The government is shut down, treaties need to be ratified by the Senate.

The executive branch is shut down. The Senate is still in session.

(The House of Representatives is effectively shut down, but only because the Speaker of the House has been unilaterally putting it into recess at the beginning of every session. The House Republicans all voted to grant the Speaker the power to do this whenever he wants, at the beginning of their current term.)

I am the Senate.

> The government is shut down, treaties need to be ratified by the Senate.

The President isn't shut down, and only the President is needed to sign a treaty; it is submitted for ratification later and that, absent a deadline in the treaty, can take as long as it takes.

Also, even if the Senate was required to sign a treaty, the Senate isn't shutdown, and is in session and doing business.

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