The Qubit on the Two Rings: Rotation Gates, Pauli Phases and the Live Checksum

A qubit is one setting of the two sliding rings. The rotation gate Rz(θ) has entries cos(θ/2) and sin(θ/2), and the instrument's law is d = 2cos(θ/2) between the centres and e = 2sin(θ/2) between the crossings at crossing separation θ: the gate matrix is read off the setting, the Bloch amplitudes are the two half-distances, and Born's rule is the two shares of one unit. Composing rotation gates is adding positions, certified by the ring's addition law. The spinor's sign, −1 after a 360° rotation and +1 after 720°, is the ring's half turn, digits summing to ten, and its lap. The Pauli phases i, −1, −i are the quarter-turn orbit 7, 9, 3, and the phase exponent of any Pauli product is carried by the digits, verified on two hundred random products. The hidden symmetry of the Jordan–Wigner transformation that Davis and Freericks (arXiv:2512.24589) use to halve measurement circuits is the ring turning, its two angles are the quarter turn and the eighth turn, and its identities are the fold law averaged round the ring, whose coefficients 3/8 and 1/8 are exact averages over the sixty positions. Their equation 24 is verified on random fixed-number states of four qubits. From this one picture the paper places the live checksum where it can live in a quantum computer: on the classical phase registers that drive virtual-Z gates and merge rotations, on the classical Pauli frame, and on measured expectation values through the symmetry identities, which flag every number-changing error, while the law of readings of the companion paper catches the number-conserving ones. And it states where it cannot live, on the amplitudes themselves, by the no-cloning theorem. The four operator-mechanics papers of Freericks and colleagues, the operator Rodrigues formula, the momentum measurement, the free expansion of a Gaussian and the squeezed-state wavefunction, are then worked side by side as examples of the same picture: one slide, one quarter turn, one shear, one rotating ellipse. Every number is verified by the script in the appendix.