  We have previously established that $\Pi^1_1$-comprehension is equivalent to
the statement that every dilator has a well-founded Bachmann-Howard fixed
point, over $\mathbf{ATR_0}$. In the present paper we show that the base theory
can be lowered to $\mathbf{RCA_0}$. We also show that the minimal
Bachmann-Howard fixed point of a dilator $T$ can be represented by a notation
system $\vartheta(T)$, which is computable relative to $T$. The statement that
$\vartheta(T)$ is well-founded for any dilator $T$ will still be equivalent to
$\Pi^1_1$-comprehension. Thus the latter is split into the computable
transformation $T\mapsto\vartheta(T)$ and a statement about the preservation of
well-foundedness, over a system of computable mathematics.
