We present CARMA 2.9 mm dust continuum emission observations of a sample of 14 Herschel-detected Class 0 protostars in the Orion A and B molecular clouds, drawn from the PACS Bright Red Sources (PBRS) sample (Stutz et al.). These objects are characterized by very red 24 micron to 70 micron colors and prominent submillimeter emission, suggesting that they are very young Class 0 protostars embedded in dense envelopes. We detect all of the PBRS in 2.9 mm continuum emission and emission from 4 protostars and 1 starless core in the fields toward the PBRS; we also report 1 new PBRS source. The ratio of 2.9 mm luminosity to bolometric luminosity is higher by a factor of $sim$5 on average, compared to other well-studied protostars in the Perseus and Ophiuchus clouds. The 2.9 mm visibility amplitudes for 6 of the 14 PBRS are very flat as a function of uv-distance, with more than 50% of the source emission arising from radii $<$ 1500 AU. These flat visibility amplitudes are most consistent with spherically symmetric envelope density profiles with $rho$~$propto$~R$^{-2.5}$. Alternatively, there could be a massive unresolved structure like a disk or a high-density inner envelope departing from a smooth power-law. The large amount of mass on scales $<$ 1500 AU (implying high average central densities) leads us to suggest that that the PBRS with flat visibility amplitude profiles are the youngest PBRS and may be undergoing a brief phase of high mass infall/accretion and are possibly among the youngest Class 0 protostars. The PBRS with more rapidly declining visibility amplitudes still have large envelope masses, but could be slightly more evolved.
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