Modal Amplitude Function

The Modal Amplitude Function is the form of a single standing-wave mode on the closed toroidal boundary, a time-varying amplitude times a phase that winds around both periodic directions of the torus. The book uses it to analyze discrete standing-wave patterns on the Zinf torus. Standing-wave eigenmodes separate along the toroidal and poloidal coordinates, so each mode is labeled by two integers, and the torus's spatial periodicity becomes a structured spectrum of oscillations governed by the causal lightspeed bound. Each mode's frequency follows the torus's changing radii through the Instantaneous Modal Frequency ω_{m,n}(t) = 𝒞→ × √((m/a(t))² + (n/b(t))²), making the boundary a quantized resonator whose geometry decides the permitted modes. The formal statement is Ψ_{m,n}(θ,φ;t) = A_{m,n}(t) × exp(i(mφ + nθ − ω_{m,n}(t)t)), where Ψ_{m,n} is the modal amplitude function, A_{m,n}(t) the time-dependent mode amplitude, m the toroidal mode number and n the poloidal mode number (both integers), φ the toroidal angle, θ the poloidal angle, ω_{m,n}(t) the instantaneous modal frequency and t time. Dimensions: Ψ_{m,n}, A_{m,n}, m, n, φ, θ [∅]; ω_{m,n} [𝕋⁻¹]; t [𝕋]; the book's check is consistent. It connects to the Instantaneous Modal Frequency, Modal Quantization, Minimal Radius Constraints, Toroidal Boundary and Standing Wave Patterns.