Novel $z\sim~10$ auroral line measurements extend the gradual offset of the FMR deep into the first Gyr of cosmic time

The mass assembly and chemical enrichment of the first galaxies provide key insights into their star-formation histories and the earliest stellar populations at cosmic dawn. Here we compile and utilize new, high-quality spectroscopic JWST/NIRSpec Prism observations from the JWST archive. We extend the wavelength coverage beyond the standard pipeline cutoff up to 5.5$\mu$m, enabling a detailed examination of the rest-frame optical emission-line properties for galaxies at $z\approx 10$. The improved calibration allows us to detect H$\beta$ and the [OIII]$\lambda\lambda 4959,5007$ doublet and resolve the auroral [OIII]$\lambda 4363$ line for the 11 galaxies in our sample ($z=9.3-10.0$) to obtain direct $T_e$-based metallicity measurements. We find that all galaxies show high ionisation fields and electron temperatures, with derived metallicities in the range $12+\log {\rm (O/H)} = 7.1 - 8.3$, consistent with previous strong-line diagnostics at high-z. We derive an empirical relation for $M_{\rm UV}$ and 12+log(O/H) at $z\approx 10$, useful for future higher-z studies, and show that the sample galaxies are `typical' star-forming galaxies though with relatively high specific star-formation rates and with evidence for bursty star formation. Combining the rest-frame optical line analysis and detailed UV to optical SED modelling, we determine the mass-metallicity relation and the fundamental-metallicity relation of the sample, pushing the redshift frontier of these measurements to $z=10$. These results, together with literature measurements, point to a gradually decreasing MZR at higher redshifts, with a break in the FMR at $z\approx 3$, decreasing to metallicities $\approx 3\times$ lower at $z=10$ than observed during the majority of cosmic time at $z=0-3$, likely caused by massive pristine gas inflows diluting the observed metal abundances during early galaxy assembly at cosmic dawn.