Polarization transfer observables in elastic electron-proton scattering at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>Q</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>2.5</mml:mn></mml:mrow></mml:math>, 5.2, 6.8, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>8.5</mml:mn><mml:mo> </mml:mo><mml:msup><mml:mrow><mml:mi>GeV</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math>

Background: Interest in the behavior of nucleon electromagnetic form factors at large momentum transfers has steadily increased since the discovery, using polarization observables, of the rapid decrease of the ratio ${G}_{E}^{p}/{G}_{M}^{p}$ of the proton's electric and magnetic form factors for momentum transfers ${Q}^{2}\ensuremath{\gtrsim}1$ ${\mathrm{GeV}}^{2}$, in strong disagreement with previous extractions of this ratio using the traditional Rosenbluth separation technique.Purpose: The GEp-III and $\mathrm{GEp}\text{\ensuremath{-}}2\ensuremath{\gamma}$ experiments were carried out in Jefferson Laboratory's (JLab's) Hall C from 2007 to 2008, to extend the knowledge of ${G}_{E}^{p}/{G}_{M}^{p}$ to the highest practically achievable ${Q}^{2}$ given the maximum beam energy of 6 GeV and to search for effects beyond the Born approximation in polarization transfer observables of elastic $\stackrel{P\vec}{e}p$ scattering. This article provides an expanded description of the common experimental apparatus and data analysis procedures, and reports the results of a final reanalysis of the data from both experiments, including the previously unpublished results of the full-acceptance dataset of the $\mathrm{GEp}\text{\ensuremath{-}}2\ensuremath{\gamma}$ experiment.Methods: Polarization transfer observables in elastic $\stackrel{P\vec}{e}p\ensuremath{\rightarrow}e\stackrel{P\vec}{p}$ scattering were measured at central ${Q}^{2}$ values of 2.5, 5.2, 6.8, and 8.54 ${\mathrm{GeV}}^{2}$. At ${Q}^{2}=2.5\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{2}$, data were obtained for central values of the virtual photon polarization parameter $\ensuremath{\epsilon}$ of 0.149, 0.632, and 0.783. The Hall C High Momentum Spectrometer detected and measured the polarization of protons recoiling elastically from collisions of JLab's polarized electron beam with a liquid hydrogen target. A large-acceptance electromagnetic calorimeter detected the elastically scattered electrons in coincidence to suppress inelastic backgrounds.Results: The final GEp-III data are largely unchanged relative to the originally published results. The statistical uncertainties of the final $\mathrm{GEp}\text{\ensuremath{-}}2\ensuremath{\gamma}$ data are significantly reduced at $\ensuremath{\epsilon}=0.632$ and 0.783 relative to the original publication.Conclusions: The final GEp-III results show that the decrease with ${Q}^{2}$ of ${G}_{E}^{p}/{G}_{M}^{p}$ continues to ${Q}^{2}=8.5\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{2}$, but at a slowing rate relative to the approximately linear decrease observed in earlier Hall A measurements. At ${Q}^{2}=8.5\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{2}$, ${G}_{E}^{p}/{G}_{M}^{p}$ remains positive but is consistent with zero. At ${Q}^{2}=2.5\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{2}$, ${G}_{E}^{p}/{G}_{M}^{p}$ derived from the polarization component ratio $R\ensuremath{\propto}{P}_{t}/{P}_{\ensuremath{\ell}}$ shows no statistically significant $\ensuremath{\epsilon}$ dependence, as expected in the Born approximation. On the other hand, the ratio ${P}_{\ensuremath{\ell}}/{P}_{\ensuremath{\ell}}^{\mathrm{Born}}$ of the longitudinal polarization transfer component to its Born value shows an enhancement of roughly 1.7% at $\ensuremath{\epsilon}=0.783$ relative to $\ensuremath{\epsilon}=0.149$, with $\ensuremath{\approx}2.2\ensuremath{\sigma}$ significance based on the total uncertainty, implying a similar effect in the transverse component ${P}_{t}$ that cancels in the ratio $R$.