So what does it mean that someone other than me can touch my funds on Layer 2? It means that if the previous owner of a VTXO I hold (essentially, bitcoin on Layer 2) colludes with the ASP, a double spend becomes possible. I may think I've received the money, but in reality it could have been double-spent so that someone else receives it too, and if that person withdraws onchain first, I have no recourse.
Some might conclude from this that it isn't self-custody at all, but that's not the case. A VTXO can have its history reset through a process called "refresh," either periodically or whenever its owner wants. Put simply, a refresh is an onchain transaction, and through it the VTXO becomes something only I can withdraw. This is cryptographically guaranteed.
And since the only party who can wrongfully touch my VTXO is its previous owner, once I've refreshed, every change VTXO that comes back to me as I spend from then on belongs to me alone, because every previous owner is me.
Some might conclude from this that it isn't self-custody at all, but that's not the case. A VTXO can have its history reset through a process called "refresh," either periodically or whenever its owner wants. Put simply, a refresh is an onchain transaction, and through it the VTXO becomes something only I can withdraw. This is cryptographically guaranteed.
And since the only party who can wrongfully touch my VTXO is its previous owner, once I've refreshed, every change VTXO that comes back to me as I spend from then on belongs to me alone, because every previous owner is me.
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