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I have a doubt, how the miner starts to count the timestamp to insert in the block.

An example the timestamp insert inside the genesis block is 1/3/2009, 7:15:05 PM GMT+1. If the genesis block would have been mined the time is the moment to the miner have started to resolve the proof-of-work or is the moment to the miner win the proof-of-work?

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The timestamp is set by the miner and is part of the data hashed for the Proof of Work. So it is set at the time the miner begins hashing a particular header and searching for a nonce for it.

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