  We present the current state of models for the $z\sim3$ carbon monoxide (CO)
line-intensity signal targeted by the CO Mapping Array Project (COMAP)
Pathfinder in the context of its early science results. Our fiducial model,
relating dark matter halo properties to CO luminosities, informs parameter
priors with empirical models of the galaxy-halo connection and previous CO(1-0)
observations. The Pathfinder early science data spanning wavenumbers
$k=0.051$-$0.62\,$Mpc$^{-1}$ represent the first direct 3D constraint on the
clustering component of the CO(1-0) power spectrum. Our 95% upper limit on the
redshift-space clustering amplitude $A_{\rm clust}\lesssim70\,\mu$K$^2$ greatly
improves on the indirect upper limit of $420\,\mu$K$^2$ reported from the CO
Power Spectrum Survey (COPSS) measurement at $k\sim1\,$Mpc$^{-1}$. The COMAP
limit excludes a subset of models from previous literature, and constrains
interpretation of the COPSS results, demonstrating the complementary nature of
COMAP and interferometric CO surveys. Using line bias expectations from our
priors, we also constrain the squared mean line intensity-bias product,
$\langle{Tb}\rangle^2\lesssim50\,\mu$K$^2$, and the cosmic molecular gas
density, $\rho_\text{H2}<2.5\times10^8\,M_\odot\,$Mpc$^{-3}$ (95% upper
limits). Based on early instrument performance and our current CO signal
estimates, we forecast that the five-year Pathfinder campaign will detect the
CO power spectrum with overall signal-to-noise of 9-17. Between then and now,
we also expect to detect the CO-galaxy cross-spectrum using overlapping galaxy
survey data, enabling enhanced inferences of cosmic star-formation and
galaxy-evolution history.
