In this paper we identify a new regularity in the systematics of galactic rotation curves, namely we find that at the last detected points in galaxies of widely varying luminosity, the centripetal acceleration is found to have the completely universal form $v^2/R=c^2(γ_0+γ^{*}N^{*})/2$ where $γ_0$ and $γ^{*}$ are new universal constants and $N^{*}$ is the amount of visible matter in each galaxy. This regularity points to a role for the linear potentials associated with conformal gravity, with the galaxy independent $γ_0$ term being found to be generated not from within individual galaxies at all, but rather to be of cosmological origin being due to the global Hubble flow of a necessarily spatially open Universe of 3-space scalar curvature $k=-(γ_0/2)^2=-2.3 \times 10^{-60}$cm$^{-2}$.

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