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Is there a set rule for how long a low fee transaction will sit in the mempool before being purged?
The transaction in question has a fixed fee of 45.2 s/vb so it’s above the purging limit yet below the low priority limit according to mempool.space, no RBF or CPFP.
Also, what is the process or mechanism that initiates dropping a transaction? Am I waiting on the transaction to drop out of various miners individual mempools.
There is no single mempool. Nodes evict a TX from their mempool if the mempool is full with higher fee rate TXs. As an example, if the node has a mempool of up to 300 MByte (the default, I think), and around 300 MByte are consumed by TXs worth 100 sat/vByte (or more), the TXs with 99 sat/vByte (and below) are dropped.
However, keep in mind that nodes re-broadcast their transactions. It might disappear from some nodes' mempools, but it might end up back in them once the fee rates are low again.
You can't be sure that the TX will not be mined, some "malicious" miner might still have access to the TX and include it despite the low fee rate. Be careful.
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I had a low fee txn that recently got picked up anyway even amidst high fees after being purged by typical mempool settings: #355345
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