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High-Precision Radio and Infrared Astrometry of LSPM J1314+1320AB - II: Testing Pre--Main-Sequence Models at the Lithium Depletion Boundary with Dynamical Masses

We present novel tests of pre$-$main-sequence models based on individual dynamical masses for the M7 binary LSPM J1314+1320AB. Joint analysis of our Keck adaptive optics astrometric monitoring along with Very Long Baseline Array radio data from a companion paper yield component masses of $0.0885\pm0.0006$ $M_{\odot}$ and $0.0875\pm0.0010$ $M_{\odot}$ and a parallactic distance of $17.249\pm0.013$ pc. We also derive component luminosities that are consistent with the system being coeval at an age of $80.8\pm2.5$ Myr, according to BHAC15 evolutionary models. The presence of lithium is consistent with model predictions, marking the first time the theoretical lithium depletion boundary has been tested with ultracool dwarfs of known mass. However, we find that the average evolutionary model-derived effective temperature ($2950\pm5$ K) is 180 K hotter than we derive from a spectral type$-$$T_{\rm eff}$ relation based on BT-Settl models ($2770\pm100$ K). We suggest that the dominant source of this discrepancy is model radii being too small by $\approx$13%. In a test that mimics the typical application of evolutionary models by observers, we derive masses on the H-R diagram using the luminosity and BT-Settl temperature. The estimated masses are $46^{+16}_{-19}$% (2.0$σ$) lower than we measure dynamically and would imply that this is a system of $\approx$50 $M_{\rm Jup}$ brown dwarfs, highlighting the large systematic errors possible when inferring masses from the H-R diagram. This is first time masses have been measured for ultracool ($\geq$M6) dwarfs displaying spectral signatures of low gravity. Based on features in the infrared, LSPM J1314+1320AB appears higher gravity than typical Pleiades and AB Dor members, opposite the expectation given its younger age. The components of LSPM J1314+1320AB are now the nearest, lowest mass pre$-$main-sequence stars with direct mass measurements.