Sorry about the confusion in the post...

I've never had the pool go below 5ppm for chlorine.

I meant to say that the free chlorine and combined chlorine have never read below 5ppm.

I'm using a test kit from Leslie's Pool Supply (Basic 4 Test Kit from the Pool Shop). I take the free chlorine reading within 2 - 3 seconds of adding the solution and then the combined chlorine reading after 1 - 2 minutes. Both of these readings come in above the top end of the range (5 ppm).

The normal source of chlorine is 3" tablets placed in the skimmer basket. I shock the pool 1 - 2 times per month.

My test kit is only good for Cl, pH, Alkalinity, and Acid Demand, so I haven't measured CYA.