Thanks very much for the details Hyung!
I understand that in practice Uniswap is path dependent because of the fees, but it seems straight forward to reason about how that’s in favour of the LPs (“monotonic increasing” as you say). However it’s less clear to me how to conclude that the ESPMM (before fees) is also monotonic increasing. Is there an argument for that somewhere I can follow? I will look at the slides you linked.
Otherwise I think the design you have here of marrying the AMM with an Order Book is really interesting, and adjusting the AMM design to facilitate order book execution so that we can have the best of both worlds is very clever!