Pocket Watch Database
Access Date: December 05, 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:
Elgin National Watch Co.
|Manufacturer Location:||Elgin, Illinois|
|Movement Serial Number:||Unserialized|
|Estimated Production Year:||1864|