Modal Quantization

Modal Quantization is the restriction of waves on the closed toroidal boundary to a discrete set of standing-wave modes, labeled by two integers and bounded by the computational lightspeed and the discreteness of time. It is the bridge the book draws between geometry and resonance on the Zinf torus. Because the torus is periodic in two directions, standing-wave eigenmodes separate along the toroidal and poloidal coordinates, and spatial periodicity becomes a structured spectrum of oscillations. Each mode has the form Ψ_{m,n}(θ,φ;t) = A_{m,n}(t) × exp(i(mφ + nθ − ω_{m,n}(t)t)), with integer mode numbers m and n, and its frequency follows the torus's changing radii through ω_{m,n}(t) = 𝒞→ × √((m/a(t))² + (n/b(t))²), where 𝒞→ is the computational lightspeed and a(t), b(t) the major and minor radii. The Minimal Radius Constraints guarantee at least one nontrivial mode in each direction, and the Temporal Alias Bound caps the fastest mode. The book concludes that the boundary is a quantized resonator: geometry decides the permitted modes, frequencies scale with the radii, and propagation speed enforces coherence. It connects to the Modal Amplitude Function, Instantaneous Modal Frequency, Minimal Radius Constraints, Maximum Modal Frequency, Nyquist Constraint and Toroidal Boundary.