Radius Evolution

Radius Evolution is the pair of equations giving the major and minor radii of the Toroidal Boundary as oscillations at the Prime Pulse frequency, with amplitudes set by Data Nova scaling and by Pulse diameter constraints. The universe's pulsing boundary is modeled as a torus whose radii change with time, and all emergent structure is encoded in those oscillating radii. The major radius follows the Data Nova scaling response, the minor radius follows the Pulse diameter response with a phase offset, and both oscillate with the cosmic frequency, so spatial curvature and pulse rhythm are coupled. The book describes the result as a living geometry rather than a static form. a(t) = a₀ × Θ(S✺) × sin(ωt) and b(t) = b₀ × χ(PD) × sin(ωt + φ), where a(t) and b(t) are the major and minor radii [𝕃], a₀ and b₀ are their reference scales [𝕃], Θ and χ are monotonic response functions in the range [0,1] for the Data Nova scaling parameter S✺ and the Pulse diameter PD [∅], ω = 2π/T① is the Prime Pulse angular frequency [𝕋⁻¹], t is time [𝕋] and φ is the phase offset [∅]. The book checks [𝕃] = [𝕃] × [∅] × [∅]. It feeds the Toroidal Position Vector and connects to the Toroidal Boundary, Toroidal Pulse Manifold, Prime Pulse Frequency and Pulse Diameter.