  Our goal is to systematically compute the $\mathbb{C}P^2$-genus of all prime
knots up to 8-crossings. We obtain upper bounds on the $\mathbb{C}P^2$-genus
via coherent band surgery. We obtain lower bounds by obstructing homological
degrees of potential slice discs. The obstructions are pulled from a variety of
sources in low-dimensional topology and adapted to $\mathbb{C}P^2$. There are
27 prime knots and distinct mirrors up to 7-crossings. We now know the
$\mathbb{C}P^2$-genus of all but 2 of these knots. There are 64 prime knots and
distinct mirrors up to 8-crossings. We now know the $\mathbb{C}P^2$-genus of
all but 9 of these knots. Where the $\mathbb{C}P^2$-genus was not determined
explicitly, it was narrowed down to 2 possibilities. As a consequence of this
work, we show an infinite family of knots such that the $\mathbb{C}P^2$-genus
of each knot differs from that of it's mirror.
