Send it by mail with a not guilty plea. Do not wait for it to be entered online bc it could take a while and might be too late for you. Get a lawyer to help you fight the ticket
In theory, your thinking is correct. When a client hires me to fight their NYC moving violation, my main practise, I never ha e the tickets mailed in. Although ver small, there is a percentage of times that the PO does not hand in his ticket. However, a week is WAY too early to even think that in this case. None appear on the computer before 3 weeks generally, and most take longer. The 15 days is not true. You have 35 and it must be on the system. That being said, you will need to be vigilant and keep checking or hire an attorney who does this regularly.
When you hire a lawyer they do all of that for you so you don't have to, plus go to court for you as well.