Minimal Radius Constraints

Minimal Radius Constraints are the conditions that each circumference of the Zinf torus be long enough to hold at least N⥥ Zinf cells, so that at least one nontrivial standing mode fits in each periodic direction under discrete sampling. A torus has two periodic directions, around its major radius and around its minor radius. Space at the Zinf limit is discretized into causal quanta of length ℓ_z, and a loop with too few cells cannot carry a proper sinusoid. The book sets the minimum at four cells per loop. These constraints make spatial quantization a structural necessity rather than a mathematical convenience: they anchor geometric stability and coherence at the most fundamental scale of recursion, and they lead into Modal Quantization, where standing-wave eigenmodes separate along both periodic coordinates. The formal statements are 2π a⥥ ≥ N⥥ × ℓ_z and 2π b⥥ ≥ N⥥ × ℓ_z, where a⥥ and b⥥ are the minimal major and minor radii, N⥥ the minimum cells per loop for stable discretization (4) and ℓ_z the Zinf-limit spatial quantum. Dimensions: a⥥, b⥥, ℓ_z [𝕃]; 2π, N⥥ [∅]; the book's check is consistent. It connects to Minimal Radii, Minimal Surface Area, Zinf-Limit Spatial Quantum, Modal Quantization and Nyquist Constraint.