This story has been bugging me all week.

A letter to Israel from Egypt’s new president hoping for regional peace kicked up a stir Tuesday when the Egyptian leader’s Islamist movement denied he sent it. Israel insisted the letter was genuine…

In it, Morsi appeared to write in English, “I am looking forward to exerting our best efforts to get the Middle east Peace Process back to its right track in order to achieve security and stability for all peoples of the region, including that Israeli people.” The Israeli president’s name was spelled “Perez.”…

An official in Peres’s office — speaking anonymously because the issue concerned sensitive diplomatic relations between the two countries — said the president’s aides received the official communique Tuesday from the Egyptian ambassador to Israel, both by registered mail and by fax from the embassy in Tel Aviv.

Peres’s office asked the Egyptian ambassador if it could publicize the letter or if it should be kept secret, the official said. The Egyptian envoy phoned Morsi’s office to inquire, the official said, and then told Peres’s aides that Morsi’s staff had given the green light to make the letter public.

Israel made the letter public, whereupon Morsi’s spokesman promptly denied he had sent it and called it a “fabrication.” As literally everyone in the world expected he would.

Only in the Middle East could something this simple be this complicated. Three obvious possibilities about what really happened, and I’m eager to hear from readers who know more about Israeli/Egyptian relations than I do (which is pretty much everyone) which theory they think is right. One: Egypt’s ruling military junta sent the letter under Morsi’s name, possibly with an eye to reassuring Israel that the peace will hold and possibly to weaken Morsi with his own Islamist base. The Egyptian ambassador doubtless answers to the generals right now, not to Morsi, so there’s no contradiction in him “confirming” for Peres’s staff that Morsi really did send it if the junta is behind this. Two: Israel concocted the letter themselves. No doubt they could do it if they wanted to, but, er, why would they want to? All it would do is irritate Morsi and, possibly and more importantly, Gen. Tantawi and the military leadership. My sense is that Israel’s hypercautious right now about upsetting the fragile status quo with Egypt, especially when they’re focused elsewhere, on Iran. Making up a fake message from Morsi defies that strategy.

Three, the likely answer: Morsi really did send it and is now lying his ass off. It’s obvious why he might lie — maybe he meant for it to be secret and had to retreat after his staff accidentally let it be published, or maybe he underestimated the grief he would get from the Brotherhood once they found it. It’s not obvious to me, though, why he would have sent the message in the first place. It’s the military that’s at peace with Israel, not so much the Egyptian public; railing against Zionism would be an easy way for Morsi to rally popular sentiment behind him and against the generals. So then … why the outreach? Is he doing it as theater for American viewers, in hopes of building some relationships in the U.S. government that might potentially rival the generals’? Is it some kind of faux outreach to Israel such that, when the peace eventually crumbles, the Brotherhood can turn around and say, falsely, “Hey, we tried”? In that case, why deny that Morsi sent the letter? Any insights are welcome.

Exit question: Even if he did sent it, does it matter?