I’ve forgotten most of the (little) Pali I ever knew, so please feel free to ignore the following…
In situations like this, when words are used in different ways, I find myself looking for an underlying metaphor that unites the different meanings.
It seems Indians had a sense of eons being cyclical, and so I find myself thinking about thinking, or cogitation, as a cyclical process. And in fact the word cogitate is from the Latin agitare, which means to stir or churn, which are cyclical movements. That doesn’t mean that the Pali kappa necessarily has the same meaning, but I find some pleasure in thinking about thoughts being like milk churning in, well, a churn.
It seems kappa can also mean a horse’s harness, which at its core is a number of cyclical/circular elements that fasten around various parts of the animal’s body.
So I wonder if “examining a kappa” is a way of talking about examining both the cyclical/cogitative processes of the mind and the cycling through birth and death it gives rise to.
I think most of the passages quoted here would support the understanding of a kappa as being a cyclical process of some kind.
“Not entering a kappa” makes me think more about not being reborn than about stepping out of (unskillful) cogitation, but it could similarly mean both things simultaneously.
While you wouldn’t “go into” a mental construct, you would “go into” an endless cycle of cogitation. The image of a cycle of thinking is more dynamic than Ven. Bodhi’s “mental constructs” that you say only gets halfway there.
The passage that gives me most pause is Paññāṇavā so uda paññakappī, unless perhaps kappa in this context is a more abstract understanding of “thinking” so that it’s that he thinks he’s wise, or perhaps the meaning of kappa here is more like “process,” so that he’s in process of becoming wise. It bothers me that I have to try so hard to make that example fit, while with the others the underlying image of a cycle fits more naturally.