Positive and Negative Particle Masses in the Bicubic Equation Limiting Particle Velocity Formalism

The interest in the negative particle mass here got encouraged by the Rachel Gaal July 2017 APS article (Gaal, 2017)
describing Khamehchi et al. (2007) observation of an effective negative mass in a spin-orbit coupled Bose-Einstein
condensate. Hence, since in the bicubic equation limiting particle velocity formalism (Soln, 2014, 2015, 2016, 2017)
positive m+ = m ≻ 0 and negative m− = −m ≺ 0 masses with m2+ = m2− = m2 are equally acceptable, then from a purely
theoretical point of view, the evaluation of particle limiting velocities for both m+ and a m− masses should be done.
Starting with the original solutions for particle limiting velocities c1; c2 and c3, given basically for a positive particle
mass m+ (Soln, 2014, 2015, 2016, 2017), now also are done for a negative particle mass m− This is done consistent with
the bicubic equation mathematics, by solving for c1; c2 and c3 not only form+ but also for m−. Hence, in addition to
having the limiting velocities of positive mass m+ primary, obscure and normal particles, now one has also the limiting
velocities of negative mass m− primary, obscure and normal particles, however, numerically equal to limiting velocities,
respectively of m+ masses obscure, primary and normal particles, forming the m+ and m− masses of equal limiting velocity
value doublets : c1(m−) = c2(m+), c2(m−) = c1(m+) , c3(m−) = c3(m+). Now, one would like to know as to which particle
with a negative mass m− = −m ≺ 0, obtained from the positive mass m+ = m ≻ 0 with the substitution m − −m, can
have a real limiting velocity? It turns out that it is the obscure particle limiting velocity c2(m+) that changes from the
imaginary value, c22(m+) ≺ 0, into the real limiting velocity value c22(m−) ≻ 0 when the change m+ − m− is made and,
at the same time, retaining the same energy. Similar procedure applied to the original primary particle limiting velocity
starting with c21(m+) ≻ 0 , keeping the total energy the same,with the change m − −m one ends up with c21
(m−) ≺ 0 that is, imaginary c1. The procedure of changing m+ − m− in normal particle limiting velocity causes no change, it remains the same realc3. Because m2 (= m2+ = m2−), E2 and v2 remain the same , these mass regenerations, m+ − m− and m− − m+ could in principle also occur spontaneously.