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227 live, computable functions in one portable, fully offline file. The complete AlchemyCalc G2 base (CGB depositions, attributed open-science corpus, instruction editor, AutoPhi math accelerator, Fractal Lab, database loader) plus a 177-instruction Fortuna pack: Medicine (48) · Energy (46) · Carbon (45) · Omnia (18) · Probability & Chance + Games, Wagers & Fortune (20).

What the Fortuna pack computes

everything the Omnia edition carries — the full clinical, power, carbon and Earth/information/finance libraries and the library search — plus the probability pack: Bernoulli's binomial, Poisson, the birthday problem, coupon collector, gambler's ruin, the Kelly criterion, Turing & Good's deciban evidence engine, Pearson's random walk, exact dice sums, lottery odds, Monty Hall, streaks, Benford's law, surprisal and the Drake equation. And the capability no deterministic calculator has: the Monte Carlo Lab — give any input of any instruction a distribution instead of a number, run ten thousand seeded trials, and read the outcome as it actually is: mean, spread, bad-luck 5th percentile, good-luck 95th, drawn as a histogram. Same seed, same fortune — even the randomness is reproducible, so even the randomness can be audited. A distribution sampler completes the tab.

227

live functions — every one attributed

177

pack instructions across five packs

12.2 M/s

log-gamma evaluations — accelerator, measured

10^±308

numerical range, honestly displayed

Learn it right — a suggestion. These are professional-grade instructions, and most people need instructional help the first time through. Our suggestion: pair this edition with an AI tutor, such as Claude by Anthropic (claude.ai) — paste an instruction's 📜 origin line, its formula, and your numbers, and ask Claude to walk you through what it means, the units, and how to read the result before you rely on it. The calculator computes; a tutor explains.

Provenance — who and what

Formulas and laws of nature are unpatentable — no one owns them. Every calculation names its origin, in-app and in the shipped ledgers (PROVENANCE.md + PROVENANCE-FORTUNA.md). The ten CGB Mathematical Depositions are © Christopher Gabriel Brown; every borrowed formula credits its originator and year.

In the box

The AlchemyCalc Fortuna app — one file (271 KB), no install, no internet, phone-ready
The Proof button — the calculator verifies itself against known values, live on your device
The Professor button — composes a full lesson prompt for an AI tutor such as Claude (claude.ai)
Plain-English guide (laymen notes), quick-start, license & IP notice
Both provenance ledgers and a fictional demo dataset

Medicine content: educational mathematics only — not medical advice. Energy content: engineering estimates — verify against codes. Carbon content: emission factors are averages — use official factors for reporting. Fortuna content: randomized results are seeded simulations — reproducible, and still simulations. Probability is not prophecy; never wager what you cannot lose.

Christopher Gabriel Brown — Inventor · Author · Visionary

christopher@cri-one.com · crioneaka@outlook.com · 1341 Wellington Cove, Lawrenceville, GA 30043-5255, USA

Complete instruction list — all 227 functions

Every instruction in this edition, with its formula and its origin. Mathematical formulas and laws of nature are unpatentable; each borrowed formula credits its originator and year, and the ten CGB Mathematical Depositions are © Christopher Gabriel Brown.

CGB depositions (10)

Instruction	Formula	Origin — who and what
1 · Dimensional Fold	V(n-sphere) = π^(n/2) / Γ(n/2+1) · rⁿ ⟹ peaks at n* = 5, → 0 as n → ∞	CGB Deposition 1 © Christopher Gabriel Brown — builds on the classical n-sphere volume via Leonhard Euler's Gamma function (18th c.)
2 · Entropic Bridge	S = −k_B Σ pᵢ ln pᵢ ≥ ∮dQ/T (Shannon = Clausius = Helmholtz)	CGB Deposition 2 © Christopher Gabriel Brown — builds on Claude Shannon (1948), Ludwig Boltzmann & Rudolf Clausius (19th c.), Rolf Landauer (1961)
3 · Golden Spiral Convergence	lim F(n+1)/F(n) = φ = (1+√5)/2 ⟹ r_opt = φ⁻² ≈ 0.382 wire/gate area	CGB Deposition 3 © Christopher Gabriel Brown — builds on Leonardo of Pisa “Fibonacci” (1202) and Euclid's extreme-and-mean ratio (~300 BC)
4 · Harmonic Decay	Σ 1/n² = π²/6 ⊗ ∫₀^∞ e^(−λt)cos(ωt)dt = λ/(λ²+ω²)	CGB Deposition 4 © Christopher Gabriel Brown — builds on Leonhard Euler's Basel solution (1734) and Pierre-Simon Laplace's transform (c. 1785)
5 · Photon Chromosome Encoding	C = ⌊log₂(Δλ/δλ)⌋ + ⌊log₂(2π/δθ)⌋ + ⌊log₂(I_max/I_min)⌋ bits/photon	CGB Deposition 5 © Christopher Gabriel Brown — builds on Claude Shannon's information theory (1948) and Max Planck's quantum (1900)
6 · Quantum Counting Paradox	2ⁿ states ≠ 2ⁿ computations ⟹ P = sin²((2k+1)θ), k* = ⌊(π/4)√N⌋	CGB Deposition 6 © Christopher Gabriel Brown — builds on Lov Grover's quantum search algorithm (1996)
7 · Recursive Growth Bound	T(n) = a·T(n/b) + n^c ⟹ compare c against log_b(a)	CGB Deposition 7 © Christopher Gabriel Brown — builds on the Master Theorem of Jon Bentley, Dorothea Haken & James B. Saxe (1980)
8 · Thermal Noise Floor	E_bit ≥ k_B·T·ln2 ⟹ FLOPS_max = P_budget / (k_B·T·ln2)	CGB Deposition 8 © Christopher Gabriel Brown — builds on Rolf Landauer's limit (1961)
9 · Voxel Resonance	Ψ(v) = Π[αₖ sin(2πfₖ/Nₖ) + βₖ e^(−γₖd²)] · det(M_seed)	CGB Deposition 9 © Christopher Gabriel Brown — original composite formulation
10 · Zero-Point Fabrication	Σ ½ħω → ∞ ⟹ ζ(−1) = −1/12 ⟹ E_reg = −ħcπ²/(720d³)	CGB Deposition 10 © Christopher Gabriel Brown — builds on Hendrik Casimir (1948) and Euler–Riemann zeta regularization ζ(−1) = −1/12

