Yes, lye will react with carbon dioxide (CO2) in the air to form soda ash (Na2CO3). See Post 17 above. For a brief time, there will be a bit of free NaOH on the surface of the soap, but the NaOH doesn't stay intact for long, because NaOH wants to react badly with something else, and the carbon dioxide in the air volunteers for that duty.
The reaction of NaOH with fats to make soap is a "non-linear decay" reaction, meaning the saponification reaction doesn't happen at a constant rate. The reaction starts fast and furious and then slowly dies down over time, similar to the curve shown below.
This "start fast and stop slow" pattern means MOST of the NaOH will be consumed in the first few hours, but a small amount of the lye can remain unreacted for some hours or days afterwards. How much and how long will depend on the recipe and the way the soap is made. That lingering small amount of NaOH is what can react with CO2 to form soda ash. The amount of lye lost to ash appears to be fairly small in most cases. If the loss was large, I would expect to see soap bars entirely covered with a thick layer of furry crystals, rather than the usual speckling of spots here and there.
Yes, the loss of NaOH to ash will increase the "hidden superfat" in a soap, but I personally think this is a small loss. Even if I did want to worry about it, I don't think it would be easy to calculate -- ash formation varies so much from batch to batch, so how can we accurately predict this?
I am much more concerned about the "hidden superfat" built into most soap recipes because most soap calcs assume NaOH is 100% pure. We all know it's not, even right from the supplier. And as we open and close the lye container, the NaOH purity rapidly drops just because we're allowing CO2 and water from the air into the container. Every soap recipe made with an online soap calc automatically has at least 5% of "hidden superfat" built in. That, to me, is a much larger concern than the loss of NaOH to ash.
"...the density of soap molecules that helps or not the soda-ash formation. It states that if we pour our soap in the mold in light trace the density of the soap molecules is low so the water can walk through more easily to reach soap's surface than in soap that are poured while thick trace in the mold where soap's molecules density is high. The video also states that it is also for the % of humidity in the environment where the soap cures that helps or not soda-ash...."
There are bits of truth in these statements, but I'd say the level of trace when the soap is poured is honestly not an issue. Every soap goes through "thin trace", so whether that happens in the soap pot or in the mold is a moot point. Quite frankly, any time NaOH is exposed to air (really, exposed to CO2), it is forming soda ash. This happens in the NaOH storage container, in the lye pitcher, in the soap pot, in the mold, and on the bars after they are cut. The more concentrated the NaOH is, the more soda ash is made in a given amount of time. You might not see the ash formed when NaOH reacts in the lye pitcher or soap pot, but it's still there -- it's just mixed into the soap rather than visible on the surface.
What I'd say is more of an issue about ash formation on the soap's surface is the total amount of water in the soap after it is poured into the mold. More water can transport more NaOH to the soap's surface. More NaOH on the surface of the soap means more ash can form.
The humidity while the soap cures might be an issue for the first few days of cure, but doesn't seem to be a big issue after that time, in my experience. If ash is going to form after the bars are cut, it usually happens within the first few days of cure.