I argue it does exist, and bitcoinium is bitcoinium. You're just missing a property of bitcoinium which is the electricity that went into it must have no other use, and that the use must be cryptographically verifiable in a decentralised way, and behave like a statistical poisson process (people around the world trying to do it and on average one mining a block of coins every 10mins).
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Its a bit more than a Joule though. Its a time-adjusted Joule that you can cryptographically prove and that has no side-uses. That really is bitcoinium and that has amazing implications.
The property of "energy went into bitcoinium having no other use" is compatible with neutrality of money system and, I believe, is essential for its universality:
https://bitcointalk.org/index.php?topic=855520.0However, it being "cryptographically verifiable in a decentralised way" is where many questions arise. The measure of decentralization is a tricky one. If an entity creates an alt-coin and runs a single node mining it - is it decentralized? If there are two or five nodes? How many nodes is decentralized enough? In the end it boils down to adoption curves and network effects, which contribute a great portion of the value:
https://bitcointalk.org/index.php?topic=873102.msg9656213#msg9656213But I agree, that the universal cryptographic AI with non-local communication would be able to erase the boundaries between different alt-coins and their network effects, essentially melting them all into the same one bitcoinium. The question remains - what boundaries will still be left? Can brain activity be thought of as cryptographic and therefore be transfused to bitcoinium as well? Can matter be turned into energy and then bitcoinium? Following that path we might all end up in the same one room where it all started, where differences didn't exist.
Maybe we should seek sameness-in-difference instead, maybe that's what bitcoinium is for.