[
 {
  "id": "C01",
  "http_status": 200,
  "recorded_at": "2026-11-03T23:09:34+00:00",
  "ms": 0,
  "reply": {
   "slug": "derivation-copilot",
   "title": "Independently check a claimed derivative (one right, one wrong)",
   "status": "ok",
   "input": "3*x**2*sin(x) + x**3*cos(x)",
   "total_ms": 953,
   "steps": [
    {
     "call": "POST /v1/derivation/check/derivative",
     "request": {
      "original_expr": "x**3*sin(x)",
      "claimed_derivative": "3*x**2*sin(x) + x**3*cos(x)"
     },
     "status": 200,
     "ms": 504,
     "reply": {
      "grade": "PROVEN",
      "passed": true,
      "method": "derivative",
      "detail": "symbolic simplification confirms exact equality"
     }
    },
    {
     "call": "POST /v1/derivation/check/derivative",
     "request": {
      "original_expr": "x**3*sin(x)",
      "claimed_derivative": "3*x**2*cos(x)"
     },
     "status": 200,
     "ms": 430,
     "reply": {
      "grade": "PROVEN",
      "passed": false,
      "method": "derivative",
      "detail": "residual is nonzero and numeric sampling confirms a real mismatch, not float noise",
      "counterexample": {
       "x": -0.857864,
       "residual": -3.527021
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     }
    }
   ],
   "note": "Real calls to the running service with the public demo identity; replies are unedited."
  }
 },
 {
  "id": "C02",
  "http_status": 200,
  "recorded_at": "2026-11-03T23:09:34+00:00",
  "ms": 0,
  "reply": {
   "slug": "derivation-copilot",
   "title": "Independently check a claimed derivative (one right, one wrong)",
   "status": "ok",
   "input": "3*x**2*sin(x) + x**3*cos(x)",
   "total_ms": 953,
   "steps": [
    {
     "call": "POST /v1/derivation/check/derivative",
     "request": {
      "original_expr": "x**3*sin(x)",
      "claimed_derivative": "3*x**2*sin(x) + x**3*cos(x)"
     },
     "status": 200,
     "ms": 504,
     "reply": {
      "grade": "PROVEN",
      "passed": true,
      "method": "derivative",
      "detail": "symbolic simplification confirms exact equality"
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    },
    {
     "call": "POST /v1/derivation/check/derivative",
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      "original_expr": "x**3*sin(x)",
      "claimed_derivative": "3*x**2*cos(x)"
     },
     "status": 200,
     "ms": 430,
     "reply": {
      "grade": "PROVEN",
      "passed": false,
      "method": "derivative",
      "detail": "residual is nonzero and numeric sampling confirms a real mismatch, not float noise",
      "counterexample": {
       "x": -0.857864,
       "residual": -3.527021
      }
     }
    }
   ],
   "note": "Real calls to the running service with the public demo identity; replies are unedited."
  }
 },
 {
  "id": "C03",
  "http_status": 200,
  "recorded_at": "2026-11-03T23:09:35+00:00",
  "ms": 0,
  "reply": {
   "slug": "derivation-copilot",
   "title": "Independently check a claimed derivative (one right, one wrong)",
   "status": "ok",
   "input": "3*x**2*sin(x) + x**3*cos(x)",
   "total_ms": 953,
   "steps": [
    {
     "call": "POST /v1/derivation/check/derivative",
     "request": {
      "original_expr": "x**3*sin(x)",
      "claimed_derivative": "3*x**2*sin(x) + x**3*cos(x)"
     },
     "status": 200,
     "ms": 504,
     "reply": {
      "grade": "PROVEN",
      "passed": true,
      "method": "derivative",
      "detail": "symbolic simplification confirms exact equality"
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    },
    {
     "call": "POST /v1/derivation/check/derivative",
     "request": {
      "original_expr": "x**3*sin(x)",
      "claimed_derivative": "3*x**2*cos(x)"
     },
     "status": 200,
     "ms": 430,
     "reply": {
      "grade": "PROVEN",
      "passed": false,
      "method": "derivative",
      "detail": "residual is nonzero and numeric sampling confirms a real mismatch, not float noise",
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     }
    }
   ],
   "note": "Real calls to the running service with the public demo identity; replies are unedited."
  }
 }
]