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 cumulative weight (unless we're in a longer blockchain fork across a difficulty reset).

If it were individual difficulty scores, this would open the door for all sorts of gaming, e.g. selfish mining after finding a particularly good block because you're sure you can orphan another, or attempting to orphan an already discovered block because it doesn't exceed the difficulty statement by much. In other words, the blockchain convergence would be a lot less stable if it were individual block scores.

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