Toroidal Position Vector

The Toroidal Position Vector is the parametrization of the pulsing Toroidal Boundary as a standard torus with time-dependent major and minor radii, giving the position of each boundary point from its poloidal and toroidal angles. It formalizes the recursive pulse boundary as a closed two-dimensional manifold on which all physical information resides, while the interior keeps null conditions. All emergent structure is encoded in the oscillating radii, which follow the Radius Evolution equations at the Prime Pulse frequency, so the position vector couples spatial curvature to pulse rhythm. x(θ,φ;t) = ((a(t) + b(t)cos θ)cos φ, (a(t) + b(t)cos θ)sin φ, b(t)sin θ), where x(θ,φ;t) is the boundary position vector [𝕃], θ is the poloidal angle and φ the toroidal angle, each from 0 to 2π [∅], a(t) is the time-dependent major radius [𝕃], b(t) is the time-dependent minor radius [𝕃] and t is time [𝕋]. The book checks [𝕃] = ([𝕃] + [𝕃] × [∅]) × [∅]. It parametrizes the Toroidal Boundary using Radius Evolution, and connects to Strict Interior Nulling, Toroidal Pulse Manifold and Instantaneous Modal Frequency.