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A review is given of hypothetical faster-than-light tachyons and thedevelopment of the author's $3+3$ model of the neutrino mass states, whichincludes one tachyonic mass state doublet. Published empirical evidence for themodel is summarized, including an interpretation of the mysterious Mont Blancneutrino burst from SN 1987A as being due to tachyonic neutrinos having$m^2=-0.38 eV^2.$ This possibility requires an 8 MeV antineutrino line from SN1987A, which a new dark matter model has been found to support. Furthermore,this dark matter model is supported by several data sets: $\gamma-$rays fromthe galactic center, and the Kamiokande-II neutrino data on the day of SN1987A. The KATRIN experiment should serve as the unambiguous test of the $3+3$model and its tachyonic mass state.

Review of the empirical evidence for superluminal particles and the $3+3$ model of the neutrino masses