We report the first measurement of the target-normal single-spin asymmetry in deep-inelastic scattering from the inclusive reaction $^3$He$^{uparrow}left(e,e right)X$ on a polarized $^3$He gas target. Assuming time-reversal invariance, this asymmetry is strictly zero in the Born approximation but can be non-zero if two-photon-exchange contributions are included. The experiment, conducted at Jefferson Lab using a 5.89 GeV electron beam, covers a range of $1.7 < W < 2.9$ GeV, $1.0<Q^2<4.0$ GeV$^2$ and $0.16<x<0.65$. Neutron asymmetries were extracted using the effective nucleon polarization and measured proton-to-$^3$He cross section ratios. The measured neutron asymmetries are negative with an average value of $(-1.09 pm 0.38) times10^{-2}$ for invariant mass $W>2$ GeV, which is non-zero at the $2.89sigma$ level. Our measured asymmetry agrees both in sign and magnitude with a two-photon-exchange model prediction that uses input from the Sivers transverse momentum distribution obtained from semi-inclusive deep-inelastic scattering.
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