  Pulsars are detected over the whole electromagnetic spectrum, from radio
wavelengths up to very high energies, in the GeV-TeV range. Whereas the radio
emission site for young pulsars is well constrained to occur at altitudes about
several percent of the light-cylinder radius and $\gamma$-ray emission is
believed to be produced in the striped wind, outside the light-cylinder, their
non-thermal X-ray production site remains unknown. The aim of this letter is to
localize the non-thermal X-ray emission region based on multi-wavelength pulse
profile fitting for PSR J2229+6114, a particularly good candidate due to its
high X-ray brightness. Based on the geometry deduced from the joint radio and
$\gamma$-ray pulse profiles, we fix the magnetic axis inclination angle and the
line of sight inclination angle but we leave the region of X-ray emission
unlocalised, somewhere between the surface and the light-cylinder. We localize
this region and its extension by fitting the X-ray pulse profile as observed by
the NICER, NuSTAR and RXTE telescopes in the ranges 2-7 keV, 3-10 keV and
9.4-22.4 keV, respectively. We constrain the non-thermal X-ray emission to
arise from altitudes between $0.2\,r_L$ and $0.55\,r_L$ where $r_L$ is the
light cylinder radius. The magnetic obliquity is approximately $\alpha \approx
45\deg-50\deg$ and the line of sight inclination angle $\zeta \approx
32\deg-48\deg$. This letter is among the first works to tightly constrain the
location of the non-thermal X-ray emission from pulsars. We plan to apply this
procedure to several other good candidates to confirm this new result.