Mathematics (8)

Instruction	Formula	Origin — who and what
Quadratic roots	ax² + bx + c = 0 ⟹ x = (−b ± √(b²−4ac)) / 2a	Classical mathematics — Babylonian tablets (~1800 BC); general solution tradition of al-Khwarizmi (c. 820). Unpatentable, public domain
Combinations & permutations	C(n,k) = n! / k!(n−k)! · P(n,k) = n!/(n−k)! (log-gamma — no overflow)	Classical combinatorics — Pingala (~200 BC), Blaise Pascal (1654); computed via Cornelius Lanczos's log-gamma approximation (1964). Public domain
Bayes' theorem	P(A\|B) = P(B\|A)·P(A) / [P(B\|A)·P(A) + P(B\|¬A)·P(¬A)]	Rev. Thomas Bayes (published 1763), generalized by Pierre-Simon Laplace (1774). Public domain
Normal distribution	φ(x) = e^(−(x−μ)²/2σ²) / σ√2π · Φ(x) via erf	Abraham de Moivre (1733), Carl Friedrich Gauss (1809); erf approximation from Abramowitz & Stegun's Handbook 7.1.26 (Hastings, 1964, US-Gov public domain)
Logistic growth	P(t) = K / (1 + ((K−P₀)/P₀)·e^(−rt))	Pierre François Verhulst (1838). Public domain
Compound & continuous growth	A = P(1 + r/n)^(nt) · A = P·e^(rt)	Classical finance mathematics; continuous compounding via Jacob Bernoulli (1683) and Euler's e. Public domain
Shannon channel capacity	C = B · log₂(1 + S/N)	Claude Shannon (1948), with Ralph Hartley (1928) — the Shannon–Hartley theorem. Public domain
Prime counting estimate	π(n) ≈ n/ln n · sharper: n/(ln n − 1)	Carl Friedrich Gauss & Adrien-Marie Legendre (1790s); Prime Number Theorem proved by Hadamard & de la Vallée Poussin (1896). Public domain

Physics & Energy (12)

Instruction	Formula	Origin — who and what
Mass–energy E = mc²	E = m·c²	Albert Einstein (1905). A law of nature — unpatentable, public domain
Kinetic energy & momentum	KE = ½mv² · p = mv	Classical mechanics — Isaac Newton (1687), Gottfried Leibniz's vis viva; modern ½mv² formalized 19th c. Public domain
Ohm's law solver	V = I·R · P = V·I (leave exactly one of V/I/R blank)	Georg Simon Ohm (1827); electrical power after James Prescott Joule. Public domain
Coulomb's law	F = k·q₁·q₂ / r²	Charles-Augustin de Coulomb (1785). Public domain
Ideal gas PV = nRT	P·V = n·R·T (leave exactly one blank)	Émile Clapeyron (1834), combining Robert Boyle (1662), Jacques Charles, Joseph Gay-Lussac and Amedeo Avogadro. Public domain
Carnot efficiency	η = 1 − T_cold / T_hot	Sadi Carnot (1824). Public domain
Battery C-rate & runtime	E = Ah·V · I = C·Ah · runtime = 1/C	Standard electrical-engineering C-rate convention (20th c. industry practice). No single originator; unpatentable arithmetic
Solar PV yield	E = A · η · H_sun · PR	Standard photovoltaic yield estimation (industry practice, late 20th c.). Unpatentable arithmetic
Wind turbine power	P = ½·ρ·A·v³·Cp (Betz limit Cp ≤ 16/27 ≈ 0.593)	Kinetic flux ½ρAv³ (classical); Betz limit — Albert Betz (1919), independently Frederick Lanchester (1915). Public domain
Radioactive decay	N(t) = N₀·e^(−λt), λ = ln2 / t½	Ernest Rutherford & Frederick Soddy (1902). A law of nature — public domain
Photon energy E = hc/λ	E = h·c / λ	Max Planck (1900) and Albert Einstein (1905). Public domain
Faraday electrolysis	m = Q·M / (z·F), Q = I·t	Michael Faraday's laws of electrolysis (1834). Public domain

Chemistry & Carbon (10)

