  We consider the cohomology groups of line bundles $\mathcal{L}$ on the
\emph{incidence correspondence}, that is, a general hypersurface $X \subset
\mathbb{P}^{n-1} \times \mathbb{P}^{n-1}$ of degrees $(1,1)$. Whereas the
characteristic $0$ situation is completely understood, the cohomology in
characteristic $p$ depends in a mysterious way on the base-$p$ digits of the
degrees $(d, e)$ of $\mathcal{L}$. Gao and Raicu (following Linyuan Liu) prove
a recursive description of the cohomology for $n = 3$, which relates to Nim
polynomials when $p = 2$. In this paper, we devise a suitable generalization of
Nim polynomials, which we call \emph{even-carry polynomials,} by which we can
solve the recurrence of Liu--Gao--Raicu to yield an explicit formula for the
cohomology for $n = 3$ and general $p$. We also make some conjectures on the
general form of the cohomology for general $n$ and $p$.
