There are two methods the President's supporters and other rational people frequently mention. One is to invoke the 14th amendment to say that the President has the right and obligation to honor the national debt, whatever the Congress's debt ceiling law says. In other words, the law itself is unconstitutional and the President needn't abide by it.
The second is to mint a trillion dollar platinum coin and pay the obligations from that single coin. If you haven't gone down the platinum coin rabbit hole, it's not entirely crazy. Here's a primer.
Republicans have even more outlandish ideas about paying some people and not others. So pay the Chinese but no Americans, so it's all good. Or just stop paying Social Security checks but pay off the foreign debt holders.
In themselves these theories range from insane to actually pretty sensible. For what it's worth, I actually think the debt ceiling legislation probably is unconstitutional and thus the President would be within his rights, actually under an obligation, to proceed. But that's the point, it's not worth anything. And it's really not worth anything what you think either. In fact, it doesn't much matter what anyone thinks unless the President and the Congress and the Supreme Court and everybody on Wall Street all together say it's totally kosher. And of course nullifies the whole aim because the goal here is to find a path around a kamikaze Congress.
The whole point of a reserve currency, a debt obligation that is regarded as having essentially zero risk, is that it has regularity, certainty a total lack of surprises. Let's say the President takes the 14th Amendment route, which I guarantee you he won't. Wall Street types could give us the exact amount. But those treasuries would need to be heavily discounted because they would certainly be the focus of a series of lawsuits that would take time to settle since the move is at best constitutionally questionable. How much less will you pay for that 14th Amendment bond since there's a chance the Supreme Court might eventually decide it's invalid and needn't be honored? What risk do you factor in that the President gets impeached over the move and Congress and the next President expressly renounce the obligations? And if you're wondering, is it even remotely possible that on the day those securities go on auction the airwaves won't be filled with House Tea Partiers warning people not to buy them because they're illegal and lack the full faith and credit of the USA?
So sure you'll buy the treasuries that the US government has been selling and making good on for over two centuries since they come with the full faith and credit of the United States of America. But how much will you pay for the ones the President says he's behind but part of the Congress doesn't? Same with the platinum coin. Even more so with cockamamie ideas about paying some people and not others.
Solidity is about confidence. Your creditors - banks and foreign nations - evaluate you not only by your ability to pay (which the US clearly has) but also the regularity with which you actually do pay - your propensity and willingness to pay. Once you start paying some obligations and not others, investors start to get freaked out, if even you're still paying them. Because who's to say they're not next to get stiffed. This is what Wall Street is working very hard to tell Washington right now. Confidence is the simple assurance that the full force of the US government is beyond the obligation; and it's going to be done by the book. Payments will be made right on time. Once you issue bonds that the government of the United States itself isn't sure are legitimate, you've punctured the balloon and there's no going back. Best case you'd have the 'real' Treasury bonds and then a few months of heavily discounted US question mark bonds. So two classes of US bonds. US junk bonds if you will.
Not only does it not matter what you or I think. It doesn't even really matter what the President thinks. It's about what markets themselves think, or to put it more concretely, global market opinion. You may think you're pulling a fast one on John Boehner or Mitch McConnell. But once you're into pulling a fast one mode or not paying some creditors or citizens so you can pay others, the confidence the reserve currency relies on starts to melt away.
So no matter how good an argument you come up with, no matter how clever or righteous. It doesn't matter. There's no escape hatch.