Pocket Watch Database
Access Date: December 01, 2023
"auth" could possibly indicate the grade of the watch and not the serial number of your movement. Serial numbers are usually more than a few digits.
From the American Horology Archive:
American Waltham Watch Co.
Grade: A.T. & Co.
|Manufacturer Location:||Waltham, Massachusetts|
|Movement Serial Number:||Unserialized|
|Grade:||A.T. & Co.|
|Estimated Production Year:||1852|