Instruction	Formula	Origin — who and what
Dilution C₁V₁ = C₂V₂	C₁·V₁ = C₂·V₂ (leave exactly one blank)	Standard laboratory chemistry (C₁V₁ = C₂V₂ conservation). No single originator; unpatentable arithmetic
pH / pOH	pH = −log₁₀[H⁺] · pH + pOH = 14	Søren Peder Lauritz Sørensen (1909). Public domain
Henderson–Hasselbalch	pH = pKa + log₁₀([A⁻]/[HA])	Lawrence Joseph Henderson (1908) and Karl Albert Hasselbalch (1917). Public domain
Arrhenius rate	k = A·e^(−Ea/RT)	Svante Arrhenius (1889). Public domain
Gibbs free energy	ΔG = ΔH − T·ΔS · K = e^(−ΔG/RT)	Josiah Willard Gibbs (1873–1878); equilibrium link after Jacobus van 't Hoff. Public domain
Nernst equation	E = E° − (RT/zF)·ln Q	Walther Nernst (1889). Public domain
Beer–Lambert	A = ε·l·c	Pierre Bouguer (1729), Johann Heinrich Lambert (1760), August Beer (1852). Public domain
Hydrocarbon combustion — CO₂ ledger	CₓHᵧ + (x+y/4)O₂ → xCO₂ + (y/2)H₂O	Stoichiometry on Antoine Lavoisier's conservation of mass (1770s–80s). Public domain
Carbon-14 dating	t = (t½/ln2)·ln(N₀/N), t½ = 5,730 yr	Willard Libby (1946–49, Nobel 1960); 5,730-yr “Cambridge half-life” (1962). Public domain
Percent yield & atom economy	yield% = actual/theoretical · 100 · AE% = M(product)/ΣM(reactants) · 100	Standard chemistry; atom economy after Barry Trost (1991, concept — the arithmetic is unpatentable). Public domain

Medicine (10)

Instruction	Formula	Origin — who and what
Body mass index	BMI = kg / m²	Adolphe Quetelet (1832); the name “body mass index” after Ancel Keys (1972); WHO bands. Public domain
Body surface area (Mosteller)	BSA = √(cm·kg / 3600)	R. D. Mosteller (1987, NEJM). A published clinical formula — unpatentable
Creatinine clearance (Cockcroft–Gault)	CrCl = (140−age)·kg·(0.85 if female) / (72·SCr)	Donald Cockcroft & Matthew Gault (1976, Nephron). Public domain
Half-life & steady state	t½ = 0.693/k · steady state ≈ 5·t½	Standard pharmacokinetics, foundational work of Torsten Teorell (1937). Public domain
Loading dose	LD = C_target · Vd · kg / F	Standard clinical pharmacokinetics (Rowland & Tozer convention). Unpatentable arithmetic
Maintenance dosing	rate = Cl · C_ss / F	Standard clinical pharmacokinetics (Rowland & Tozer convention). Unpatentable arithmetic
Clearance ↔ Vd ↔ t½	Cl = k·Vd · t½ = 0.693·Vd/Cl	Standard pharmacokinetics — clearance/volume/half-life identities. Public domain
Mass dose → molar dose	n = dose / M(formula) — molar mass from the built-in engine	Standard chemistry applied to dosing; molar masses from IUPAC values. Unpatentable arithmetic
IV drip rate	gtt/min = volume·dropFactor / minutes	Standard clinical/nursing formula. No single originator; unpatentable arithmetic
Cardiac output (Fick)	CO = VO₂ / [(CaO₂ − CvO₂) · 10]	Adolf Eugen Fick's principle (1870). Public domain

Pharmacokinetics (10)

Instruction	Formula	Origin — who and what
Michaelis–Menten kinetics	v = V_max·C / (K_m + C)	Leonor Michaelis & Maud Menten (1913). Public domain
Hill dose–response	θ = Cⁿ / (EC50ⁿ + Cⁿ)	Archibald Vivian Hill (1910). Public domain
Accumulation & fluctuation	R = 1 / (1 − e^(−k·τ)), k = 0.693/t½	Standard pharmacokinetics (superposition principle). Unpatentable arithmetic
Steady-state infusion level	C_ss = R₀ / Cl · time to 90% ≈ 3.32·t½	Standard pharmacokinetics (Rowland & Tozer convention). Unpatentable arithmetic
Infusion rise curve	C(t) = (R₀/Cl)·(1 − e^(−k·t))	Standard pharmacokinetics — first-order rise to steady state. Public domain
AUC from dose & clearance	AUC = Dose·F / Cl	Standard pharmacokinetics — dose/clearance identity. Public domain
Volume of distribution	V_d = Dose / C₀	Standard pharmacokinetics — volume of distribution definition. Public domain
Protein-binding level correction (Sheiner–Tozer form)	C_corr = C_measured / (0.2·albumin + 0.1)	Sheiner & Tozer form (1979) — generic protein-binding correction. Public domain
Renal dose adjustment	Dose_adj = Dose × CrCl / 120	Standard renal dosing convention. Unpatentable arithmetic
Absolute bioavailability	F = (AUC_oral·Dose_IV) / (AUC_IV·Dose_oral)	Standard pharmacokinetics — AUC ratio definition. Public domain

M_RB (12)

