Could the 2PM, for example, continue building their block after being notified that a block has already been made? They could then complete their block (seconds or minutes later), which will be valid but not part of the blockchain. They then use proof of building this block as proof of work (“late” work), giving them authority to be a second-layer agent (hereafter referred to as “Layer Two Server” or “L2S”, but can refer to operator, validator, facilitator, sequencer, etc).
I think this shows a quite fundamental misunderstanding of what is actually happening during mining. A miner is just guessing random numbers, there's no specific work involved other than constructing the block (which is trivial).
For a miner, the chance of finding a block is the same 10 seconds after a new block has been found or 50 minutes after a block has been found. You can't know who the second best miner is. No sane miner would continue building on a stale branch since they would be losing out on revenue.
I'm sure I have fundamental misunderstandings but also I have a different way of looking at this math:
For any given miner, the chance of finding a block is the same 10 seconds after a new block or 50 minutes later (you are right). However, for ALL miners, some miner would complete a "late" or "second" valid block very quickly. Then once that second (what I'm calling "late") block is found and announced, everyone could go back to looking for the next block.
In reality, this would be opt-in, so not all miners would play this game. Miners who don't would be more likely to win the next block prize, but forego any chance of earning fees in the L2. And those who do participate would be less likely to win the next block (though not impossible), but would potentially earn fees on the L2.
I don't think the value proposition is absent; what they would get if they "win" is the status of earning fees on the L2. These fees would likely be dynamic, so if few miners were early adopters, they would make the highest fees. It would also smooth out their income (of course, most mining is pooled anyway, but this could smooth out the revenue for the pool operator).
I know this is abstract because it isn't a proposal for a specific L2, and would depend entirely on the incentive structure therein.
I also failed to make clear that running an L2S to gain fees would require minimal computation power and wouldn't reduce the miner's ability to mine as usual (they would be doing both). In fact, it's really stupid of me to not point that out from the onset.
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