  We develop the abstract framework for a proof-theoretic analysis of theories
with scope beyond ordinal numbers, resulting in an analog of Ordinal Analysis
aimed at the study of theorems of complexity $\Pi^1_2$. This is done by
replacing the use of ordinal numbers by particularly uniform, wellfoundedness
preserving functors in the category of linear orders.
  Generalizing the notion of a proof-theoretic ordinal, we define the
functorial $\Pi^1_2$ norm of a theory and prove its existence and uniqueness
for $\Pi^1_2$-sound theories. From this, we further abstract a definition of
the $\Sigma^1_2$- and $\Pi^1_2$-soundness ordinals of a theory; these quantify,
respectively, the maximum strength of true $\Sigma^1_2$ theorems and minimum
strength of false $\Pi^1_2$ theorems of a given theory. We study these
ordinals, developing a proof-theoretic classification theory for recursively
enumerable extensions of $\mathsf{ACA}_0$
  Using techniques from infinitary and categorical proof theory, generalized
recursion theory, constructibility, and forcing, we prove that an admissible
ordinal is the $\Pi^1_2$-soundness ordinal of some recursively enumerable
extension of $\mathsf{ACA}_0$ if and only if it is not parameter-free
$\Sigma^1_1$-reflecting. We show that the $\Sigma^1_2$-soundness ordinal of
$\mathsf{ACA}_0$ is $\omega_1^{ck}$ and characterize the $\Sigma^1_2$-soundness
ordinals of recursively enumerable, $\Sigma^1_2$-sound extensions of
$\Pi^1_1{-}\mathsf{CA}_0$.
