Where are all these stories about a split between Jeremy Corbyn and John McDonnell coming from? In the week following the Budget, the newspapers had reports about divisions between the Labour leader and his long-time friend and ideological ally. The Times reported that the shadow chancellor was at odds with Jeremy Corbyn’s policy chief, Andrew Fisher, over the party’s plans to match the Conservative tax cut for people earning £45,000 and above. The Financial Times reported that McDonnell was making announcements in interviews without clearing them via the leader’s office. The Sunday Times went further, reporting that Corbyn’s son Seb had been insultingly demoted from his post as chief of staff to McDonnell.

The Labour leader’s inner circle think they know the source of these stories: those aides are have either just left, or are about to depart the Corbyn project. One frontbencher joked that a recent escapee was “on a list of people to be hunted down with an ice pick”. (References to the assassination of Leon Trotsky are never far away in Labour circles.) This is hotly denied by the departed.

The more important question is this: are the stories true? Yes, the Labour leadership (and indeed the wider party) is divided, because of three decisions made before the last general election. The first is, of course, the pledge to match the Conservatives’ planned tax cuts. The second is the pledge to abolish tuition fees. The third is the party’s fiscal rule, which commits Labour to run a day-to-day budget surplus, although it can borrow for infrastructure projects.

No one who matters in today’s Labour Party wants to overturn the tuition fee pledge; but many privately chafe at its £11.2bn cost. It was the single most expensive manifesto commitment in the 2017 programme, and it ate up almost all the new revenues generated by the tax hikes for people earning £80,000 and above, according to Labour’s own figures.

That means there would be little money left over to pause or undo the ongoing move from multiple benefits to Universal Credit, a policy which has caused so much pain that Philip Hammond was forced to act in his Budget. Labour has long flirted with scrapping the Universal Credit rollout; however, it is hard to see where it would get the money to do so. Nonetheless, the abolition of tuition fees is safe: it is both popular and totemic. Staffers might grumble about it but the disquiet is not serious.

Yes, there are, too, occasional squalls over media strategy. It is the business of successful politicians to try to turn themselves into legends, and McDonnell has worked hard to present himself as a “sensible bank manager” (I’m told he initially favoured the term “sensible bureaucrat” before being told that bureaucrats are too unpopular). Staffers in the leader’s office are known to roll their eyes about the shadow treasury team’s hyperactivity, with one of them recently telling me: “I love these interviews. Sometimes I learn about them on TV.”

Part of McDonnell’s routine involves talking about how he failed to win his seat, Hayes and Harlington, in 1992, and how it shaped his thinking on taxation and spending. That narrative got another airing on BBC Newsnight on 5 November, and is often a feature of stories about divisions between the shadow chancellor and his old friend.

However, the story owes more to McDonnell’s myth-making than to the observable facts. During the Blair years, he frequently criticised New Labour’s timidity, and he accepted the job of shadow chancellor in 2015 knowing that Corbyn had pledged to increase the basic rate of income tax.

What about Seb Corbyn’s apparent demotion? Here, it should be remembered that the stories of Jeremy Corbyn’s courtesy and kindness are true: he will often make tea for even the most transitory of interns. Successive Islington organisers have plenty of euphemisms – “Corbyn Mean Time”, “the Jeremy tax” – for the amount of time he will take to find out about the health of each and every person he meets when knocking on doors. It’s true, too, that his sense of loyalty to those who have stuck by him has protected more than one underperforming shadow cabinet minister from the sack.

But – and this is important – Corbyn’s kindness isn’t the same as softness or stupidity. His image as a cross between the Werther’s Original Grandpa and Clement Attlee obscures a streak of ruthlessness. When he feels that people have been disloyal, he is perfectly capable of action: just ask any of the shadow cabinet ministers he has sacked. Any deliberately unfavourable treatment of his son would result in public and private conflict, no matter how long he had known the person who had done it. The truth is that McDonnell’s office expanded and there was a reorganisation. Demoting Corbyn’s son would have been provocative. As for the Sunday Times story, one shadow minister described briefing against Seb Corbyn as a “dangerous button to press, whoever did it”.

The reality is that McDonnell is not haunted by his defeat in 1992 and Corbyn is not a soft touch. Any stories that take these assumptions as their starting point can be safely dismissed, even if the protagonists do not contest them. After all, the image of one as a cautious bank manager and the other as an avuncular grandad might yet get them both into office.