Our key results are: (I) Absorbing galaxies with $\log _{10} [\mathrm{M}_\star ~(\mathrm{M}_\odot)] \gtrsim 10$ have star formation rates that are lower than predicted for the main sequence of star formation. (II) The distribution of impact parameter with column density and with absorption-metallicity for absorbing galaxies at $z\sim 2-3$ extends to $z\sim 0.7$ and to lower column densities. (III) A robust mean metallicity gradient of $\langle \Gamma \rangle = 0.022 \pm 0.001~[\mathrm{dex~kpc}^{-1}]$. (IV) By correcting absorption metallicities for $\langle \Gamma \rangle$ and imposing a truncation-radius at $12~\mathrm{kpc}$, absorbing galaxies fall on top of predicted mass-metallicity relations, with a statistically significant decrease in scatter.]{}

Introduction {#sec:intro}
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Scaling relations between galaxy observables (direct measurements and derived quantities) allow us to probe what drives galaxy evolution, and act as standard tests for simulations. Historically, scaling relations have been derived for luminosity-selected samples. In particular, such selection criteria have revealed a redshift-dependent relation between the galaxy stellar mass (M$_\star$) and the gas-phase metallicity (Z$_\text{gas}$) in the redshift range $z \sim$0.1–3.5 [@Tremonti2004; @Savaglio2005; @Erb2006; @Maiolino2008]; and a relation between M$_\star$ and star formation rate (SFR) [@Noeske2007]. These relations were combined as projections of a fundamental redshift-invariant relation tying M$_\star$, Z$_\text{gas}$, and SFR together [@Mannucci2010].

Complementing 