The multi-planet system TOI-5624: Four transiting sub-Neptunes with an outer companion revealed by transit-timing variations

Following the 2022 alert of a TESS object of interest transiting TOI-5624 (a G7 V star $\sim$100 pc away), a CHEOPS campaign in 2023 detected four planetary signals at $P_b\approx3.4$, $P_c\approx7.9$, $P_d\approx13.7$, and $P_e\approx21.5$ days, later confirmed by additional TESS and CHEOPS photometry in 2024-2025. After analysing the TESS & CHEOPS photometric data, we extracted and modelled the HARPS-N & SOPHIE RV time series using two independent methodologies both within an MCMC framework. We further integrated the N-body equations of motion, while simultaneously fitting the transit times and the detrended RVs, to dynamically characterise the system. We present the discovery of four transiting sub-Neptunes with radii of $R_b=2.314\pm0.035 R_{\oplus}$, $R_c=2.474\pm0.042 R_{\oplus}$, $R_d=3.584_{-0.050}^{+0.051} R_{\oplus}$, and $R_e=3.247_{-0.043}^{+0.042} R_{\oplus}$ and masses of $M_b=9.4\pm1.4 M_{\oplus}$, $M_c=4.8\pm1.9 M_{\oplus}$, $M_d=4.9\pm2.2 M_{\oplus}$, and $M_e=8.9_{-3.0}^{+2.9} M_{\oplus}$. Our photometric analysis reveals that the outermost transiting planet TOI-5624 e shows significant TTVs. We find a robust Keplerian signal in the RV time series close to the 2:1 period commensurability with TOI-5624 e, which explains the TTV pattern exhibited by TOI-5624 e according to our dynamical analysis. We label this non-transiting planet as TOI-5624 f and find its minimum mass to be $M_f\sin{i_f}=13.0\pm3.7 M_{\oplus}$. Among the known systems hosting more than four planets, the remarkable precision with which the radii have been measured (<1.7%) and the firm assessment (>3$\sigma$) of the mass for at least three planets has been previously reached only for TRAPPIST-1. Additional photometric observations will enable a better sample of the TTV modulation and a more robust dynamical determination of the masses.