We present Atacama Large Millimeter Array (ALMA) observations of $^{12}$CO2$-$1 emission from circumstellar material around the massive star $eta$~Carinae. These observations reveal new structural details about the cool equatorial torus located $sim$4000 au from the star. The CO torus is not a complete azimuthal loop, but rather, is missing its near side, which appears to have been cleared away. The missing material matches the direction of apastron in the eccentric binary system, making it likely that $eta$~Cars companion played an important role in disrupting portions of the torus soon after ejection. Molecular gas seen in ALMA data aligns well with the cool dust around $eta$~Car previously observed in mid-infrared (IR) maps, whereas hot dust resides at the inner surface of the molecular torus. The CO also coincides with the spatial and velocity structure of near-IR H$_2$ emission. Together, these suggest that the CO torus seen by ALMA is actually the pinched waist of the Homunculus polar lobes, which glows brightly because it is close to the star and warmer than the poles. The near side of the torus appears to be a blowout, associated with fragmented equatorial ejecta. We discuss implications for the origin of various features northwest of the star. CO emission from the main torus implies a total gas mass in the range of 0.2-1 $M_{odot}$ (possibly up to 5 $M_{odot}$ or more, although with questionable assumptions). Deeper observations are needed to constrain CO emission from the cool polar lobes.
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