  The current ordinal analysis provides the proof-theoretic ordinal of a
theory, which calibrates the robustness of the $\Pi^1_1$-consequences of the
theory. We can ask whether there is an ordinal characteristic capturing more
complex consequences, and it turns out that we can define the
$\Sigma^1_2$-proof-theoretic ordinal capturing the robustness of the
$\Sigma^1_2$-consequences of a theory. In this paper, we study the behavior of
$\Sigma^1_2$-proof-theoretic ordinal, and it turns out that
$\Sigma^1_2$-proof-theoretic ordinal also follows an analogue of Walsh's
characterization of proof-theoretic ordinal.
