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1.“What will happen when circulating supply = Total supply? Will emissions stop?”

Yes. Once the max supply of ESHARE (28K) is reached, emissions stop. This is going to be in a year from when the ESHARE farms start December 24th. ESHARE will always print EMP in the Boardroom, though, as long as EMP is above peg.
(Note: There is a proposed extension of ESHARE emissions being considered. Please inquire in our Telegram/Discord Channels for more information)
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2.“Why is ‘at-peg’ (TWAP between 1.00 and 1.01) the best time to provide liquidity?”

When EMP is pegged or close to being pegged to 4000 EMP: 1 ETH, it is more akin to having exposure to a single asset (single staking) than to your traditional LP-ing experience, where you would run the risk of impermanent loss if one of the tokens went up in value and the other did not.

3. "Wait, you mean that if ETH pumps, EMP pumps with it?"

Yes! Let's take an example: If ETH pumps in price, it won't 'outrun' EMP. The APR will vary in terms of its USD value, but emissions won’t. This is something that wouldn't be possible with another 1:1 pegged asset like a stable coin LP position, where the USD value is directly tied to the emissions. If ETH rises in USD value, EMP goes with it. Same if ETH falls in USD value, EMP will be worth less in USD, but it won’t affect the peg. Peg is solely determined based on buying and selling volume of EMP.