The behavior of higher proof theory I: Case Σ¹₂

The current ordinal analysis provides the proof-theoretic ordinal of a theory, which calibrates the robustness of the Π¹₁-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 Σ¹₂-proof-theoretic ordinal capturing the robustness of the Σ¹₂-consequences of a theory. In this paper, we study the behavior of Σ¹₂-proof-theoretic ordinal, and it turns out that Σ¹₂-proof-theoretic ordinal also follows an analogue of Walsh's characterization of proof-theoretic ordinal.