{
  "schema_version": 1,
  "fixture_id": "qft.fixture.conformal-bootstrap.rational-half-line-certificate",
  "arithmetic": "exact rational",
  "purpose": "Exercise the cone, functional, polynomial, Gram-matrix, serialization, and independent-verification stages without conformal-block or floating-point ambiguity.",
  "claim_ceiling": "This certificate excludes one target from the declared toy half-line moment cone. It is not a conformal-field-theory bound.",
  "problem": {
    "basis": [
      "1",
      "x",
      "x^2"
    ],
    "domain": {
      "variable": "x",
      "constraint": "x >= 0"
    },
    "generator": [
      "1",
      "x",
      "x^2"
    ],
    "finite_control_points": [
      "0",
      "1",
      "2"
    ],
    "finite_control_generators": [
      [
        "1",
        "0",
        "0"
      ],
      [
        "1",
        "1",
        "1"
      ],
      [
        "1",
        "2",
        "4"
      ]
    ],
    "feasible_control_target": [
      "1",
      "1",
      "1"
    ],
    "feasible_control_weights": [
      "0",
      "1",
      "0"
    ],
    "excluded_target": [
      "1",
      "1",
      "0"
    ]
  },
  "dual_certificate": {
    "functional": [
      "1",
      "-2",
      "1"
    ],
    "excluded_target_action": "-1",
    "generator_polynomial_coefficients": [
      "1",
      "-2",
      "1"
    ],
    "sos_basis": [
      "1",
      "x"
    ],
    "sos_factor": [
      "1",
      "-1"
    ],
    "gram_matrix": [
      [
        "1",
        "-1"
      ],
      [
        "-1",
        "1"
      ]
    ]
  },
  "one_dimensional_cft_certificate": {
    "external_dimension": "1",
    "assumed_nonidentity_gap": "9/2",
    "functional_basis": [
      "f'(1/2)",
      "f'''(1/2)"
    ],
    "functional_coefficients": [
      "-49/32",
      "1/32"
    ],
    "identity_derivatives": [
      "-32",
      "-1536"
    ],
    "identity_action": "1",
    "shifted_variable": "x = Delta - 9/2 >= 0",
    "analytic_positivity_input": "For Delta > 0, g_Delta(1/2) > 0 and r_Delta = g_Delta'(1/2)/g_Delta(1/2) >= 8 Delta/3.",
    "lower_envelope_identity": "alpha(F_Delta) = 2 g_Delta(1/2) {K(x) [r_Delta - 8 Delta/3] + P(x)}",
    "k_polynomial_coefficients": [
      "133/8",
      "8",
      "1"
    ],
    "k_gram_representation": {
      "basis": [
        "1",
        "x"
      ],
      "gram_matrix": [
        [
          "133/8",
          "0"
        ],
        [
          "0",
          "1"
        ]
      ],
      "x_times_constant_gram": [
        [
          "8"
        ]
      ]
    },
    "p_polynomial_coefficients": [
      "11",
      "133/3",
      "64/3",
      "8/3"
    ],
    "p_gram_representation": {
      "basis": [
        "1",
        "x"
      ],
      "gram_matrix": [
        [
          "11",
          "0"
        ],
        [
          "0",
          "64/3"
        ]
      ],
      "x_times_gram_matrix": [
        [
          "133/3",
          "0"
        ],
        [
          "0",
          "8/3"
        ]
      ]
    },
    "claim_ceiling": "Together with the stated analytic positivity input, the exact rational data exclude a nonidentity gap Delta >= 9/2 in the declared reflection-positive one-dimensional crossing problem. They do not validate a generic block approximation or a solver implementation."
  }
}