Instantaneous Modal Frequency

The Instantaneous Modal Frequency is the frequency of a standing-wave mode on the pulsing toroidal boundary at a given moment, fixed by the mode numbers, the torus's current radii and the computational lightspeed. The book treats the toroidal boundary as a quantized resonator. Waves on it are written as modes Ψ_{m,n}(θ,φ;t) = A_{m,n}(t) × exp(i(mφ + nθ - ω_{m,n}(t)t)), labelled by a toroidal number m and a poloidal number n, both integers. Because the torus pulses, its major and minor radii change with time, and each mode's frequency follows them, while the propagation speed bound keeps the dynamics causal. Geometry thus decides which modes are allowed, frequencies scale with the evolving radii, and coherence is enforced by the speed limit. The formal statement is ω_{m,n}(t) = 𝒞→ × √((m/a(t))² + (n/b(t))²), where ω_{m,n}(t) is the instantaneous modal frequency, 𝒞→ the computational lightspeed limit, m the toroidal mode number, n the poloidal mode number, a(t) the time-dependent major radius and b(t) the time-dependent minor radius. Dimensions from the book's Where-list: ω_{m,n} [𝕋⁻¹], 𝒞→ [𝕃·𝕋⁻¹], m and n [∅], a(t) and b(t) [𝕃], t [𝕋]. It connects to the Modal Amplitude Function, Maximum Modal Frequency, Modal Quantization, Toroidal Boundary and Computational Lightspeed.