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Lye
  KOH₁₀₀  = Σ(oilᵢ × SAPᵢ ÷ 1000)                  # SAP in mg KOH per g
  KOH_sf  = KOH₁₀₀ × (1 − superfat ÷ 100)
  NaOH    = KOH_sf × 39.997 ÷ 56.106 ÷ (purity ÷ 100)
  KOH     = KOH_sf ÷ (purity ÷ 100)
  Dual: KOH share f of the saponification → KOH = KOH_sf × f, NaOH = KOH_sf × (1 − f) × 0.7129

Acids in the pot
  NaOH_add = mass × purity × (n × 39.997 ÷ M_acid)   # citric 0.6246 g/g, acetic 0.6660, lactic 0.4440

Water
  % of oils:          water = oils × p ÷ 100
  lye concentration:  water = lye × (100 − c) ÷ c
  water : lye ratio:  water = lye × r
  liquids:            plain water = max(0, water − Σ liquidᵢ × water%ᵢ)

Qualities (community convention)
  hardness = lauric + myristic + palmitic + stearic
  cleansing = lauric + myristic
  conditioning = oleic + linoleic + linolenic + ricinoleic
  bubbly = lauric + myristic + ricinoleic
  creamy = palmitic + stearic + ricinoleic
  INS = Σ wᵢ × (SAPᵢ − IVᵢ)

Fragrance
  % of oils = fragrance ÷ oils × 100;  PPO (oz per lb of oils) = fragrance ÷ oils × 16

Mold
  batch = V × ρ × fill;  oils = batch ÷ (1 + lye/oil + water/oil + additives/oil + fragrance/oil)

• AOCS Official Method Cd 3-25, "Saponification Value of Fats and Oils". American Oil Chemists' Society. Defines SAP as mg KOH required to saponify 1 g of sample.
• IUPAC Commission on Isotopic Abundances and Atomic Weights, "Atomic weights of the elements 2013" (Pure Appl. Chem. 88(3), 2016). Na 22.990, K 39.098, O 15.999, H 1.008 → NaOH 39.997 g/mol, KOH 56.106 g/mol.
• Dunn, Kevin M. Scientific Soapmaking: The Chemistry of the Cold Process. Clavicula Press, 2010. Chapters on saponification stoichiometry, lye discount and lye purity.
• Community convention: "full water" = 38 % of oil weight, the default that most hobby calculators have used since the 2000s; roughly a 27–28 % lye solution for a typical bar recipe. — A convention, not a chemical constant. Lye concentration is the better control variable because it does not drift with the oil blend.
• Neutralisation stoichiometry: citric acid is triprotic (3 OH⁻ per molecule); acetic and lactic acids are monoprotic; disodium EDTA takes 2, sodium bicarbonate 1 (to carbonate) and magnesium sulfate 2 (to magnesium hydroxide). g NaOH per g = n × 39.997 ÷ M.
• Community convention for soap qualities, in continuous use in hobby lye calculators since c. 2005: hardness = lauric + myristic + palmitic + stearic; cleansing = lauric + myristic; conditioning = oleic + linoleic + linolenic + ricinoleic; bubbly lather = lauric + myristic + ricinoleic; creamy lather = palmitic + stearic + ricinoleic. Recommended bands: hardness 29–54, cleansing 12–22, conditioning 44–69, bubbly 14–46, creamy 16–48, and iodine value 41–70. — A convention, not a measurement. It ignores superfat, water, cure and additives, and treats every fatty acid in a group as equal. The iodine value itself is measured (AOCS Cd 1-25); the 41–70 band beside it belongs to this convention, not to the method.
• McDaniel, Robert S. Essentially Soap. Krause Publications, 2000. INS = saponification value (mg KOH/g) − iodine value; recommended range 136–165, ideal 160.
• Empirical default. Fresh cold-process soap batter has a density close to water, ≈ 0.95–1.0 g/mL; the community rule "in³ × 0.4 = oz of oils" implies about 0.69 g/mL of oils, which is ≈ 1.05 g/mL of batter at 38 % water and 5 % superfat. Calibrate by weighing a filled mold once. — Density varies with whipping, fragrance, temperature and how full the mold is poured. Treat the default as a starting point and calibrate.