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  "assumptions": [
    "Concentric reducer: h = K v₂² / (2g); K = 0.5 (1 − A₂/A₁); K referenced to downstream speed; K is reported, not an input",
    "A₂/A₁ in (0, 1); default g=9.80665 m/s²; default ρ=1000 kg/m³; incompressible; geometry from area_ratio or A1+A2 or D1+D2",
    "Not sudden expansion (v1−v2)², not gradual contraction α, not vena-contracta C_c, not catalog minor-loss K given, not a four-way cross, not Q=Av"
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    "Concentric reducer: h = K v₂² / (2g); K = 0.5 (1 − A₂/A₁); K referenced to downstream speed; K is reported, not an input",
    "A₂/A₁ in (0, 1); default g=9.80665; default ρ=1000 for Δp; incompressible; geometry from area_ratio or A1+A2 or D1+D2",
    "Not sudden expansion (v1−v2)², not gradual contraction α, not vena-contracta C_c, not catalog minor-loss K given, not a four-way cross, not Q=Av"
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    "missing v_m_s, h, or area_ratio when required → MISSING_REQUIRED_INPUT",
    "v_m_s ≤ 0, h ≤ 0, or area_ratio ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "area_ratio ≥ 1 or K=0 on velocity invert → VALUE_OUT_OF_RANGE",
    "mixed area_ratio with A1+A2 or D1+D2, C_c, α, v1, K given, Q, f, kind, sigma, beta → MIXED_INPUT_ENCODING",
    "unknown mode → INVALID_MODE"
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      "title": "ISO 80000-4 — Mechanics",
      "section": "Length, speed, pressure",
      "url": "https://www.iso.org/standard/31895.html",
      "supports": "h is a head; v is a speed; Δp is a pressure",
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    {
      "title": "Idelchik — Handbook of Hydraulic Resistance",
      "section": "Concentric reducer; K referenced to the downstream section",
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