Proof of Transaction (PoT lol) Possible?

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I'm not going to pretend to understand this but was curious if this would be possible.

IOTA has instant and zero fee transactions. These transactions they say require work to combine in tangle. Could you make a crypto that works based on the transactions in Iota. Essentially mining by moving IOTA around over and over?

Obviously if this was possible this would only work on IOTA as it has "free" and "instant" transactions. And storing all that wouldn't be your problem.

Thanks