Every formula, and where it comes from

SAP_NaOH (mg/g) = SAP_KOH × 39.997 ÷ 56.106                # molar masses, IUPAC 2013
KOH₁₀₀ (g)      = Σ oilᵢ (g) × SAP_KOHᵢ ÷ 1000               # SAP defined as mg KOH per g fat
KOH_sf          = KOH₁₀₀ × (1 − superfat ÷ 100)
NaOH to weigh   = KOH_sf × (1 − f_KOH) × 39.997 ÷ 56.106 ÷ (purity_NaOH ÷ 100)
KOH to weigh    = KOH_sf × f_KOH ÷ (purity_KOH ÷ 100)
superfat from a given lye = (1 − lye_supplied_KOHeq ÷ KOH₁₀₀) × 100

• 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.

water = oils × p ÷ 100                    # p = water as % of oils (38 is the old default)
water = lye × (100 − c) ÷ c               # c = lye concentration % (33 is our default)
water = lye × r                           # r = water : lye ratio
c = lye ÷ (lye + water) × 100;  r = water ÷ lye;  p = water ÷ oils × 100

Master batch at strength s %:
solution = lye ÷ (s ÷ 100);  water_in = solution − lye;  water_add = water − water_in

Liquids replacing water:
plain water = max(0, water − Σ liquidᵢ × water%ᵢ ÷ 100)

• Dunn, Kevin M. Scientific Soapmaking: The Chemistry of the Cold Process. Clavicula Press, 2010. Lye solution concentration and its effect on trace and gel.
• 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.

NaOH consumed (g) = mass × purity × n × 39.997 ÷ M_acid
  citric acid (anhydrous, triprotic, M 192.124):  0.6246 g NaOH per g   (0.8760 g KOH)
  citric acid monohydrate (M 210.139):            0.5710
  acetic acid (M 60.052):                         0.6660   → 5 % vinegar: 0.0333 per g
  lactic acid (M 90.078):                         0.4440
effective superfat = (oils × sf ÷ 100 + fat from milks) ÷ (oils + fat from milks) × 100

• IUPAC 2013 atomic weights. Molar masses: citric acid C₆H₈O₇ 192.124 g/mol (monohydrate 210.139), acetic acid C₂H₄O₂ 60.052, lactic acid C₃H₆O₃ 90.078, NaOH 39.997, KOH 56.106.
• Neutralisation stoichiometry: citric acid is triprotic (3 OH⁻ per molecule); acetic and lactic acids are monoprotic. g NaOH per g acid = n × 39.997 ÷ M_acid.
• USDA FoodData Central, SR Legacy: Milk, whole, 3.25 % milkfat (FDC 171265): water 88.1 %, fat 3.25 %. Milk, goat, fluid (FDC 171278): water 87.0 %, fat 4.14 %. Cream, heavy whipping (FDC 170859): water 57.7 %, fat 36.1 %.

M̄_fatty acid   = Σ(fraction_i × M_i)            # mass-weighted, renormalised to 100 %
M_triglyceride = 3 × M̄_fatty acid + 38.03      # glycerol 92.094 less three waters
SAP (mg KOH/g) = 3 × 56 106 ÷ M_triglyceride    # three KOH per triglyceride

free fatty acids (stearic acid, lauric acid and the like) take one KOH each:
SAP = 56 106 ÷ M̄_fatty acid

Validated on the 35 Codex oils where the saponification range and the
fatty-acid ranges come from the same standard:
  median absolute error  0.90 %
  90th percentile        2.74 %
  worst                  5.18 %
  bias                  −0.01 %
A derived value therefore carries a ±3 % band, and is offered only where the
published profile sums to between 95 % and 115 %.

• Saponification stoichiometry: one mole of triglyceride consumes three moles of KOH. SAP (mg KOH/g) = 3 × 56 106 ÷ M_triglyceride, where M_triglyceride = 3 × M̄_fatty acid + 38.03 (glycerol 92.09 less three waters at 18.015).
• Fatty-acid molar masses from their molecular formulae with IUPAC 2013 atomic weights (C 12.011, H 1.008, O 15.999).
• AOCS Official Method Cd 3-25: the saponification value of a specific lot is measured by titration. A certificate of analysis replaces the published range with a figure for the fat in front of you.

