The back-and-forth of the debt limit negotiations can be perplexing. But the facts driving the debate are quite straightforward. Any deal, or any workaround to a deal, will need to be approved by a majority of the House of Representatives. The Republicans in the House of Representatives are extremely conservative on fiscal matters and are significantly out of step with the public as a whole. Given that Democrats continue to control the Senate and the White House, and that their votes may be necessary in the House as well, this makes compromise nearly impossible.

Consider, for example, the poll that Gallup released today, which asked Americans whether they prefer tax increases or spending cuts as part of a deal. Preferences were weighted toward spending cuts:

Few Americans, however, take the view that spending cuts alone should be made in a deal, with no tax increases at all. In fact, only 26 percent of the Republican voters surveyed in Gallup’s poll took that position, along with 20 percent of voters overall.

We can also use the Gallup poll to tease out what mix of tax increases and spending cuts the public would like to see in a deal. Assume that the people who told Gallup that they wanted “mostly” cuts would prefer a 3-to-1 ratio of spending reductions to tax increases, and that those who said they wanted mostly tax increases would prefer a 3-to-1 ratio in the opposite direction. (The other choices that Gallup provided in the poll — an equal mix of tax increases and spending cuts or a deal that consisted entirely of one or the other — are straightforward to interpret.)

The average Republican voter, based on this data, wants a mix of 26 percent tax increases to 74 percent spending cuts. The average independent voter prefers a 34-to-66 mix, while the average Democratic voter wants a 46-to-54 mix:

Now consider the positions of the respective parties to the negotiation. One framework that President Obama has offered, which would reduce the debt by a reported $2 trillion, contains a mix of about 17 percent tax increases to 83 percent spending cuts. Another framework, which would aim for twice the debt reduction, has been variously reported as offering a 20-to-80 or 25-to-75 mix.

With the important caveat that the accounting on both the spending and tax sides can get tricky, this seems like an awfully good deal for Republicans. Much to the chagrin of many Democrats, the mix of spending cuts and tax increases that Mr. Obama is offering is quite close to, or perhaps even a little to the right of, what the average Republican voter wants, let alone the average American.

However, all but 7 Republicans in the House of Representatives, or 97 percent of them, have signed the pledge of Grover Norquist’s Americans for Tax Reform stating that any net tax increases are unacceptable. One might have believed this to be simply a negotiating position. But the proposal that Senate Republican leader Mitch McConnell floated yesterday, which would give up on striking a deal and instead rely on some procedural gymnastics to burden Mr. Obama with having to raise the debt ceiling, suggests otherwise. Republicans in the House really may be of the view that a deal with a 3:1 or 4:1 or 5:1 ratio of spending cuts to tax increases is worse than none at all.

If we do take the Republicans’ no-new-taxes position literally, it isn’t surprising that the negotiations have broken down. Consider that, according to the Gallup poll, Republican voters want the deal to consist of 26 percent tax increases, and Democratic voters 46 percent — a gap of 20 percentage points. If Republicans in the House insist upon zero tax increases, there is a larger ideological gap between House Republicans and Republican voters than there is between Republican voters and Democratic ones.

It would be foolish, in my view, to render any overly specific predictions about how the negotiations are likely to be resolved. But I would put greater weight on scenarios that would involve House Republicans not having to violate the pledge they signed to Mr. Norquist, including an end-around like Mr. McConnell’s — or even a failure to raise the debt ceiling at all, resulting in some combination of a debt default, a government shutdown, and a Constitutional crisis.

If a deal were to occur, it would probably have to be revenue-neutral. There is still some possibility of this because there are some tax reductions that many Democrats in Congress would support on their own merits — for example, cuts to the payroll tax, which could have a stimulative effect, in exchange for the closing of tax loopholes elsewhere.

But the House Republicans are very unlikely to capitulate on their no-tax pledge. And Democrats have little reason to capitulate either: they are on the right side of public opinion.