{
  "exact_crossing": {
    "identities_zero": true,
    "h_half": "4132146251/8",
    "h_one": "-185540038",
    "positive_root_bracket": [
      "620773913352840272667917871074087/649037107316853453566312041152512",
      "2483095653411361090671671484296349/2596148429267413814265248164610048"
    ],
    "descartes_signs_ascending": [
      1,
      1,
      1,
      -1
    ],
    "transversality": "t*h_prime=3*h-(8477280+485224392*t+3570069381*t^2), so h_prime<0 at the positive root; alpha_prime>0 by the exact Hurwitz derivative identity.",
    "equilibrium_det_numerator": "216000*k5^4",
    "scope": "Exactly one positive crossing; stable before, two unstable roots after. Determinant of a positive equilibrium never vanishes."
  },
  "structure": {
    "rank": 4,
    "positive_kernel_vector": [
      "2",
      "3",
      "2",
      "2",
      "2"
    ],
    "four_column_minors": [
      "48",
      "-72",
      "48",
      "-48",
      "48"
    ],
    "deletion_covectors": [
      [
        "7/2",
        "-1/4",
        "0",
        "1/2"
      ],
      [
        "-2",
        "2/3",
        "0",
        "1/2"
      ],
      [
        "7/2",
        "-17/24",
        "11/12",
        "-5/12"
      ],
      [
        "-2",
        "-1/4",
        "0",
        "1/2"
      ],
      [
        "-2",
        "5/24",
        "-11/12",
        "-4/3"
      ]
    ],
    "scope": "With at least one retained positive rate, each deleted system has a strictly increasing covector observable on the positive orthant. All-zero rates instead give constant solutions."
  },
  "children": {
    "child_count": 24,
    "covers": [
      {
        "species_zero_based": [
          2,
          3
        ],
        "reactions_zero_based": [
          1,
          0
        ],
        "weights": [
          1,
          2
        ],
        "energy_matrix": [
          [
            "4",
            "-4"
          ],
          [
            "-4",
            "8"
          ]
        ],
        "all_principal_minors": [
          "4",
          "8",
          "16"
        ]
      },
      {
        "species_zero_based": [
          0,
          2,
          3
        ],
        "reactions_zero_based": [
          0,
          1,
          3
        ],
        "weights": [
          10,
          2,
          7
        ],
        "energy_matrix": [
          [
            "20",
            "-8",
            "4"
          ],
          [
            "-8",
            "8",
            "-2"
          ],
          [
            "4",
            "-2",
            "28"
          ]
        ],
        "all_principal_minors": [
          "20",
          "8",
          "28",
          "96",
          "544",
          "220",
          "2608"
        ]
      },
      {
        "species_zero_based": [
          0,
          1,
          2,
          3
        ],
        "reactions_zero_based": [
          0,
          1,
          2,
          3
        ],
        "weights": [
          100,
          35,
          40,
          140
        ],
        "energy_matrix": [
          [
            "200",
            "0",
            "-160",
            "180"
          ],
          [
            "0",
            "280",
            "80",
            "-210"
          ],
          [
            "-160",
            "80",
            "320",
            "-320"
          ],
          [
            "180",
            "-210",
            "-320",
            "560"
          ]
        ],
        "all_principal_minors": [
          "200",
          "280",
          "320",
          "560",
          "56000",
          "38400",
          "79600",
          "83200",
          "112700",
          "76800",
          "9472000",
          "13468000",
          "9088000",
          "14560000",
          "1524480000"
        ]
      },
      {
        "species_zero_based": [
          1,
          3
        ],
        "reactions_zero_based": [
          1,
          0
        ],
        "weights": [
          1,
          2
        ],
        "energy_matrix": [
          [
            "8",
            "0"
          ],
          [
            "0",
            "8"
          ]
        ],
        "all_principal_minors": [
          "8",
          "8",
          "64"
        ]
      },
      {
        "species_zero_based": [
          2,
          3
        ],
        "reactions_zero_based": [
          2,
          0
        ],
        "weights": [
          1,
          2
        ],
        "energy_matrix": [
          [
            "8",
            "-8"
          ],
          [
            "-8",
            "8"
          ]
        ],
        "all_principal_minors": [
          "8",
          "8",
          "0"
        ]
      },
      {
        "species_zero_based": [
          1,
          2,
          3
        ],
        "reactions_zero_based": [
          1,
          2,
          0
        ],
        "scaled_characteristic_factorization": "lambda*(lambda+4*d1)*(lambda+4*d2+2*d3)"
      }
    ],
    "conclusion": "Every positive diagonal scaling of every child has nonpositive spectral real parts; singular children are not claimed strictly stable."
  },
  "crossing_orientation": {
    "tH": 0.956453654736546,
    "omegaH": 0.10099875775976236,
    "periodH": 62.21052066922343,
    "alpha_prime": 0.004291967001236513
  },
  "midpoint_refinements": [
    {
      "steps": 128,
      "parameter": 0.9565066092713799,
      "period": 62.22261435282665,
      "return_residual": 1.0091489229864156e-15,
      "mean_flux_ratios_midpoint": [
        1.0,
        1.0,
        1.0000000000000004,
        1.0,
        1.0
      ],
      "mean_z4_midpoint": 0.9999999993590207
    },
    {
      "steps": 256,
      "parameter": 0.9565066547192029,
      "period": 62.213242163441485,
      "return_residual": 9.472119168108556e-16,
      "mean_flux_ratios_midpoint": [
        1.0000000000000004,
        0.9999999999999999,
        1.0000000000000007,
        0.9999999999999998,
        1.0
      ],
      "mean_z4_midpoint": 0.99999999935873
    },
    {
      "steps": 512,
      "parameter": 0.9565066660839242,
      "period": 62.21089964527928,
      "return_residual": 1.1231121851245984e-15,
      "mean_flux_ratios_midpoint": [
        1.0,
        1.0000000000000004,
        0.999999999999999,
        0.9999999999999994,
        1.0
      ],
      "mean_z4_midpoint": 0.9999999993586571
    }
  ],
  "continuous_shooting": {
    "amplitude": 0.02,
    "parameter": 0.9565066698726885,
    "period": 62.2101188529162,
    "stable_coordinates": [
      0.0008740500369441779,
      -5.6663722006039966e-05
    ],
    "return_residual_in_basis": 2.550140970103233e-13,
    "independent_BDF_relative_difference": 6.828881460307911e-11,
    "minimum_relative_concentration": 0.9800185698357848,
    "mean_flux_ratios_trapezoidal": [
      0.9999999999999997,
      1.0000000000000022,
      0.9999999999999983,
      0.9999999999999993,
      1.0
    ],
    "mean_z4_minus_one": -6.413670488273517e-10,
    "variance_z4": 1.2827340934070287e-09,
    "mean_square_identity_defect": -3.836322853521655e-18,
    "scope": "Numerical continuous shooting and quadrature, not a validated orbit at this decimal parameter."
  },
  "scope": "Exact child/deletion/crossing identities and a rational interval Lyapunov sign; local Hopf attraction uses the conventional theorem. No global attraction, explicit certified parameter neighborhood, or laboratory realization. Lean not rerun."
}