Isn't this just a digital check? People can tamper with their app and their checks will bounce once reality catches up.
The double spend problem is not solved.
And how could it be if it works offline. If I have a phone in some state s. Then I can just do the transaction and revert the phone's state back to s and spend the money again.
Every time you create Euros, you simply create a negative balance. If you offline sign multiple I Owe You contracts, you accumulate debt. Just like in the real world, if you sell your house to 12 different banks, you have a (felony-level) problem. We ensure that double spending is detected with certain guarantees. Each micro-block with a transaction within Trustchain needs to be signed by two parties, so each copy is valid. Block hiding attacks wont work then.
The proof-of-work system is very elegant. The emergent properties of mining have created an amazing ecosystem. But slow finality might be a showstopper for mass uptake. Our alternative with instant finality using multiple legally binding signatures might offer a way out.
Various mechanism can be used to ensure integrity. What we implemented specifically is that various witnesses can inspect your Trustchain blocks and co-sign your balance. So any node can act as a digital notary. You can also use a reputation or trust function.
Here is a master thesis from Mathematics on distributed accounting systems. It contains numerous new mathematical proofs around integrity without strong identity assumption; .PDF "On the Sybil-Proofness of Accounting Mechanisms in P2P Networks", https://repository.tudelft.nl/islandora/object/uuid:6b4011c6...
Reputation and trust can be gamed. Also requiring reputation will slow down adaptation.
I will read this paper in the evening but I don't think you can have Sybil-Proof Accounting on P2P without some tradeoffs.