It occurred to me overnight that we can actually do a bit better. In fact, no additional physical postulate or argument, such as causality or my "physical regularity", are needed. We can determine the sign of ##\lambda_a## simply because the case ##\lambda_a > 0## does not yield a mathematically well-defined nontrivial group of velocity boosts.
To see this in more detail, note first that a nontrivial 1-parameter group of velocity boosts must have a parameter space ##V## which is at least an open set containing 0. Elementary group properties require that for any two velocities ##v, v' \in V##, the composition of those velocities must also be in ##V##, else we do not have a good group. Now consider the velocity addition formula in post #37, specialized to the case where##0 < \lambda_a =: \zeta^{-2}##, for some real ##\zeta > 0##. The velocity addition formula becomes $$v'' ~=~ \frac{v + v'}{1 - v'v/\zeta^2} ~.$$The value ##\,v = \zeta\,## then cannot be an allowed parameter value in ##V##, since composition with itself gives ##v''\sim\infty##, i.e., undefined. For convenience, let us introduce a new variable ##\omega := v/\zeta##. Then the velocity addition formula can be written as $$\omega'' ~=~ \frac{\omega + \omega'}{1 - \omega'\omega} ~.$$Any pair ##\omega'## and ##\omega## satisfying ##\,\omega\,\omega' = 1\,## yields an undefined ##\omega'' \sim\infty##. Moreover, we cannot solve this problem by restricting to ##|\omega| < W##, for some constant ##W##, since multiple boosts can eventually yield a resultant velocity greater than ##W##, contradicting our attempted restriction. Only the trivial case ##w \in \{0\}## remains mathematically valid, but this is useless for physics.
Therefore we can discard ##\lambda_a > 0## simply because on a nontrivial velocity domain it gives a mathematically invalid group. There is no need to invoke any additional physical postulate or argument. Merely requiring mathematical consistency is sufficient.
So if ##\lambda_a \ne 0##, only 1 possibility remains: $$\boxed{~ \lambda_a ~<~ 0 \;,~}$$ with velocity addition formula: $$v'' ~=~ \frac{v + v'}{1 + v'v \, |\lambda_a|} ~.$$This gives a mathematically well-defined group, with ##V## containing only those ##v## such that ##\,|v|^2 \le -\lambda_a^{-1}##.
I want to understand the nature as much as I can.
