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  "assumptions": [
    "Three-phase path uses balanced line-to-line RMS voltage.",
    "PF on the V×I path is total power factor λ = |P|/S in [0, 1]; blank PF defaults to 1.",
    "I²R path assumes resistive power dissipation (I + R inputs). V²/R is the Ohm’s-law identity, not a third tab.",
    "Energy path is average power P = E/Δt with Δt > 0."
  ],
  "limitations": [
    "V×I path is real power P, not apparent power S (use amps-to-va)",
    "Three-phase is balanced sinusoidal L-L only",
    "I²R tab is I+R only; V²/R is Ohm’s law, not a third tab",
    "PF blank → 1; PF=0 → P=0; PF outside [0,1] → INVALID_POWER_FACTOR",
    "Δt ≤ 0 → INVALID_TIME_INTERVAL (not a soft 0 W)",
    "Sinusoidal PF teaching model; true PF with harmonics is not computed"
  ],
  "boundary_conditions": [
    "V×I path: zero V or I → 0 W",
    "Blank / omitted PF defaults to 1; PF = 0 is valid and yields P = 0",
    "PF < 0 or PF > 1 → INVALID_POWER_FACTOR",
    "I²R path: zero R or I → 0 W; negative R → VALUE_MUST_BE_NON_NEGATIVE",
    "Energy path: Δt ≤ 0 → INVALID_TIME_INTERVAL (not a soft 0 W)",
    "Zero energy with Δt > 0 → 0 W",
    "Non-finite inputs → INVALID_NUMBER"
  ],
  "sources": [
    {
      "title": "Watt’s law / power identity",
      "section": "Real power P = V × I × PF (1φ); P = √3 × V × I × PF (3φ L-L)",
      "supports": "P from V×I path with phase and PF",
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    },
    {
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      "section": "P = I²R = V²/R for resistive dissipation",
      "supports": "P from I²R mode",
      "kind": "definition"
    },
    {
      "title": "Energy–power relation",
      "section": "Average power P = E / Δt (J/s → W)",
      "supports": "P from energy mode",
      "kind": "definition"
    },
    {
      "title": "IEC 60050 — International Electrotechnical Vocabulary",
      "section": "Active power (IEV 131-11-42)",
      "url": "https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-11-42",
      "supports": "P is active power; unit watt (W). V×I path computes P, not apparent power S.",
      "kind": "standard",
      "accessed": "2026-09-05",
      "clause": "IEV 131-11-42"
    },
    {
      "title": "IEC 60050 — International Electrotechnical Vocabulary",
      "section": "Power factor (IEV 131-11-46)",
      "url": "https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-11-46",
      "supports": "λ = |P| / S; AC V×I path multiplies by λ",
      "kind": "standard",
      "accessed": "2026-09-05",
      "clause": "IEV 131-11-46"
    },
    {
      "title": "IEEE Std 1459-2025",
      "section": "Power factor and real vs apparent power",
      "url": "https://standards.ieee.org/ieee/1459/7578/",
      "supports": "PF on AC V×I path; power triangle context",
      "kind": "standard",
      "accessed": "2026-09-05",
      "publisher": "IEEE",
      "published": "2025-05-16",
      "errata_checked": true
    },
    {
      "title": "BIPM SI Brochure (9th edition, version 4.01)",
      "section": "SI units for watt, volt, ampere, and joule",
      "url": "https://www.bipm.org/en/publications/si-brochure",
      "supports": "W = J/s; DC P = V·I; P = I²R = V²/R for ohmic dissipation",
      "kind": "reference",
      "accessed": "2026-09-05",
      "published": "2026-06",
      "doi": "https://doi.org/10.59161/AUEZ1291"
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    {
      "title": "NIST Guide to the SI (SP 811)",
      "section": "Watt as joule per second",
      "url": "https://www.nist.gov/pml/special-publication-811",
      "supports": "W = J/s; SI relationships for V, A, Ω, W",
      "kind": "reference"
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