Dear Hippolyte,
sorry for taking so long to respond. I'm afraid I've somewhat overstretched my time budget in starting so many threads of correspondence in this essay contest.
One thing I've been thinking about, which I think needs some more thinking about (?), is how one could formulate these self-referential 'chains' of observables within my framework. Basically, I construct an 'inconsistent' observable by means of Lawvere's theorem---something not too dissimilar from Russell's set that contains itself iff it does not contain itself. It would be interesting to see whether one could extend this to yield something like the 'liar-cycles' which have no consistent assignment of truth values. Perhaps one could 'daisy-chain' the Lawvere argument.
Another question, it seems, might be whether there's an analogue to something like Yablo's infinite set of paradoxical sentences, too. This concerns sentences of the form:
(S_n): For each i > n, S_i is not true
Assuming S_n to be true, we get some later S_k, k > n, such that it is both true and not true; but then, assuming that each S_i is not true, yields the conclusion that S_n must be true, because that S_i for i > n is not true is exactly what it asserts. Hence, we obtain a contradiction.
This is an 'indirect' sort of self-reference, in that each sentence does not refer to itself, either directly or via a circle of intervening propositions, but rather, to the whole set of sentences, with a contradiction arising from that. I'm not sure, however, how one would go about finding an analogue of this in terms of observables.
Anyway, that's still sorta open-ended speculation on my part. I'd be very interested to hear your thoughts on my essay!
Cheers
Jochen