  We compare the redshifts of neutral carbon and carbon monoxide in the
redshifted sources in which the fine structure transition of neutral carbon,
[CI], has been detected, in order to measure space-time variation of the
fundamental constants. Comparison with the CO rotational lines measures gives
the same combination of constants obtained from the comparison fine structure
line of singly ionised carbon, [CII]. However, neutral carbon has the distinct
advantage that it may be spatially coincident with the carbon monoxide, whereas
[CII] could be located in the diffuse medium between molecular clouds, and so
any comparison with CO could be dominated by intrinsic velocity differences.
Using [CI], we obtain a mean variation of dF/F = (-3.6 +/- 8.5) x 10^-5, over z
= 2.3 - 4.1, for the eight [CI] systems, which degrades to (-1.5+/- 11) x
10^-5, over z = 2.3 - 6.4 when the two [CII] systems are included. That is,
zero variation over look-back times of 10.8-12.8 Gyr. However, the latest
optical results indicate a spatial variation in alpha, which describes a dipole
and we see the same direction in dF/F. This trend is, however, due to a single
source for which the [CI] spectrum is of poor quality. This also applies to one
of the two [CII] spectra previously used to find a zero variation in
alpha^2/mu. Quantifying this, we find an anti-correlation between |dF/F| and
the quality of the carbon detection, as measured by the spectral resolution,
indicating that the typical values of >50 km/s, used to obtain a detection, are
too coarse to reliably measure changes in the constants. From the fluxes of the
known z > 1 CO systems, we predict that current instruments are incapable of
the sensitivities required to measure changes in the constants through the
comparison of CO and carbon lines. We therefore discuss in detail the use of
ALMA for such an undertaking ... ABRIDGED
