  Although Deligne's theorem classifies all symmetric tensor categories (STCs)
with moderate growth over algebraically closed fields of characteristic zero,
the classification does not extend to positive characteristic. At the forefront
of the study of STCs is the search for an analog to Deligne's theorem in
positive characteristic, and it has become increasingly apparent that the
Verlinde categories are to play a significant role. Moreover, these categories
are largely unstudied, but have already shown very interesting phenomena as
both a generalization of and a departure from superalgebra and supergeometry.
In this paper, we study $\mathrm{Ver}_4^+$, the simplest non-trivial Verlinde
category in characteristic $2$. In particular, we classify all isomorphism
classes of non-degenerate symmetric bilinear forms and non-degenerate quadratic
forms and study the associated Witt semi-ring that arises from the addition and
multiplication operations on bilinear forms.
