Inflow-driven galaxy evolution - I. Revealing the physics of the fundamental metallicity relation

We present a unified physical framework for the fundamental metallicity relation (FMR), based on the mass-continuity equations. The FMR is not merely the anti-correlation between star formation rate (SFR) and gas metallicity ($Z_{\rm g}$) at fixed stellar mass ($M_\star$); it is a redshift-invariant surface in the $(M_\star,{\rm SFR},Z_{\rm g})$ space. We construct a minimal cosmological gas flow model, calibrated to reproduce the mass-metallicity relation, star-forming main sequence, and stellar-to-halo mass relation at $z=0-3$, and show that the FMR emerges as a prediction of the calibrated physics. Through controlled experiments that progressively simplify the model, we reveal that in a universe where both the star formation efficiency ($\epsilon$) and mass-loading factor ($\eta$) are constants, the FMR reduces to a universal scaling between $Z_{\rm g}$ and $M_\star/$SFR, whose shape traces the transition from inflow-driven regime to equilibrium. The specific parameterisation of the observed FMR is not a fundamental symmetry but a contingent consequence of how $\epsilon$ and $\eta$ depend on stellar mass and redshift. We show that the gaseous FMR (gFMR), defined in the $(M_\star,M_{\rm g},Z_{\rm g})$ space, is more fundamental than the standard FMR: in the inflow-driven limit, $Z_{\rm g}$ is proportional to $M_\star/M_{\rm g}$, and the approach to equilibrium is governed by $M_\star/M_{\rm g}$ and $\eta$ alone. We derive an analytic solution for an idealised version of the model that provides closed-form expressions relating $Z_{\rm g}$, $M_{\rm g}/M_\star$, and $\eta$, and show this framework accurately reproduces the cosmological gas flow model. By establishing the physical origin of the FMR and its connection to the more fundamental gFMR, we provide the theoretical foundation to turn metallicity scaling relations into precision probes of the baryon cycle over cosmic history.