Those who switched off with a sigh of relief in July may not have noticed. But something big is slowly stirring in the undergrowth of British politics. Fact by fact, announcement by announcement, the case for Britain to remain in the European Union’s single market and customs union is growing stronger and more irresistible by the day. Such an outcome is most definitely not this government’s policy. But, this autumn, something will have to give.

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Over the past 10 days David Davis’s Brexit department has published seven so-called partnership papers: important documents covering a wide range of subjects, from customs and Northern Ireland to civil justice and, most recently, disputes mechanisms, including the role of the European Court of Justice. According to the introductory blurb inside each, these papers are all about forming a bespoke post-Brexit partnership with the EU. Yet, by intention or accident, they do something very different. Together they make a case for sticking with the existing partnership as it stands, or at least with its key arrangements, such as the single market and customs union.

In every case, the papers start from the reality of the Brexit vote and then gently proceed to undermine it. None makes the case that Britain should turn its back on the EU, as the Brexiteers would like. None heads off into the fantasy world in which nations, dazzled by British exceptionalism, queue up to make bilateral deals with Liam Fox. Instead, all seek to retain large parts of the cooperation and openness that Europe has given this country. The trajectory has shifted from go-it-alone – sometimes unbelievably so, as in the dogged refusal to recognise that the commitments to leaving the EU and maintaining an open border in Ireland are almost impossible to combine – towards the status quo.

None of them quite say this, of course. The politics of the Tory party do not permit it. The relevant ministers continue to believe that Britain can have its cake and eat it. But as this week’s U-turn on the role of the ECJ exemplified, the hard Brexit policy that has sustained Theresa May for a year is now visibly crumbling around her.

Facebook Twitter Pinterest Liam Fox, the trade secretary, has a ‘fantasy world in which nations, dazzled by British exceptionalism, queue up to make bilateral deals with him’. Photograph: Paul J Richards/AFP/Getty Images

Today’s announcements about immigration are not, in a formal sense, part of this slow crescendo of government pragmatism towards Europe. In a political sense, however, they are cut from precisely the same cloth, not least in the way they brutally undermine some of May’s most trenchantly repeated mantras about Brexit – that the 2016 vote was centrally about a genuine migration problem, that migration can be slashed to the tens of thousands to the national good, and that foreign students must be treated as part of the migration issue.

Amber Rudd’s blizzard of announcements and statistical documents today take a lot of this ground from under May’s feet. The prime minister may be tempted to say that the fall in net migration, down 81,000 from the 327,000 recorded in 2016, shows that Brexit is already clearing out foreigners who either are, or feel themselves to be, unwanted.

More thoughtful people, however, see that the departed have taken significant skills, spending, taxes and goodwill with them. More departures and failures are likely, especially with the status of EU migrants after Brexit remaining unclear. The impact is already being felt in the growth of the eurozone: double that of the UK economy in both the first and second quarters of 2017.

Rudd’s paper assessing the border exit checks programme is also subversive. Along with HMRC statistics on tax credit claims, also published today, the documents confront several of the myths and exaggerations that fire the immigration argument. Once again, the effect is to reduce May’s authority by undermining claims she has made her own.

While the HMRC figures mock the idea that benefit tourism is a big problem (less than 2% of tax credit spending goes to recently arrived Europeans), the Rudd paper shows that at least 96% of the 1.34 million people to enter the UK on visas expiring in 2016-17 left before the expiry date. The share is more than 97% in the case of those on study visas. In other words, May’s insistence that students be treated like all other migrants in the figures does not hold water. All but a handful of students come to study, and then leave. Counting them as migrants simply inflates the figures and causes public anxiety.

That such public anxiety exists is beyond question. This reality shapes the way that many in both the Conservatives and Labour have responded to the Brexit vote. However, the mass of evidence that is now being published ought to permit both parties to look again, with less panic and more thought, at the anxiety over immigration, at its context and drivers, and at what can be done about it.

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May is in danger of being trapped by this. She has treated the Brexit vote as a vote on migration. But was it? Voters were not asked about migration. They were asked whether or not they wanted to be in the EU. Migration was indisputably part of that, being especially prominent in the campaign waged by Arron Banks and Nigel Farage. But the vote was about many things: and in the end the only thing about which May or anyone else can be certain is that it was a vote to leave.

Opinion polls suggest, by and large, that the majority to leave is still there. But some things have changed since June 2016. False claims, like the NHS dividend and the total collapse of the economy, have been discredited. May and Ukip have both had a bad year. And the case for engagement with the EU after Brexit now looks much more solid than the fantasies about global Britain.

The logic of this summer’s developments, including David Davis’s papers and Amber Rudd’s figures, points clearly to the possibility that Britain could embrace the Norway option of single market and customs union membership while standing clearly outside the EU. This is a question that both Labour and the Tories are going to be forced to wrestle with in the coming weeks. Neither of them is particularly well prepared for the exercise. But the outcome will decide whether Theresa May has a future.