> Nobody was there to observe the conditions...
Pre-biotic conditions on Earth cannot be known with certainty, but they can be known *generally* based on conditions foudn in the rest of the universe. Free oxygen, for example, is astonishingly rare. Somewhere around 95% of the visible mass of the universe is hydrogen; thus hydrogen dominates. Free oxygen very rapidly chemicaly bonds with hydrogen and forms water. Same with methane. And there are no known non-biological processes which will both produce and sustain an oxidizing atmosphere. If you were to kill off *all* life on Earth, Earth would have a carbon dioxide/nitrogen atmosphere relatively soon.
Looking at the rest of the universe, we can make a reasonable approximation of wha tthe pre-biotic atmosphere of the Earth was like. It it not "circular reasoning" to conclude that what holds true most places in the universe would also hold true on Earth. Consequently, a CO2/methane/nitrogen atmosphere on the early Earth is entirely reasonable and consistent with all available information.
Under what temperature, pressure, and concentration?
Oxygen molecules, atoms, or ions?
Hydrogen molecules, atoms, or ions?
Double-check your rate constants, not just equilibrium constants...