And the formal document transmitting the articles of impeachment to the Senate is the equivalent of the signed deed. The "statute of frauds" is of very recent vintage...the authors of the Constitution would not recognize it. The functioning of Congress is and must be more formalized (and always has been).
There are many things in the internal rules of the branches of the Congress that are not specifically mentioned in the Constitution, which wisely leaves those up to the membership of those bodies.
The House has voted on articles of impeachment. Statutory impeachment, however, has NOT happened until the Senate receives the official documents of that process.
Where does it say that? Does failure to deliver the documents to the Senate then negate the will of the House? This would convey great power to whoever has the ability to prevent delivery.
There is a saying that "ignorance of the law is no excuse".
If the House passes a bill, which is approved by the Senate, and signed by the President, then I am expected to know its effect and can be held to account if I don't.
I see no reason not to expect the Senate to recognize the official acts of the House in a similar manner.