  The origin of Galactic cosmic rays remains unconfirmed, but promising
candidates for their sources are found in star-forming regions. We report a
series of X-ray observations, with Suzaku, toward the nearby star-forming
region of Cygnus X. They aim at comparing diffuse X-ray emissions on and off
the $\gamma$-ray cocoon of hard cosmic rays revealed by Fermi LAT. After
excluding point sources and small-scale structures and subtracting the
non-X-ray and cosmic X-ray backgrounds, the 2--10~keV X-ray intensity
distribution is found to monotonically decrease with increasing Galactic
latitude. This indicates that most of the extended emission detected by Suzaku
originates from the Galactic ridge. In two observations, we derive upper limits
of $3.4 \times 10^{-8}~{\rm erg~s^{-1}~cm^{-2}~sr^{-1}}$ and $1.3 \times
10^{-8}~{\rm erg~s^{-1}~cm^{-2}~sr^{-1}}$ to X-ray emission in the 2--10 keV
range from the gamma-ray cocoon. These limits exclude the presence of
cosmic-ray electrons with energies above about 50 TeV at a flux level capable
of explaining the gamma-ray spectrum. They are consistent with the emission
cut-off observed near a TeV in gamma rays. The properties of Galactic ridge and
local diffuse X-rays are also discussed.
