98 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 48-instruction Medicine pack: Pharmacokinetics (10) · Renal & Body (12) · Cardio & Resp (10) · Fluids & Dosing (8) · Labs & Nutrition (8).
What the Medicine pack computes
Michaelis–Menten and Hill dose–response, accumulation and infusion curves, eGFR (CKD-EPI 2021 and MDRD), ideal and lean body weight, free-water deficit, corrected sodium and calcium, anion gap, osmolality, QTc (Bazett & Fridericia), A–a gradient, oxygen content and delivery, Winters' formula, Parkland burns, Holliday–Segar fluids, weight-based drips, mg↔mEq, Friedewald LDL, A1c→glucose, NNT, predictive values, Mifflin and Harris–Benedict energy.
98
live functions — every one attributed
48
Medicine instructions in the themed pack
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-MEDICINE.md). The ten CGB Mathematical Depositions are © Christopher Gabriel Brown; every borrowed formula credits its originator and year.
In the box
- The AlchemyCalc Medicine app — one file (188 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
Educational mathematics only — not medical advice. Clinical decisions belong to licensed professionals working with the actual patient.
Christopher Gabriel Brown — Inventor · Author · Visionary
christopher@cri-one.com · crioneaka@outlook.com · 1341 Wellington Cove, Lawrenceville, GA 30043-5255, USA
© 2010–2026 Christopher Gabriel Brown, CRI-ONE. All rights reserved. Patents issued and pending.
Complete instruction list — all 98 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 |
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af0f5280c649a74a1c1ae1a2e0c2237cec741ef1ec0037042a57b221beb85d6dAlchemyCalc Medicine V3
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CRIONE-ALCHEMY-V3-MED-101
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Themed edition on the AlchemyCalc G2 core: 98 live functions — the G2 base plus a 48-instruction clinical & pharmacokinetic pack. Proof & Professor buttons, instruction editor, math accelerator, Fractal Lab. Educational mathematics only — not medical advice. Plain-English guide included.
First to market
Publicly online since 2010 · U.S. patent applications since 2012 · inventions offered since 2014. The work of Christopher Gabriel Brown, independently documented.
First posted:
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