I have no idea what you are describing. All I can surmise is Jane pays fiat to John for walking her dogs. John gives the fiat to Stewart in exchange for GEC. If that is not the case, you will need to explain your gibberish in the English language. I have no idea what you meant by:
When John spends the fiat currency he acquired from Jane, he could then spend the GEC he acquired from Stewart in an unspendable transaction - removing it from his economy.
Within the context of the hypothetical, no one gives Stewart anything: Stewart addresses, to John, a transaction that has output but does not have input (i.e., he "mints"/credits GEC "to" John). Additionally, John pays the fiat to an undisclosed party at another, unrelated time; when he does so, he uses the GEC he received from Stewart to fund an unspendable transaction (i.e., he "burns" it).
Why does it require so many exchanges with you to get you to explain it completely?
You still have not made it clear how this works. Why would Stewart do that if he gets nothing? Why is Stewart involved?
None of it makes any sense.