Quantum Dynamics and Entanglement on the Two Rings: Larmor Precession, Rabi Oscillations, Ramsey Fringes, Two Rings That Share One Radius, and the Line Between Classical and Quantum Correlation

Quantum mechanics does two things with a two-level system. It turns it, and it entangles it. Both live on the two-ring instrument. Dynamics first. The Schrödinger equation for any two-level Hamiltonian is precession. The Bloch vector rotates about the axis of the energy splitting at the rate the splitting sets, so time evolution on the rings is the setting moving, certified at every instant. Larmor precession is the pure turn at γB, 42.577 million turns a second for a proton in one tesla. The Rabi oscillation is the energy lever swept in time, the ground and excited shares cos² and sin² of one accumulated setting. The Ramsey fringe is two settings and a fold, the most precise frequency measurement in physics read as d/2. The spin echo is the half turn against drift. Entanglement second. For any pure two-qubit state the two qubits' rings carry the same radius, and that radius is set by the concurrence, r = √(1 − C²), full radius for a product state and zero for a Bell pair. Entanglement is two rings forced to share one radius. Generating it is the energy-sloshing lever of the companion papers read between rings. Under the exchange interaction the concurrence grows as sin Ωt while both radii shrink as cos Ωt, and the pair is a Bell state at the quarter period, when both radii vanish. The second part is then made into instruments. The entanglement monitor reads the concurrence of a running experiment from one qubit's statistics, one measurement setting for the exchange family and three for any pure state, against nine for tomography. The certified entanglement clock reads the slosh tick by tick, the calibration, the diagnostics and the ledger of an entangling gate. And the classical line is drawn where correlation stops being classical. The radius is blind on the whole Werner family, and the witness |⟨XX⟩| + |⟨YY⟩| + |⟨ZZ⟩| = 1, the fold read beside the correlation in the pair's own frame, draws the line the radius cannot. The same identity passed a real machine's clean run at 1.01 standard errors and flagged an injected fault at 112. Every number is verified by the script in the appendix, and the document is not built unless every check prints True.