I see. So I guess there is some benefit gained from this, but it is very minor. It seems to me that the simplest rule that explains why Harry’s integer factorization is not okay, but, for example, the “silver on the tree” password from the end of TSPE is okay, is the following: if you would gain information at time T, and send information from time T to any time S < T, then it must be the case that you would have gained that same information even if you hadn’t learned it at time S.
Now consider your “keying” scenario. We have clones 0-5, and at time 1 clone x goes back to time 0 and becomes clone x+1. When clones give a “time key,” it will be a number between 0 and 6, identifying a clone/wall-clock pair. Now suppose at wall-clock time T clone 1 says: “the thought I had at time P doesn’t work.” Assume for simplicity that time P refers to clone 0 at a wall-clock time S > T (though it would work out the same anyway). Now at time S, when clone 0 has the thought, he has two choices. On the one hand, he can continue working out why it doesn’t work, but in this case he gains only the minor benefit of knowing in advance that it will not work out. On the other hand, he can move on and not consider the thought, but at time T as clone 1 just repeat (without knowing why), the fact that it doesn’t work. In that case, he gained information that he would not have learned had he not told himself. Or, in your terminology, recursion was required in order to reach a stable state.
I completely agree. Recall also Draco’s speech about muggles scratching in the dirt, and his reaction to Harry’s estimate of the lunar program budget. It’s not just wizards not paying attention to relative values of gold and silver in the muggle world—for the most part, the possibility that there could be a substantial amount of either in the muggle world doesn’t occur to them. Now you might expect muggleborns to know better, even after making allowances for the fact that they enter the wizarding world at age 11. On the other hand, if a muggleborn is clever enough to see the potential for profit, they might also be clever enough to see what Harry apparently does not—that calling attention to the fact that the muggles are ripe for exploitation is a Bad Idea.