  We determine the A(1)-homotopy of the topological cyclic homology of the
connective real K-theory spectrum ko. The answer has an associated graded that
is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le
* \le 30. The calculation is achieved by using prismatic and syntomic
cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of
Bhatt-Morrow-Scholze from the case of classical commutative rings to E_\infty
rings. A new feature in our case is that there are nonzero differentials in the
motivic spectral sequence from syntomic cohomology to topological cyclic
homology.
