  We present measurements of the resolving power of a soft x-ray spectrometer
consisting of 200 nm-period lightweight, alignment-insensitive critical-angle
transmission (CAT) gratings and a lightweight slumped-glass Wolter-I focusing
mirror pair. We measure and model contributions from source, mirrors, detector
pixel size, and grating period variation to the natural line width spectrum of
the Al and Mg K$_{\alpha_1 \alpha_2}$ doublets. Measuring up to 18$^{\rm th}$
diffraction order at characteristic Al-K wavelengths we consistently obtain
small broadening due to gratings corresponding to a minimum effective grating
resolving power $R_g > 10,000$ with 90\% confidence. Upper limits are often
compatible with $R_g = \infty$. Independent fitting of different diffraction
orders, as well as ensemble fitting of multiple orders at multiple wavelengths,
gives compatible results. Our data leads to uncertainties for the
Al-K$_{\alpha}$ doublet line width and line separation parameters 2-3 times
smaller than values found in the literature. Data from three different gratings
are mutually compatible. This demonstrates that CAT gratings perform in excess
of the requirements for the Arcus Explorer mission and are suitable for
next-generation space-based x-ray spectrometer designs with resolving power
5-10 times higher than the transmission grating spectrometer on the Chandra
X-ray Observatory.
