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So for Obama to win the fight, it needs to be reframed, and at his press conference, Obama has done a decent job doing so with his line about a "deadbeat nation."
Republicans like to say that breaking the debt ceiling would not lead to a debt default. And that's possibly true. But there's no question that in a debt ceiling break, we would see some obligations (that are budgeted) go unpaid. The US government would be legally deadbeat.
And it's interesting that Obama himself, while talking about the consequences of a debt ceiling break, said that things like Social Security and Medicare and the FAA would go unpaid. He did not say the US would miss an interest payment.