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      "corrected_minor": "-54",
      "sign_enclosures": {
        "tau": [
          "-8/27",
          "-8/27"
        ],
        "c": [
          "-128/19683",
          "-128/19683"
        ],
        "D": [
          "16",
          "16"
        ]
      },
      "root_interval": [
        "0",
        "0"
      ],
      "rates": [
        "8/27",
        "8/27",
        "2/27",
        "8/27",
        "1/27"
      ]
    },
    {
      "ordinal": 43,
      "degree": 1,
      "tau": "-212/165",
      "c": "-800/70543",
      "trace_scaled_corrected_minor": "351/265",
      "raw_minor": "-2772/2809",
      "corrected_minor": "-11583/11236",
      "sign_enclosures": {
        "tau": [
          "-212/165",
          "-212/165"
        ],
        "c": [
          "-800/70543",
          "-800/70543"
        ],
        "D": [
          "351/265",
          "351/265"
        ]
      },
      "root_interval": [
        "0",
        "0"
      ],
      "rates": [
        "4/33",
        "4/11",
        "9/55",
        "52/165",
        "2/55"
      ]
    },
    {
      "ordinal": 44,
      "degree": 1,
      "tau": "-140/111",
      "c": "-1960/151959",
      "trace_scaled_corrected_minor": "306/175",
      "raw_minor": "-6327/4900",
      "corrected_minor": "-16983/12250",
      "sign_enclosures": {
        "tau": [
          "-140/111",
          "-140/111"
        ],
        "c": [
          "-1960/151959",
          "-1960/151959"
        ],
        "D": [
          "306/175",
          "306/175"
        ]
      },
      "root_interval": [
        "0",
        "0"
      ],
      "rates": [
        "14/111",
        "14/37",
        "6/37",
        "34/111",
        "1/37"
      ]
    },
    {
      "ordinal": 45,
      "degree": 1,
      "tau": "-4/3",
      "c": "-25/2916",
      "trace_scaled_corrected_minor": "9/10",
      "raw_minor": "-27/40",
      "corrected_minor": "-27/40",
      "sign_enclosures": {
        "tau": [
          "-4/3",
          "-4/3"
        ],
        "c": [
          "-25/2916",
          "-25/2916"
        ],
        "D": [
          "9/10",
          "9/10"
        ]
      },
      "root_interval": [
        "0",
        "0"
      ],
      "rates": [
        "1/9",
        "1/3",
        "1/6",
        "1/3",
        "1/18"
      ]
    },
    {
      "ordinal": 46,
      "degree": 3,
      "tau": "-27013*t**2/788 + 2864*t/197 - 347/197",
      "c": "195771744*t**2/23566913 - 80487600*t/23566913 + 2269824/23566913",
      "trace_scaled_corrected_minor": "-21992441*t**2/195424 + 1912067*t/48856 + 48565/48856",
      "raw_minor": "6622680735*t**2/24232576 - 654557821*t/6058144 + 10193501/6058144",
      "corrected_minor": "43339051337*t**2/96930304 - 4360783643*t/24232576 + 98213675/24232576",
      "sign_enclosures": {
        "tau": [
          "-114787615605/88834477892",
          "-114787609877/88834477892"
        ],
        "c": [
          "-36164572259544/2656794938935517",
          "-36164532015744/2656794938935517"
        ],
        "D": [
          "49178198128831/22030950517216",
          "49178201952965/22030950517216"
        ]
      },
      "root_interval": [
        "614/17443",
        "455/12926"
      ],
      "rates": [
        "19295*t**2/788 - 1905*t/197 + 79/197",
        "-3859*t**2/788 + 184*t/197 + 63/197",
        "3859*t**2/394 - 959*t/197 + 71/197",
        "-11577*t**2/394 + 2483*t/197 - 16/197",
