Interesting, my serial number begins with 667. Any ideas on the star and scalene(I think,it's been a minute since I took geometry) triangle on the barrel?According to the Remington barrel codes, K would be the month of May. The barrel codes, however, don't start until the year of 1921, so I don't see where you're getting May of 1920.
There is a serial number list for the model 12, but it starts in January of 1930 with the number 66366.
There appears to be no record of the guns made before that 66000 number.