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Preimage is known to every node on the path of the payment as it settles. Receiver gets it in the end so this is actual proof-of-payment.

Does that mean that one of those nodes on the path could say that they were payer of the invoice?

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TLDR: no, those nodes in the path are only "forwarders", they don't know too much or nothing.

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(The intermediate nodes all see the pre-image but cannot know the exact destination node (unless the next-to-last node knows the last node is a dead end with only one channel)).

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