A Pauli-matrix spin register for proton–neutron structure:unifying the Pauli spin matrices and the Pauli exclusion principle in a 3×3 boundary condition for nucleon structure

We propose a diagonal spin register, M-hat, that compactly records the single-quark spin projections of the proton and neutron in a chosen quantization basis. The physical spin content of each quark pair is instead determined by rotationally invariant two-quark spin projectors together with the color, flavor, and spatial exchange symmetries of the ground-state nucleon. The resulting spin-dependent chromomagnetic interaction is reduced to an effective one-body potential and inserted into individual relativistic Dirac equations with confinement imposed through the MIT bag boundary condition. The quark magnetic moments are then combined through the standard S U ( 6 ) spin-flavor recoupling. The revised formulation reproduces the negative sign of the neutron magnetic moment at a representative numerical parameter point, while making explicit that the negative sign in the degenerate S U ( 6 ) limit is already fixed by the quark charges and spin-flavor recoupling. The present one-body reduction is not intended as a precision treatment; the full two-body color-magnetic matrix element is identified as the well-defined quantitative extension.