        "t"
      ]
    },
    {
      "ordinal": 47,
      "degree": 2,
      "tau": "5*t/4 - 5/3",
      "c": "17248/83205 - 58328*t/83205",
      "trace_scaled_corrected_minor": "129*t/10 - 6/5",
      "raw_minor": "558/25 - 387*t/5",
      "corrected_minor": "558/25 - 387*t/5",
      "sign_enclosures": {
        "tau": [
          "-63245/49827",
          "-289825/228336"
        ],
        "c": [
          "-684198/43978465",
          "-21499808/1381951845"
        ],
        "D": [
          "240906/83045",
          "110397/38056"
        ]
      },
      "root_interval": [
        "5280/16609",
        "6049/19028"
      ],
      "rates": [
        "1/3 - 3*t/4",
        "t/12 + 1/3",
        "1/3 - 5*t/12",
        "t",
        "t/12"
      ]
    },
    {
      "ordinal": 48,
      "degree": 3,
      "tau": "-1050*t**2/71 + 456*t/71 - 116/71",
      "c": "-28496*t**2/19525 + 41456*t/58575 - 848/19525",
      "trace_scaled_corrected_minor": "10500*t**2/781 - 10240*t/781 + 1728/781",
      "raw_minor": "-6000*t**2/121 + 2630*t/121 - 274/121",
      "corrected_minor": "-229200*t**2/8591 + 147656*t/8591 - 17146/8591",
      "sign_enclosures": {
        "tau": [
          "-4931750863/3687093190",
          "-986350127/737418638"
        ],
        "c": [
          "-209266236/20279012545",
          "-15694957336/1520925940875"
        ],
        "D": [
          "204322537/130832339",
          "6333999159/4055802509"
        ]
      },
      "root_interval": [
        "453/8635",
        "631/12028"
      ],
      "rates": [
        "750*t**2/71 - 346*t/71 + 22/71",
        "-150*t**2/71 - 16*t/71 + 24/71",
        "300*t**2/71 - 181*t/71 + 23/71",
        "-900*t**2/71 + 472*t/71 + 2/71",
        "t"
      ]
    },
    {
      "ordinal": 49,
      "degree": 2,
      "tau": "24*t - 6",
      "c": "15184*t/3721 - 2960/3721",
      "trace_scaled_corrected_minor": "3050*t/27 - 260/27",
      "raw_minor": "185/27 - 244*t/3",
      "corrected_minor": "1730/243 - 20435*t/243",
      "sign_enclosures": {
        "tau": [
          "-13242/9911",
          "-1292/967"
        ],
        "c": [
          "-92176/36878831",
          "-26980/10794621"
        ],
        "D": [
          "3297440/267597",
          "1930355/156654"
        ]
      },
      "root_interval": [
        "1926/9911",
        "2255/11604"
      ],
      "rates": [
        "2*t",
        "2 - 10*t",
        "1 - 5*t",
        "12*t - 2",
        "t"
      ]
    },
    {
      "ordinal": 50,
      "degree": 2,
      "tau": "17*t - 2",
      "c": "139218320/845360439 - 4017888760*t/845360439",
      "trace_scaled_corrected_minor": "5739*t/77 + 60/77",
      "raw_minor": "210156/5929 - 11954337*t/11858",
      "corrected_minor": "210156/5929 - 11954337*t/11858",
      "sign_enclosures": {
        "tau": [
          "-17569/12941",
          "-16563/12200"
        ],
        "c": [
          "-3844580359/257834933895",
          "-163123324520/10939809441099"
        ],
        "D": [
          "511833/142351",
          "3377679/939400"
        ]
      },
      "root_interval": [
        "489/12941",
        "461/12200"
      ],
      "rates": [
        "2/9 - 40*t/9",
        "4/9 - 17*t/9",
        "1/3 - 11*t/3",
        "9*t",
        "t"
      ]
    },
    {
      "ordinal": 51,
      "degree": 3,
      "tau": "-5286*t**2/169 + 1946*t/169 - 330/169",
      "c": "-29534344*t**2/42433365 + 14251168*t/42433365 - 1184656/42433365",
