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  "assumptions": [
    "Right triangle; positive side lengths.",
    "c > a and c > b when solving for a leg.",
    "Exactly two fields are inputs when solving. Explicit solve_for selects the output. If all three are supplied without solve_for, they are verified rather than overwritten.",
    "Interactive evaluation, shared URLs, and REST use the same engine, solve_for contract, and error codes.",
    "Share URL is canonical: solve_for=a|b|c forces solve even if a leftover third side is present; mode=verify is required to stay in Verify after a tab change."
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    "Four operations:c,a,b,verify. solve_for selects the output;all supplied side tokens still validate. All three without solve_for verify;mode=verify conflicts with solve_for.",
    "Verification checks nondegenerate geometry and squared relative residual |a²+b²-c²|/c²<=1e-6 with scaled squares. Approximate binary64 diagnostics may differ by2e-15 near the threshold; no exact-equality proof.",
    "Solved sides must fit positive finite binary64. Unrepresentable positive area/perimeter are null with explicit range status;invalid verify has no right-angle area. Values display12significant digits. Exact radical only for exactly representable safe-integer radicands.",
    "Diagram is schematic,not to scale;no right-angle diagram for failed verification. Browser calculations run locally;shared URLs and REST/MCP transmit inputs and may enter logs."
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    "Invalid solve_for → INVALID_SOLVE_FOR",
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    "Non-positive side → SIDE_NOT_POSITIVE",
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    "All three without solve_for that do not satisfy Pythagoras → status invalid (not overwritten)"
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      "title": "Euclid, Elements, Book I, Proposition 47",
      "section": "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle",
      "url": "https://mathcs.clarku.edu/~djoyce/elements/bookI/propI47.html",
      "supports": "Classical statement of the Pythagorean theorem used as the geometric identity a² + b² = c².",
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      "accessed": "2026-08-28",
      "publisher": "Clark University"
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      "section": "A = ab/2",
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