  The metal content of galaxies is a direct probe of the baryon cycle. A
hallmark example is the relationship between a galaxy's stellar mass, star
formation rate (SFR), and gas-phase metallicity: the Fundamental Metallicity
Relation (FMR). While low-redshift ($z\lesssim4$) observational studies suggest
that the FMR is redshift-invariant, recent JWST data indicate deviations from
this model. In this study, we utilize the FMR to predict the evolution of the
normalisation of the mass-metallicity relation (MZR) using the cosmological
simulations Illustris, IllustrisTNG, EAGLE, and SIMBA. Our findings demonstrate
that a $z = 0$ calibrated FMR struggles to predict the evolution in the MZR of
each simulation. To quantify the divergence of the predictions, we introduce
the concepts of a ''static'' FMR, where the role of the SFR in setting the
normalization of the MZR does not change with redshift, and a ''dynamic'' FMR,
where the role of SFR evolves over time. We find static FMRs in Illustris and
SIMBA and dynamic FMRs in IllustrisTNG and EAGLE. We suggest that the
differences between these models likely points to the subtle differences in the
implementation of the baryon cycle. Moreover, we echo recent JWST results at $z
> 4$ by finding significant offsets from the FMR in IllustrisTNG and EAGLE,
suggesting that the observed FMR may be dynamic as well. Overall, our findings
imply that the current FMR framework neglects important variations in the
baryon cycle through cosmic time.
