  At the Relativistic Heavy-Ion Collider (RHIC), the Polarized Atomic Hydrogen
Gas Jet Target polarimeter (HJET) is employed for the precise measurement of
the absolute transverse (vertical) polarization of proton beams, achieving low
systematic uncertainties of approximately $\sigma^\text{syst}_P/P\leq0.5\%$.
The acquired experimental data not only facilitated the determination of single
$A_\text{N}(t)$ and double $A_\text{NN}(t)$ spin analyzing powers for 100 and
255 GeV proton beams but also revealed a non-zero Pomeron spin-flip
contribution through a Regge fit. Preliminary results obtained for forward
inelastic $p^{\uparrow}p$ and elastic $p^{\uparrow}A$ analyzing powers will be
discussed. The success of HJET at RHIC suggests its potential application for
proton beam polarimetry at the upcoming Electron-Ion Collider (EIC), aiming for
an accuracy of 1\%. Moreover, the provided analysis indicates that the RHIC
HJET target can serve as a tool for the precision calibration, with the
required accuracy, of the $^3$He beam polarization at the EIC.
