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Astrophysical Distance Scale The JAGB Method: I. Calibration and a First Application

J-Branch Asymptotic Giant Branch (JAGB) stars are a photometrically well-defined population of extremely red, intermediate-age AGB stars that are found to have tightly-constrained luminosities in the near-infrared. Based on JK photometry of some 3,300 JAGB stars in the bar of the Large Magellanic Cloud (LMC) we find that these very red AGB stars have a constant absolute magnitude of <M_J> = -6.22 mag, adopting the Detached Eclipsing Binary (DEB) distance to the LMC of 18.477 +/- 0.004 (stat) +/- 0.026 (sys). Undertaking a second, independent calibration in the SMC, which also has a DEB geometric distance, we find <M_J> = -6.18 +/- $ 0.01 (stat) +/- 0.05~(sys) mag. The scatter is +/-0.27 mag for single-epoch observations, (falling to +/-0.15~mag for multiple observations averaged over a window of more than one year). We provisionally adopt <M_J> = -6.20 mag +/- 0.01 (stat) +/- 0.04 (sys) mag for the mean absolute magnitude of JAGB stars. Applying this calibration to JAGB stars recently observed in the galaxy NGC 253, we determine a distance modulus of 27.66 +/- 0.01(stat) +/- 0.04 mag (sys), corresponding to a distance of 3.40 +/- 0.06 Mpc (stat). This is in excellent agreement with the averaged TRGB distance modulus of 27.68 +/- 0.05 mag, assuming M_I = -4.05 mag for the TRGB zero point.