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Let $\mathbf J$ be the spin- j angular-momentum operator and $\boldsymbol\sigma^{(n)}$ the Pauli vector for probe n . A candidate rotationally invariant interaction is $H_{\mathrm{int}}^{(n)} =\hbar\Omega\, \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}, \qquad U_n=\exp\!\left(-i\kappa \frac{\mathbf J\cdot\boldsymbol\sigma^{(n)}}{2j}\right)$. where $\Omega$ has units of inverse seconds, interaction time $\tau$ has seconds, and $\kappa$=$\Omega$$\tau$ is dimensionless. The factor 2j is a declared normalization choice, not an inherited law. Alternative normalizations must be compared because they change the large- j limit if coupling strength is held fixed under different conventions.

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