Dear Jochen,
No problem, I myself struggle to find the time to read and discuss other essays.
Concerning building a "Liar cycle" within your framework, I guess an idea would be to try to retrieve a proof of the Kochen-Specker theorem (e.g. Mermin-Peres square or the KCBS inequality) using Lawvere's theorem as you did with Bell inequality ? The major difference being that in general, these proofs (but not KCBS) are state-independent.
Concerning the Yablo paradox. I also happened to ask myself the same question. Indeed, the Yablo paradox is a kind of indirect Liar, and I also wondered if one could find some proof of quantum contextuality that would share a similar structure. Unfortunately, so far, I also have no idea how such referential structure could be formulated with observables.
I have finally found the time to read your essay, I'll leave a comment on your section !
Best,
Hippolyte