The PADME experiment at the INFN Laboratori Nazionali di Frascati performed a dedicated positron-beam energy scan (Run III, end of 2022) to search for the hypothetical
$X_{17}$, with expected mass of 17 MeV. The beam energy was varied between 262 and 296 MeV, with step size smaller than half of the expected resonance line-shape width, corresponding to a centre-of-mass energy scan from 16.4 to 17.4 MeV. The search targets a localized excess in the two-body final-state yield from $e^+e^-$ annihilation on target electrons. We describe the Run III setup and analysis strategy and present the final uncertainty budget together with the “blind unblinding" procedure, which was used to validate the error model before opening the signal box. The data are consistent with the expected background over most of the explored energy range, and limits are set in previously unexplored regions of parameter space. An excess is observed at $M_X=16.90$ MeV, with local and global significances of $2.5\sigma$ and $1.77\sigma$, respectively.