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    "Equal-area sharp wye: h = K v_c² / (2g); K referenced to common-channel speed; α ∈ (0, 90°); K is reported, not an input",
    "Branch K=1+q²−2q cos α. Dividing run K=0.4(1−q)². Combining run K=1−q²−2q(1−q)cos α. q=Q_path/Q_c in [0,1]; default g=9.80665 m/s²; default ρ=1000 kg/m³",
    "Not a 90° tee, 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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  "limitations": [
    "Equal-area sharp wye: h = K v_c² / (2g); K referenced to common-channel speed; α ∈ (0, 90°); K is reported, not an input",
    "Branch K=1+q²−2q cos α. Dividing run K=0.4(1−q)². Combining run K=1−q²−2q(1−q)cos α. q=Q_path/Q_c in [0,1]; default g=9.80665; default ρ=1000 for Δp",
    "Not a 90° tee, 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, q_ratio, or alpha_deg when required → MISSING_REQUIRED_INPUT",
    "v_m_s ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE",
    "q_ratio > 1, α ∉ (0, 90), or K=0 on velocity invert → VALUE_OUT_OF_RANGE",
    "K, rd, n_miters, θ, f, Q, A, v1/v2, ri_d, α=90° → MIXED_INPUT_ENCODING",
    "flush, circular, or unknown kind/path/mode → INVALID_MODE"
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  "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"
    },
    {
      "title": "Idelchik — Handbook of Hydraulic Resistance",
      "section": "Diagrams 7-21 / 7-29 (dividing branch) and 7-1 (combining); 7-20 No.1 dividing run; equal-area wye",
      "supports": "Branch K=1+q²−2q cos α; dividing run K=0.4(1−q)²; combining run K=1−q²−2q(1−q)cos α; K referenced to v_c. Not a 90° tee, not catalog K, not a bend",
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      "detail": "dividing run K=0.4(1−q)²",
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