HESS J1303631 is an evolved, TeV-bright pulsar wind nebula (PWN) powered by the energetic radio pulsar PSR~J13016305, offset by northwest of the TeV centroid, with an energy-dependent gamma-ray morphology. We present new, deep joint Chandra (108 ks) and XMM-Newton (102 ks) observations of this field. High-resolution Chandra ACIS imaging resolves the pulsar from a neighboring point source that XMM-Newton cannot separate. The pulsar’s 2-10 keV spectrum is well described by an absorbed power law with and an unabsorbed luminosity of at kpc. Chandra also reveals faint diffuse emission east of the pulsar: a clumpy compact nebula on arcsecond scales (, ) and a fainter extended nebula on arcminute scales (, ). This morphology, the hard GeV counterpart, and the energy-dependent TeV emission together support an evolved, synchrotron-underluminous PWN whose X-ray nebula is far more compact than its TeV nebula. We also serendipitously discovered a transient X-ray source, CXOU~J130131.0625919, in one of the nine new Chandra observations. Its hard spectrum () and 0.5-7,keV luminosity of at a fiducial distance of 3 kpc, together with a plausible near-infrared counterpart, most closely match a flare from a young pre-main-sequence star, though an active binary or compact-object binary is not excluded. Its large offset from HESS J1303631 makes association with the TeV source unlikely.

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