WASHINGTON — When he released his budget Tuesday, Rep. Paul Ryan, R-Wis., did not win membership to President Barack Obama’s Common Sense Caucus. That’s the name the president uses to describe his yet-to-be-defined group of lawmakers who are open to supporting a grand budget bargain. Membership in the caucus is built around a basic agreement about how to shrink the deficit: Republicans will accept some revenue from the tax code in exchange for Democrats agreeing to cuts in entitlement spending.

That formula is what puts Ryan out of the running: His budget will balance in 10 years, without tax increases. His plan is also premised on the repeal of Obamacare.

To achieve balance, Ryan’s budget uses magic beans and cuts funding for unicorn husbandry. That’s the tone of administration officials responding to Ryan’s budget. Despite the mockery, the president invited Ryan to lunch last week, and White House press secretary Jay Carney said the president thinks Ryan is a “thought leader in the Republican party.” So what’s going on here?

The president’s outreach to Paul Ryan shows just how flexible he is trying to appear in his overtures to Republicans. He’s not just going after the low hanging fruit — senators with a history of bipartisanship — he’s going for the top budget banana. The approach also marks at least the third stage in the Obama/Ryan relationship, which has gone from warm to chilly to late-night Siberian ice fishing in January. The icicles are melting a bit, and while they may never disappear, finding the right temperature may be a key to determining whether a grand budget bargain can be struck.

In the early days of the administration, the Obama and Ryan relationship started out warmly. In January 2010, when Obama attended the House Republican retreat in Baltimore, he tried to show how much respect he had for those with an opposing ideology by talking about Ryan. He called him a “sincere guy” whose roadmap for balancing the budget was a “serious proposal.”

In the meeting, the president several times said he didn’t want to oversimplify Ryan’s plan for fear of mischaracterizing it. He then used Ryan’s plan to make a general appeal for removing politics from either side’s proposals for reforming Medicare. “We’re not going to be able to do anything about any of these entitlements if what we do is characterized, whatever proposals are put out there, as, well, you know, that’s — the other party is being irresponsible; the other party is trying to hurt our senior citizens.”

A year and a half later, Obama was using Ryan as a political foil. At a speech at George Washington University in April 2011, with the budget chairman in the audience, the president said Ryan was trying to change the basic social compact in America by destroying entitlements that had protected Americans. Ryan became a regular target for White House attacks, which continued as he signed up as Mitt Romney’s running mate. Obama’s second term kicked off with an attack on the budget chairman in the president’s inaugural address. “They do not make us a nation of takers,” said the president referring to the entitlement programs of Medicare, Medicaid and Social Security. “They free us to take the risks that make this country great.” It was a shot at Ryan, whose talk about “takers who feed off the government and don’t pay taxes” became a white-hot issue during the presidential campaign.