  According to a conjecture of H. Clemens, the dimension of the space of
rational curves on a general projective hypersurface should equal the number
predicted by a na\"ive dimension count. In the case of a general hypersurface
of degree 7 in $\mathbb{P}^5$, the conjecture predicts that the only rational
curves should be lines. This has been verified by Hana and Johnsen for rational
curves of degree at most 15. Here we extend their results to show that no
rational curves of degree 16 lie on a general heptic fourfold.
