So I'm not sure if measuring time periods (inaccurately) are part of quantum information (I couldn't find any relevant tags). Would this question be on topic on this site?
As far as I can tell, the subject of the question would be on-topic here the same way it is in physics.SE. The problem with the question, and the reason it was closed (it might not be the stated reason, but I'd wager is what really led to it being closed), is that it's not sufficiently clear. And this is site-independent.
A few notes/observations/suggestions in this regard:
You don't sufficiently flesh out the context. What kind of system is this? What's the underlying dynamics (i.e. the Hamiltonian), if the underlying dynamics is relevant? You say "$U$ is the unitary operator". What unitary operator are you referring to? Or are you referring to a generic unitary operator?
The actual problem is not sufficiently clear. This is the most crucial aspect of any question. You say "you can bound the time elapsed by ...". What time are you referring to? That's not explicitly stated. Also, why do you care about bounding such time? Saying that helps reader understand the question itself.
Questions in the check-my-proof format are never very good in my opinion. Is the answer supposed to be "yes" or "no"? Such questions will hardly ever be useful to anyone else.
Saying things along the lines of "I can prove non-standard result X" is a red flag, and will make many read the question under the assumption that you are just trying to promote some non-mainstream personal theory, regardless of that being the case or not. Avoid this sort of language at all costs.
It's loooooooong. That's not an issue in and of itself, but it certainly makes the question way less likely to get answers. It's still fine if a reader can get what is needed to answer from the summary/text at the beginning of the question though. Just don't assume most (if anyone) will actually read the whole thing.
There is more than one question. The is there any realistic experiment one can do to realise this quantum mechanical watch-stop? is a separate question as far as I can tell. This should be asked separately.
Note that I'm not asking you for clarifications on this points here. These are things that should be addressed by directly editing the question accordingly.
If it was me, I would ask this type of question in a more standard format of "Can X be inferred from information Y in context Z?". Be very careful that this problem is well-defined and stated clearly and concisely. You can then if you want sketch the approach you think works to solve the problem. Even so, I'd avoid spelling out the whole thing if it makes the post very long. Just briefly state the relevant ideas and intuitions. This is of course completely up to you though.