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We present the evolution of the mass-metallicity (MZ) relations at z=4-10derived with 135 galaxies identified in the JWST/NIRSpec data taken from thethree major public spectroscopy programs of ERO, GLASS, and CEERS. Becausethere are many discrepancies between flux measurements reported by early EROstudies, we first establish our NIRSpec data reduction procedure for reliableemission-line flux measurements and errors successfully explaining Balmerdecrements with no statistical tensions via thorough comparisons of the earlyERO studies. Applying the reduction procedure to the 135 galaxies, we obtainemission-line fluxes for physical property measurements. We confirm that 10 outof the 135 galaxies with [OIII]4363-lines have electron temperatures of(1.1-2.3)*10^4K, similar to lower-z star-forming galaxies, that can beexplained by heating of young massive stars. We derive metallicities of the 10galaxies by the direct method and the rest of the galaxies with strong lines bythe metallicity calibrations of Nakajima et al. (2022) applicable for theselow-mass metal-poor galaxies, anchoring the metallicities with thedirect-method measurements. We thus obtain MZ relations and star-formation rate(SFR)-MZ relations over z=4-10. We find that there is a small evolution of theMZ relation from z~2-3 to z=4-10, while interestingly that the SFR-MZ relationshows no evolution up to z~8 but a significant decrease at z>8 beyond theerror. This SFR-MZ relation decrease at z>8 may suggest a break of themetallicity equilibrium state via star-formation, inflow, and outflow, whilefurther statistical and local-baseline studies are needed for a conclusion.

JWST Census for the Mass-Metallicity Star-Formation Relations at z=4-10  with the Self-Consistent Flux Calibration and the Proper Metallicity  Calibrators