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We performed a measurement of the inclusive missing-mass spectrum of the |$^{12}$| C |$(K^-, K^+)$| reaction at an incident beam momentum of 1.8 GeV/ c. This measurement was carried out by using the Superconducting Kaon Spectrometer (SKS) and the K1.8 beamline spectrometer at the Hadron Experimental Facility in J-PARC. From the missing-mass of the |$^{12}$| C |$(K^-, K^+)$| reaction, the binding energy of a |$\Xi ^-$| hyperon in a core |$^{11}$| B nucleus, |$B_{\Xi ^-}$|⁠ , can be calculated. Our experimental setup yielded a good energy resolution of 8.2 MeV (full width at half maximum), which allowed us to observe significant enhancements in the proximity of the |$^{12}_{\Xi }$| Be production threshold region. In order to extract information from the missing-mass spectrum, we employed several fitting parameters assumptions. A good agreement with the spectrum shape was obtained by adding two Gaussian functions, with the constant experimental resolution for the |$\Xi$| -hypernuclear states, to the background distribution. The peak positions were obtained to be |$B_{\Xi ^-} = 8.9 \pm 1.4$| (stat.) |$^{+3.8}_{-3.1}$| (syst.) MeV and |$B_{\Xi ^-} = -2.4 \pm 1.3$| (stat.) |$^{+2.8}_{-1.2}$| (syst.) MeV. Another model assumption, one Breit–Wigner function with |$B_{\Xi ^-} = -2.7 \pm 2.2$| (stat.) |$^{+0.5}_{-0.7}$| (syst.) MeV and |$\Gamma = 4.1 \pm 2.1$| (stat.) |$^{+1.2}_{-0.7}$| (syst.) MeV, also yielded a similar |$\chi ^2$| value. [ABSTRACT FROM AUTHOR]