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 Maybe it's possible, but it is quite complex because the mint would have to track how many times a "chain of banknotes" is moved around, and I am not sure if that's compatible with "pay to npub" types of tx, which work differently.

Even if possible, that's too complex imo. 
 It seems strange that zaps are free, and that cashu, charges.  I agree it's going to be tough on the UX.  My point is more around letting people try the system first, get used to it (like with zaps), before adding the complexity of transaction fees.   
 Yup agreed! 
mSat fees are a lot more elegant + align incentives better