Water, and the discount
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The ingredient that does not end up in the bar, and the one that decides how the batch behaves.
What the water is for
Dissolving the lye, so it can meet the oils.
It is not a reactant. Most of it leaves during the cure, which is why a bar loses weight for weeks.
Full water and a discount
"Full water" is a conventional amount, often expressed as a percentage of the oil weight or as a lye concentration.
A water discount uses less, giving a stronger lye solution.
Which is a real formulation decision rather than a preference, and it is usually expressed as a lye concentration because that is the number that actually describes it.
What a discount changes
Faster trace, which is helpful for a plain bar and unhelpful if you intend a swirl.
A harder bar sooner, and a shorter cure to a salable weight.
Less shrinkage over the cure, because there is less water to lose.
More heat in the mold, which makes gel more likely and overheating more likely with it.
And a stronger, more hazardous solution to handle.
The trade is speed and hardness against working time and safety margin, and beginners should not discount until they are comfortable with the process at full water.
Which water
Distilled or deionized, as standard.
Tap water carries minerals that react with soap to form insoluble scum, which shows as dull bars, reduced lather and sometimes spotting.
It is cheap and it removes a variable that changes seasonally and by region, which is the same argument this estate makes for every water-based product.