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 one of the GEC addresses John controls). 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).