= Solution
The <baryon hyperfine mass formula> is a phenomenological <constituent-quark model>. The three <constituent quark masses> include effects of strong-interaction dressing and an approximate spin-independent confinement energy; they are not the small running <quark masses> in a short-distance <Quantum chromodynamics> calculation.
The residual correction has the form of a colour-magnetic <spin-spin interaction>. A nonrelativistic reduction of one-<gluon> exchange produces a contact contribution proportional to
$$
\alpha_s\frac{\mathbf S_i\cdot\mathbf S_j}{m_im_j}\,\delta^3(\mathbf r_i-\mathbf r_j).
$$
After taking its expectation in the common spatial <wavefunction> and the <colour singlet>, the colour factor, coupling, and contact density are absorbed into $A'$. Summing the three quark pairs gives the supplied formula. Positive $A'$ raises aligned-spin configurations relative to lower-spin configurations. In <natural units>, the <spin angular momentum> operators are dimensionless, so $A'$ has units of <energy> cubed, as in the given $\mathrm{GeV^3}$.
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