As Boris Johnson arrived in Luxembourg today to meet Jean-Claude Juncker for further Brexit talks, the prime minister and his negotiators were professing growing optimism about the prospects for a last-minute deal.
The European side - including the European Commission president - is much more cautious, noting the immense political challenges facing Johnson.
These lie not just in Brussels - where talks about the Irish border are still a long way from resolution - but also in Westminster where his majority has collapsed and MPs have boxed him in with legislation forcing him to seek an extension if no deal is achieved by October 19.
If Johnson is to succeed in his pledge to leave the EU on October 31, "do or die", he needs to find a deal that is both negotiable in Brussels and saleable in a bitterly divided House of Commons.
Threading this needle will not be easy, since Johnson will need to satisfy the technocrats in Brussels while demonstrating a majority for that deal in Parliament before the EU makes any concessions.
So here we look at the possible deals available to Johnson and rate them on their chances of success.
The Johnson dream: 'Slice n dice' the Irish backstop
Convince the EU to abolish the Irish backstop and replaced it with a piecemeal deal. The UK argues this could avoid a return to the borders of the past in Ireland using a mix of technology and political goodwill. To achieve this, Northern Ireland would agree to follow EU rules on plant and animal products while other issues relating to the border - like customs checks and VAT - are addressed via trusted trader schemes and small trader exemptions. Crucially, Northern Ireland would remain outside the EU's customs territory. The rest of the UK would therefore be free to do a 'no strings' Free Trade Agreement with Brussels that would not impinge on the UK's freedom to strike free trade deals all around the world. This could mean tariffs, checks and controls at the EU-UK border.
Negotiability in Brussels? Looks impossible, unless the EU side commits a massive last-minute U-turn on its determination to defend the EU single market and its continued support of Irish demands that any solution delivers a "fully open" border. The UK suggests create a trade border, just one set back from the line itself. Privately, EU officials and diplomats are scathing about British proposals, describing the technological solutions for customs as "flimsy", "ad hoc" and failing to understand the nature of EU law and borders. ULTRA LOW
Saleability in London? Tough. Such a deal would be a great victory for Johnson and would delight Brexiteers in his own party. It would, however, lead to a very 'hard' Brexit for the UK mainland, possibly including tariffs and checks at the EU border which would hit UK manufacturers and traders. This 'hard' exit is a long way from the Labour vision for a "worker-friendly" Brexit, which is based on the UK staying in a customs union with the EU to protect jobs. Some 30-40 Pro-Brexit Labour MPs might potentially back such a deal just to get Brexit over and done with, but if they did so they would be delivering their Conservative opponent a great political victory in the process by enabling him to meet his October 31 pledge. Despite their shared views on Brexit, tribal political loyalties make it unlikely Labour MPs would do this, rather than reject a deal and force Johnson into the humiliation of seeking an extension, as the Benn Act demands. LOW
The Northern Ireland-only backstop
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If the slice-n-dice deal Johnson advocates is rejected, he could pivot to a 'Northern Ireland-only' backstop that would leave Northern Ireland entirely in the EU's customs territory and aligned on rules and regulations as necessary to maintain a "fully open" border in Ireland. This would address the Irish issue by effectively moving the border to the Irish Sea - or more accurately the ports of Northern Ireland.
Negotiability in Brussels? No problem. EU diplomats are clear this deal is on offer. After all, this was the original proposal from Brussels back in spring 2018 and the text of this deal remains in Michel Barnier's bottom drawer. It was rejected by Theresa May because it divided up the United Kingdom with an internal trade border. Johnson has also rejected the idea for this reason. The DUP who currently still support the Tory government are implacably opposed. But European diplomats wonder whether, when faced with more unpalatable choices - asking for an extension, breaking the law by ignoring Parliament or sucking up a 'no deal' - Johnson might reluctantly accept the offer as a least-worst option. He could claim he has open the Withdrawal Agreement and the EU could add some sweeteners to the deal as they did so. HIGH
Saleability in London? Tricky. This deal would be met with fury by the DUP whose 10 MPs are in a confidence and supply arrangement with the Tories. That said, hardline Brexiteers might be persuaded to accept Northern Ireland-only deal because it would open the door to a hard, 'buccaneers' Brexit for the rest of the UK, essentially by hiving off Northern Ireland into the regulatory orbit of the EU. But this would be a huge problem for labour which - as noted above - has no interest in a super-hard Brexit that economists warn will see businesses forced to suppress wages to remain competitive as tariff and regulatory trade barriers are erected between the UK and the EU. Pro-Brexit Labour MPs have hinted they might back such a deal if Johnson committed to "a future relationship that protects jobs and livelihoods", but that would mean giving up on the idea of a hard, 'buccaneers' Brexit, with consequent fallout from Brexiteers. VERY LOW
Go back to kicking the can?
