  Many authors noted that the principle of relativity, together with space-time
symmetries, suffices to derive Lorentz-like coordinate transformations between
inertial frames. These contain a free parameter, $k$, (equal to $c^{-2}$ in
special relativity) which is usually claimed to be empirically determinable, so
that special relativity does not need the postulate of constancy of the speed
of light. I analyze this claim and find that all methods destined to measure
$k$ fail without further assumptions, similar to the second postulate.
Specifically, measuring $k$ requires a signal that travels identically in
opposite directions (this is unrelated to the conventionality of
synchronization, as the one-postulate program implicitly selects the standard
synchronization convention). Positing such a property about light is logically
weaker than Einstein's second postulate but suffices to recover special
relativity in full.
