Now the battle of interpretation begins, and to the naked ear the next 24 hours in Westminster might sound a lot like an argument about international law. In fact, the debate over the “legally binding changes” that Theresa May announced, her voice hoarse with exhaustion, as the Strasbourg clocks chimed midnight will not, ultimately, be a legal one. For all the talk of “instruments” and “protocols” and “arbitration panels”, this will come down to politics.

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All eyes will, naturally, be on the Brexiters who voted down May’s deal in January and whose minds she needs to change before the second meaningful vote comes before the House of Commons on Tuesday (unless of course it gets postponed again – always a possibility with this government). Will they find enough in the new language May extracted from the EU to switch sides and endorse her withdrawal agreement? The early signals were mixed, even from within the hardline European Research Group: the faction’s leader, Jacob Rees-Mogg, made emollient noises about steps in the right direction; his deputy, Steve Baker, said May’s concessions fell short.

On the face of it, Baker’s view seems the more compelling. Think of what the ERG and the Democratic Unionists object to about the key stumbling block: the Northern Irish backstop, the insurance policy designed to avoid a hard border on the island of Ireland. They don’t like the fact that it has no time limit, that it could, theoretically, go on forever. And yet the best that May’s new motion laid before parliament could say is that the new legally binding joint instrument “reduces the risk that the UK could be held in the Northern Ireland backstop indefinitely”. “Reduces the risk” is not the same as “eliminates the risk” – and it’s that that many of those Brexiters wanted to hear. (Put aside the fact that it was always an unrealistic demand: you could say the same about the entire case for Brexit.)

A second demand of the Brexiters, one bizarrely endorsed in January by May herself and a majority of the Commons, was that the backstop be replaced by “alternative arrangements.” Gamely, May tried to pretend that she’d won an EU concession on that too, and that those alternative arrangements will be in place by December 2020. As indeed they will – if they exist by then. But for now, the technological wizardry so great that it would render the backstop redundant does not exist. And so this was another hollow victory.

Finally, the Brexit crowd wanted the UK to have the unilateral right to exit the backstop whenever it liked. May did indeed get something unilateral – the right to issue her own unilateral declaration, in which she could freely state that “it is the position of the United Kingdom that there would be nothing to prevent the UK instigating measures that would ultimately dis-apply the backstop.” This is rather like my son winning the right to declare that it is his position that he should get more pocket money. It doesn’t mean I’ve agreed to give him more pocket money. The clue is in the word “unilateral.” The EU is not bound by this UK declaration and has, in fact, conceded nothing.

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There will be others who take a different, rosier view. But the opinion that will matter most on Tuesday will be that delivered by the attorney general, Geoffrey Cox. It’s said that he plans to issue a genuine legal judgement, rather than acting simply as a May loyalist delivering for his boss.

Even so, few MPs will cast their votes on a pure assessment of the legal merits. Rather they will make a calculation as to whether Cox has given them sufficient cover to make whatever move they want to make – either to back or to oppose May’s deal. In other words, what will settle this is what was always going to settle it: politics.

• Jonathan Freedland is a Guardian columnist