If for some reason no automatic method can be applied because it requires L2 specific data, then some address can be hard-coded/stored on L1, and only that address is allowed to call the init function.
At a high level, this could simply be the Deploying EOA? which is already going to match across all EVM L2s.
Small thing that isn’t clicking for me is Factory contracts. A booster rollup sounds great for de-fragmenting liquidity. But it doesn’t really work for example for liquidity pools where the same canonical tokens aren’t given the same address across instances (e.g., USDC). (Repeat Perseverance’s point on init-like behavior).
If the L1 state needs to be updated, where are the cost reductions for the booster roll-up (L2 is cheaper explicitly b/c it is fragmented)?