The Hot-Neptune Initiative (HONEI) I. Two hot sub-Neptunes on a close-in, eccentric orbit (TOI-5800 b) and a farther-out, circular orbit (TOI-5817 b)

Neptune-sized exoplanets are key targets for atmospheric studies, yet their formation and evolution remain poorly understood due to their diverse characteristics and limited sample size. The so-called "Neptune desert", a region of parameter space with a dearth of short-period sub- to super-Neptunes, is a critical testbed for theories of atmospheric escape and migration. The HONEI program aims to confirm and characterize the best Neptune-sized candidates for composition, atmospheric and population studies. By measuring planetary masses with high precision, we want to provide the community with optimal targets whose atmosphere can be effectively explored with the JWST or by ground-based high-resolution spectroscopy. For this purpose, we started a radial velocity follow-up campaign, using the twin high-precision spectrographs HARPS and HARPS-N, to measure the masses of TESS Neptune-sized candidates and confirm their planetary nature. In this first paper of the series, we confirm the planetary nature of two candidates: TOI-5800b and TOI-5817b. TOI-5800b is a hot sub-Neptune ($R_p=2.44\pm0.29$ $R_\oplus$, $M_p=9.4\pm1.8$ $M_\oplus$, $T_{eq} = 1108\pm20$ K) located at the lower edges of the Neptune desert ($P=2.628$ days) and is the most eccentric planet ($e\sim0.3$) ever found within $P<3$ d. TOI-5800b is expected to be still in the tidal migration phase with its parent star, a K3 V dwarf ($V=9.6$ mag), although its eccentricity could arise from interactions with another object in the system. Having a high-transmission spectroscopy metric ($TSM\sim103$), it represents a prime target for future atmospheric characterization. TOI-5817b is a relatively hot sub-Neptune ($R_p=3.08\pm0.14$ $R_\oplus$, $M_p=10.3\pm1.4$ $M_\oplus$, $T_{eq}=950\pm21$ K) located in the Neptune savanna ($P=15.610$ d) [...]