What makes millisatoshi real?

What makes millisatoshi real?

In the Lightning Network, it's possible to pay in units a thousandth of a satoshi, a millisatoshi.

But you can't redeem them directly on the main chain when closing a channel, it will be subject to a round error, your value will be 'trimmed' of some millisatoshis.

But how they can exist before that? Is that part of the LN based on trust? The nodes store the number somehow, but what if the decided to change it? How can the digital signatures 'cover' the value, if they sign only the data of a on-chain transaction?

http://bit.ly/2CWRA7c

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