Instruction	Formula	Origin — who and what
eGFR — CKD-EPI 2021 (race-free)	eGFR = 142·min(SCr/κ,1)^α·max(SCr/κ,1)^−1.200·0.9938^age·(1.012 if female)	Inker et al., CKD-EPI 2021 race-free equation (NEJM 2021). Published clinical formula — unpatentable
eGFR — MDRD (re-expressed)	eGFR = 175·SCr^−1.154·age^−0.203·(0.742 if female)	Levey et al., MDRD Study equation (1999; re-expressed 2006). Public domain
Measured creatinine clearance	CrCl = (U_Cr·V) / (P_Cr·t)	Standard renal physiology — clearance definition. Public domain
Fractional excretion of sodium	FENa = (U_Na·P_Cr) / (P_Na·U_Cr) × 100	C. H. Espinel (1976, JAMA). Public domain
Ideal & adjusted body weight (Devine)	IBW = 50 (♂) / 45.5 (♀) + 2.3·(inches over 60) · AdjBW = IBW + 0.4·(TBW − IBW)	Ben J. Devine (1974); adjusted-weight convention standard practice. Public domain
Body surface area (Du Bois)	BSA = 0.007184·H^0.725·W^0.425	Delafield Du Bois & Eugene F. Du Bois (1916). Public domain
Lean body weight (Janmahasatian)	♂ 9270·W/(6680+216·BMI) · ♀ 9270·W/(8780+244·BMI)	Janmahasatian et al. (2005, Clin Pharmacokinet). Published clinical formula — unpatentable
Free-water deficit	deficit = TBW_fraction·kg·(Na/140 − 1)	Standard clinical water-balance arithmetic. Unpatentable
Sodium corrected for glucose	Na_corr = Na + 1.6·(glucose − 100)/100	Katz correction (1973, NEJM). Public domain
Anion gap (albumin-corrected)	AG = Na − Cl − HCO₃ · corrected +2.5·(4 − albumin)	Standard acid–base chemistry; albumin correction after Figge et al. (1998). Public domain
Calcium corrected for albumin	Ca_corr = Ca + 0.8·(4 − albumin)	Payne et al. (1973, BMJ). Public domain
Serum osmolality & gap	calc = 2·Na + glucose/18 + BUN/2.8	Standard formula (Smithline & Gardner 1976 review of conventions). Public domain

M_CR (10)

Instruction	Formula	Origin — who and what
Mean arterial pressure	MAP = DBP + (SBP − DBP)/3	Standard hemodynamics. Public domain
Corrected QT (Bazett & Fridericia)	QTc_B = QT/√RR · QTc_F = QT/RR^(1/3), RR = 60/HR	Henry Cuthbert Bazett (1920) and Louis Sigurd Fridericia (1920). Public domain
Shock index	SI = HR / SBP	Allgöwer & Burri (1967). Public domain
Alveolar gas & A–a gradient	PAO₂ = FiO₂·(P_atm − 47) − PaCO₂/0.8 · A–a = PAO₂ − PaO₂	Standard alveolar gas equation (Fenn, Rahn & Otis lineage, 1946). Public domain
PaO₂ / FiO₂ ratio	P/F = PaO₂ / FiO₂	Standard ratio; severity bands per the Berlin definition (2012). Public domain
Oxygen content & delivery	CaO₂ = 1.34·Hb·SaO₂ + 0.003·PaO₂ · DO₂ = CO·CaO₂·10	Standard respiratory physiology — oxygen-content equation. Public domain
Systemic vascular resistance & stroke volume	SVR = 80·(MAP − CVP)/CO · SV = CO/HR	Standard hemodynamics (Ohm's-law analogy after Georg Ohm 1827). Public domain
Winters' formula	expected PaCO₂ = 1.5·HCO₃ + 8 (± 2)	R. W. Winters and colleagues (1967). Public domain
Minute & alveolar ventilation	V_E = RR·V_T · V_A = RR·(V_T − V_D)	Standard respiratory physiology. Public domain
Rapid shallow breathing index	RSBI = RR / V_T(L)	Karl Yang & Martin Tobin (1991, NEJM). Published clinical index — unpatentable

M_FD (8)

Instruction	Formula	Origin — who and what
Parkland burn resuscitation	24-h volume = 4 mL × kg × %TBSA · half in first 8 h	Charles R. Baxter, Parkland Memorial Hospital (1968). Public domain
Maintenance fluids (Holliday–Segar)	100/50/20 mL·kg⁻¹·day⁻¹ by 10-kg band · 4-2-1 hourly	Malcolm Holliday & William Segar (1957, Pediatrics). Public domain
Weight-based dose splitter	per-dose = (mg/kg/day × kg) / doses per day	Standard weight-based dosing arithmetic. Unpatentable
BSA-based dose	dose = mg/m² × BSA (Mosteller)	Standard oncology-style BSA dosing; BSA per R. D. Mosteller (1987). Public domain
Weight-based drip rate	mL/h = dose(µg/kg/min) × kg × 60 / conc(µg/mL)	Standard critical-care infusion arithmetic. Unpatentable
Milligrams ↔ milliequivalents	mEq = mg × valence / molecular weight	Standard chemistry — equivalent-weight arithmetic. Public domain
Infusion time & rate	hours = volume / rate	Standard clinical arithmetic. Unpatentable
Nutrition calories (TPN arithmetic)	kcal = 3.4·g_dextrose + 4·g_protein + 9·g_fat	Standard nutrition constants (Atwater-system lineage, 1890s). Public domain

M_LN (8)

Instruction	Formula	Origin — who and what
LDL — Friedewald	LDL = TC − HDL − TG/5 (mg/dL; invalid when TG > 400)	William Friedewald, Robert Levy & Donald Fredrickson (1972). Public domain
A1c → estimated average glucose	eAG = 28.7 × A1c − 46.7	Nathan et al., ADAG study (2008, Diabetes Care). Published formula — unpatentable
Number needed to treat	NNT = 1 / (CER − EER)	Laupacis, Sackett & Roberts (1988, NEJM). Public domain
Odds ratio & relative risk (2×2)	OR = ad/bc · RR = [a/(a+b)] / [c/(c+d)]	Standard epidemiology; odds ratio after Jerome Cornfield (1951). Public domain
Predictive values from prevalence	PPV = sens·p / [sens·p + (1−spec)(1−p)]	Bayes' theorem applied to diagnosis (Thomas Bayes 1763). Public domain
Resting energy (Mifflin–St Jeor)	10·kg + 6.25·cm − 5·age + 5 (♂) / − 161 (♀) · TDEE = BMR × activity	Mifflin & St Jeor (1990, Am J Clin Nutr). Published formula — unpatentable
Resting energy (Harris–Benedict)	♂ 66.5+13.75·kg+5.003·cm−6.755·age · ♀ 655.1+9.563·kg+1.850·cm−4.676·age	James Arthur Harris & Francis Gano Benedict (1919). Public domain
Body-fat estimate (Deurenberg)	BF% = 1.2·BMI + 0.23·age − 10.8·(1 if ♂) − 5.4	Paul Deurenberg et al. (1991, Br J Nutr). Published formula — unpatentable

