There’s a letter. That’s what I was told back in May. Dominic Cummings, the brilliant but unorthodox chief strategist of Vote Leave once called “a career psychopath” by David Cameron, had told people that the Electoral Commission had given Vote Leave permission to donate to other campaigns – and it had it in writing.

Those donations are now the subject of a new investigation by the Electoral Commission. They came in the form of a gift of £625,000 to a fashion student, Darren Grimes, and £100,000 given to an anti-EU pressure group, Veterans for Britain. Both used the funds during the weeks running up to last year’s referendum to campaign for the UK to leave the European Union. The commission now concedes that these donations should be investigated – did Vote Leave make them to avoid exceeding strict campaign spending limits determined by law?

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And the letter? The letter that Cummings said granted them permission? It doesn’t appear to exist – though it took me and Paul-Olivier Dehaye, a data researcher based in Switzerland, six months to establish this through freedom of information (FoI) requests.

In September, five months after the first one was filed, as I was appealing against the commission’s decision that “in some circumstances we cannot responsibly release requested information”, Cummings broke cover on Twitter.

Q&A Can the Electoral Commission inquiry undermine Brexit? Show Hide Campaigners on either side of the EU debate will have differing opinions on whether any adverse finding on spending calls the result of the referendum into question. However, Jo Maugham, the barrister behind the Good Law Project, said any illegal spending would undermine the result, adding to concerns about Russian meddling and Brexiters failing to deliver promises they made during the campaign. The commission can impose a fine of up to £20,000 for individual rule breaches. Any evidence of more serious offences uncovered in the course of the investigation can be referred to the police if warranted.

“FYI u seem unaware (not blaming u no reason u wd know) of a crucial fact: the EC gave us written permission in advance for what we did… ” he told Jolyon Maugham, the head of the campaigning group the Good Law Project, in response to his announcement that he would be seeking a judicial review of the Electoral Commission’s decision not to investigate these donations. Cummings added: “When they suddenly told us we cd make donations we were so shocked we asked for written confirmation & got it. Extremely surprising…” And finally: “so it wd be pointless them having an investigation/reporting us to cops cos wed [sic] just hand over their own letter giving permission. Hope helpful.”

It was. I FoI-ed the commission again. Cummings says there’s a letter, I wrote. And finally, on 27 October, I got the definitive answer: the commission had reviewed all communications between it and Cummings/Vote Leave. And “we can’t find any record of any exchange with us on the subject of donations between them from that period”.

So there is no letter: that’s what the Electoral Commission says. Either it’s mistaken, or Cummings – the man credited with pulling off the greatest campaign triumph of a generation, who gave Britain Brexit – is in the uncomfortable position of appearing to have been rather economical with the truth. Has he? And if so, what does that mean for the campaign? Because this was the official campaign, sanctioned by the Electoral Commission, which received £600,000 of taxpayers’ money and is now facing an increasing number of troubling and unanswered questions, questions that the commission has for months refused to ask. And some of which it is still refusing to ask.

The new investigation – which the Electoral Commission denies is prompted by the Good Law Project’s judicial review – focuses on the money Vote Leave gave to 22-year-old Grimes, and to Veterans for Britain. They purported to be completely separate campaigns, in which case such donations would have been allowable. But it is Cummings, the campaign’s mastermind, who belongs in the hot seat, a place it turns out he’s been in before.

Because Cummings is not a newcomer to the dark arts. During his time as a special adviser at the Department of Education, he was described by one commentator as “the paramilitary wing of Michael Gove”. And working with another adviser, Henry de Zoete – who also took a senior role in Vote Leave – he left a trail of headlines: the time he was sanctioned by the Information Commissioner’s Office for failing to respond to FoI requests; the time an email was leaked in which he said he used Gmail rather than a departmental address expressly to avoid such requests; and most notoriously, the time the Observer revealed he and De Zoete were associated with an anonymous Twitter account, @toryeducation, that savagely attacked anyone who dared criticise Michael Gove or his policies.

Its operation eventually led to questions in parliament. Guardian columnist Suzanne Moore was “a liar”, Kevin Brennan, a Labour MP, “chubby”, and Tim Loughton, a Tory MP, a “lazy incompetent narcissist”.

“What is unquestionably true is that Cummings and De Zoete were heavily involved with the account,” claims a former political correspondent. “It was a shitbagging account of anyone who opposed Michael Gove run from within the Department of Education. What’s interesting, looking back, is that it was ahead of its time. Cummings has always been very, very clever about knowing how to operate on this line between what is acceptable and what is not.”

A line that, in terms of our electoral law, is increasingly impossible to police. The technology platforms have usurped a hundred years of legislation designed to ensure free and fair elections – a situation that Cummings acknowledged in an email to me: “The law/regulatory agencies are such a joke the reality is that anybody who wanted to cheat the law could do it easily without people realising.”

Facebook Twitter Pinterest Darren Grimes the student who received a £625,000 gift from Vote Leave.

A 133,000-word essay on education that Cummings wrote proved to be a key incident in the story of the referendum campaign. The essay led to an invitation to Sci Foo, an exclusive invite-only ideas conference hosted by Larry Page, the co-founder of Google, at the Googleplex in August 2014.

And this led him to meet certain people, including Steve Hsu, a physicist at Michigan State University he quotes liberally in his essays, blogs and Twitter account. Hsu’s research into IQ and intelligence, and in particular his project at a state-run genomics lab in China, has troubled some people with what Daniel HoSang, a professor at the University of Oregon, describes as its “Eugenicist overtones”. But in his blog, Hsu writes how Cummings encouraged him to study Bismarck and he urged others to take note of Cummings’s work in politics.

