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I have a transaction ID here, recipient address and sender address which I thought to be a transaction from some Bitcoin purchase platform such as LocalBitcons.com but I cannot find my purchase receipt.

Is there some way to find out which party is responsible for the transaction by its transaction ID?

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Bitcoin transactions are (intentionally) devoid of extraneous information such as 'sender' and 'recipient'. The inputs/outputs do not contain any information about who owns them, and neither does the ordering of those inputs outputs (eg, FIFO is meaningless in terms of analyzing bitcoin transactions).

That said, while this information is not available publicly, the users participating in a transaction will (privately) know at least some of this information (eg, which coins are theirs). But they will not necessarily even know the entirety of the ownership mapping for any given transaction.

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