Electrical (11)

Instruction	Formula	Origin — who and what
Resistors in series & parallel	R_series = ΣR · 1/R_parallel = Σ(1/R)	Circuit laws of Gustav Kirchhoff (1845) on Georg Ohm's law (1827). Public domain
Voltage divider	V_out = V_in · R₂/(R₁+R₂)	Standard circuit analysis (Ohm/Kirchhoff lineage). Public domain
Capacitor energy & charge	E = ½CV² · Q = CV	Classical electrostatics (Leyden-jar lineage, 18th c.; field energy after Maxwell). Public domain
Inductor energy	E = ½LI²	Classical electromagnetism (Faraday/Maxwell lineage, 19th c.). Public domain
RC charging curve	V(t) = V₀·(1 − e^(−t/RC))	Standard first-order transient analysis (19th-c. telegraphy lineage). Public domain
LC resonant frequency	f = 1 / (2π·√(LC))	William Thomson, Lord Kelvin (1853). Public domain
Transformer ratios	V_s/V_p = N_s/N_p · I_s/I_p = N_p/N_s	Michael Faraday's induction (1831); practical transformer 1880s. Public domain
Three-phase power	P = √3·V_L·I_L·pf	Standard AC power engineering (three-phase after Dolivo-Dobrovolsky & Tesla era, 1880s–90s). Public domain
Wire resistance & voltage drop	R = ρL/A · round-trip drop = 2·I·R	Pouillet's law form of Ohm's law (Claude Pouillet, 1837). Public domain
Power-factor correction	Q_c = P·(tanφ₁ − tanφ₂)	Standard AC power engineering. Unpatentable arithmetic
Ohmic (I²R) heating	P = I²R	Joule's first law — James Prescott Joule (1841). Public domain

E_GS (11)

Instruction	Formula	Origin — who and what
Hydroelectric power	P = ρ·g·Q·H·η	Classical hydraulics (Bernoulli lineage, 18th c.). Public domain
Pumped-storage energy	E = ρ·V·g·h·η	Classical mechanics — gravitational potential energy. Public domain
Flywheel energy	E = ½·I·ω²	Classical rotational mechanics (Euler lineage). Public domain
Capacity factor	CF = actual MWh / (rated MW × 8760)	Standard utility-industry metric. Unpatentable arithmetic
Solar array sizing	array kW = daily kWh / (sun-hours × PR)	Standard photovoltaic sizing practice. Unpatentable arithmetic
Battery bank sizing	Ah = kWh/day × autonomy days × 1000 / (V × DoD × η)	Standard off-grid sizing practice. Unpatentable arithmetic
Peukert battery runtime	t = H·(C/(I·H))^k	Wilhelm Peukert (1897). Public domain
Battery pack mass from density	mass = kWh × 1000 / (Wh/kg)	Arithmetic on published energy densities. Unpatentable
Generator EMF (Faraday)	EMF_peak = N·B·A·ω	Michael Faraday's law of induction (1831). Public domain
Heat-pump COP (Carnot bound)	COP_heat ≤ T_h/(T_h−T_c) · COP_cool ≤ T_c/(T_h−T_c)	Carnot bound — Sadi Carnot (1824). Public domain
Fuel mass → electricity	kWh = kg × MJ/kg / 3.6 × η	Standard calorimetry and plant-efficiency arithmetic. Public domain

E_TH (9)

Instruction	Formula	Origin — who and what
Sensible heat Q = mcΔT	Q = m·c·ΔT	Joseph Black's specific heat (18th c.). Public domain
Latent heat Q = mL	Q = m·L (water: fusion 334, vaporization 2257 kJ/kg)	Joseph Black's latent heat (1761). Public domain
Heat conduction (Fourier)	Q̇ = k·A·ΔT / d	Jean-Baptiste Joseph Fourier (1822). Public domain
Radiation (Stefan–Boltzmann)	P = ε·σ·A·T⁴	Josef Stefan (1879) and Ludwig Boltzmann (1884). Public domain
Newton's cooling	T(t) = T_a + (T₀−T_a)·e^(−kt)	Isaac Newton (1701). Public domain
Building U-value loss	Q̇ = U·A·ΔT · R = 1/U	Standard building physics (Fourier lineage). Public domain
Heating degree-day energy	E = UA·HDD·24 / 1000	Standard HVAC degree-day method (20th c. practice). Unpatentable arithmetic
Thermal expansion	ΔL = α·L·ΔT	Classical thermophysics. Public domain
Water-heating time	t = m·c·ΔT / P	Q = mcΔT applied (Joseph Black lineage). Public domain

E_FM (10)

