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Is it possible to have a "unlucky" transaction that can never be part of block? Meaning grouping it with other transactions never results in a valid block?
[Edit] By not being in valid block, I meant no valid nonce is found with given target difficulty. It seems mathematically, this is possible and what happens to transaction (if there is no expirey) in this case?
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No, if the transaction is valid then it can always be included in a block if a valid nonce/extra nonce is found.
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No. By the definition, any valid transaction can be placed in block. If something in block invalidates it - this is only an array of random bytes, not a valid bitcoin transaction.
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If you mean that the transaction may be pending for hours, or even days, yes: it can happen and it's part of Bitcoin Core protocol.
You may also want to have a look at this question
The number of possible blocks including that transaction is a number with tens of thousands of digits. You're imagining there would be no possible nonce for any of them? – David Schwartz – 2017-11-26T10:58:49.320