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Double-spin asymmetries and absolute cross sections were measured at largeBjorken $x$ (0.25 $ \le x \le $ 0.90), in both the deep-inelastic and resonanceregions, by scattering longitudinally polarized electrons at beam energies of4.7 and 5.9 GeV from a transversely and longitudinally polarized $^3$He target.In this dedicated experiment, the spin structure function $g_2$ on $^3$He wasdetermined with precision at large $x$, and the neutron twist-three matrixelement $d_2^n$ was measured at $\left< Q^2\right>$ of 3.21 and 4.32GeV$^2$/$c^2$, with an absolute precision of about $10^{-5}$. Our results arefound to be in agreement with lattice QCD calculations and resolve thedisagreement found with previous data at $\left< Q^2\right> =$ 5 GeV$^2$/$c^2$.Combining $d_2^n$ and a newly extracted twist-four matrix element, $f_2^n$, theaverage neutron color electric and magnetic forces were extracted and found tobe of opposite sign and about 30 MeV/fm in magnitude.

A Precision Measurement of the Neutron Twist-3 Matrix Element $d_2^n$: Probing Color Forces