Large sub-Neptunes are uncommon around the coolest stars in the Galaxy and are rarer still around those that are metal-poor. However, owing to the large planet-to-star radius ratio, these planets are highly suitable for atmospheric study via transmission spectroscopy in the infrared, such as with JWST. Here we report the discovery and validation of a sub-Neptune orbiting the thick-disk, mid-M dwarf star TOI-2406. We first infer properties of the host star by analysing the stars near-infrared spectrum, spectral energy distribution, and Gaia parallax. We use multi-band photometry to confirm that the transit event is on-target and achromatic, and we statistically validate the TESS signal as a transiting exoplanet. We then determine physical properties of the planet through global transit modelling of the TESS and ground-based time-series data. We determine the host to be a metal-poor M4V star, located at a distance of 56 pc, with a sub-solar metallicity $(mathrm{[Fe/H] = -0.38 pm 0.07})$, and a member of the thick disk. The planet is a relatively large sub-Neptune for the M-dwarf planet population, with $mathrm{R_p = 2.94 pm 0.17} mathrm{R_oplus}$ and $mathrm{P = 3.077}$ d, producing transits of 2% depth. We note the orbit has a non-zero eccentricity to 3$mathrm{sigma}$, prompting questions about the dynamical history of the system. This system is an interesting outcome of planet formation and presents a benchmark for large-planet formation around metal-poor, low-mass stars. The system warrants further study, in particular radial velocity follow-up to determine the planet mass and constrain possible bound companions. Furthermore, TOI-2406 b is a good target for future atmospheric study through transmission spectroscopy, particularly in the category of warm sub-Neptunes.
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