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Background: The Unproven Paradox involves sending a mathematical proof to individuals in the past that is obtained from the fact that there is a recipient in the past that publicly reveals the proof. David Deutsch asks the question: Where did the proof originally come from? Another interpretation of this scenario simply leads to the conclusion that "information" cannot be sent back in time (probability of self-inconsistent events is 0).

Scenario: Suppose, in the future, we generate a one-time pad from a source of perfectly random bits and encrypt (bitwise XOR) a desired mathematical proof. Assuming its existence, we use a communications channel (ex. CTCs) to send this ciphertext back to ourselves in the past. From the perspective of participants in the past, this information should be structurally equivalent to and indistinguishable from random data.

Note that this communications channel can be noisy. Assume that this noise factor makes it so that the probability of a successful transmission of "unproven" information is zero. For example, sending a proof in the clear that has not been proven should not be possible due to noise. This is intended to be a "direct" way of combating the Unproven Theorem paradox.

Question 1. Should ciphertexts be classified as information that could violate the unproven theorem paradox? What various models of randomness are most appropriate for this situation? Should the generation of truly random bits be conditioned on events that occur in the future? If so, does this not technically provide information about the future itself?

Now consider the following modification of the protocol:

Extension: Let us instead generate the pad in the past, then wait on the other end of the communications channel for a transmission. When we have the desired proof in the future (from whatever source), we can encrypt it and send it back through our communications channel to the past.

Our past selves decrypt the information using the pad to recover the original plaintext. Assume that the participants in the protocol operate in good faith and that we have sufficient error correction abilities. Depending on the theorem itself, we may be able to formally verify its correctness.

This situation may be alternatively viewed as us generating two random strings and XORing them together to wishfully produce any desired proof that exists.

Question 2. Are there problems with this protocol that are not dependent on our model of randomness and the mere existence of our communications channel? Does this situation technically violate acceptable modern interpretations of the self consistency of time? I prefer that our analysis focus on a single timeline rather than some multiple worlds perspective.

If desired, let me know what I need to clarify in the question via a comment.

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  • Perhaps people could suggest how I could improve the question?
    – mdxn
    Nov 19, 2013 at 17:53
  • I have a question about the premises. If I look in a math history book, and it says that Fred Bloggs proved Blogg's theorem in 1823; and Fred Blogg's private notebook says: "I'm glad I found this proof from the future under the old oak tree. But I'd have rather had next week's stock prices," then we are to take this as evidence of time travel? What is the mechanism of verifying the facts of the situation?
    – user4894
    Jan 29, 2014 at 1:53
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    Just to make my usual terminological amendment to these issues, the existence of closed temporal loops of intensional information concerning an object doesn't form a paradox about the object's existence. The proof existed even before the information was received in the "past" because extensional identification of proofs is a matter of mathematical fact, not a contingent aspect of human mathematical history; what is new and threatens temporal paradox is this particular information about the proof (that it can be written thus).
    – Paul Ross
    Mar 2, 2014 at 22:37

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Yes perfectly random bits would violate the unproven theorem paradox because even if it changed nothing else about the time line it would no longer be random the second loop. The randomness itself is a self inconsistent event. How can we ever have a ONE time pad that loops throughout time way more than one time.

Furthermore the paradox will always remain whenever you create something and send it back in time regardless of the way in witch way the information gets sent back. Trying to cheat by converting in and out of randomness in no way changes anything about the general thrust of this HYPOTHETICAL paradox.

This being said. If you want to treat time travel scientifically there could always be some quantum mechanics principle that allows for your theory to work. You also mentioned parallel worlds witch easily reason away from the problem. If we could test around with this stuff I'm sure we could figure something out but in the meantime just pop on back to the future and consider how cheating with the almanac worked out for Biff in the end.

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    Really interesting point about the "one-timeness" of the pad. Also, I do realize that certain notions of randomness are going to fail here. I am looking for considerations of the scenario using alternative models of randomness, too, rather than just one. You do seem to identify the underlying issue I was concerned about, though.
    – mdxn
    Jan 31, 2014 at 20:31
  • I just realized that key details about the communications channel were removed while editing. I have re-added them.
    – mdxn
    Jan 31, 2014 at 20:50

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