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Is this proportional to time required to process? Is a high gas transaction more expensive for the miner to include in a block and does it add more to blockchain size than a low gas transaction?

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Gas is directly proportional to the cost of computation of the transaction. More operations or interactions with stored data means higher gas cost for the sender. Also, more expensive operations mean more gas. So yes, the op code matters a great deal (e.g. SHA3 is naturally going to be more expensive than EQ). The gas is calculated by the client of the sender.

The gas is paid to the miner because yes, a high gas transaction is naturally more expensive to process (because it is more computationally expensive).

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