  Vardanyan's Theorems state that $\mathsf{QPL}(\mathsf{PA})$ - the quantified
provability logic of Peano Arithmetic - is $\Pi^0_2$ complete, and in
particular that this already holds when the language is restricted to a single
unary predicate. Moreover, Visser and de Jonge generalized this result to
conclude that it is impossible to computably axiomatize the quantified
provability logic of a wide class of theories. However, the proof of this fact
cannot be performed in a strictly positive signature. The system
$\mathsf{QRC}_1$ was previously introduced by the authors as a candidate
first-order provability logic. Here we generalize the previously available
Kripke soundness and completeness proofs, obtaining constant domain
completeness. Then we show that $\mathsf{QRC}_1$ is indeed complete with
respect to arithmetical semantics. This is achieved via a Solovay-type
construction applied to constant domain Kripke models. As corollaries, we see
that $\mathsf{QRC}_1$ is the strictly positive fragment of $\mathsf{QGL}$ and a
fragment of $\mathsf{QPL}(\mathsf{PA})$.
