Do I qualify for visa free entry?

I am traveling with my family of 6 to Beijing in August. We arrive to Beijing from Thailand at 6:30am and depart 3 days later at 1:00pm to return home to the US. Could we wait at the airport until 1:00 to obtain our transit papers and still qualify? When exactly does the 72 hour period start counting? the tickets have already been booked.

Thank you.