  We use observations with the infrared-optimized MagAO system and Clio camera
in 3.9 $\mu$m light to place stringent mass constraints on possible undetected
companions to Sirius A. We suppress the light from Sirius A by imaging it
through a grating vector-apodizing phase plate coronagraph with 180-degree dark
region (gvAPP-180). To remove residual starlight in post-processing, we apply a
time-domain principal-components-analysis-based algorithm we call PCA-Temporal
(PCAT), which uses eigen-time-series rather than eigen-images to subtract
starlight. By casting the problem in terms of eigen-time-series, we reduce the
computational cost of post-processing the data, enabling the use of the fully
sampled dataset for improved contrast at small separations. We also discuss the
impact of retaining fine temporal sampling of the data on final contrast
limits. We achieve post-processed contrast limits of $1.5 \times 10^{-6}$ to
$9.8 \times 10^{-6}$ outside of 0.75 arcsec which correspond to planet masses
of 2.6 to 8.0 $M_J$. These are combined with values from the recent literature
of high-contrast imaging observations of Sirius to synthesize an overall
completeness fraction as a function of mass and separation. After synthesizing
these recent studies and our results, the final completeness analysis rules out
99% of $\ge 9 \ M_J$ planets from 2.5-7 AU.
