Everything’s coming up Corbyn: today is turning out to be quite a good day for the Labour leader’s preferred Brexit outcome, though the man himself can take little credit for either development.

The first is that Conservative Remainers are going off the idea of voting for the customs union on Wednesday. As I explained in greater detail here, Jeremy Corbyn will only face significant pressure to change Labour’s stance on the European Economic Area once the number of Tory rebels exceeds 25: the number you would need to have any realistic prospect of exceeding the number of Labour MPs who are certain to vote to take the United Kingdom out of the EEA come what may. Up until that point, Team Corbyn can defend their preferred Brexit outcome – one in which they are freed from a significant number of the European Union’s rules – simply on mathematical grounds: there aren’t the votes to sustain any other alternative.

That’s been further boosted by an amendment laid down by Yvette Cooper and Hilary Benn, to take the United Kingdom in to the EEA “with appropriate safeguard measures” on the free movement of people.

This amendment has some very important things in common with the Labour leadership’s internal market amendment, which would mandate the government to seek a deal that makes the United Kingdom members of the European Union’s “internal market” with shared institutions and rules.

The most important area of commonality is that both are dead on arrival with the European Union. To take the Labour leadership’s internal market idea first: the European Union is not going to negotiate a bespoke set of rule-making institutions for one nation. It has resisted doing so with Switzerland and will resist doing so with the United Kingdom as well. Something like the shared market could work as a set of proposals to reform the European Union’s perimeter nations – the nations of the EEA plus the United Kingdom – but as a solution to the British withdrawal negotiations, they’re nothing doing.

As for the idea of the EEA “with safeguards”, the idea that you can use Articles 112 and 113 as an “emergency brake” from the free movement of people has spread like wildfire in parts of the parliamentary Labour party. But the problem is, while there is no doubt in my mind that this belief is sincere, it doesn’t make it correct. Articles 112-3 don’t work as an emergency brake because the United Kingdom is not suffering from serious environmental, economic or societal difficulties as a result of migration from the European Union. (If anything, quite the reverse.)

Labour whips will make this argument and they will be bolstered by the fact that every non-partisan wonk will agree with them. But the amendment hands the Labour leadership another weapon to argue against the EEA, which is that even some of its advocates don’t like it in its current form and that their proposed changes wouldn’t fly.

What both Labour’s internal market and “the EEA with safeguards” idea are trying to grapple with (with varying degrees of sincerity) is the problem that to the extent that the vote to leave the European Union related to any specific institutional qualities of the European Union it was opposition to the free movement of people. But if you don’t have free movement of people, you don’t have single market membership which means you have fairly drastic breach from the European Union.

At the moment, the ascendant faction in the Labour leader’s office doesn’t care about free movement but they are happy to accept ending it as the price of the wider relationship they want with the European Union’s institutions. The majority opinion in the parliamentary Labour party wants a close relationship with the European Union but without free movement, which is impossible to achieve.

The only way for British pro-Europeans to get the relationship they seek is to win the argument here in the United Kingdom for the free movement of people within the EU: otherwise, the only flavour of Brexit capable of winning parliamentary and public support will continue to be a hard one.