We make the most precise determination to date of the number density of extragalactic 21-cm radio sources as a function of their spectral line widths - the HI velocity width function (HIWF) - based on 22832 sources from the final 7000 deg$^2$ data release of the Arecibo Legacy Fast ALFA (ALFALFA) survey. The number density of sources as a function of their neutral hydrogen masses - the HI mass function (HIMF) - has previously been reported to have a significantly different low-mass slope and knee mass in the two sky regions surveyed during ALFALFA. In contrast with this, we find that the shape of the HIWF in the same two sky regions is remarkably similar, consistent with being identical within the confidence intervals implied by the data (but the overall normalisation differs). The spatial uniformity of the HIWF implies that it is likely a stable tracer of the mass function of dark matter haloes, in spite of the environmental processes to which the measured variation in the HIMF are attributed, at least for galaxies containing enough neutral hydrogen to be detected. This insensitivity of the HIWF to galaxy formation and evolution can be exploited to turn it into a powerful constraint on cosmological models as future surveys yield increasingly precise measurements. We also report on the possible influence of a previously overlooked systematic error affecting the HIWF, which may plausibly see its low-velocity slope steepen by $sim$40 per cent in analyses of future, deeper surveys.
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