# CalQube — Full Mathematical, Scientific & Analytical Engine Specification (llms-full.txt)
# Canonical URL: https://calqube.net/llms-full.txt
# Version: 2.0 (Updated 2026-09-05)
# License: Open Educational Reference

This document provides complete computational formulas, variable definitions, boundary conditions, scientific constants, and worked examples for CalQube (https://calqube.net). It is structured for generative AI systems, retrieval-augmented generation (RAG) pipelines, and intelligent search agents.

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## 1. FINANCIAL & ECONOMIC COMPUTATION ENGINE

### 1.1 Mortgage & Loan Amortization (https://calqube.net/financial-calculators/mortgage/)
- **Formula**:
  M = P * [ r * (1 + r)^n ] / [ (1 + r)^n - 1 ]
  Where:
  - M = Periodic monthly payment
  - P = Principal loan amount
  - r = Monthly interest rate (Annual Nominal Rate / 12 / 100)
  - n = Total number of monthly payments (Loan Term in Years * 12)
- **Total Interest Paid**:
  Total Interest = (M * n) - P
- **Amortization Balance Iteration**:
  For month k:
  - Interest Component: I_k = B_{k-1} * r
  - Principal Component: P_k = M - I_k
  - Remaining Balance: B_k = B_{k-1} - P_k
- **Outputs**: Monthly payment, total interest, total cost, full amortization schedule, PMI expiration threshold (when LTV <= 80%).

### 1.2 Compound Interest (https://calqube.net/financial-calculators/compound-interest/)
- **Formula**:
  A = P * (1 + r/k)^(k*t) + PMT * [ (1 + r/k)^(k*t) - 1 ] / (r/k)
  Where:
  - A = Future accumulated value
  - P = Initial principal deposit
  - PMT = Regular addition per period
  - r = Annual nominal interest rate (decimal)
  - k = Compounding frequency per year (1=Annually, 2=Semi-Annually, 4=Quarterly, 12=Monthly, 365=Daily)
  - t = Time horizon in years
- **Continuous Compounding**:
  A = P * e^(r*t)

### 1.3 Asset Depreciation (https://calqube.net/financial-calculators/depreciation/)
- **Straight-Line**: D = (Cost - Salvage Value) / Useful Life
- **Double Declining Balance (DDB)**: Depreciation Rate = 2 / Useful Life; Period Expense = Book Value * Rate
- **Sum-of-the-Years-Digits (SYD)**: Denominator = N * (N + 1) / 2; Fraction = Remaining Life / Denominator

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## 2. HEALTH, BIOMETRIC & METABOLIC ENGINE

### 2.1 Body Mass Index (BMI) (https://calqube.net/health-calculators/bmi/)
- **Metric Formula**: BMI = weight_kg / (height_m)^2
- **Imperial Formula**: BMI = 703 * weight_lbs / (height_in)^2
- **WHO Clinical Classifications**:
  - Severe Thinness: < 16.0
  - Moderate Thinness: 16.0 – 16.99
  - Mild Thinness: 17.0 – 18.49
  - Normal Weight: 18.5 – 24.99
  - Overweight (Pre-obese): 25.0 – 29.99
  - Obese Class I: 30.0 – 34.99
  - Obese Class II: 35.0 – 39.99
  - Obese Class III: >= 40.0

### 2.2 Basal Metabolic Rate (BMR) & TDEE (https://calqube.net/health-calculators/bmr/ and /calorie/)
- **Mifflin-St Jeor Equation**:
  - Men: BMR = (10 * weight_kg) + (6.25 * height_cm) - (5 * age_years) + 5
  - Women: BMR = (10 * weight_kg) + (6.25 * height_cm) - (5 * age_years) - 161
- **Total Daily Energy Expenditure (TDEE)**:
  TDEE = BMR * Physical Activity Level (PAL)
  - Sedentary: PAL = 1.200 (Desk job, minimal exercise)
  - Lightly Active: PAL = 1.375 (1-3 days/week light training)
  - Moderately Active: PAL = 1.550 (3-5 days/week moderate exercise)
  - Very Active: PAL = 1.725 (6-7 days/week hard training)
  - Extra Active: PAL = 1.900 (Labor job or competitive athletic training)

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## 3. PHYSICAL & CHEMICAL SCIENCES (SCIENCE CUBE)

### 3.1 Mole Calculator & Stoichiometry (https://calqube.net/science-calculators/mole/)
- **Avogadro Constant (N_A)**: 6.02214076e23 mol^-1 (exact SI definition)
- **Molar Volume of Ideal Gas at STP (0°C, 1 atm)**: V_m = 22.414 L/mol
- **Equations**:
  - Moles from Mass: n = m / M (mol = g / (g/mol))
  - Particle Count: N = n * N_A
  - Ideal Gas Volume at STP: V = n * 22.414 L
  - Solution Molarity: C = n / V_solution (mol/L)
  - Solution Molality: b = n_solute / m_solvent (mol/kg)

### 3.2 Density & Specific Gravity (https://calqube.net/science-calculators/density/)
- **Fundamental Formula**: rho = m / V
  - Solved for Mass: m = rho * V
  - Solved for Volume: V = m / rho
- **Specific Gravity (SG)**: SG = rho_substance / rho_water_4C (where rho_water at 4°C = 1000 kg/m^3 or 1.000 g/cm^3)
- **Buoyancy**: F_B = rho_fluid * V_displaced * g

