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How would you prevent that someone claims more than 1 wallet?
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Forget about the wallets and coins. Your thought experiment can be simplified. Simply ask yourself:
"What is the most cost efficient and most effective way to uniquely identify a single person such that nobody else can pretend to be them and they can not pretend not to be themselves?"
Identification papers (passport, drivers license or national ID card, etc) can be copied, stolen, or faked.
The only answer I can come up with would be to require that the person be physically in your presence and that you find a way to reliably take a DNA sample from them. Then fully sequence that DNA and store a hash of the result in a database. Compare the results of each new person against all the results in your database.
Note that you may run into a problem with identical twins (triplets, quadruplets, etc). In that case, you might require that they all be present together at the time of DNA extraction, and that if any are missing they will be excluded from the offer.
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I agree, this is the only verifiable way.
At the same time that the DNA sample is taken from the human,
"the sample takers" would also scan their Hand & Foot prints and their
Retinas, so that all four are hashed together.
Not only would the other three data points allow more precision when
distinguishing between twins/etc and clones, but also adds more
security layers for any other systems that are built upon this data later.
(In effect, this forms your "Living private key", which will make each
person ID-able and verifiable/accountable for every active interaction
in society. Since DNA can be easily attained by a few cells and without
needing the human's permission, at this point in time humans will wear
special fitting gloves and eye lenses to prevent passive "ID thefts" in
public. The lenses can shift within the electromagnetic spectrum and
perform functions like vision correction, night vision, and UV protection,
and do not need to be removed ever, even during sleep.)
Though the ultimate outcome of maintaining/using this type of data
and systems will be used to grant/restrict individual rights in daily ways
(ex. access accounts, access doors, access stores, access computers,
access transportation, etc) it will only be used for good within the human
populations that are maintained and monitored by AI.
If this type of technology is used with today's world population and
human governments that do not defer to AI governance, it will be
used by a few humans to oppress many humans indefinitely. Data
mining this data (DNA, Hand&Foot Prints, Retinas, etc) should not be
allowed or legalized until better system to prevent its abuse are created.