  Galactic cosmic ray (GCR) particles have a significant impact on the
particle-induced background of X-ray observatories, and their flux exhibits
substantial temporal variability, potentially influencing background levels. In
this study, we present one-day binned high-energy reject rates derived from the
Chandra-ACIS and XMM-Newton EPIC-pn instruments, serving as proxies for GCR
particle flux. We systematically analyze the ACIS and EPIC-pn reject rates and
compare them with the AMS proton flux. Our analysis initially reveals robust
correlations between the AMS proton flux and the ACIS/EPIC-pn reject rates when
binned over 27-day intervals. However, a closer examination reveals substantial
fluctuations within each 27-day bin, indicating shorter-term variability. Upon
daily binning, we observe finer. temporal structures in the datasets,
demonstrating the presence of recurrent variations with periods of $\sim$ 25
days and 23 days in ACIS and EPIC-pn reject rates, respectively, spanning the
years 2014 to 2018. Notably, during the 2016--2017 period, we additionally
detect periodicities of $\sim$13.5 days and 9 days in the ACIS and EPIC-pn
reject rates, respectively. Intriguingly, we observe a time lag of $\sim$ 6
days between the AMS proton flux and the ACIS/EPIC-pn reject rates during the
second half of 2016. This time lag is not visible before 2016 and aftern2017.
The underlying physical mechanisms responsible for this time lag remain a
subject of ongoing investigation.
