Mr Sharon's hopes of staying as Likud leader rest on his well-based claim to have a much better chance of winning a general election than his rival, Benjamin Netanyahu. But a now deeply ideological majority of Likud members may yet decide that this remains a price worth paying for punishing its leader for a Gaza disengagement which it opposed from the start.
Mr Netanyahu's best hope of winning a general election is probably a big upsurge in violence, reinforcing his right-wing stance. Otherwise his strategy appears to be making an unabashed pitch to the hard right until the leadership contest is over, and then with nods and winks to the centre seek to acquire a broader electoral base, rather as he did before becoming Prime Minister in 1996. But according to Professor Yaron Ezrahi from the Hebrew University, who is among the country's leading political scientists, Mr Netanyahu is unlikely to benefit from such a move nine years later. Being seen by many Israelis as "something between a politician and a crook", he could be seriously damaged by such obvious "opportunism".
Which is what could make a new party, or at least a Likud Mark II, potentially attractive to Mr Sharon. Professor Ezrahi says Mr Sharon might just agree to a party condition, however humiliating, that he would not withdraw from any more West Bank settlements without the approval of Likud's Central Committee - provided he was sure he could then beat Mr Netanyahu. But that otherwise he might found a new party which would secure a Knesset membership of perhaps 15 to 18 and form a coalition with Labour and the secular centrist party Shinui, and preferably one religious party.
Professor Ezrahi believes that driven increasingly by "necessity" rather than ideology, Mr Sharon now shares with the Labour leader Shimon Peres a desire to return to negotiations with the Palestinians. That view of Mr Sharon is strongly contested by many. But if Mr Sharon is determined to stop Mr Netanyahu at any cost, he could still opt for a breakaway. Mr Peres, sensing the possibility of winning an election for the first time in his career - or at least leading the biggest single party - might be reluctant to join a party led by Mr Sharon, especially as such a party might win fewer seats overall than if it fought an election as the separate but component parts of a future coalition. But Professor Ezrahi suggests that Mr Peres, needing Mr Sharon's support to sell further peace moves to Israeli voters, might even rotate the premiership with Mr Sharon as he did with Yitzhak Shamir in the 1980s.
All this is now the stuff of intense political debate here. But there is one certainty and one other possibility. The latter is that at 77, Mr Sharon, having secured disengagement from Gaza, may simply decide to fight it out with Mr Netanyahu and if he loses, withdraw magisterially from political life to his ranch in the Negev. And perhaps not take it too badly if Mr Netanyahu makes a mess of the job. The certainty is that Mr Sharon will never join a government led by the man who resigned this month in an 11th-hour protest at a piece of disengagement he had himself voted for. Asked recently if he could contemplate such a thing, he said just three words: "Of course not."