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.
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