Nyquist Constraint

The Nyquist Constraint is the requirement that each loop of the Zinf torus contain at least N⥥ = 4 cells, so that at least one nontrivial standing mode can form without spatial or temporal aliasing. The book treats the minimal closed torus like a sampled signal. A standing wave needs enough cells per loop to be represented; with too few, it collapses into a trivial or aliased state, just as a signal sampled below the Nyquist rate aliases. The book adds that the substrate's clock also sets a floor on the product N⥥ · κ_z · ①⥂⧖, and a torus specified below that floor would have to turn faster than the substrate can represent, the same failure as aliasing arriving from the angular rather than the temporal side. The constraint fixes the Minimal Radii a⥥ = b⥥ = (N⥥/(2π)) × ℓ_z, the Minimal Surface Area A⥥ = (N⥥)² × (ℓ_z)² and the count of boundary cells, and so defines the smallest closed torus that can sustain nontrivial oscillations. In time, its partner is the Temporal Alias Bound ω⥣ × ①⥂⧖ ≤ π, with ①⥂⧖ the Pulse Rhythm. The formal statement is N⥥ ≥ 4, where N⥥ is the minimum number of cells per loop for a stable sinusoidal mode, dimensionless [∅]; ℓ_z, a⥥ and b⥥ are [𝕃]. It connects to Minimal Radii, Minimal Radius Constraints, Minimal Surface Area, Maximum Modal Frequency, Zinf-Limit Spatial Quantum and Computational Degrees of Freedom.