I thought of a weird hack based on ambiguity about "information". Here's how it works:
In the present, you have a problem you need to solve but will only be able to solve a long time in the future.
Set up a random process to guess an answer and write it to a sheet of paper A.
Do something with the answer, then hide the paper somewhere safe where it won't be disturbed until you later solve the problem properly.
Memory charm yourself to forget what was written to A.
Much later, solve the problem properly. Also learn the Imperius curse, and then Imperius yourself to do the following protocol. Cast all sorts of other advanced protection spells to prevent anyone or anything else interrupting the protocol.
At time T, you expect to receive a sheet of paper B with a correct answer to the problem. You then look in the place where you hid paper A and see if the answers match. If they match, you write "Success" on a third sheet of paper C. If they don't match, you write "Fail" on paper C, add a completely different answer from either A or B to paper C, and put it in your pocket. If you don't receive a paper B at all, then write "Fail" on paper C, add a random answer, and put it in your pocket. Memory charm yourself to forget what was written on A and B, and what you wrote on C.
At time T+1 hour, you look in your pocket and do the following. If you find a paper C starting with "Fail" then copy the rest of its contents to a paper B, and send it back in time one hour to yourself. If you have a paper C starting with "Success" then write down the correct answer to the problem on a sheet of paper B and send it back in time one hour to yourself.
It seems like the only consistent loop is where the guessed answer written on paper A matches the correct answer later written on to paper B. But since it's "only a lucky guess", it's not strictly information about the future. Would that work?
This doesn't seem significantly different from the loop Harry already tried, that didn't work. Don't summon Azathoth.
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