For Mitt Romney, it was the number that proved everything. Since the very first speech of his campaign, the Republican candidate has used a simple figure to bolster his argument that President Obama couldn’t fix the U.S. economy: 8 percent.

In this campaign, begun in the midst of a staggering downturn, monthly unemployment reports have been a running scorecard. They distill a vast and complicated economy down to terms simple enough for a stump speech: a number and a direction, up or down.

For Romney, any number above 8 percent proved he was right and Obama was wrong.

Obama had promised, Romney told audiences repeatedly, never to let unemployment get that high. Instead, Romney said, the jobless rate blew past 8 percent and got stuck there.

Until Friday.

The 0.3 percent dip in unemployment in September, from 8.1 to 7.8 percent, deprived Romney of one of his central campaign themes.

It was enough to put him on the defensive just as he was basking in the afterglow of his debate performance Wednesday, the best moment of his campaign against Obama so far. It wasn’t because the figures showed a healthy economy — they didn’t — but because the economy had crossed a threshold that Romney had implied it would never cross without him.

“We can do better,” Romney said Friday at a rally in the Virginia coal-country town of Abingdon. It was the same argument he has used throughout the campaign, but without the number he’d always used to hammer it home. “There were fewer new jobs created this month than last month. And the unemployment rate . . . has come down very, very slowly, but it’s come down nonetheless.”

The political importance of the 8 percent threshold was driven home, in a backhanded way, by a few conservatives who floated a conspiracy theory that Friday’s dip had been engineered to give Obama a boost.

Former General Electric chief executive Jack Welch wrote on Twitter: “these Chicago guys will do anything. can’t debate so change numbers.”

The Bureau of Labor Statistics said the data were worked out the same way as always, with no interference. And Welch later conceded that he had no evidence of a conspiracy.

There is no special economic magic to 8 percent. A truly healthy economy, experts say, would have a rate far lower.

“Eight is bad, 7.9 is bad, 8.1 is bad,” said Douglas Holtz-Eakin, a former director of the Congressional Budget Office and an adviser to GOP nominee John McCain in 2008. “We want to be at six.”

But the figure assumed its political significance in early 2009, before Obama had taken office, in a report written by a pair of his advisers, Christina Romer and Jared Bernstein. That report projected, with caveats, that if Congress passed a large stimulus package, unemployment would peak at 8 percent.

The stimulus passed. But the rate kept going up.

It reached 10 percent in October 2009 and then fell only slowly, despite the billions pouring in from the government. Before last month, the rate had hovered between 8.3 and 8.1 percent. Obama’s advisers later said they had not understood the depth of the country’s economic troubles when they made their projection.

The figure became one of the constants in Romney’s stump speeches and fundraising talks: It meant that Obama had failed, even by his own standards.

“We’ve had 43 straight months with unemployment above 8 percent,” Romney said in closing in the Denver debate.

Obama appeared Friday at campaign rallies in Cleveland and at George Mason University in Fairfax, evidently still smarting from his poor performance in Wednesday’s debate. He seemed to be throwing out comebacks to Romney that he wished he’d thought of on the debate stage back in Denver.

“Someone is finally getting tough on Big Bird!” Obama said, responding a day and a half later to Romney’s promise to take federal funding from PBS.

But he was clearly heartened by the job numbers, citing them as proof that he is the right leader to guide the economy.

“Today’s news should give us some encouragement. It shouldn’t be an excuse for the other side to try to talk down the economy just to try to score a few political points,” Obama said in Cleveland. “It’s a reminder that this country has come too far to turn back now.”

The dip in the jobless number was caused, in part, by a surprising jump in one measure of employment.

The Bureau of Labor Statistics, a branch of the Labor Department, uses two main sources. One is a survey of 141,000 businesses. The other looks at 60,000 households, asking if the people in those households were working or looking for work in the last month. The household survey captures data that the business survey doesn’t, such as people who are self-employed or who work on farms.

The September survey of businesses indicated a relatively modest gain in hiring: Payrolls rose by about 114,000. But the household survey indicated a much greater boost in hiring, with about 456,000 people no longer unemployed.

The number of newly employed people in that survey jumped by the largest amount in nearly three decades. On Friday, economists said that it was probably an exaggeration, an outlier made possible by a relatively small sample.

“The numbers just seem too big to be real real. But everything moved in the right direction,” said Chad Stone of the Center for Budget and Policy Priorities, a liberal think tank. “These numbers really do jump around a lot. You never want to read too much into one month’s unemployment data, because the next month could look really different.”

The data for October will be released Nov. 2, four days before Election Day.

On Friday, Welch’s suggestion that the numbers were fixed was picked up by a few conservatives. The most prominent was Rep. Allen B. West (R-Fla.), who posted on his Facebook page: “Somehow by manipulation of data, we are all of a sudden below 8 percent unemployment, a month from the presidential election.”

That assertion was denied by the Bureau of Labor Statistics, which compiles its reports in an intense and highly secretive process,carefully guarding access to the data before the official release. The bureau currently has no political appointees; its interim director is a career civil service employee.

On CNBC on Friday morning, Labor Secretary Hilda L. Solis said it was “ludicrous” to suggest that the data have been manipulated to boost Obama.

“I’m insulted when I hear that, because we have a very professional civil service organization where you have top, top economists that work at the BLS,” Solis said. “They’ve been doing these calculations. These are our best-trained and best-skilled individuals working in the BLS, and it’s really ludicrous to hear that kind of statement.”

Later in the day, Welch told MSNBC’s Chris Matthews that he had no hard evidence that the data had been fudged.

But he said he stood by his suspicions.

“I don’t want to take back one word in that tweet,” Welch said. “These numbers defy logic.”

Rucker reported from Abingdon, Va. David Nakamura and Nia-Malika Henderson contributed to this report.