I got it. But how can we check that the private key on this gadget really belongs to the address the balance of which we can check on the blockchain without revealing it? In other words, is it possible to "authenticate" the wallet on the device without "compromising" it? If it is not possible to do without showing the private key, you can't accept it from someone without risking a double-spend or give it to someone else unless you give it "blindly", i.e. without knowing the private key and whether the wallet is authentic. Thereby, you can pass this wallet only once offchain, then you will have to use the blockchain again (at first to check and then to reset the stick)...
There is a whole FAQ on authenticity and validation here
https://opendime.com/#faq . It also ships with a few open source validation python scripts for advance users. But better yet, the Electrum plugin will be merged soon and you will be able to do "paranoid" check right from Electrum!
I read the FAQ on your page, and this part clearly states that Opendime units are expected to be used only once:
We expect most Opendime units to be loaded once, probably with a "round number" of bitcoins, and unloaded exactly once in their lifetime. It's impossible to know what's happened in the meantime—just like a gold coin that has passed through many hands over the years
That means that you can't make n-transactions offchain as you at first stated. What did I miss?