User blog comment:PsiCubed2/Question: General breakdown of levels up to Second Order Arithmetic/@comment-30118230-20180204192313/@comment-30118230-20180205211318

+PsiCubed2 Sorry for the late reply. Just some notes to add to the conversation.

I guess you could define a less arbitrary list of subsystems for $$\Pi^1_2-\text{CA}_0$$. One could define some relation for "nth order" nonprojectible ordinals,forming a sequence of "nth order" nonprojectible ordinals with a supremum for n less than omega being the supremum of the nonprojectible universe,although that would be a bit difficult.

$$\Delta^1_2+\text{CA}+\text{BI}+(K)$$ has a PTO of \(\Psi_\Omega(\varepsilon_{K+1})\).

If I had to guess,$$\Delta^1_2-\text{TR}_0$$ may be the same theory as $$\Pi^1_1-\text{TR}_0$$ and therefore have a PTO of $$\psi(\psi_I(0))$$ but don't take my word for it. "Is there a place were I can find a more comprehensive list of subsystems of Z2 up to Π12-CA0, ordered by strength, including systems we don't yet know how to fully analyze?"

I'm afraid I can't exactly answer that,but I do know papers (mostly by Michael Rathjen) that contain analysis of $$\Pi^1_2-\text{CA}_0$$ (as well as some foundations to get you started with this.) Try these: https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf ; https://www1.maths.leeds.ac.uk/~rathjen/BULF.pdf