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    "Equal-area sharp 90° four-way cross: h = K v_c² / (2g); K referenced to common-channel speed; two opposite laterals are equal; K is reported, not an input",
    "Dividing: branch K=(1+q)², run K=0.4(1−q)². Combining: branch K=1+q²−4(1−2q)²/(2−q), run K=1+q²−q²/[1+q(0.75+0.25q)²]. Branch q≤0.5; default g=9.80665 m/s²; default ρ=1000 kg/m³",
    "Not a 90° tee, not a non-90° wye, not catalog minor-loss K given, not circular-arc R/D, not mitered n, not pipe-entrance geometry, not pipe-exit K=1, not Q=Av"
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    "Equal-area sharp 90° four-way cross: h = K v_c² / (2g); K referenced to common-channel speed; two opposite laterals are equal; K is reported, not an input",
    "Dividing: branch K=(1+q)², run K=0.4(1−q)². Combining: branch K=1+q²−4(1−2q)²/(2−q), run K=1+q²−q²/[1+q(0.75+0.25q)²]. Branch q≤0.5; default g=9.80665; default ρ=1000 for Δp",
    "Not a 90° tee, not a non-90° wye, not catalog minor-loss K given, not circular-arc R/D, not mitered n, not pipe-entrance geometry, not pipe-exit K=1, not Q=Av"
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    "missing v_m_s, h, or q_ratio when required → MISSING_REQUIRED_INPUT",
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    "q_ratio > 1, branch q>0.5, or K≤0 on velocity invert → VALUE_OUT_OF_RANGE",
    "K, rd, n_miters, θ, α, f, Q, A, v1/v2, ri_d, q_opp, n_legs → MIXED_INPUT_ENCODING",
    "flush, circular, perpendicular, colliding, or unknown kind/path/mode → INVALID_MODE"
  ],
  "sources": [
    {
      "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",
      "kind": "standard"
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    {
      "title": "Idelchik — Handbook of Hydraulic Resistance",
      "section": "Diagram 7-21 welded equal-area 90° cross; dividing through 7-20 No.1; combining recovery 0.75+0.25q",
      "supports": "Dividing branch K=(1+q)² referenced to v_c; dividing run K=0.4(1−q)²; combining run uses the welded-cross recovery term. Not a 90° tee, not catalog K, not a wye",
      "kind": "reference"
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