...It's odd that the PH was so high with my fist try, and it was slippery and had strong cleaning properties, but wouldn't saponify the oils.
Like I said in one of my earlier posts in this thread, wood ash lye is
not potassium hydroxide, KOH. It is mostly potassium
carbonate, K2CO3. Potassium carbonate can be used directly to make soap, but only if you heat and stir it sufficiently long enough. You cannot do a cold process or even a hot process soap with wood ash lye. The saponification must be done as a boiled process.
The pH measurement tells you nothing about concentration -- pH is not remotely accurate enough. For example, a 1% NaOH solution has a pH over 13, so if your pH test, assuming it's accurate, shows 13+, the only thing you know is the concentration is over 1%. The relationship between pH and concentration for KOH is similar . K2CO3 has a lower maximum pH at saturation concentration, but again the trend is the same.
You can measure the solution density to get a better idea of concentration (aka floating an egg or using a hydrometer) or you can do a titration with an acid solution to test the concentration directly. But pH isn't useful.
Somewhere I have a bar of her soap and it is not rancid. It got brown and is very hard....
I rest my case. When soap turns brown with time, that is evidence it is oxidizing and becoming rancid. That's no indictment of your mother's soap -- it's just a sign of the natural decomposition process of fats and soap. Since soap is a natural product, it's not immune to decomposition.
My brother has some of my grandmother's lard soap. It started out white but over the decades it has also become brown and very hard, just like your mother's. It's no shame that their soap is gradually deteriorating -- it's just a fact of nature.
All I'm saying is it's not a good idea to use equipment or utensils made of copper, brass, or other copper containing metals to make soap, because copper accelerates the rate of oxidation and rancidity. If people want their soap to have the longest shelf life, one effective rule is to avoid copper.
We have a lot of newcomers reading this forum, and it's important to pass on good information to those people.