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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}, Comment: Paper 5 of 7 in series. 17 pages + appendix and bibliography (30 pages total); 15 figures, 6 tables; accepted for publication in ApJ; v3 reflects the accepted version with minor changes and additions to text

We present the power spectrum methodology used for the first-season COMAP analysis, and assess the quality of the current data set. The main results are derived through the Feed-feed Pseudo-Cross-Spectrum (FPXS) method, which is a robust estimator with respect to both noise modeling errors and experimental systematics. We use effective transfer functions to take into account the effects of instrumental beam smoothing and various filter operations applied during the low-level data processing. The power spectra estimated in this way have allowed us to identify a systematic error associated with one of our two scanning strategies, believed to be due to residual ground or atmospheric contamination. We omit these data from our analysis and no longer use this scanning technique for observations. We present the power spectra from our first season of observing and demonstrate that the uncertainties are integrating as expected for uncorrelated noise, with any residual systematics suppressed to a level below the noise. Using the FPXS method, and combining data on scales $k=0.051-0.62 \,\mathrm{Mpc}^{-1}$ we estimate $P_\mathrm{CO}(k) = -2.7 \pm 1.7 \times 10^4\mu\textrm{K}^2\mathrm{Mpc}^3$, the first direct 3D constraint on the clustering component of the CO(1-0) power spectrum in the literature., Comment: Paper 4 of 7 in series. 18 pages, 11 figures, as accepted in ApJ