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These dedicated networks will still use the Ethereum mainnet to perform any major adjudication. The mainnet will be a "fallback of last resort", like a supreme court. This will allow the app-dedicated chains to enjoy the security of eth, but will barely consume any tx volume
This is the vision of most L2 solutions, as well as of Etheruem Swarm Team @ethswarm, with their "service agreement games".

swarm-gateways.net/bzz:/ca5f4684b…
swarm-gateways.net/bzz:/theswarm.…
The point is that Web 3.0 will run on agreements signed off-chain. In normal circumstances, the network will just run and money will pay for service. If anyone tries to cheat, the matter gets gradually escalated. Only the final arbitrating contracts will be on the eth mainchain.
I expect the same to hold in app-specific chains. The chain will be self-contained and self-running under normal circumstances. Cheating will escalate gradually through a "litigation hierarchy". Each level of the hierarchy will be more expensive to litigate in, but more secure
This beautifully mirrors real-world arbitration schemes, as well as new cryptoeconomic arbitration schemes (where the hierarchy of courts are made of people, not of blockchains).
Interestingly, the hierarchical court system of mainchains-and-sidechains was not designed intending emulate the real world, but just arose naturally from cost-benefit analysis in bounded-throughput blockchains, i.e. from trying to use scarce expensive resources most frugally. >>
>> this makes me think there might be some inherent property or "truth" about complex systems, which causes such hierarchcal solutions to arise naturally as optimal solutions. Just as the math of special relativity follows from a few simple axioms, maybe hierarchcal courts do too
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