Gold does not have a universal directory of past transactions. Nor does cash.
Yeah, but cash does have a unique serial number. I should know because I have this little beauty in my possession:
Of course, its intrinsic value is only one Canadian dollar.

Right, but it does not have a universal directory of past transactions. Cash with a serial number is very similar to crypto with opaque blockchain. It can be verified to be authentic, but you can not determine its previous owners by using its serial number.
edit: 1111111