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  "assumptions": [
    "Rectangular yn=yc: Sc=[n Q/(A R^{2/3})]^2 at yc=(q²/g)^{1/3}; A=b yc; R=b yc/(b+2 yc); SI k=1; Sc is unknown in slope mode",
    "yc/b∈[0.2, 10]; n∈[0.008, 0.15]; Sc∈[1e-6, 0.1]; geometry is b, not A and R; default g=9.80665",
    "Not Manning Q with given A,R,S; not normal-depth yn invert; not critical-depth yc as the product; not Chezy C; not Q=Av; not trapezoid z"
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  "limitations": [
    "Rectangular critical slope: Sc such that yn=yc; SI Manning at y=yc; yc=(q²/g)^{1/3}; Sc=[n Q/(A R^{2/3})]^2 with A=b yc, R=b yc/(b+2 yc)",
    "yc/b∈[0.2, 10] so Sc(yc) is one-to-one; n∈[0.008, 0.15]; Sc∈[1e-6, 0.1]; SI k=1; default g=9.80665",
    "Not Manning Q from A,R,S; not normal-depth yn invert; not critical-depth yc as the product; not Chezy C; not Q=Av; not US k=1.486; not trapezoid z"
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  "boundary_conditions": [
    "missing b, n, Q, or Sc when required → MISSING_REQUIRED_INPUT",
    "yc/b, n, or Sc outside envelope → VALUE_OUT_OF_RANGE",
    "A/R, yn, yc/q, S, E, Chezy C, v, B, z, y1, H, 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"
    },
    {
      "title": "Chow, V.T. Open-Channel Hydraulics",
      "section": "Critical slope and critical depth",
      "supports": "Sc is the bed slope at which uniform yn equals critical yc. Not Manning with given A,R as a product page, not yc as the unknown, not Chezy C",
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