You could call it swing Saturday or crossover day, for this Saturday, 19 January, marks an important moment. This is the day, in theory, when the country turns remain. Even if not a single person has changed their mind since the referendum, the demographic shift alone will have done the heavy lifting. Enough old leavers will have died and enough young remainers will have come on to the electoral register to turn the dial on what the country thinks about Brexit.

The psephologist and founding YouGov president, Peter Kellner, calculates that the leave vote has been declining by about 1,350 a day, taking into account the differential turnout: the young turn out to vote much less often than the old. By using exactly the same proportion of every age group turning out to vote exactly as they did in 2016, demographics alone will have transformed the UK into a remainer nation.

The true “will of the people” looks considerably more questionable if it turns out to be the will of dead people – not the will of those who have the most life ahead of them to face the consequences. Does this guarantee remain would win a referendum this year? Of course not. People are changing their minds, though polls show predominant movement in the pro-remain direction. But once a ferocious campaign gets under way there’s no knowing what might swing opinion. No one expected the leavers’ two toxic lies to spring out of the wicked imagination of the likes of Dominic Cummings and Michael Gove – £350m a week extra for the NHS and 70 million Turks destined for our shores.

Brexit: MPs to debate and vote on Brexit 'plan B' options on Tuesday 29 January - Politics live Read more

It’s right to fear how much further they might go in the case of a second referendum, always wrong-footing a painfully fact-based gentler remain campaign. I like the remain campaigners’ self-mocking and unshrill homemade placards on their upbeat Ode to Joy demonstrations: “Brexit – is it worth it?”, “This doesn’t seem very well thought through”, “Let’s call the whole thing off”, “CakeNotHate”,”I want to have my gateau and share it”, “I want my continent back”. Will the remainers be up to the bare-knuckle bloody fistfight of a bitter second round? Factual rebuttals are so much harder to explain than crude bare-faced lies. Can they ensure young fresh faces lead, and muzzle Tony Blair, whose Today programme contribution this morning was singularly unhelpful.

One placard outside Westminster on Tuesday night read: “Who’s afraid of a people’s vote?” The answer, of course, is the leavers. Why would it anger them so much if they didn’t fear losing? They say another vote would “disrespect” democracy. But what a curious argument to suggest that when all else fails in parliament, asking the people to decide between options not available last time is somehow undemocratic.

The long-time Eurosceptic Charles Moore writes in the Telegraph that Brexiters should boycott any such referendum: that can only be because they fear remain is on the rise. As for the first result being fixed in the firmament for all time, they never say for how long exactly. It will be three years since the last, so should it be five, 10, or “never, never, never!”, as old Ian Paisley Sr would be bellowing? Democracy is kinetic, on the move, as mercifully people do change their minds with changing circumstances – and populations change.

Quick guide Brexit and backstops: an explainer Show Hide A backstop is required to ensure there is no hard border in Ireland if a comprehensive free trade deal cannot be signed before the end of 2020. Theresa May has proposed to the EU that the whole of the UK would remain in the customs union after Brexit, but Brussels has said it needs more time to evaluate the proposal. As a result, the EU insists on having its own backstop - the backstop to the backstop - which would mean Northern Ireland would remain in the single market and customs union in the absence of a free trade deal, prompting fierce objections from Conservative hard Brexiters and the DUP, which props up her government. That prompted May to propose a country-wide alternative in which the whole of the UK would remain in parts of the customs union after Brexit. “The EU still requires a ‘backstop to the backstop’ – effectively an insurance policy for the insurance policy. And they want this to be the Northern Ireland-only solution that they had previously proposed,” May told MPs. Raising the stakes, the prime minister said the EU’s insistence amounted to a threat to the constitution of the UK: “We have been clear that we cannot agree to anything that threatens the integrity of our United Kingdom,” she added.

Theresa May’s inexplicable intransigence makes a referendum more likely as the only resolution left available. Reaching out? With her Paisleyite steel-trap mentality, she refuses to cross any of the fatal red lines that set her on the wrong course from the start: no customs union, no free movement, no European court to oversee agreements. She cleaves to her hard Brexiters, more afraid of splitting her party than sinking her country.

Jeremy Corbyn has rightly laid down just one essential red line: no no-deal Brexit. That’s the only fixed point that parliament agrees is the disaster to be averted at all costs, cabinet ministers ready to resign to stop it. But a soft Brexit where Britain obeys all rules but loses its voice and its veto over them doesn’t have his party’s support. The will of his party will before long oblige him to opt for a vote: his members overwhelmingly back one, including Momentum’s great heft. His voters are in favour, and they are mainly remainers.

It’s now or never for May. Time to compromise on Brexit | Martin Kettle Read more

Labour is caught between a small number of Labour leave voters, who are often swing voters in certain seats, and the overwhelming number of Labour remainers. Kellner warns that “it would be suicidal if Labour were to facilitate Brexit”, causing a stampede against it in all the metropolitan and university seats that are also Labour heartlands. Labour needs to win in both the Mansfields and its new turf in the Kensingtons and Canterburys. Brexit is an entirely Tory project: Labour must always be internationalist remainers. Corbyn and his close circle will find the only way out of the conundrum is to ask the people. Parliament will need to force this on May – and secretly, even if she doesn’t know it, it will be a relief to her, solving her dilemma too.

No one sensible bets on what happens next. Parliament must seize control in parliament when there is, in effect, no government. But logic – if logic were to prevail – suggests the only answer that will rescue both party leaders is to have a great national consultation and a referendum imposed upon them.

• Polly Toynbee is a Guardian columnist