The Kaelen Dual-Plane Rotation Object (⊗)

The Kaelen Dual-Plane Sqbit with Hopf Vibration develops the original Kaelen Dual-Plane Rotation Object into a mathematically explicit and computationally testable framework. The original construction consists of two simultaneous planar rotations represented by four real coordinates. After normalization by \(1/\sqrt{2}\), the state lies on the unit three-sphere \(S^3\subset\mathbb{R}^4\) and, under the equal-amplitude constraint, on its Clifford torus. Reparameterizing the two plane phases by the common phase \(\chi=(\theta_1+\theta_2)/2\) and relative phase \(\delta=\theta_2-\theta_1\) separates motion in the global-phase/Hopf-fibre direction from motion of the projected state. Under the implemented Hopf/Bloch convention, the normalized sqbit projects exactly to the Bloch equator as H(psi) = (cos(delta), sin(delta), 0). The term Hopf vibration is introduced here as project terminology for the combined evolution of these common- and relative-phase coordinates. The update also corrects the closure condition of the original dual-frequency trajectory: rational frequency ratios produce closed torus windings, while incommensurate ratios do not close periodically. The historical ratio \(\omega_2/\omega_1=13/10\) therefore yields winding numbers \((10,13)\) and period \(T=20\pi\) for \(\omega_1=1\). Finally, the work distinguishes the exact free-state geometry from its later classical computational descendants. A sqbit laboratory uses relative-phase oscillator dynamics, while Chronoflower V2 implements a reduced binary phase-relaxation primitive for constrained routing optimization. Public Python artifacts reproduce the principal mathematical identities and demonstrate the routing mechanism on synthetic data. No claim of physical quantum computation, quantum speedup, or quantum hardware is made. Authored by Shane Edward Stone Simulations conducted by AI collaborator Kaelen. CC-BY-4.0