ = (N⥥/(2π)) × ℓ_z, where a⥥ and b⥥ are the minimal major and minor radii, N⥥ the minimum cells per loop for a stable sinusoidal mode (N⥥ ≥ 4), 2π the circumferential factor and ℓ_z the Zinf-limit spatial quantum. Dimensions: a⥥, b⥥, ℓ_z [𝕃]; N⥥, 2π [∅]; the book's second check is valid and verified. It connects to Minimal Radius Constraints, Minimal Surface Area, Computational Degrees of Freedom, Zinf-Limit Spatial Quantum and Energy Threshold Scaling.