Odd-even staggering for production cross sections ofnuclei near the neutron drip-line

In our previous studies Phys. Rev. C 97, 044619 (2018); Phys. Rev. C 103, 044610 (2021)], a universal odd-even staggering (OES) has been observed in extensive cross sections of isotopes not far from stability, measured for different fragmentation and spallation reactions. Four OES relations have been proposed on the basis of this OES universality. However, it is still unclear whether this OES universality and OES relations are applicable to many isotopes near the drip-lines. Here, the OES in recent experimental cross sections of very neutron-rich nuclei approaching the drip-line (from \begin{document}$^{76}$\end{document}Ge,\begin{document}$^{82}$\end{document}Se+\begin{document}$^{9}$\end{document}Be) is quantitatively investigated, to further validate the OES universality and OES relations. The OES magnitudes in these experimental data approaching the neutron drip-line generally agree with those evaluated previously, mainly from experimental data near stability. New OES evaluations derived from these experimental data are also recommended for more exotic nuclei near the neutron drip-line, which extends the conclusions of our previous OES studies. In addition, the OES relation calculations are consistent with these experimental data of very neutron-rich nuclides according to their comparisons in this work. Finally, comparisons with additional experimental data (from \begin{document}$^{238}$\end{document}U+\begin{document}$^{9}$\end{document}Be) also support that new OES evaluations and OES relation calculations can be applied for exotic nuclei near the neutron drip-line.