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Why not digitial "bearer" instruments be non-negotiable without
a given signature?

I suppose these wouldn't be "bearer" but whatever we call them,
doesn't this solve the double spending problem somewhat?

For example.  Why not have the bank issue the note to an anonymous
entity who has a public key on record with the bank.

In the absence of a signature from the related secret key, the
instrument will not be honored.

The instrument can be converted to a bearer instrument by the holder
at any time by signing it over to noone as opposed to signing it over
to a named party or key.  (Much like making a check payable to "cash")

The double spending problem is solved to the degree the key of the
intended payee is secure.  No?