Instruction	Formula	Origin — who and what
Vehicle road-load power	P = C_rr·m·g·v + ½·ρ·C_dA·v³	Standard vehicle dynamics (rolling + aerodynamic drag, Rayleigh drag lineage). Public domain
EV range & charge time	range = kWh×1000 / (Wh/km)	Standard EV engineering arithmetic. Unpatentable
Fuel economy conversion & cost	L/100km = 235.215 / mpg(US)	Unit conversion + arithmetic. Unpatentable
Stop-and-go energy & regen	E = ½mv² per stop · recovered = E×regen	Classical kinetic energy (Leibniz/Coriolis lineage). Public domain
Rocket equation (Tsiolkovsky)	Δv = v_e·ln(m₀/m_f)	Konstantin Tsiolkovsky (1903). Public domain
Escape velocity	v_e = √(2GM/r)	Newtonian gravitation (Isaac Newton, 1687). Public domain
Orbital period (Kepler)	T = 2π·√(a³/GM)	Johannes Kepler's third law (1619), form via Newton. Public domain
Power ↔ torque ↔ rpm	P = τ·ω, ω = rpm·2π/60	Classical mechanics; horsepower after James Watt (1780s). Public domain
Gravity storage E = mgh	E = m·g·h	Classical mechanics — potential energy. Public domain
Appliance running cost	cost = W × h/day × 365 × rate / 1000	Billing arithmetic. Unpatentable

E_NO (5)

Instruction	Formula	Origin — who and what
Binding energy from mass defect	E = Δm·c² · 1 u = 931.494 MeV	Albert Einstein (1905); mass-spectrograph defect work of Francis Aston (1919). Public domain
Fission energy per kilogram	E = N_atoms × ~200 MeV	Fission discovered by Hahn & Strassmann, explained by Meitner & Frisch (1938); ~200 MeV/fission standard nuclear engineering. Public domain
Radioactivity from mass	A = λN, λ = ln2/t½	Rutherford & Soddy decay law (1902). Public domain
Inverse-square intensity	I = I₀·(r₀/r)²	Classical geometry of radiation (Kepler/Newton lineage). Public domain
Photon flux from power	n = P·λ / (h·c)	Max Planck (1900) and Albert Einstein (1905). Public domain

Combustion & Fuels (11)

Instruction	Formula	Origin — who and what
Oxygenated-fuel combustion CₓHᵧO₂	CₓHᵧO_z + (x + y/4 − z/2)·O₂ → x·CO₂ + (y/2)·H₂O	Stoichiometry on Antoine Lavoisier's conservation of mass (1770s–80s). Public domain
Stoichiometric air–fuel ratio	AFR = (x + y/4 − z/2) × 137.9 / M_fuel (mass air per mass fuel)	Standard combustion engineering (stoichiometric air requirement). Public domain
Excess air & lambda	λ = AFR_actual / AFR_stoich · %EA = (λ−1)×100	Standard combustion engineering (lambda convention). Unpatentable arithmetic
Higher heating value (Dulong)	HHV ≈ 0.338·C% + 1.428·(H% − O%/8) + 0.095·S% MJ/kg	Pierre Louis Dulong's heating-value formula (19th c.). Public domain
LHV from HHV	LHV = HHV − 2.442 × 9·H%/100 (water-of-combustion penalty)	Standard calorimetry — latent-heat correction. Public domain
Combustion air volume	V_air = mol_fuel × (x+y/4−z/2) × 4.76 × 22.414 L (STP)	Standard combustion stoichiometry. Public domain
Biogas energy content	E = CH₄% × 35.8 MJ/m³	Standard fuel-gas calorimetry. Unpatentable arithmetic
Grid emission factor from fuel	kgCO₂/kWh = CO₂-per-kg ÷ (MJ/kg ÷ 3.6 × η)	Standard power-plant emission arithmetic. Unpatentable
Wood fuel vs moisture	LHV_as-received = LHV_dry·(1−M) − 2.442·M	Standard biomass-fuel calorimetry. Public domain
Volumetric energy density	MJ/L = MJ/kg × density	Arithmetic on published fuel properties. Unpatentable
Fuel cost per useful kWh	$/kWh = price ÷ (kWh-per-unit × η)	Billing arithmetic. Unpatentable

C_CC (12)

Instruction	Formula	Origin — who and what
Carbon ↔ CO₂ mass	CO₂ = C × 44.009/12.011 = C × 3.664	Molecular mass ratio 44.009/12.011 — a fact of chemistry. Public domain
Tree biomass → CO₂ stored	CO₂ = green mass × (1−moisture) × C-fraction × 3.664	Standard forestry carbon accounting (IPCC-style default fractions). Public domain
Forest sequestration rate	annual = area × rate	Standard forestry accounting arithmetic. Unpatentable
ppm ↔ gigatonnes	1 ppm CO₂ = 2.13 GtC = 7.81 GtCO₂	Standard carbon-cycle conversion (2.13 GtC per ppm). Public domain
Emissions → atmospheric rise	Δppm = GtCO₂ × AF / 7.81	Standard carbon-cycle bookkeeping (airborne fraction after Keeling-era studies). Public domain
CO₂ radiative forcing	ΔF = 5.35 × ln(C/C₀) W/m²	Myhre, Highwood, Shine & Stordal (1998, GRL) simplified forcing expression. Published formula — unpatentable
Doubling forcing & warming	ΔF₂ₓ = 5.35·ln2 ≈ 3.71 W/m² · ΔT = λ·ΔF	Svante Arrhenius (1896) lineage; modern coefficient after Myhre et al. (1998). Public domain
CO₂ concentration projection	C(t) = C₀ + rate × years (linear trend)	Linear projection on the record begun by Charles David Keeling (1958). Public domain
Dissolved CO₂ (Henry's law)	[CO₂(aq)] = k_H × pCO₂	William Henry (1803). Public domain
Carbonate speciation vs pH	pK₁ = 6.35, pK₂ = 10.33 — fractions of H₂CO₃*/HCO₃⁻/CO₃²⁻	Standard aquatic chemistry (Bjerrum speciation, after Niels Bjerrum). Public domain
Greenhouse gases → CO₂e	CO₂e = mass × GWP₁₀₀ (CH₄ 27.9 · N₂O 273)	GWP₁₀₀ factors per IPCC Sixth Assessment Report (2021) — public scientific record
Per-capita emissions	t/person = national MtCO₂ × 10⁶ / population	Arithmetic. Unpatentable

