I think that the depth of the hypothetical that tdees postulates has eluded you...

it would not be Obama who has a Constitutional/Legal crisis...

it would be the United States.

It would be, quite literally, US Law mandating the violation of ...

US Law.

And then what? The vice president would become the president, and he would have the exact same problem. The executive branch can't just miraculously disappear, even if congress is acting like a schizophrenic moron.

The most likely outcome, if the executive branch has no legal course of action, is to choose the "least illegal" course of action [1]. The linked article from the Columbia Law Review has a detailed analysis of the three options. Here's the most plausible picks:

- Selective repayment. Pick and choose which payment obligations to discharge. There's obvious political issues here. Who gets paid, and who doesn't? Social security over Medicare? Military contractors over NASA?

- Increase taxation without the authorization of congress. Again, many inherently political questions. Who do we increase taxes on? In many ways, this is a mirror image of option #1.

- Issue new debt not authorized by congress. The biggest issues here are market factors, not political questions. How will the market react to this unauthorized debt? If it refuses to buy at low interest rates, this option is not likely to solve the underlying problem.

Several humerous non-options are also debunked in [1], to illustrate what "more unconstitutional" looks like:

- Sell Alaska back to the Russians. Use the proceeds to buy time.

- Mint coins, and by fiat appraise them for massive amounts. Sell them to the fed, and pay off the debt with the new funds.

Just read the article. It's a bit biased, but its analysis of options appears on the whole quite reasonable (I am not a constitutional scholar).

1. http://www.columbialawreview.org/wp-content/uploads/2012/10/...