In order to for the custodian to be solvent liabilities have to be less

than reserves, so for every Bitcoin "claim" I have there must be one

real Bitcoin on chain able to be redeemed.

It's not too difficult to prove how much Bitcoin you have on chain actually - You can just publish your addresses somewhere. Several exchanges and ETF:s already do this. For example Bitwise, Fidelity, Kraken.

The sticky part is proving liabilities, I.E how much Bitcoin is owed to your customers. This info is not on the chain itself so it's harder to prove. The best way at the moment to do this is a 3rd party audit, but then you have to trust the auditor is not colluding with the custodian to hide some liabilities.

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Shouldn't is be possible to instantly lock sats equivavent to orders as soon as they came in? That would make it possible to see the balance. Only orders that locked sats would be valid. By continually publishing a list of the balance, one would see if they had a deficit or a surplus of bitcoin holdings