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I want to know about what happens to the commitment transactions as the HTLC's are sent and cleared along to peers in a route.

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When a HTLC is created, a new commitment transaction is created. The HTLC is an additional output (script) in this new commitment transaction, in addition to the RSMC and “regular” outputs, and represents the amount that is pending “routing confirmation”.

When the HTLC is cleared, another commitment transaction is created, absorbing the HTLC output amount into the RSMC/“regular” outputs.

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