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      "limitations",
      "sources",
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  "assumptions": [
    "Rectangular alternate depths: E = y + Q²/(2g A²) with A=b y; both y_sub>yc and y_super<yc at the same E, Q, b; SI; default g=9.80665",
    "Unknown is the pair (depths mode) or E (energy invert from one y); E>Ec=1.5 yc; Fr is reported not an input",
    "Not specific energy E from y+v; not specific force M; not hydraulic jump y2 from Fr₁>1; not critical-depth yc as the product; not Manning n; not Q=Av"
  ],
  "limitations": [
    "Rectangular alternate depths: E = y + Q²/(2g A²) with A=b y; both y_sub>yc and y_super<yc at the same E, Q, b; SI; default g=9.80665",
    "Unknown is the pair (depths mode) or E (energy invert from one y); E>Ec=1.5 yc; Fr is reported not an input",
    "Not specific energy E from y+v; not specific force M; not hydraulic jump y2 from Fr₁>1; not critical-depth yc as the product; not Manning n; not Q=Av"
  ],
  "boundary_conditions": [
    "missing b, Q, E, or y when required → MISSING_REQUIRED_INPUT",
    "E ≤ Ec or y/b outside [0.01, 10] → VALUE_OUT_OF_RANGE",
    "A/v, y1/y2, M, Fr/L, n/C/S/R/Sc, yn/yc/q, z, H/Cd → MIXED_INPUT_ENCODING",
    "unknown mode → INVALID_MODE"
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  "sources": [
    {
      "title": "ISO 80000-4 — Mechanics",
      "section": "Energy and volume flow",
      "url": "https://www.iso.org/standard/31895.html",
      "supports": "E and y are lengths; Q is volume flow",
      "kind": "standard"
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    {
      "title": "Chow, V.T. Open-Channel Hydraulics",
      "section": "Specific energy and alternate depths",
      "supports": "The same E gives two depths when E>Ec. E from y and v is a different product. yc from q is a different product.",
      "kind": "reference"
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  "fingerprint": "sha256:beddff820a76f12f72f1628ea53870ee40ec6fdc11c09bc78efbf6e8c97bef4c",
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      "ok": true,
      "kind": "property",
      "detail": "y_sub from E,Q,b",
      "expect": 1,
      "actual": 1.0000046650505459
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      "id": "property-alt-ysuper",
      "ok": true,
      "kind": "property",
      "detail": "y_super<yc<y_sub",
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      "id": "property-alt-not-yc",
      "ok": true,
      "kind": "property",
      "detail": "not yc as the product",
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    {
      "id": "property-alt-roundtrip-E",
      "ok": true,
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      "kind": "property",
      "detail": "alternate_depth_rectangular",
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      "golden": 0,
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