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What is calculated towards the sum of PoW of the chain? Is it the target difficulty that has been met or is it the actual block header hash?

Say the target is 000000ABC00000000... and I mine block with hash 00000012300000000... - do I count 000000ABC00000000... amount of work for my block or do I count 00000012300000000... ?

In other words: if someone else mines a block at the same time, extending the same chain as my block (i.e. natural fork) and his hash is 00000002300000000... is his chain's cumulative PoW greater than mine or are we same?

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It's the sum of difficulty targets, not the individual difficulty scores. Therefore, two blocks at the same blockchain height are always the same cumulative difficulty¹.

If it were individual difficulty scores, it would make chaintips more gameable, e.g. a block with a particularly low hash might prompt a selfish mining attack because the attacker could expect that their chaintip will outdo another even when they reveal the block later, or miners might continue mining at the same height because an already discovered block barely met the difficulty statement. In other words, the blockchain would converge less reliably if it was based on individual block scores.


¹ Unless we're in a longer blockchain fork across a difficulty reset which changed the difficulty by different amounts for the two chaintips.

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