Johnson realises that neither of the above options flies, but notes there is a majority in Parliament now for getting a compromise deal over the line. Accordingly he pivots back to two ideas that he ruled out in July - namely a 'time-limit' to the Irish backstop, or an exit mechanism from the backstop. This could enable the attorney Geoffrey Cox to change the legal advice warning that the UK risked being trapped indefinitely in the Irish backstop. This was the issue that blocked many Brexiteers from voting for the deal. The EU could also find some ways to sugar what would be a bitter pill for Johnson if he determines this is the only way to turn the page on Brexit.
Negotiability in Brussels? Officially these ideas are not on the table, and to re-heat them would require EU leaders to give Michel Barnier a fresh mandate to negotiate. It is not clear there is sufficient time to do this. This would also require EU member states to pressure Ireland into a deal it does not want, since Dublin warns that such ideas just create a "countdown to a border poll" and will destablise Ireland. The current attitudes in London towards Ireland do not inspire confidence in Dublin. Nor does the political instability in London inspire confidence in Europe. That said, if the choice is between 'no deal' chaos tomorrow and a time-limit that pushes that chaos into the middle distance - say seven to ten years hence - Ireland might find it harder to convince other EU leaders not to back such a compromise. MEDIUM-LOW
Saleability in London? This approach might find broad support in the House of Commons, but it would be a humiliating climbdown for Johnson to return to Westminster with a reheated version of Theresa May's thrice-rejected deal. It would, to use the American phrase, be putting "lipstick on the pig". Johnson and the EU could try to dress up this volte-face as an act of statesmanship, but the Brexit wing of his party would cry betrayal and the move would surely risk turbocharging Nigel Farage's Brexit Party on the Tories' electoral right flank. It is also still not clear if Labour would back even this compromise deal, at least without further concessions and the government agreeing to a confirmatory Second Referendum. MEDIUM-LOW
The extended transition, sleight-of-hand deal
It becomes clear that the slice-n-dice model does not work, but the two sides have made progress in talks on issues like aligning on plant and animal product regulations, the single electricity market and the common travel area for people. Big issues like customs and VAT remain, but enough momentum has been created to enable Johnson to say that he is now convinced that these will be addressed in the future relationship trade talks. To create more time, the EU agrees to extend the transition period to avoid the hated backstop ever kicking in. The backstop would remain, but Johnson would say much more convincingly than May that it will never be used, because of the progress made.
Negotiability in Brussels? No problem. As long as the backstop stays, the EU would be open to fiddling with the transition period mechanisms if that's what the UK wants. Legally speaking the transition cannot be open-ended, so the EU would need to put an aspirational date on the completion of trade talks and resolution of the Irish border via other means. Johnson could then point to this as a statement of intent - while making clear that if the UK would extend rather than ever having the backstop imposed. HIGH
Saleability in London? It is hard to see how Brexiteers would accept this route, which is essentially an extended version of one May tried, but failed dismally to sell to MPs. The UK would effectively be in double purgatory, risking being stuck both in transition and/or the Irish backstop indefinitely. Pro-Brexit Labour MPs would be equally upset, and as in all the scenarios above, it is not clear why the rest of the Labour Party will move to get Johnson out of his current bind by helping him to deliver Brexit on October 31. VERY LOW
As every day goes by and the clock ticks down to the European Council on October 17, Johnson looks increasingly snookered. What is hypothetically negotiable in Brussels, looks increasingly unsaleable in London. As always with Brexit, taking a step in one direction leads to losing support in another.
Still, Johnson wants to look like he's trying hard, so he can blame the EU of intransigence if a deal is not done; meanwhile the EU will continue to "keep listening" up until the final day of the talks, so that it can say it did all that it could.
But for all the talk of optimism from the British side, it is increasingly apparent to both sides the Brexit circle remains a long way from being squared - even if, at the moment, it does not suit either to be explicit.