Mr. Ryan’s ties to that world began with a job at Empower America, a group founded by Mr. Kemp that ran “candidate schools” for aspiring conservatives and advocated for a flat tax and lower spending. As a rank-and-file congressman during the presidency of George W. Bush, Mr. Ryan advocated for the privatization of Social Security, helping push the idea toward the Republican mainstream and cementing his reputation as a conservative intellectual.

Privately, Mr. Ryan would later say, he was also stewing over what he and other conservatives viewed as the Bush administration’s fiscal profligacy and ideological drift, including the addition of a drug benefit to Medicare and, later, a bank bailout plan, the Troubled Asset Relief Program. (Mr. Ryan voted for both.)

That dissatisfaction was shared by the Kochs, who in the middle of the last decade began organizing conferences of like-minded donors and founded Americans for Prosperity.

Mr. Ryan, who became House budget chairman in 2006, began attending and speaking at Americans for Prosperity events. In 2008, the Wisconsin chapter gave Mr. Ryan its annual “Defender of the American Dream” award. Mr. Ryan also began attending the Kochs’ annual donor seminars. Last spring, Mr. Ryan was a speaker at a “Hands Off My Health Care” rally organized by Tea Party leaders outside the Capitol, drawing enthusiastic applause.

In Congress, he emerged as a skeptic of mainstream climate change theory — opposition to which has been a top priority of Koch-affiliated activists and research groups — and a reliable vote against energy efficiency standards, including a House vote to prohibit the Environmental Protection Agency from regulating greenhouse gases.

The relationship helped Mr. Ryan’s campaign coffers as well as his career: the Koch Industries PAC has donated more than $100,000 to Mr. Ryan’s campaigns and his leadership PAC, more than has any other corporate PAC, according to a New York Times analysis of campaign records.

Mr. Ryan has also developed relationships with other people in the Koch orbit, like Mr. Asness, a libertarian-minded financier known for his open letters blasting Mr. Obama, and Kenneth Griffin, a Chicago hedge-fund executive: wealthy donors whose taste for number-crunching and policy minutiae match Mr. Ryan’s own.