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It was a week of backstopology: an EU summit, cabinet meetings, threatened resignations and lurid language from the backbenches about knifing or noosing the prime minister.

But for many there was one highlight, the anti-Brexit demonstration in London on Saturday. The huge turnout showed “remainers are also angry and determined, also patriotic and more convinced than ever that there is no available Brexit, on paper or in anyone’s wildest dreams”.

The organisers were expecting more than 100,000 people, but about 700,000 turned up.

Witty banners proclaiming Brexit lies, national divisions and democracy deficits coupled with all-day sunshine kept spirits up during the four-hour march.

Buses and trains brought marchers from across the UK. Summing up the mood, one demonstrator said: “I want to tell my grandkids that at least I tried.”

Back in the House of Commons, Theresa May gave her verdict: the people already had their vote and there would be no second referendum.

The prime minister has survived much longer than predicted and is into what has been described as Hell Week 3. she again calmed the turbulent waters after a weekend of assaults on her leadership in which one Tory backbencher was quoted as saying she was finished: “The moment is coming when the knife gets heated, stuck in her front and twisted. She’ll be dead soon.”

Standing in the Commons on Monday afternoon, she gratefully accepted all the messages of support in another attempt to hold off a backbench rebellion over the Irish border. May urged MPs to hold their nerve while she negotiates the last elements of the Brexit deal.

She told MPs that the “Brexit talks are not about my interests. They are about the national interest” but admitted that the last hurdle – the Irish border – was still proving a significant obstacle. Otherwise, the withdrawal agreement deal is 95% settled, she said.

Earlier in the day she faced a bid by the Brexiter MP Steve Baker to try and kill off the backstop, which May signed up to last December as part of a joint UK-EU report. This marked the end of the first phase of Brexit negotiations on EU citizens, the divorce bill and the Irish border.

Baker’s attempt was dropped in the afternoon when he withdrew his amendment to Northern Ireland legislation, tweeting that he had been “persuaded this would not be in the public interest”.

The Irish border has dominated talks and was the key focus of May’s briefing to the Commons and the EU summit last week.

It was confirmed last Wednesday that the EU was open to extending the transition period by a year if it would help May win support back home for an Irish border backstop. Ireland also supports this.

The backstop has been likened to an insurance policy in the event of no Brexit deal emerging, but has caused deep divisions in the government because the EU’s proposal is to keep Northern Ireland in the customs union and single market if the UK crashes out.

In the feverish hours that followed, May appeared to confirm this was something she was considering as she entered the Europa building on Wednesday night to speak to the EU 27.

By Thursday morning, however, she had backtracked, presumably after seeing the reaction from Brexiter MPs who were furious that anyone could contemplate staying in the EU a day longer than necessary.

It was clear from her statement in the HCommons on Monday that the EU proposal for an extension was still an option, but she attempted to “de-dramatise” the proposal explaining it would only last for, say, three months.

It might be needed, she said, if the trade deal was completed by the end of the transition period on 31 December 2020, but European parliaments needed more time to ratify the deal.

May has pushed back further, laying down four new red lines for the backstop including a legally binding UK-wide customs arrangement, and the option to swap the backstop for the extension in a deal that the UK could pull out anytime. Got that? If it was designed to baffle, it was successful.

What next?

The Brexit secretary, Dominic Raab, said he believes a deal can be done by the end of November. Even if a deal can be struck though, May is not home and dry.

She must get approval for the deal from parliament and that is far from a given. Even if she gets support she faces more trouble down the line. This timeline gives the key junctures over the next six months.

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As the chancellor, Philip Hammond, puts the finishing touches to next year’s budget, Philip Inman say May’s conference pledge to end austerity was not deliverable.

The chancellor is keen to keep the lid on extra money for public services, according to his own guidelines, at least until a spending review next year and any post-Brexit bounce materialises.

In their latest dispatch from Anywhere but Westminster, the Guardian’s John Harris and John Domokos tell how the future of politics lies in towns.