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  "assumptions": [
    "Trapezoid uniform flow: Q=(1/n) A R^{2/3} S^{1/2}; A=(b+z yn) yn; P=b+2 yn √(1+z²); SI k=1; z>0 is run/rise; yn is unknown in depth mode",
    "yn/b∈[0.01, 10]; n∈[0.008, 0.15]; S∈[1e-6, 0.1]; z∈(0, 10]; geometry is b and z, not A and R",
    "Not rectangular yn; not trapezoid yc; not Manning Q with given A,R; not Chezy C; not Q=Av"
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    "Trapezoid uniform-flow normal depth: invert SI Manning Q=(1/n) A R^{2/3} S^{1/2} with A=(b+z yn) yn, P=b+2 yn √(1+z²), R=A/P; yn is unknown; n, S, b, z given",
    "yn/b∈[0.01, 10]; n∈[0.008, 0.15]; S∈[1e-6, 0.1]; z∈(0, 10]; SI k=1; geometry is b and z, not A and R",
    "Not rectangular yn; not trapezoid yc; not Manning Q with given A,R; not Chezy C; not Q=Av"
  ],
  "boundary_conditions": [
    "missing b, z, n, S, Q, or yn when required → MISSING_REQUIRED_INPUT",
    "yn/b, n, S, or z outside envelope → VALUE_OUT_OF_RANGE",
    "z=0, A/R, yc/q, E, Chezy C, v, B, theta, g/Fr → MIXED_INPUT_ENCODING",
    "unknown mode → INVALID_MODE"
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  "sources": [
    {
      "title": "ISO 80000-4 — Mechanics",
      "section": "Volume flow rate and hydraulic radius",
      "url": "https://www.iso.org/standard/31895.html",
      "supports": "Q is volume flow; hydraulic radius R = A/P",
      "kind": "standard"
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    {
      "title": "Chow, V.T. Open-Channel Hydraulics",
      "section": "Normal depth in a trapezoidal section",
      "supports": "Uniform-flow yn from SI Manning with trapezoid A=(b+z y)y and P=b+2 y √(1+z²). Rectangular A=b yn is a different page. Not Manning with given A,R as a product page, not yc, not Chezy C",
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