Today, the government publishes its Brexit plan, two long years too late. The Chequers plan in this white paper should be a Magna Carta to lay out the country’s economic future and the destiny of Britain’s place in the world. In practice this poor specimen is dead on arrival. Never has there been a white paper of such profound potential importance and simultaneous absolute insignificance.

No battle plan survives first contact with the enemy, said Helmuth von Moltke the Elder, Prussian chief of staff; this plan has so many enemies it should never have made it past the drawing board. Least threatening to it are the hard Brexiters putting down killer amendments to kill off its central idea, Theresa May’s facilitated customs arrangement (FCA).

If their mistaken aim is a show of strength, their amendments’ failure will diminish their power, revealing possibly only some 30 or so backers. A useful display of their weakness will help remind everyone – broadcasters take note – that the likes of Jacob Rees Mogg, Boris Johnson, Owen Paterson and Priti Patel are extreme mavericks: telegenic for their eccentricities, vanities and foibles, but a sideshow to the existential crisis facing the nation. (As for Nigel Farage, there is no political excuse for giving a man of no standing twice the airtime of, say, Sir Vince Cable, who leads actual MPs.)

The EU is not the enemy, but our closest friend: the Nato summit with Trump should have shocked us into remembering who our real friends are. But not even May expects her plan to survive first encounter with the EU27, who will reject her FCA. In evidence yesterday to the international trade select committee, Liam Fox admitted another obstacle: even the World Trade Organisation may reject the legality of the FCA plan. May will be pushed to further softening compromises, inching towards a Norway-lite solution.

But what then? She has eventually brought to parliament a final Brexit offering that still pleases no one: far too soft for the fanatics yet still too menacing to the country’s good to command enough MPs’ support. Whatever new damage limitation it attempts, the deal will still leave us as rule-takers, budget contributors, with no MEPs and no seat on EU councils, brutally exposing how much we lose compared with what we have now.

What then? Gridlock. The head-counters see no majority in the Commons for accepting the deal: Labour and other parties will vote against, along with the Moggites. That will be a painful decision for some remainers, fearing rejection of a Norway-type deal will lead to something even worse. But they will still vote against, because the deal is still so bad for Britain. Barring the madmen, there will also be absolutely no majority in the House for an economy-killing no-deal crash-out. Then what?

This will be an unprecedented constitutional crisis, a logjam, an impasse with the deadline timebomb ticking. Labour will clamour for a general election, but why would a government – even this fractious ragbag – vote for no confidence in itself? They can throw out their leader, but that solves nothing. Do they go for a moderate compromiser – in which case, why not hold on to the one they have – or a Brexit ideologue, when there would still be no majority for either the dead-in-the-water deal or for crashing out?

This is the point where the only majority might be for asking the people to break the deadlock in a new referendum, as William Hague advocates and Labour doesn’t rule out.

The Brexiters will protest that the will of the people has already been fixed for all time, but the mood in the country suggests people are already sick of the whole shebang. However, polling shows a majority would back a vote on the final deal – (48% compared with 25% against the idea (asked if they want a vote, people tend to say yes on most issues). Polling numbers point increasingly towards a victory for remain, by some five percentage points: some people are changing their mind, but more of the pro-EU young are reaching voting age as old Brexit-voters drop off the perch. Though once another virulently mendacious “take back control” ,“80m Turks are coming” campaign gets going, backed by 80% of our press, dubious funding and subterranean foreign intervention, the result is unknowable.

Don’t imagine deciding on a people’s vote is a simple matter either. Parliament will choose the crucial wording of the question – and what will a gridlocked Commons choose? Question one – should the UK accept the deal or remain in the EU? The Brexiters would scream betrayal and demand instead question two – accept the deal or crash out without one?

Others suggest a three-way question, with all three options, which sounds fair. Except that offers the lethal likelihood that the moderate vote splits between remainers and deal-accepters, letting a minority of crash-outers win. So not even the people’s vote is a simple answer to this poisoned constitutional conundrum. No one knows what comes next – and the time is short.

• Polly Toynbee is a Guardian columnist