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A complete census of dusty star-forming galaxies (DSFGs) at early epochs isnecessary to constrain the obscured contribution to the cosmic star formationrate density (CSFRD), however DSFGs beyond $z \sim 4$ are both rare and hard toidentify from photometric data alone due to degeneracies in submillimeterphotometry with redshift. Here, we present a pilot study obtaining follow-upAtacama Large Millimeter Array (ALMA) $2\,$mm observations of a complete sampleof 39 $850\,\rm\mu m$-bright dusty galaxies in the SSA22 field. Empiricalmodeling suggests $2\,$mm imaging of existing samples of DSFGs selected at$850\,\rm\mu m - 1\,$mm can quickly and easily isolate the "needle in ahaystack" DSFGs that sit at $z>4$ or beyond. Combining archival submillimeterimaging with our measured ALMA $2\,$mm photometry ($1\sigma \sim0.08\,$mJy$\,$beam$^{-1}$ rms), we characterize the galaxies' IR SEDs and usethem to constrain redshifts. With available redshift constraints fit via thecombination of six submillimeter bands, we identify 6/39 high-$z$ candidateseach with $>50\%$ likelihood to sit at $z > 4$, and find a positive correlationbetween redshift and $2\,$mm flux density. Specifically, our models suggest theaddition of $2\,$mm to a moderately constrained IR SED will improve theaccuracy of a millimeter-derived redshift from $\Delta z/(1+z) = 0.3$ to$\Delta z/(1+z) = 0.2$. Our IR SED characterizations provide evidence forrelatively high emissivity spectral indices ($\langle \beta \rangle =2.4\pm0.3$) in the sample. We measure that especially bright ($S_{850\rm\mum}>5.55\,$mJy) DSFGs contribute $\sim10$% to the cosmic-averaged CSFRD from$2<z<5$, confirming findings from previous work with similar samples.

Searching Far and Long I: Pilot ALMA 2mm Follow-up of Bright Dusty Galaxies as a Redshift Filter