A month into his vice presidential candidacy, it’s clear Rep. Paul Ryan has had an impact — elevating Medicare into the national debate, putting more of his fellow House Republicans on defense and now explaining his own math skills, after he dramatically oversold his marathon-running speed.

What is yet to be determined is whether the Wisconsin Republican’s impact helps or hurts the Romney ticket on Election Day.

During that month he and Mitt Romney, the man at the top of the GOP ticket, closed the gap in polls with President Obama, then fell behind again.

Republicans say he has been good for the ticket, boosting enthusiasm and giving them a reason to talk about big ideas, though they acknowledge they are having to answer tougher questions about the budgets Mr. Ryan wrote as House Budget Committee chairman.

But Democrats said he has been even better for them, giving them a firm target to fire at — particularly on Medicare, where his budgets proposed turning it into more of a defined spending plan rather than the defined benefit plan it is today.

“I think Ryan does something for [the Republican] base and doesn’t do anything else,” said Rep. Gerald E. Connolly, a Virginia Democrat who served two years on the Budget Committee with Mr. Ryan earlier in his career. “I think he risks the senior vote. I think he puts into more serious play 29 electoral votes in Florida, which would be disastrous for the Romney ticket. I think he is the semi-affable radical that he is.”

Mr. Romney tapped Mr. Ryan to be his running mate Aug. 11, while Congress was in the middle of a monthlong summer vacation.

Now, as his colleagues stream back to town for a brief burst of legislative activity, he is a major topic for those on both sides.

“I see members very excited about Paul,” said House Majority Whip Kevin McCarthy, California Republican. “I think Paul was a great choice. It’s not very often members know somebody personally, worked with them, and worked with their issues, and I also think it pushes a lot of the issues the House has been working on the last two years to the forefront as well.”

Even in the Senate, where collegiality reigns and the lower chamber is often ignored, Majority Leader Harry Reid led off Monday’s session by mocking Mr. Ryan for saying he ran a marathon in a little less than three hours — when the true time was more than four hours.

Mr. Ryan explained on CBS this weekend that the race was more than 20 years ago and he made an “honest mistake.”

But Mr. Reid dubbed it “Ryan math,” and said it exposed a deeper problem with the GOP ticket’s No. 2 man.

“His math doesn’t work for running a marathon or anything else,” the Nevada Democrat said. “The Ryan math doesn’t work with his budgets, it doesn’t work with Medicare, and it doesn’t work with his tax plan.”

Picking Mr. Ryan did seem to boost Mr. Romney initially. Polling showed the race tightened to a dead heat in the run-up to Republicans’ convention in Tampa, Fla.

Introducing himself to the nation with his address the middle day of the convention, Mr. Ryan had the line of the week when he joked about college students still living at home “staring up at fading Obama posters and wondering when they can move out and get going with life.”

But fact checkers poked Mr. Ryan for suggestions that the Obama administration was responsible for a manufacturing plant closing in his hometown of Janesville, Wis.

Democrats blasted him for criticizing Mr. Obama for failing to heed the recommendations of the Bowles-Simpson deficit commission — even though Mr. Ryan himself was on the commission and voted against the final report.

Now the polling shows Mr. Obama has once again opened up a significant lead over the Romney-Ryan ticket.

Jim Manley, a Democratic strategist who spent years as a staffer on Capitol Hill, said selecting Mr. Ryan has brought more scrutiny to the two budgets he wrote for House Republicans, and also to social issues such as abortion, where Mr. Ryan’s pro-life record of opposing abortion in all cases except where the life of the mother is at stake has drawn scrutiny.

“He had a reputation — I don’t think well-deserved — within the Washington press corps as being a serious thinker,” Mr. Manley said. “Now that people have had a chance to get beyond the editorial page of the Wall Street Journal and look at specifics, they see his budget doesn’t add up.”

Mr. Ryan wrote two budgets that passed the House, but both stalled in the Senate. Senate Democrats defeated Mr. Ryan’s plan but never offered one of their own.

His plans included a call for a major overhaul of the tax code and of Medicaid and Medicare. On that latter program, he proposed a voucherlike system that would apply to seniors 10 years from now, and would give them a choice of traditional Medicare or a private plan paid for by the government.

Rep. Michael K. Simpson, an Idaho Republican who serves on the Budget Committee with Mr. Ryan, said picking him means congressional Republicans will face even more questions about that Medicare plan, but he said that was inevitable. He said Mr. Ryan gives the GOP a good spokesman on the issue.

“It’s a debate that’s going to happen. The question is whether the American people are ready to have that debate or not,” he said.

Mr. Simpson said picking Mr. Ryan has been good for Mr. Romney and the ticket — though he was uncertain whether it had been good for the prospects for Mr. Ryan himself.

Indeed, Mr. Ryan has gone from a man of big ideas and bold votes — including regularly voting last decade against the GOP’s own spending bills — to a cautious vice presidential candidate who has to run his words and stances by the Romney campaign.

In the latest example, the House will vote later this week on a stopgap spending bill to keep the government running into 2013. Funding would otherwise run out Sept. 30, resulting in a government shutdown.

On Monday, Mr. McCarthy told reporters Mr. Ryan would be back at the Capitol for the vote and said Mr. Ryan will support the legislation — then stepped back from those comments and said the vice presidential nominee should answer for himself.

After first saying they didn’t have an answer, Mr. Ryan’s staffers late Tuesday said he will vote for the spending bill.

That vote does have some peril.

Congressional Republican leaders wrote the stopgap bill at a year-long discretionary spending rate of $1.047 trillion, which is higher than Mr. Ryan’s budget called for. Voting for that higher level will mean Mr. Ryan is caving on an issue where he and his House colleagues had drawn a line in the sand earlier this year.

But if he had voted against the stopgap measure, he could have been seen to be supporting a government shutdown.

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