Urg. I just spent the last half our typing a response to your post only to hit the back button and erase everything. My post included calculations using the ideal gas law to back my hypothesis. So long for that.
Anyway, Camtheman you cannot over simplify it to "the air shrinks as it cools" because when you seal the lid on it is basically at the same temperature as where it will be when it cools. Therefore causing little to no change in volume or pressure as both are state properties.
And Work2fish, ALL organic materials including matches, paper, and methane produce CO2 and Water (as steam) when they burn. Some impure organics will produce other oxides when they burn as well. By definition an organic material is composed of Carbon and Hydrogen atoms and when ignited produce CO2 and Water.
I'm in Materials Engineering, I should know this stuff.
You should know this stuff, but apparently you don't
The small amount of air in the jar gets heated by the flame before you can get the lid on. When the jar goes back to room temperature that air cools causing the vacuum. It's not much of a vacuum, but it's enough to draw the lid of the jar down. The only change that matters here in creating the vacuum is the temperature change of the air in the jar. What happens after the lid is put on doesn't matter since it's now a closed system. So yes, you kinda can simplify it with "the air shrinks as it cools", or a little bit more complicated with your ideal gas law, PV=nRT, since V,n, and R remain constant, and T changes, and P is unknown it must also change in response to the temperature decrease, by decreasing. Now so long as the air in the jar is not heated to the temperature it was when it was sealed, there will be a small vacuum.
The burning that occurs once you seal the jar doesn't is really irrelevant ,as well as the lid heating up. It's only the heated air that matters.
While it's true co2 and water are produced, what I was hinting at is that you would never get perfect combustion from paper. It wouldn't be the "ONLY" thing you'd get. You wouldn't get perfect combustion with methane either, but again I was talking about a magic jar where anything is possible. CO2 and water are the "only" byproducts in "perfect" combustion which rarely occurs, and wouldn't be occurring in this situation. CO, NO2, and "soot" are produced along with other impurities in the paper, and the air. But thanks for the attempt at a grade 8 level science lesson.