Israel's President Isaac Herzog has begun consultations with political parties on whom to task with forming a new coalition government, after he received the official results of a general election last week he said had no clear winner.

Herzog told party leaders that "not every commitment and promise can be fulfilled" if they want to avoid a new election. The warning was aimed at parties whose campaign pledges make the arithmetic of a coalition difficult.

Israel's election commission on Wednesday confirmed the final results of last week's vote that stripped Benjamin Netanyahu's far-right coalition of its governing majority, rejecting claims of mass fraud promoted by the prime minister.

The seat arithmetic

In the election on 27 October, Netanyahu's right-wing Likud party and its far-right and ultra-Orthodox Jewish allies won 53 seats in the 120-seat Knesset. Eisenkot's centrist Yashar party and three other Jewish opposition parties in his bloc won 48.

Neither bloc reached the 61 seats needed for a majority, which is why the result went to the president rather than to a victor. Two Arab-led alliances, the United Arab List and the Joint List, together won an unprecedented 19 seats following a surge in turnout among Arab voters.

That makes them potential kingmakers. However, all but one of the Jewish parties pledged before the polls not to work with them to form a government, which is the promise Herzog was referring to when he spoke about commitments that cannot all be kept.

The fraud claims rejected

The Central Elections Committee said in a statement: "The review found no evidence of irregularities that compromised the integrity of the election or affected its results."

Its chair, Noam Sohlberg, said some suspicions were still under investigation but that they did not cast doubt on the election as a whole or justify delaying the election results. The committee had faced pressure from Netanyahu's camp over allegations that Arab-led parties had "stolen" the vote.

The Shin Bet security service is investigating allegations of foreign interference in the election. Officials have said that inquiry would not fundamentally change the outcome, a judgment that matters because it removes the last procedural route to reopening the count.

The Shin Bet request

Hours before the results were certified, Israel's attorney general said she had blocked a demand from the Shin Bet to interrogate senior politicians from the Palestinian-led parties because she had not been presented with any evidence of wrongdoing.

"In a country governed by the rule of law, neither citizens nor elected officials can be interrogated without evidence that justifies doing so," Gali Baharav-Miara said in a statement, after Haaretz reported on the attempt to target four politicians.

The main challenger, Gadi Eisenkot, said Netanyahu had launched a "pogrom against democracy" in attempting to hold on to power by discrediting valid votes and weaponising security forces against parties led by Palestinian citizens of Israel.

Where this can go

The president's consultations are the formal start of a process that has few easy exits. A government can be formed by the Netanyahu bloc only if it finds partners outside its 53 seats. The opposition bloc can reach a majority only if it breaks the pre-election promise about the Arab-led parties.

A third path is a wider unity government, which would require both men to accept a arrangement that neither described during the campaign. A fourth is another election, which is the outcome Herzog explicitly asked the parties to avoid.

The numbers make the first two paths hard and the last two plausible. What has changed this week is that the dispute over the count is closed, so the argument now has to be about seats rather than about ballots.

Herzog's role is to pick the candidate with the best chance of assembling 61 seats, and the consultations are where each party states its price. The president has said he wants to avoid another election, which gives him an incentive to push the two blocs toward an arrangement neither has described publicly.

For the parties, the calculation is about survival. A politician who breaks a pre-election promise about the Arab-led alliances will be accused of doing exactly that, and one who refuses will be blamed for another campaign. That is the trade Herzog is asking them to make in the coming days. The timetable is short: the president's office has not set a deadline for a mandate, but every day without one increases the chance that a Knesset member files a dissolution bill. The parties that ran against Netanyahu have more to lose from a new vote than from an awkward coalition, and he has more to lose from either than from opposition. The consultations are expected to continue through the week, with the mandate to form a government likely to follow once the president has heard every party.

Related reporting on the same day includes the deadlock that followed the 27 October vote, the US decision to halt military aid to Israel and the US strikes on Qeshm Island that reshaped the regional agenda.