I have this sentence
I guess it comes from 付く which would be in my opinion a similar form to 付いて来る. Is my guess OK, or is has the ついて here a different origin? Thanks a lot!
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Yes, the verb is 付く【つく】, the same 付く as in 付いてくる. It has many meanings (jisho.org gives 14 definitions and デジタル大辞泉 gives 28), and one of the meanings is "to be settled/determined/resolved". Similar usages include:
Unsurprisingly, the transitive equivalent つける means "to settle/determine/resolve".
On the other hand, we do not say ×結論がつく or ×議論がつく. Verbs like つく, かける, とる and する have many possible meanings, and it's often more practical to remember which noun goes with which verb as a set phrase, one by one. English also has a large number of unpredictable verb usages. (English speakers say "do the dishes", but not "do the glasses". Japanese people never say 皿をする. This is something a learner has to remember.)