But in his counteroffer, Obama had reversed the formulation so that the tax revenue figure — now at $1.16 trillion — would be the minimum that rewriting the code could achieve (a floor), rather than a maximum (a ceiling). With a slight turn of phrase, he rejected Boehner’s entire premise that growth could be counted on to deliver some of the revenue. Boehner could seek all the macro estimates he wanted if it made him feel better, but he wouldn’t be able to use those estimates to lower the amount of new tax revenue that Congress would need to collect.

White House aides would later insist that, despite their rough agreement on a framework the previous Sunday, the discussion about tax reform had always been fluid and unsettled, an ongoing negotiation in which both sides were still feeling out each other’s limits. The Gang of Six briefing had no doubt complicated this negotiation, they agreed, but it wasn’t as if they had signed on to something and then taken it back. If this is true, though, then it’s true only in the technical sense. If you shake hands with a guy on the price of a car, and you agree to talk again after the car has been inspected and the loan has been approved, you don’t really expect to show up and find out that car now costs $5,000 more. This is essentially what happened to Boehner. What both Tuesday’s panicky calls from the White House and the subsequent counteroffer make clear is that Obama knew he was changing the terms and felt he had no choice.

The other remaining area of contention had to do with the problem of enforcement provisions, or “triggers,” in the deal. Because tax reform would take some time for Congress to puzzle out, while the spending cuts were relatively straightforward, the White House had been concerned from the start about being double-crossed. How could Democrats be assured that the Republican-controlled House wouldn’t simply announce a deal, enact only the spending cuts they wanted and then sabotage the revenue piece? The answer, Obama’s team decided, were a couple of “triggers” — something both sides really hated — that would automatically kick in if they didn’t come up with a version of tax reform that each party could stomach.

Specifically, Obama had two triggers in mind. The first, for Democrats, would have rescinded the Bush tax cuts for the highest earners. Boehner rejected this idea. He pointed out that Democrats themselves would have little incentive to pass tax reform if, by not passing it, they could achieve one of their most cherished policy objectives — the elimination of the Bush tax cuts.

The second trigger, to appease Republicans, would include an automatic $425 billion in cuts to Medicare and Medicaid over 10 years. But Boehner repeatedly said that he wanted his own “political trophy” as a trigger, something that had the same resonance for the right as the Bush tax cuts had for the left — namely the elimination of the “individual mandate,” the central plank in Obama’s health-care law that required every American to be insured. Striking down the provision was a top priority for the Tea Partiers in Congress, who saw it as evidence of Obama’s tyrannical tendencies. Obama wouldn’t entertain the possibility. The argument had been going on since the first round of negotiations between the two men and their staffs, but now that a deal seemed imminent, the question of how to enforce it had taken on a new urgency. At its core, the trigger debate was a matter of trust; each man had to be assured that the other wasn’t going to let his party renege on the tax-reform agreement when the inevitable arguments arose. And because they hadn’t worked together much and barely knew each other on a personal level, the only way for Obama and Boehner to feel reassured was if the political cost of pulling out was intolerably high to both of them.

Obama and Boehner argued heatedly but respectfully over both sticking points — the revenue number and the triggers — during a two-hour meeting in the Oval Office on Wednesday, July 20. By the next morning, both men were facing rebellions on the Hill. The Times’s Carl Hulse and Jackie Calmes had written a front-page article disclosing the existence of the new round of talks and asserting that a deal was very near. Arriving for the weekly lunch of the Democratic Senate caucus, Jack Lew found himself berated by senators who were angered by the talk of entitlement cuts in exchange for the relatively paltry $800 billion in tax money, and livid at having heard about it from The Times. Senator Harry Reid, the majority leader, had been fully briefed (along with the House leader, Nancy Pelosi) only the night before. He remained stonily and pointedly silent in the meeting, while Lew absorbed one verbal blow after another.

At that very moment, Boehner was dialing Rush Limbaugh’s radio show, unbidden, in an effort to quell the eruption on the right. It wasn’t only the additional revenue that conservatives hated. Having campaigned in 2010 against Obama’s health-care plan, which included future Medicare cuts, conservatives in Congress were no more eager than Democrats to give the issue away in advance of 2012. (Their resistance to this part of the grand bargain highlighted what is perhaps the central paradox of budget politics on the right: Republicans have defined themselves almost entirely by their determination to reduce debt, but virtually every means of actually getting there — taxes, defense cuts, restructuring entitlements — strikes them as politically unpalatable.) “There is absolutely no deal,” Boehner assured Limbaugh on air.