John Boehner (AP Photo)

The numbers that matter in Washington are not the ones tossed around in discussions of debt ceilings or continuing resolutions.

The numbers that matter are found in the polls of public reaction to the ongoing government shutdown, and to the prospect that a bad circumstance could grow dramatically worse with the undermining of the “full faith and credit” of the federal government.

Those poll numbers explain why there has been at least some movement on the part of House Republicans—who engineered the shutdown as part of a scheme to derail implementation of the Affordable Care Act—to back down from their most hardline positions.

The latest data from NBC News and The Wall Street Journal is devastating for the GOP.

Fifty-three percent of Americans surveyed blame the GOP for the shutdown, while just 31 percent blame President Obama. Overall, approval ratings for the president are far better than those for the Republicans, and approval of the Affordable Care Act has spiked since the standoff began.

To the extent that this is a blame game, the Republicans are shouldering the burden. Indeed, their circumstance is far worse than in the shutdown of the mid-1990s. Then, only 44 percent of Americans blamed the Gingrich-led Republicans, while 33 percent blamed then-President Bill Clinton.

But this isn’t merely a blame game.

This is politics. And politics is a competition for power.

So the numbers that really matter are those that suggest there might have been some truth in the July 2013, observation by Congressman Tom Cole, the savvy senior Republican from Oklahoma, that “the only way Republicans will lose the House is to shut down the government or default on the debt.”

To wit:

Forty-seven percent of Americans surveyed for NBC now say they would like to see Democrats take charge of the US House of Representatives.

A mere 39 percent want the Republicans to retain control.

Some recent surveys suggest an even greater swing, with a Quinnipiac University National Poll putting Democrats up nine points.

In the fresh NBC poll, the lean toward the Democrats has increased five points over the past month, as the shutdown fight has evolved. In other words, as this fight has progressed, the Democratic position has improved at a rate of better than one percentage point per week. In politics, that’s a dramatic shift.

Make no mistake, Democrats need just such a shift to even begin talking about taking back control of the US House that they lost in 2010.

To achieve a majority in 2014, Democrats do not just need to win, they need to win big.

The political dynamic in the current House is defined by a combination of gerrymandering, a skewed first-past-the-post election system and big money influence. That dynamic favors the Republicans. Indeed, in 2012, Democrats won 1.7 million more votes than Republicans in House races across the country, yet the Republicans finished the election with a reasonably comfortable 234-201 majority. (With current vacancies, the split is now 232 Republican to 200 Democrats.)

To take the House in 2014—an off-year election where Republicans (as the opposition party to a president who is in his second term) begin not just with structural advantage but with what is traditionally seen as the political upper hand—Democrats would need a substantial shift.

The Democrats had a 48.3 to 46.9 advantage in 2012. But because most districts are skewed to favor the Republicans, the Democrats moved from 193 seats after the 2010 election to 201 seats after the 2012 election.

Since the district lines will remain essentially the same in 2014 as they were in 2012, Democrats must win not just the handful of districts that favored President Obama while sending a Republican to the House. They need to be competitive in districts that leaned Republican in 2012. That’s unlikely if they merely maintain the 1.4-percent advantage the party gained in 2012. But it could happen if the Democratic advantage moves dramatically upward.

There is a long time between now and the 2014 election. A lot can change. But if the Democrats could reap just half the advantage they now have in the NBC polling—for a 3.5 percent margin over the Republicans in next year’s House races—that would begin to put the most marginal Republican House races in play. And that, despite the extreme impact of gerrymandering, increases the sense of vulnerability among Republicans in more competitive districts.

Those Republicans are starting to feel the heat.

And if the numbers keep moving in the Democratic direction, those Republicans will begin—despite all the pressures from the Tea Party and its wealthy benefactors, despite the very real threat of primary challenges—to start angling for negotiations and compromise. They won’t make a “do-the-right-thing” argument. They will make a necessity argument based on a prospect that did not exist a few weeks ago, that a majority carefully constructed through meticulous gerrymandering might not hold in the face of broad revulsion at obstruction and extremism.

John Nichols calls the Ryan plan to end the budget showdown a “Shock Doctrine” fix.