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From the Alchemy docs

Uncle and nephew block rewards: The final part of our block reward calculation is to add additional rewards for mining an Uncle block (uncle reward) or including it in the latest block (nephew reward). An uncle block occurs when two miners create blocks at almost the same time. While both blocks are valid, the network can only accept one block at a time. Therefore one block is rejected and labeled as an uncle block. Instead of letting this block go stale, a nephew reward equal to 1/32 of a block reward is issued to any miner willing to later include this uncle block inside a block they are mining. Additionally, an uncle reward is issued to the miner of the uncle block

Q: Is an uncle reward only awarded if an uncle is included in a block? In other words, if no miners decide to include an uncle (and thus all forgo a nephew reward), no reward is given to the miner of the uncle. Is that correct?

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Yes, you are correct.

If a block isn't included in the main chain, nor as an uncle then that block is like it was never mined. The miner won't receive rewards.

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