Egg in the bottle demo gas law identity so, i've taught chemistry and physics for 15 years, and yet i still need help with this.  the classic egg in a bottle demo i'm referring to is the one done with a lit object being placed in the bottle.  i get that all those demos calling for heated water that is cooled or using liquid nitrogen to cool the bottle are examples of gay-lussac's law.  however, i've seen (on NSTA's website among many many others) teachers stating that the burning a thing method is also an application of gay-lussac's law.  this does not make sense to me.  
first of all, dropping a burning object into the bottle then immediately capping the bottle with the egg means that the egg has just created a closed system.  if gay-lussac's law applied according to the reasoning i've read online, then the heated air should actually increase in pressure and push the egg out, not suck the egg in.  secondly, the burning thing (a match, a piece of paper) doesn't burn long enough to significantly change the temperature in the bottle, does it?  
thirdly, since the egg closes the system, the only thing that could cause the egg to be sucked into the bottle under these circumstances would be the loss of the pressure that was due to the oxygen being consumed, right? so, then, i have another question: what about the co2 that is produced as a result of the combustion reaction?  if burning wood based products is sort of similar to the combustion of glucose, the mole ratio of o2 to co2 should be one to one.  since co2 is so much denser than air, is it that it sinks to the bottom of the container and is unavailable for collisions at the top of the container?  it seems like a read an article on this many years ago, but i cannot find it now.  thoughts?
 A: Let us first assume a system where the fire immediately goes out when the egg seals the bottle. Before placing the egg, air rushes out so there are fewer molecules in the bottle, since pressure inside and outside is atmospheric, but hot molecules push harder*. When the egg seals the container, the air in the bottle cools down to room temperature, but since there are fewer molecules inside the bottle, there is a net pressure**.
In reality, there is another more subtle aspect as you have correctly identified. Here's a rough equation for wood burning.

Notice fewer molecules of gas on the right hand side. Does this decrease pressure while the bottle is unsealed? No, air can rush in or out at any rate it wishes to keep the pressure in the bottle at atmospheric*. When we seal the bottle with the egg, the number of molecules in the bottle decreases as the fire continues to burn. Whilst the heat of the fire may temporarily create an outward pressure, eventually, there are even fewer molecules in the bottle than before. As they decrease to room temperature, they create a much lower than atmospheric pressure inside the bottle***.
We have described these actions at a molecular level. But the model I'm using is essentially the ideal gas law $PV = nRT$ which includes Lussac's law and only works at equilibrium. (to make life easy, let's say we have instantaneous thermodynamic equilibrium at every stage). I will highlight how each stage of my attempt at explanation is predicted by the equation.
let $T_1$ be room temperature, let $T_2$ be the temperature in the bottle when it is burning, let $V$ be volume of bottle, let $P_1$ be atmospheric pressure and $P_2$ be final pressure, $n_1$ the number of molecules inside while it is unsealed and not lit, $n_2$ the number of molecules while it is unsealed and lit and $n_3$ the number of molecules at the end.
Initially, $P_1V = n_1RT_1$
.* We light it, pressure is constant, temperature increases so $n$ decreases: $P_1V = n_2RT_2$
** seal it but it stops burning, pressure is variable and temperature decreases: $P_2V = n_2RT_1$.
*** seal it but continues to burn, pressure is variable and temperature decreases after $n$ decreases slightly: $P_2V = n_3RT_1$.
Clearly $n_1>n_2>n_3$ and e.g. in the second case: $P_2 = \frac{n_3}{n_1}P_1$ so $P_2 < P_1$.
A: The paper heats the air inside the bottle before the egg is placed. Once the egg is placed it seals the bottle and the remaining oxygen is quickly consumed. Then the air inside the bottle cools decreasing the air pressure inside the bottle and sucking the egg inside.
