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We thank the anonymous referee for their corrections and help in improving the paper. We warmly thank the entire technical staff of the Observatorio Astronomico Nacional at San Pedro Martir in Mexico for their unfailing support to SAINT-EX operations, namely: E. Cadena, T. Calvario, E. Colorado, B. Garcia, G. Guisa, A. Franco, L. Figueroa, B. Hernandez, J. Herrera, E. Lopez, E. Lugo, B. Martinez, J. M. Nunez, J. L. Ochoa, M. Pereyra, F. Quiroz, T. Verdugo, I. Zavala. B.V.R. thanks the Heising-Simons Foundation for support. Y.G.M.C acknowledges support from UNAM-PAPIIT IG-101321. B.-O. D. acknowledges support from the Swiss National Science Foundation (PP00P2-163967 and PP00P2-190080). R.B. acknowledges the support from the Swiss National Science Foundation under grant P2BEP2_195285. M.N.G. acknowledges support from MIT's Kavli Institute as a Juan Carlos Torres Fellow. A.H.M.J.T acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement nffi 803193/BEBOP), from the MERAC foundation, and from the Science and Technology Facilities Council (STFC; grant nffi ST/S00193X/1). T.D. acknowledges support from MIT's Kavli Institute as a Kavli postdoctoral fellow Part of this work received support from the National Centre for Competence in Research PlanetS, supported by the Swiss National Science Foundation (SNSF). The research leading to these results has received funding from the ARC grant for Concerted Research Actions, financed by the Wallonia-Brussels Federation. TRAPPIST is funded by the Belgian Fund for Scientific Research (Fond National de la Recherche Scientifique, FNRS) under the grant FRFC 2.5.594.09.F, with the participation of the Swiss National Science Fundation (SNF). M.G. and E.J. are F.R.S.-FNRS Senior Research Associate. This publication benefits from the support of the French Community of Belgium in the context of the FRIA Doctoral Grant awarded to MT. We acknowledge the use of public TESS data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. We acknowledge the use of public TESS Alert data from pipelines at the TESS Science Office and at the TESS Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. Funding for the TESS mission is provided by NASA's Science Mission Directorate. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This paper includes data collected by the TESS mission that are publicly available from the Mikulski Archive for Space Telescopes (MAST). We thank the TESS GI program G03274 PI, Ryan Cloutier, for proposing the target of this work for 2-min-cadence observations in Sector 30. This work is based upon observations carried out at the Observatorio Astronomico Nacional on the Sierra de San Pedro Martir (OAN-SPM), Baja California, Mexico. This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF. This work includes data collected at the Vatican Advanced Technology Telescope (VATT) on Mt. Graham. This paper includes data taken on the EDEN telescope network. We acknowledge support from the Earths in Other Solar Systems Project (EOS) and Alien Earths (grant numbers NNX15AD94G and 80NSSC21K0593), sponsored by NASA. Some of the observations in the paper made use of the High-Resolution Imaging instrument Zorro (Gemini program GS-2020B-LP-105). Zorro was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Zorro was mounted on the Gemini South telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigacion y Desarrollo (Chile), Ministerio de Ciencia, Tecnologia e Innovacion (Argentina), Ministerio da Ciencia, Tecnologia, Inovacoes e Comunicacoes (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This work made use of the following Python packages: astropy (Astropy Collaboration 2013, 2018), lightkurve (Lightkurve Collaboration 2018), matplotlib (Hunter 2007), pandas (Wes McKinney 2010), seaborn (Waskom & The Seaborn Development team 2021), scipy (Virtanen et al. 2020) and numpy (Harris et al. 2020)., Context. 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. Aims. Here we report the discovery and validation of a sub-Neptune orbiting the thick-disk, mid-M dwarf star TOI-2406. The star's low metallicity and the relatively large size and short period of the planet make TOI-2406 b an unusual outcome of planet formation, and its characterisation provides an important observational constraint for formation models. Methods. We first infer properties of the host star by analysing the star's 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. Results. We determine the host to be a metal-poor M4 V star, located at a distance of 56 pc, with properties T-eff = 3100 +/- 75 K, M-* = 0.162 +/- 0.008M(circle dot), R-* = 0.202 +/- 0.011R(circle dot), and [Fe/H] = -0.38 +/- 0.07, and a member of the thick disk. The planet is a relatively large sub-Neptune for the M-dwarf planet population, with R-p = 2.94 +/- 0.17R(circle plus) and P= 3.077 d, producing transits of 2% depth. We note the orbit has a non-zero eccentricity to 3 sigma, prompting questions about the dynamical history of the system. Conclusions. 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. Although the planet's mass remains to be constrained, we estimate the S/N using amass-radius relationship, ranking the system fifth in the population of large sub-Neptunes, with TOI-2406 b having a much lower equilibrium temperature than other spectroscopically accessible members of this population., Heising-Simons Foundation, Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica (PAPIIT), Universidad Nacional Autonoma de Mexico IG-101321, Swiss National Science Foundation (SNSF), European Commission PP00P2-163967 PP00P2-190080 P2BEP2_195285, MIT's Kavli Institute as a Juan Carlos Torres Fellow, European Research Council (ERC) nffi 803193/BEBOP, MERAC foundation, UK Research & Innovation (UKRI), Science & Technology Facilities Council (STFC), Science and Technology Development Fund (STDF) nffi ST/S00193X/1, MIT's Kavli Institute as a Kavli postdoctoral fellow, Australian Research Council, Fonds de la Recherche Scientifique - FNRS FRFC 2.5.594.09.F, French Community of Belgium in the context of the FRIA Doctoral Grant, NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center, NASA's Science Mission Directorate, National Aeronautics and Space Administration under the Exoplanet Exploration Program, TESS GI program G03274, National Science Foundation (NSF), Earths in Other Solar Systems Project (EOS), Alien Earths - NASA NNX15AD94G 80NSSC21K0593, High-Resolution Imaging instrument Zorro (Gemini program) GS-2020B-LP-105, NASA Exoplanet Exploration Program, National Aeronautics & Space Administration (NASA)