it depends on what you mean by "independent". since the chain is storing its state in bitcoin blocks, it is rather dependent on bitcoin. the benefit it gains from that dependence is that it inherits full double-spend resistance from bitcoin, so re-orging the other chain is as difficult as re-orging bitcoin.
I mean Bitcoin itself is unaware of it. There'd be no way peg mechanism, so it's like a sibling chain. That's great, as people can just do it in a permissionless manner.
I think the specifics of double spend protection or not would depend on the consensus system of the sibling chain. Sounds like a cool design space to explore.
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true, the "sibling chain" as you put it could allow re-orgs independent of bitcoin. but then I'm not sure what the benefit of putting all of the chain data inside bitcoin blocks would be, as opposed to just putting a hash so that nodes could simply follow a chain of headers and make sure that matches up with the chain other nodes are feeding them. can you think of what the benefits of putting all of the chain data inside bitcoin blocks would be in this case?
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