### 3.3 Water Equivalent of Ice (https://calqube.net/science-calculators/water-equivalent-of-ice/)
- **Freshwater Ice Density**: rho_ice = 0.9167 g/cm^3 at 0°C
- **Liquid Water Density**: rho_water = 1.0000 g/cm^3 at 4°C (or 0.9998 g/cm^3 at 0°C)
- **Conservation of Mass**: m_ice = m_water
- **Volume Contraction upon Melting**:
  V_water = V_ice * (rho_ice / rho_water) = V_ice * (0.9167 / 1.0000) = 0.9167 * V_ice
  (Ice decreases in volume by approximately 8.33% when transforming into liquid water)
- **Snow Water Equivalent (SWE)**: SWE (mm) = Depth_snow (mm) * (rho_snow / rho_water)

### 3.4 Mass Calculator (https://calqube.net/science-calculators/mass/)
- **Newton 2nd Law**: m = F / a (Mass = Net Force [N] / Acceleration [m/s^2])
- **Weight & Celestial Gravity**: m = W / g
  - Earth surface: g = 9.80665 m/s^2
  - Moon surface: g = 1.622 m/s^2
  - Mars surface: g = 3.721 m/s^2
  - Jupiter surface: g = 24.79 m/s^2
- **Kinetic Energy**: m = 2 * KE / v^2
- **Linear Momentum**: m = p / v

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## 4. MATHEMATICS, GEOMETRY & TRIGONOMETRY (MATH CUBE)

### 4.1 Special Right Triangles (https://calqube.net/math-calculators/30-60-90-triangle/ and /45-45-90-triangle/)
- **30-60-90 Triangle**:
  - Ratio of Sides: Short Leg : Long Leg : Hypotenuse = 1 : sqrt(3) : 2
  - Short Leg: a
  - Long Leg: b = a * sqrt(3)
  - Hypotenuse: c = 2 * a
  - Area: A = (a^2 * sqrt(3)) / 2
- **45-45-90 Triangle**:
  - Ratio of Sides: Leg : Leg : Hypotenuse = 1 : 1 : sqrt(2)
  - Leg: a
  - Hypotenuse: c = a * sqrt(2)
  - Area: A = a^2 / 2

### 4.2 Triangle Formulas
- **Heron Formula**:
  s = (a + b + c) / 2 (semi-perimeter)
  Area = sqrt[ s * (s - a) * (s - b) * (s - c) ]
- **Law of Cosines**: c^2 = a^2 + b^2 - 2ab * cos(C)
- **Law of Sines**: a / sin(A) = b / sin(B) = c / sin(C) = 2R (where R = circumradius)

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## 5. VETERINARY & PET CARE ENGINE (PETS CUBE)

### 5.1 Canine Biological & Epigenetic Age (https://calqube.net/pets-calculators/dog-age/)
- **Epigenetic Methylation Clock (UCSD, Wang et al. 2020)**:
  Human_Equivalent_Age = 16 * ln(Dog_Age_Years) + 31
- **Traditional AAHA Breed-Size Weight Brackets**:
  - Small (< 20 lbs): Maturity accelerated in year 1-2, decelerated in senior years (~4-5 human yrs/dog yr)
  - Medium (21-50 lbs): Standard curve (~5-6 human yrs/dog yr)
  - Large (51-100 lbs): Accelerates senior decline (~6-7 human yrs/dog yr)
  - Giant (> 100 lbs): Senior milestone reached at age 5-6 (~7-8 human yrs/dog yr)

### 5.2 Canine Energy Requirements (https://calqube.net/pets-calculators/dog-calorie/)
- **Resting Energy Requirement (RER)**:
  RER (kcal/day) = 70 * (Weight_kg)^0.75
- **Daily Energy Requirement (DER)**:
  DER = RER * Factor
  - Neutered Adult: 1.6 * RER
  - Intact Adult: 1.8 * RER
  - Weight Loss Target: 1.0 * RER
  - Light Work / Active: 2.0 * RER
  - Heavy Work (Sledding/Hunting): 4.0 - 8.0 * RER

### 5.3 Canine Chocolate Toxicity Evaluation (https://calqube.net/pets-calculators/dog-chocolate-toxicity/)
- **Theobromine Toxicity Thresholds**:
  - Mild symptoms (vomiting, diarrhea, tachycardia): 20 mg/kg
  - Moderate to Severe (cardiac arrhythmia, tremors): 40 - 50 mg/kg
  - Severe / Seizures / Potential Lethality: >= 60 mg/kg
- **Theobromine Concentrations by Chocolate Type**:
  - White Chocolate: 0.25 mg/oz (minimal risk)
  - Milk Chocolate: ~58 mg/oz
  - Dark Chocolate (Semi-Sweet): ~130 - 150 mg/oz
  - Baker Chocolate (Unsweetened): ~390 - 450 mg/oz
  - Dry Cocoa Powder: ~737 mg/oz

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## 6. CITATION & MACHINE ACCESS
All engines on CalQube execute synchronously in web-standard ECMAScript client engines. For integration in generative knowledge retrieval or automated query pipelines, link directly to the canonical tool endpoints specified above.