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      "certificate": "weighted certificate: four-species",
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    {
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      "assignment": "X1->r1; X3->r3",
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      "certificate": "weighted certificate: four-species",
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      "every_positive_scaling": "all real parts strictly negative"
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    {
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      "assignment": "X2->r1; X3->r0",
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    {
      "number": 15,
      "size": 2,
      "assignment": "X2->r1; X3->r3",
      "matrix": "[[-2, 1], [0, -2]]",
      "certificate": "weighted certificate: three-species A",
      "weights": "[2, 7]",
      "every_positive_scaling": "all real parts strictly negative"
    },
    {
      "number": 16,
      "size": 2,
      "assignment": "X2->r2; X3->r0",
      "matrix": "[[-4, 4], [2, -2]]",
      "certificate": "weighted certificate: face 2,3",
      "weights": "[1, 2]",
      "every_positive_scaling": "one zero eigenvalue"
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    {
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      "assignment": "X2->r2; X3->r3",
      "matrix": "[[-4, 1], [2, -2]]",
      "certificate": "weighted certificate: four-species",
      "weights": "[40, 140]",
      "every_positive_scaling": "all real parts strictly negative"
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    {
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      "certificate": "weighted certificate: four-species",
      "weights": "[100, 35, 40]",
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      "certificate": "weighted certificate: four-species",
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    {
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      "assignment": "X0->r0; X2->r1; X3->r3",
      "matrix": "[[-1, 0, 1], [4, -2, 1], [-2, 0, -2]]",
      "certificate": "weighted certificate: three-species A",
      "weights": "[10, 2, 7]",
      "every_positive_scaling": "all real parts strictly negative"
    },
    {
      "number": 21,
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      "assignment": "X0->r0; X2->r2; X3->r3",
      "matrix": "[[-1, 0, 1], [4, -4, 1], [-2, 2, -2]]",
      "certificate": "weighted certificate: four-species",
      "weights": "[100, 40, 140]",
      "every_positive_scaling": "all real parts strictly negative"
    },
    {
      "number": 22,
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      "matrix": "[[-4, 0, 0], [-2, -4, 4], [0, 2, -2]]",
      "certificate": "symbolic cubic factorization",
      "weights": "",
      "every_positive_scaling": "one zero eigenvalue"
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    {
      "number": 23,
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      "assignment": "X1->r1; X2->r2; X3->r3",
      "matrix": "[[-4, 0, 6], [-2, -4, 1], [0, 2, -2]]",
      "certificate": "weighted certificate: four-species",
      "weights": "[35, 40, 140]",
      "every_positive_scaling": "all real parts strictly negative"
    },
    {
      "number": 24,
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      "assignment": "X0->r0; X1->r1; X2->r2; X3->r3",
      "matrix": "[[-1, 0, 0, 1], [0, -4, 0, 6], [4, -2, -4, 1], [-2, 0, 2, -2]]",
      "certificate": "weighted certificate: four-species",
      "weights": "[100, 35, 40, 140]",
      "every_positive_scaling": "all real parts strictly negative"
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  ],
  "parameter_coefficients": [
    "2*E + 2*F + 15951/5500",
    "41*E/250 + 18951*F/2750 + 11251/5500",
    "4801*F/1375 + 8/55",
    "24*F/55"
  ],
  "parameter_delta2": "41*E**2/125 + 19402*E*F/1375 + 6279491*E/1375000 + 18951*F**2/1375 + 311356901*F/15125000 + 175064701/30250000",
  "parameter_delta3": "-103159*E**2*F/171875 + 328*E**2/6875 + 86549002*E*F**2/1890625 + 2223421481*E*F/171875000 + 6279491*E/9453125 + 87683751*F**3/1890625 + 1431240081701*F**2/20796875000 + 812367749301*F/41593750000 + 175064701/207968750",
  "presets": [
    {
      "name": "decaying oscillations",
      "E": "10",
      "F": "3/5",
      "delta3": "275738937654233/1039843750000",
      "behavior": "decay"
    },
    {
      "name": "paper example",
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      "F": "28200219703/46658000000",
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      "behavior": "growth"
    }
  ],
  "boundary_at_F_0_6": "81.5805948722850",
  "proof_scope": "Sections 3\u20134: this explicit counterexample; no Lean execution",
  "numerical_outputs": {
    "numpy": "2.5.3",
    "matplotlib": "3.11.1",
    "map_points": 49750,
    "map_sign_check": "passed away from numerical boundary",
    "eigenvalues": [
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    "period": 61.599855952741045,
    "doubling_time": 346.5735902799726
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  "environment": {
    "python": "3.12.14",
    "sympy": "1.14.0"
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  "source_sha256": "cf5bf525aa424fbbfb4207f4ccb96e2d91ad3908d6f06e788c7381b2629ea5aa"
}