You still have not explained what Stewart's individual economic motivation is. And why is he involved. Nor have you explained why John has an individual economic motivation to burn GEC when he spends fiat. Nor have you adequately explained why they benefitted from involving GEC in the first place. Seems to me that GEC was never involved. Jane paid John in fiat. John spends the fiat. GEC has no involvement except for some fantasy conjured in your mind.
Both Stewart and John are interested in using GEC (Stewart more so than John); however, not many other people are. Accordingly, Stewart carries USD transactions over into the GEC economy so they (i.e., Stewart and John) can exchange GEC with each other and other GEC holders instead of exchanging USD with each other. However, in doing this, USD and GEC must be removed from their respective economies to maintain the one-to-one correspondence of the GEC they are exchanging inside the GEC economy with the USD they receive and pay outside of the GEC economy (inside the USD economy).
You've violated the fundamental tenet of crypto-currency which is decentralized trust, i.e. the abolition of requiring individual trust.
There is no way the parties can be sure that someone has retired fiat in order to use its value on GEC, then burning GEC again when it is desired to unretire the fiat.
I thought about all these mechanisms before and I came to the conclusion there is no way to have decentralized physical backing for a crypto-coin.
Sorry.