What is the name of your state (only U.S. law)? IL
Honestly, I was away for 10-15mins and the cop can prove this to the judge -> probably by showing the time of call of witness and my arrival time etc.
I will explain how your "math" is flawed on this aspect:
Bystander notices child in the car, stands around and waits 10 min before calling 911.
Unless the officer was already in the parking lot, it would have taken him a while to get there. So, we'll give him an arrival time of 5 min.
(you with me so far?)
So, we already have 15 min.
How long was the child in the car before the bystander took notice?
How long did it take you to arrive back at the car after the officer arrived?
Unless the bystander watched you walk away from the car (leaving the child), your story of "I was gone 10-15 min" is a bit shaky.
If you gave the age of the child, I missed it. Nevertheless, if the child is a baby, a toddler, or under about the age of 10? You placed the child in danger. Your ticket is justified, and the best you should hope for is some sort of plea agreement