Under section 4 of the 14th amendment, it says the "validity of the public debt of the United States ... shall not be questioned." And since the House has the explicit constitutional responsibility of control over all three functions of the key monetary issues: spending, taxation, and borrowing, it it also their constitutional duty to make sure the debt is covered. They and they alone have that responsibility as expressed in the US Constitution. Should the House threaten or actually fail to honour the debt of the USA, President Obama should sue the House to ensure they comply with the constitution and have the debt paid forthwith.
There is no out for the House to not honour the debt. They cannot arbitrarily decide not to pay what is owed by the USA. They cannot, after the fact, put stipulations of the debt of which it's creditors are unaware.
Neither the Senate nor President has any legal duty to accept budgetary changes requested by the House as a condition to raising the debt ceiling under which the debt was lawfully incurred. The House has no other option but to pay it. That is the ultimate law of the land; the constitutional law.
Therefore, I see the situation in a different light than how the Tea Party is trying to portray the debt limit increase situation. The House has no option but to raise the debt limit. They have no leverage. It is part of the oath of office the took upon being elected. Given that fact, I would urge the President to go to court now, the Supreme Court if he can go directly, to get a ruling to placate the markets and foreign debt holders.