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We conduct a line-by-line differential analysis of a sample of 125 co-movingpairs of stars (dwarfs and subgiants near solar metallicity). We obtain highprecision stellar parameters with average uncertainties in effectivetemperature, surface gravity and metallicity of 16.5 K, 0.033 dex and 0.014dex, respectively. We classify the co-moving pairs of stars into two groups,chemically homogeneous (conatal; |Delta[Fe/H]| $\le$ 0.04 dex) andinhomogeneous (non-conatal), and examine the fraction of chemically homogeneouspairs as a function of separation and effective temperature. The four mainconclusions from this study are: (1) A spatial separation of \ds = 10$^6$ AU isan approximate boundary between homogeneous and inhomogeneous pairs of stars,and we restrict our conclusions to only consider the 91 pairs with \ds $\le$10$^6$ AU; (2) There is no trend between velocity separation and the fractionof chemically homogeneous pairs in the range \dv $\le$ 4 \kms; (3) We confirmthat the fraction of chemically inhomogeneous pairs increases with increasing\teff\ and the trend matches a toy model of that expected from planetingestion; (4) Atomic diffusion is not the main cause of the chemicalinhomogeneity. A major outcome from this study is a sample of 56 brightco-moving pairs of stars with chemical abundance differences $\leq$ 0.02 dex(5\%) which is a level of chemical homogeneity comparable to that of the Hyadesopen cluster. These important objects can be used, in conjunction with starclusters and the \gaia\ ``benchmark'' stars, to calibrate stellar abundancesfrom large-scale spectroscopic surveys.

C3PO: Towards a complete census of co-moving pairs of stars. I. High precision stellar parameters for 250 stars