A United States Corporation is a company chartered not under the laws of any state, but directly under the Constitution of the United States. This is generally consitutional under the "Interstate Commerce" clause.
We expect the incentives inherent in the system to mean that most large companies operating in multiple states will re-incorporate as United States Corporations. Small companies may incorporate as USCs, in their state of operation, or in the state of Delaware (as is current practice).
One of the privileges of being a United States Corporation is that certain transactions may be denominated in their corporate stock.
One of the requirements of a United States Corporation is a re-structuring of corporate government. Stockholders elect the board by Single Transferable Vote to allow minority positions of shareholders to have direct representation. Other stakeholders (employees, government regulators) have certain board rights, though not necessarily voting rights.