  Hard-TeV BL Lacs are a new type of blazars characterized by a hard intrinsic
TeV spectrum, locating the peak of their gamma-ray emission in the spectral
energy distribution (SED) above 2-10 TeV. Such high energies are problematic
for the Compton emission, using a standard one-zone leptonic model. We study
six examples of this new type of BL Lacs in the hard X-ray band with the NuSTAR
satellite. Together with simultaneous observations with the SWIFT satellite, we
fully constrain the peak of the synchrotron emission in their SED, and test the
leptonic synchrotron self-Compton (SSC) model. We confirm the extreme nature of
5 objects also in the synchrotron emission. We do not find evidence of
additional emission components in the hard X-ray band. We find that a one-zone
SSC model can in principle reproduce the extreme properties of both peaks in
the SED, from X-ray up to TeV energies, but at the cost of i) extreme electron
energies with very low radiative efficiency, ii) conditions heavily out of
equipartition (by 3 to 5 orders of magnitude), and iii) not accounting for the
simultaneous UV data, which then should belong to a different emission
component, possibly the same as the far-IR (WISE) data. We find evidence of
this separation of the UV and X-ray emission in at least two objects. In any
case, the TeV electrons must not "see" the UV or lower-energy photons, even if
coming from different zones/populations, or the increased radiative cooling
would steepen the VHE spectrum.
