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We present 10 novel [OIII]$\lambda 4363$ auroral line detections up to $z\sim9.5$ measured from ultra-deep JWST/NIRSpec MSA spectroscopy from the JWSTAdvanced Deep Extragalactic Survey (JADES). We leverage the deepestspectroscopic observations yet taken with NIRSpec to determine electrontemperatures and oxygen abundances using the direct T$_e$ method. We directlycompare against a suite of locally calibrated strong-line diagnostics andrecent high-$z$ calibrations. We find the calibrations fail to simultaneouslymatch our JADES sample, thus warranting a self-consistent revision of thesecalibrations for the high-$z$ Universe. We find weak dependence between R2 andO3O2 with metallicity, thus suggesting these line-ratios are ineffective in thehigh-$z$ Universe as metallicity diagnostics and degeneracy breakers. We findR3 and R23 still correlate with metallicity, but we find tentative flatteningof these diagnostics, thus suggesting future difficulties when applying thesestrong-line ratios as metallicity indicators in the high-$z$ Universe. We alsopropose and test an alternative diagnostic based on a different combination ofR3 and R2 with a higher dynamic range. We find a reasonably good agreement(median offset of 0.002 dex, median absolute offset of 0.13 dex) with the JWSTsample at low metallicity. Our sample demonstrates higher ionization/excitationratios than local galaxies with rest-frame EWs(H$\beta$) $\approx 200 -300$Angstroms. However, we find the median rest-frame EWs(H$\beta$) of our sampleto be $\sim 2\text{x}$ less than the galaxies used for the local calibrations.This EW discrepancy combined with the high ionization of our galaxies does notpresent a clear description of [OIII]$\lambda 4363$ production in the high-$z$Universe, thus warranting a much deeper examination into the factors affectingproduction.

JADES: Detecting [OIII]$\lambda 4363$ Emitters and Testing Strong Line  Calibrations in the High-$z$ Universe with Ultra-deep JWST/NIRSpec  Spectroscopy up to $z \sim 9.5$