KOH_low  = Σ(oilᵢ × SAP_minᵢ ÷ 1000)       # every oil at the bottom of its published range
KOH_high = Σ(oilᵢ × SAP_maxᵢ ÷ 1000)       # ...and at the top
KOH_mid  = Σ(oilᵢ × SAP_repᵢ ÷ 1000)       # what the lye is actually weighed from

supplied (as KOH equivalent) = KOH_mid × (1 − sf ÷ 100)

superfat actually achieved:
  worst case = (1 − supplied ÷ KOH_low)  × 100   # fat needs LESS lye than was weighed
  best case  = (1 − supplied ÷ KOH_high) × 100   # fat needs MORE, so more is left unsaponified

smallest nominal superfat whose worst case clears a floor f:
  sf ≥ (1 − (1 − f) × KOH_low ÷ KOH_mid) × 100

• Codex Alimentarius CXS 210-1999, Standard for Named Vegetable Oils, and CXS 211-1999 for animal fats: saponification value is published as a range per named oil, not a single figure, because the property varies with origin, season and processing.
• Worst-case bound: every oil is taken at the same end of its published range simultaneously. Real lots are not perfectly correlated, so the achievable spread is narrower. — This is not a confidence interval and implies nothing about how likely either end is. It is the limit of what the published data permits.
• AOCS Official Method Cd 3-25: the saponification value of a specific lot is measured by titration. A certificate of analysis replaces the published range with a figure for the fat in front of you.

profile = Σ (oilᵢ ÷ total oils) × fatty acidsᵢ           # mass-weighted percent
hardness     = lauric + myristic + palmitic + stearic        (29–54)
cleansing    = lauric + myristic                             (12–22)
conditioning = oleic + linoleic + linolenic + ricinoleic     (44–69)
bubbly       = lauric + myristic + ricinoleic                (14–46)
creamy       = palmitic + stearic + ricinoleic               (16–48)
iodine value = Σ wᵢ × IVᵢ                                    (41–70)
INS          = Σ wᵢ × (SAP_KOHᵢ − IVᵢ)                       (136–165, ideal 160)

• 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. — A convention, not a measurement. It ignores superfat, water, cure and additives, and treats every fatty acid in a group as equal.
• AOCS Official Method Cd 1-25, "Iodine Value of Fats and Oils, Wijs Method". g I₂ absorbed per 100 g of fat; a measure of unsaturation.
• McDaniel, Robert S. Essentially Soap. Krause Publications, 2000. INS = saponification value (mg KOH/g) − iodine value; recommended range 136–165, ideal 160.

f = oils × p ÷ 100                          # p = % of oils (PPO)
f = base × p ÷ (100 − p)                    # p = % of finished batch; base = everything but fragrance
% of finished product = f ÷ (base + f) × 100 ≤ certificate maximum for the product's IFRA category
blends (conservative): Σ loadᵢ ÷ maxᵢ ≤ 1
allergen % in product = Σ (fragrance % × allergen % in fragrance ÷ 100) ÷ (1 − cure mass loss)
declare if > 0.001 % leave-on, > 0.01 % rinse-off

• IFRA, "Guidance for the use of IFRA Standards", 51st Amendment (June 2023), Table 11 product-category mapping and §6 on calculating the concentration of a fragrance mixture in a finished product. — IFRA Standards are voluntary; limits are maximum concentrations in the finished product; a Certificate of Conformity is issued by the fragrance supplier, not by IFRA.
• Regulation (EC) No 1223/2009, Annex III, as amended by Regulation (EU) 2023/1545: fragrance allergens must be named in the ingredient list when their concentration exceeds 0.001 % in leave-on products or 0.01 % in rinse-off products.

V (mL) = l × w × h  |  π (d ÷ 2)² h  |  cavities × volume each     # 1 in³ = 16.387064 mL
batch (g) = V × ρ × fill                                        # ρ ≈ 1.0 g/mL, calibrate
oils = batch ÷ (1 + lye/oil + water/oil + additives/oil + fragrance/oil)
for comparison: the "in³ × 0.4 oz" rule assumes full water and 5 % superfat

• Volume of a rectangular prism V = l × w × h; of a cylinder V = π × (d ÷ 2)² × h. 1 in³ = 16.387064 mL.
• 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.

solids = oils + pure KOH                 # mass is conserved in saponification
paste concentration = solids ÷ paste × 100
water to add = solids ÷ (target ÷ 100) − paste

• Conservation of mass in saponification: the soap and glycerol formed weigh exactly what the fat and hydroxide weighed. Solids in a paste are therefore taken as oils + pure lye; the remainder is water.
• Failor, Catherine. Making Natural Liquid Soaps. Storey Publishing, 2000. Paste method, dilution ratios and sequestering.