Cummings came away more convinced than ever that it was scientists doing fundamental research who held the key to the future, not politicians. And when it came to building a team for the referendum, his first action was to “hire physicists”. Reading the thousands of words that Cummings has blogged on the subject, it’s clear his political philosophy lies at the intersection of history and physics; his intellectual spirit guides are Otto von Bismarck, the Prussian statesman who united Germany, and Richard Feynman, the theoretical physicist and guru of quantum mechanics.

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In Cummings’s view, a political system is the same as any other complex system, subject to non-linear outcomes in which some relatively trivial event can cast a huge shadow. An event that sets off a cascade of “branching histories”. The country takes one course when it could have taken another.

Britain could have voted yes to Brexit. Or it could have voted no. In the end, he writes in his blog, it came down to 600,000 people. The “cold reality of the referendum is no clear story, no ‘one big causal factor’… if about 600,000 people – just over 1% of registered voters – had decided differently, IN [Remain] would have won.” It is, he says, “a small enough margin” that if “a few specific events and decisions” had been different, everything could have been different.

A small enough margin that £775,000 – more than one tenth of the campaign’s total budget – could have made a difference? Money that paid for a firehose of Facebook ads in the last days of the campaign. In his summary of where and how Vote Leave spent its money, Cummings writes that 98% of its budget went online on ads that received nearly a billion impressions. This was where the battle for Brexit was won: on the internet.

Many people, of course, think differently. The fundamental factors that led to Brexit (and Trump) – rising inequality, frustration with elites, economic uncertainty – are complex, multilayered and years, if not decades, in the making. But it’s certainly true that all campaigns look for the winnable gains, the tiny sliver of people who in the US election were swing voters in swing states and in Britain were “persuadable” – either in terms of who to vote for, or whether to vote at all.

In its official application to the Electoral Commission, the campaign noted: “While Vote Leave plans to invest significant time and resources into online activity on networks such as Facebook and Twitter, we do not regard them as appropriate primary platforms for this historic national debate.”

Appropriate or not, it’s where it occurred. Last week, Emily Las, a digital marketer in New York, found a handful of what appear to be Vote Leave “dark ads”, a tiny sample of the thousands targeted at individuals and until now invisible and unrecorded. They included a video that claimed “Turkey is joining the EU. Our schools and hospitals already can’t cope.”

Facebook Twitter Pinterest One of Vote Leave’s Facebook “dark posts” uncovered last week. Photograph: Facebook

A video that was viewed 515,000 times and that simply wasn’t in any way true. Because this is something else that Cummings and the campaign’s chief executive, Matthew Elliott, had learned along the way. They had worked together on the 2011 No to Alternative Vote campaign, and as the Financial Times noted in an interview with Cummings: “That campaign made exaggerated emotive claims [and] also recognised that the Advertising Standards Authority had no power to regulate political ads, however misleading.”

One of Cummings’s tweets on the Electoral Commission’s letter. Photograph: Twitter

In Tim Shipman’s book All Out War, the Conservative MEP Daniel Hannan describes Cummings’s physicists as “a group of west coast American academics” who were signed up “in great secrecy” from “disciplines like astrophysics”. They “found this way of scraping data off people’s Google searches and feeding it into a program to tell you, by postcode, where your voters were.”

Who were the physicists? It remains a mystery. The Electoral Commission can’t know because they’re not listed on any of Vote Leave’s spending returns. There are some invoices to a company Vote Leave calls “Advanced Skills Initiative”, otherwise known as ASI – a total of £72,018.50 – but this is for “advertising” and “market research/canvassing”. So who did the data modelling? Who were the secret “west coast academics”? And where’s the bill for their services? Was this a “gift”? And if so, was it “permissible”?

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None of these questions form part of the new investigation. Nor do they address who knew what, or how high up this goes. “You cannot overestimate how loyal Gove is to Cummings and Cummings is to Gove,” says one former political correspondent. “They are joined at the hip.” Did Gove know? Did Stephen Parkinson, now Theresa May’s most senior political aide but previously Vote Leave’s ground organiser? Did Cleo Watson, another senior Vote Leave director and now another May aide? And what of the Democratic Unionist party? It holds the balance of power in Westminster – but no questions about its role in the referendum are yet being officially asked. Where did the £435,000 it spent came from? What does May know? Or is she not asking?

Cummings, Elliott and Gisela Stuart, Vote Leave’s chair, did not respond to any of the Observer’s requests for a response to this article, but Vote Leave said: “As the Electoral Commission has previously said, Vote Leave complied with the rules on making donations to other campaigns. We will fully cooperate with the commission’s investigation, however as this process is ongoing it would be inappropriate to comment further.’

The launch of last week’s investigation is a landmark. Especially in the light of two other ongoing investigations into Leave.EU and Arron Banks, the entrepreneur who funded it (both deny any wrongdoing). The surface has been scratched. It’s what lies beneath that should concern us. What still lies buried. What isn’t being examined. And why it’s not. Whose decision is that?

Brexit is a non-linear event. The history of our nation was set in one direction, when it could have gone another. Cummings deleted his Twitter account in October but before he did, he brought his brilliant strategic mind to bear on it: “I said before REF was dumb idea, other things shdve [sic] been tried 1st. In some possible branches of the future leaving will be an error.”