On Thursday morning, Trump tweeted that Cohen “received a monthly retainer, not from the campaign and having nothing to do with the campaign … used to stop the false and extortionist accusations made by her about an affair despite already having signed a detailed letter admitting that there was no affair.”

“Money from the campaign, or campaign contributions, played no roll [sic] in this transaction,” the president insisted.

But Giuliani quickly contradicted that explanation in an interview with Fox and Friends Thursday morning, indicating that the payment to Daniels was meant to prevent damaging information from emerging in the latter days of the 2016 campaign. “Imagine if that came out on October 15, 2016, in the middle of the last debate with Hillary Clinton,” Giuliani said. “Cohen didn’t even ask. He made it go away. He did his job.”

That statement, legal experts said, appears to confirm that the payment was a campaign expenditure. “This is good circumstantial evidence this was campaign-related,” said Rick Hasen, a law professor at the University of California, Irvine. “Giuliani did Trump no favors.”

Fischer agreed. “Giuliani's reference to the October campaign debate makes the electoral purpose of this payment more clear than it already was, and it already was pretty clear,” Fischer said. “It is hard to argue that this payment was not intended to influence the 2016 election, and it should have been reported."

Cohen’s payment to Daniels could constitute an illegal over-the-limit contribution, as federal election law caps individual contributions at $2,700 per election. Giuliani’s admission could also make Trump’s reimbursement of Cohen an unreported campaign expenditure, along with the legal payments to Cohen for legal work Trump acknowledged making Thursday morning.

Convictions for campaign-finance violations, like those for corruption, have become notoriously hard to secure. The Federal Election Commission, which has the power to fine campaigns for violating the law, currently only has four of its six commissioners, two Republicans, one Democrat, and one independent, and any enforcement action would require their unanimous support.

Then there’s the ghost of the John Edwards case, which would loom large over any potential prosecution effort. The Justice Department dropped its case against the former Democratic presidential candidate over more than $1 million in payments made to his pregnant lover during the 2008 campaign, after failing to win a conviction. Prosecutors said the payments, financed by wealthy Edwards backers, were campaign contributions, meant to hide potentially damaging news of an affair from the press. Edwards’s attorneys argued that the payments were made to hide the affair from his wife, who was ill with cancer and died in 2010, not to improve his political chances.