Let's assume I can build an AMM. Now I need to understand exactly how they work in Eth, that makes this kind of thing impossible in Bitcoin. And I don't: it's not my thing.

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but the AMM on Ethereum will work completely differently than a CAT based on Bitcoin, Ethereum has been designed for these applications. So even if technically CAT allows you to build an AMM, it doesn't mean it will ever be used.

So someone needs to argue convincingly that it won't, otherwise enabling CAT is epically dumb.