C_CM (13)

Instruction	Formula	Origin — who and what
Fullerene C₆₀ arithmetic	M(C₆₀) = 60 × 12.011 = 720.66 g/mol	Buckminsterfullerene discovered by Kroto, Curl & Smalley (1985, Nobel 1996); arithmetic on molar mass. Public domain
Graphene sheet mass	mass = area × 0.77 mg/m² (single layer)	Graphene isolated by Geim & Novoselov (2004, Nobel 2010); areal density from graphite lattice — a material fact
Carbon allotrope volume	V = m/ρ (graphite 2.266, diamond 3.514 g/cm³)	Arithmetic on published densities. Unpatentable
CO₂ gas density	ρ = P·M / (R·T)	Ideal-gas density (Clapeyron 1834 applied). Public domain
Dry-ice cooling	Q = m × 571 kJ/kg (sublimation)	Standard thermophysical constants. Public domain
Limestone calcination CO₂	CaCO₃ → CaO + CO₂ (44.01/100.09 by mass)	Standard industrial chemistry (lime burning, ancient practice; stoichiometry after Lavoisier). Public domain
Bicarbonate buffer pH	pH = 6.35 + log₁₀([HCO₃⁻]/[H₂CO₃*])	Henderson–Hasselbalch (1908/1917) applied to carbonic acid. Public domain
Adsorption — Freundlich	q = K·C^(1/n)	Herbert Freundlich (1907). Public domain
Adsorption — Langmuir	q = q_m·K·C / (1 + K·C)	Irving Langmuir (1918, Nobel 1932). Public domain
CO₂ compression work (isothermal)	W = n·R·T·ln(P₂/P₁)	Classical thermodynamics — isothermal ideal-gas work (Boyle/Clapeyron lineage). Public domain
δ¹³C isotope signature	δ¹³C = (R_sample/R_VPDB − 1) × 1000 ‰ · R_VPDB = 0.011180	Standard isotope geochemistry on the VPDB scale (program of Harold Urey's school, 1950s). Public domain
Photosynthesis energy ledger	6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ · ΔG ≈ +2,870 kJ/mol	Standard biochemistry; carbon-fixation pathway after Melvin Calvin (1950s, Nobel 1961). Public domain
Carbon-fiber composite modulus	E = V_f·E_f + V_m·E_m (rule of mixtures)	Rule of mixtures after Woldemar Voigt (1889). Public domain

C_CA (9)

Instruction	Formula	Origin — who and what
Driving CO₂ from fuel economy	kg = L/100km × km/100 × 2.31 (petrol) / 2.68 (diesel)	Stoichiometric fuel factors (2.31/2.68 kg per liter) — standard emission arithmetic. Public domain
Flight CO₂ estimate	kg ≈ km × factor × class multiplier	Standard aviation emission-factor averages. Unpatentable arithmetic
Electricity CO₂	kg = kWh × grid factor / 1000	Standard grid-factor arithmetic. Unpatentable
Fuel volume → CO₂ (stoichiometric)	petrol 2.31 · diesel 2.68 kg CO₂ per liter	Stoichiometric fuel factors — standard. Public domain
Natural gas → CO₂ (computed)	CH₄: 16.04 g/mol at 22.414 L/mol → ×44.009/16.043 CO₂	Computed from molecular ratios (CH₄ → CO₂). Public domain
Carbon price cost	cost = tCO₂ × price	Arithmetic. Unpatentable
Tree-planting offset estimate	trees = annual tCO₂ ÷ ~0.021 t/tree/yr (rough average)	Common forestry average — arithmetic on published estimates. Unpatentable
Linear net-zero pathway	annual cut = current ÷ (target year − now)	Arithmetic. Unpatentable
Carbon-capture energy penalty	extra fuel = p / (1 − p)	Standard carbon-capture engineering arithmetic. Public domain

Earth & Sky (8)

Instruction	Formula	Origin — who and what
Great-circle distance (haversine)	d = 2R·asin(√(sin²(Δφ/2) + cosφ₁·cosφ₂·sin²(Δλ/2)))	Classical spherical trigonometry; haversine tables of James Andrew (1805), computing use popularized by Roger W. Sinnott (1984). Public domain
Pressure vs altitude (barometric)	P = P₀·e^(−M·g·h / (R·T)) (isothermal approximation)	Barometric formula — classical atmospheric physics (Laplace lineage, 19th c.). Public domain
Earthquake energy (Gutenberg–Richter)	log₁₀E(J) = 1.5·M + 4.8	Beno Gutenberg & Charles Richter energy–magnitude relation (1956). Public domain
Wind chill (NWS 2001)	WC = 13.12 + 0.6215T − 11.37v⁰·¹⁶ + 0.3965T·v⁰·¹⁶ (°C, km/h)	JAG/TI wind-chill formula — US National Weather Service & Environment Canada (2001). Government work, public
Heat index (Rothfusz)	NWS regression on Steadman's apparent temperature (°F, RH %)	Lans P. Rothfusz (1990, NWS), regression on Robert Steadman's model (1979). Government work, public
Dew point (Magnus)	T_d = b·γ/(a−γ), γ = ln(RH/100) + aT/(b+T) (a 17.62, b 243.12)	Magnus–Tetens approximation — Gustav Magnus (1844), Otto Tetens (1930). Public domain
Doppler shift (sound)	f' = f·v/(v − v_source) (approaching)	Christian Doppler (1842). Public domain
Decibels — combine & compare	dB = 10·log₁₀(I/I₀) · combined = 10·log₁₀(10^(a/10)+10^(b/10))	The decibel, after Alexander Graham Bell (Bell System convention, 1920s); standard acoustics. Public domain

