How does the classical boiling process produce purified neat soap?

The classical boiling process produces purified neat soap through a series of controlled steps in a kettle, designed to remove glycerin and impurities.1

• Saponification: Melted fats are boiled with caustic soda using open steam, creating a mixture of soap and glycerin.1
• Glycerin Removal: The mass is treated with brine to "salt out" the soap. Because soap is insoluble in salt solution while glycerin is soluble, the mixture separates into a curdy soap layer and a lower layer of spent lye containing the glycerin.1
• Strong Change: After removing the lye, the soap is boiled with additional strong caustic solution to ensure all remaining free fat is reacted.1
• Pitching and Settling: This final stage involves boiling the mass with water until it separates into two layers. The upper layer is the purified neat soap (approximately 70 percent soap and 30 percent water). The lower layer, called nigre, collects impurities such as dirt, coloring matter, and metal soaps, which are relatively insoluble in the neat soap.12