In that you control your UTXOs, yes. The multi-sig transactions to open and close a channel with your counterpart are the on-chain footprint, with the “layer two” Lightning protocol representing the ownership of your portion of that UTXO at any given time. So while there is always the self-custodie on-chain anchor underneath those Lightning transactions, the exchange of L2 sats is also happening via an open-source protocol where both parties cryptographically secure their own funds, while the amount that each party owns is changing and being updated off-chain. Maybe over simplified (I’m no expert) but this is how I understand it

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L2: CRYPTO SECURITY, CENTRALIZED RISK.

The fact that the amounts are updated off-chain is what makes it the same as other third-party custody

It does have a few more assurances than, for example, mints. Every L2 solution is essentially batching. There's a whole spectrum of trade-offs between speed, throughput, and security. On chain is king. On chain nostr zaps would be an incredible innovation. Mints have single points of failure. Lighting is a good middle ground. But just one of many solutions.

Good points!

Except that it’s not a third party who is updating balances or having custody of funds. It’s you and the code you’re running, in collaboration with another free user who controls their own funds.

It’s not at all the same as letting someone else hold your Bitcoin and hope that they’ll play nice when you ask for it back…