When the miners try to extract the block with the submitblock method, do they determine the amount(value) of coinbase by themselves? Or does the network automatically give them some amount as a reward?
Thanks for any advice.
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The consensus rules allow miners to claim up to a certain amount of coinbase output value, namely the sum of the fees of all (non-coinbase) transactions in the block plus the subsidy (which halves every 210000 blocks).
The miner does have to actually construct that coinbase transaction to claim these funds, and they are allowed to claim less than what is allowed. If that happens, the missed amount is permanently lost.