      "trace_scaled_corrected_minor": "1031651*t**2/20995 - 551442*t/20995 + 80044/20995",
      "raw_minor": "-1436706608*t**2/25928825 + 586636681*t/25928825 - 76683972/25928825",
      "corrected_minor": "-2593619069*t**2/51857650 + 606521129*t/25928825 - 76144548/25928825",
      "sign_enclosures": {
        "tau": [
          "-71597423238/51112161763",
          "-71597421292/51112161763"
        ],
        "c": [
          "-142810032642592/12833497136262855",
          "-47603339463808/4277832378754285"
        ],
        "D": [
          "15762809194456/6349703172865",
          "15762809745898/6349703172865"
        ]
      },
      "root_interval": [
        "918/16207",
        "1057/18661"
      ],
      "rates": [
        "4405*t**2/507 - 1678*t/507 + 106/507",
        "881*t**2/507 - 944*t/507 + 224/507",
        "881*t**2/169 - 437*t/169 + 55/169",
        "-2643*t**2/169 + 1142*t/169 + 4/169",
        "t"
      ]
    }
  ],
  "query": {
    "admits_cusp": true,
    "layer": "cusp",
    "representative": [
      0,
      5,
      12,
      18,
      22
    ],
    "transport": {
      "source_indices": [
        0,
        1,
        2,
        3,
        4
      ],
      "rate_factors": [
        "1",
        "1",
        "1",
        "1",
        "1"
      ],
      "species_exchange": false
    },
    "evidence": "Published exact classification lookup; negative obstruction payloads and Lean proof are not replayed here."
  },
  "rational_identities": {
    "scalar": "mu + nu*z - z**3",
    "equilibria": [
      {
        "x": "1 - epsilon",
        "y": "(1 - epsilon)**2",
        "determinant": "6*epsilon**2/121",
        "trace": "8*epsilon/11 - 13/11"
      },
      {
        "x": "1",
        "y": "1",
        "determinant": "-3*epsilon**2/121",
        "trace": "epsilon**2/11 - 13/11"
      },
      {
        "x": "epsilon + 1",
        "y": "(epsilon + 1)**2",
        "determinant": "6*epsilon**2/121",
        "trace": "-8*epsilon/11 - 13/11"
      }
    ],
    "negative_collision_determinant": "d*y*(3*e - f)",
    "scope": "Scalar identity describes equilibria on the y-nullcline, not a one-dimensional time evolution or invariant manifold."
  },
  "reactor_rates": [
    "9/100",
    "299/1100",
    "3/11",
    "3/11",
    "1/11"
  ],
  "equilibria": [
    {
      "x_interval": [
        "9/10",
        "9/10"
      ],
      "x": 0.9,
      "y": 0.81,
      "multiplicity": 1,
      "stability": "stable",
      "trace_midpoint": "-61/55",
      "determinant_midpoint": "3/6050"
    },
    {
      "x_interval": [
        "1",
        "1"
      ],
      "x": 1.0,
      "y": 1.0,
      "multiplicity": 1,
      "stability": "saddle",
      "trace_midpoint": "-1299/1100",
      "determinant_midpoint": "-3/12100"
    },
    {
      "x_interval": [
        "11/10",
        "11/10"
      ],
      "x": 1.1,
      "y": 1.21,
      "multiplicity": 1,
      "stability": "stable",
      "trace_midpoint": "-69/55",
      "determinant_midpoint": "3/6050"
    }
  ],
  "numerical_solver_differences": [
    2.6835622612964016e-10,
    3.7565861532584677e-10
  ],
  "scope": "Exact enumeration, all selected positive jet/sign/corrected-minor checks, and rational bistable model. Negative-layer classification uses the imported paper theorem; its 9,947 obstruction payloads and Lean/native evaluation are not rerun. No global basin or arbitrary-supernetwork claim."
}