O_IC (6)

Instruction	Formula	Origin — who and what
Nyquist sampling & signalling	f_s ≥ 2·f_max · C = 2B·log₂(M)	Harry Nyquist (1928); sampling theorem after Claude Shannon (1949) and Vladimir Kotelnikov (1933). Public domain
Amdahl's law	S = 1 / ((1−p) + p/N)	Gene Amdahl (1967). Public domain
Little's law	L = λ·W (in-system = arrival rate × time in system)	John D. C. Little (1961, proof of L = λW). Public domain
Moore's-law projection	N = N₀ · 2^(years/2) (the 1975 two-year cadence)	Gordon Moore's observation (1965; two-year cadence 1975). A published observation — unpatentable
Password entropy	bits = length · log₂(charset)	Information theory of Claude Shannon (1948) applied to passphrases. Public domain
Data-transfer time	t = size × 8 / rate	Arithmetic. Unpatentable

O_FE (4)

Instruction	Formula	Origin — who and what
Loan payment (amortization)	M = P·r·(1+r)ⁿ / ((1+r)ⁿ − 1)	Classical annuity/amortization mathematics (17th–18th c. actuarial lineage). Public domain
Compound annual growth rate	CAGR = (end/start)^(1/years) − 1	Standard compound-growth arithmetic (Bernoulli/Euler lineage). Public domain
Break-even point	units = fixed / (price − variable)	Standard cost–volume–profit accounting (20th c. practice). Unpatentable arithmetic
Inflation — future cost & real value	future = A·(1+i)ʸ · real = A/(1+i)ʸ	Compound-interest arithmetic applied to price levels. Public domain

Probability & Chance (12)

Instruction	Formula	Origin — who and what
Binomial distribution	P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ	Jacob Bernoulli, Ars Conjectandi (1713). Public domain
Poisson distribution	P(k) = λᵏ·e^(−λ) / k!	Siméon Denis Poisson (1837). Public domain
Geometric wait for first success	P(first at n) = p·(1−p)ⁿ⁻¹ · E = 1/p	Classical probability (Bernoulli-trial lineage). Public domain
Exponential waiting time	P(T > t) = e^(−t/τ)	Classical probability — the memoryless law. Public domain
Birthday problem	P(shared) = 1 − ∏(1 − i/365)	Classic problem; formal treatment popularized by Richard von Mises (1939). Public domain
Coupon collector	E(draws to collect all n) = n·(1 + ½ + ⅓ + … + 1/n)	Classical problem (De Moivre/Euler lineage; harmonic numbers). Public domain
Random walk (drunkard's walk)	RMS distance after n steps = √n · E\|d\| = √(2n/π)	Karl Pearson posed 'the problem of the random walk' (1905); recurrence after George Pólya (1921). Public domain
Normal probability of a range	P(a < X < b) = Φ(b) − Φ(a)	De Moivre (1733) and Laplace — the normal law of errors. Public domain
Surprisal — bits of surprise	I = −log₂(p) · independent surprises add	Claude Shannon's information theory (1948). Public domain
Benford's law	P(first digit d) = log₁₀(1 + 1/d)	Simon Newcomb (1881), rediscovered by Frank Benford (1938). Public domain
Evidence engine (odds & decibans)	posterior odds = prior odds × LRⁿ · decibans = 10·log₁₀(LR)	Bayes in odds form; decibans and weight-of-evidence after Alan Turing & I. J. Good (Banburismus, 1940s; Good 1950). Public domain
Streak probability	P(a run of k heads somewhere in n fair flips) — exact recursion	Runs in coin tossing — classical, after Abraham de Moivre (18th c.). Public domain

F_GW (8)

Instruction	Formula	Origin — who and what
Dice sum probability	P(sum = s) for n six-sided dice — exact convolution	Exact dice distributions via convolution — Abraham de Moivre (1718). Public domain
Lottery odds (hypergeometric)	P(match m) = C(k,m)·C(n−k, k−m) / C(n,k)	Hypergeometric distribution — classical combinatorics. Public domain
Gambler's ruin	fair game: P(ruin) = 1 − s/t · biased: ((q/p)ˢ−1)/((q/p)ᵗ−1)	Pascal–Fermat correspondence (1654); solved form by Christiaan Huygens (1657). Public domain
Expected value of a wager	EV = Σ value·probability · Var = Σ p·(v−EV)²	Expected value — Christiaan Huygens (1657). Public domain
Kelly criterion	f* = (b·p − q) / b (b = net odds, q = 1−p)	John L. Kelly Jr. (1956, Bell System Technical Journal). Published formula — unpatentable
Monty Hall	stay = 1/N · switch = (N−1)/(N·(N−k−1)) after k doors opened	Steve Selvin (1975); made famous by Marilyn vos Savant (1990). Public domain
Drake equation	N = R★ · f_p · n_e · f_l · f_i · f_c · L	Frank Drake (1961). A framework of probabilities — public domain
Fair-walk expected duration	E(bets until ruin or target) = s·(t − s) (fair game)	Classical fair-walk duration result (Markov-chain lineage). Public domain

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AlchemyCalc Fortuna V5

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