{
  "exact_active_enclosed_design": {
    "certificate": {
      "budget_exactly_one": true,
      "certified_efficiency_lower_exact": "756522130884465853615895779432349369/757286444490472008341328125000000000",
      "certified_optimality_gap_upper_exact": "764313606006154725432345567650631/9496383846007969390900000000000000000",
      "dual_metric_trace_exactly_one": true,
      "dual_optimum_upper_exact": "9693266489478041706769/121553713228902008203520",
      "dual_witness_exact": [
        [
          "4442489104/10000042505",
          "4968839948/10000042505"
        ],
        [
          "4968839948/10000042505",
          "5557553401/10000042505"
        ]
      ],
      "dual_witness_integer_factor": [
        [
          66652,
          0
        ],
        [
          74549,
          0
        ]
      ],
      "dual_witness_psd_by_exact_factorization": true,
      "primal_generalized_floor_lower_exact": "7966423252809941/100000000000000000",
      "primal_shift_positive_by_exact_ldl": true,
      "rationalized_design_gram_exact": [
        [
          "1223207477578270001/12416358528000000000",
          "-92298989294454241/5501091976800000000"
        ],
        [
          "-92298989294454241/5501091976800000000",
          "57727014750403739/609317395051250000"
        ]
      ]
    },
    "declared_source_metric": [
      [
        "1/1",
        "0/1"
      ],
      [
        "0/1",
        "1/1"
      ]
    ],
    "exact_blind_dimension": 0,
    "independent_exact_verification": {
      "dual_upper_exact": "9693266489478041706769/121553713228902008203520",
      "method": "independent exact-rational reconstruction of quotient, design, primal, dual, noise, and integer realization",
      "passed": true,
      "primal_lower_exact": "7966423252809941/100000000000000000"
    },
    "noise_performance": {
      "approximate_design_measure": {
        "amplitude_exact": "1/1",
        "certified_squared_separation_lower_exact": "7966423252809941/10000000000000000",
        "continuous_budget_exposure_multiplier": 10,
        "descriptive_exact_gaussian_error": 0.3276996276926949,
        "formula": "Phi(-amplitude*sqrt(exposure_multiplier*floor)/2)",
        "gaussian_error_rational_upper_exact": "40000000000000000/87966423252809941",
        "gaussian_error_rational_upper_method": "Q(d/2)<=exp(-d^2/8)/2<=4/(8+d^2)",
        "role": "certified consequence of the floor under the already-whitened unit-covariance Gaussian model"
      },
      "integer_realization": {
        "amplitude_exact": "1/1",
        "certified_squared_separation_lower_exact": "7966423252809941/2000000000000",
        "continuous_budget_exposure_multiplier": 50000,
        "descriptive_exact_gaussian_error": 7.337272277746456e-219,
        "formula": "Phi(-amplitude*sqrt(exposure_multiplier*floor)/2)",
        "gaussian_error_rational_upper_exact": "8000000000000/7982423252809941",
        "gaussian_error_rational_upper_method": "Q(d/2)<=exp(-d^2/8)/2<=4/(8+d^2)",
        "minimum_budget_multiplier_exactly_realizing_weights": 50000,
        "protocol_counts": [
          22945,
          21644
        ],
        "realized_budget_multiplier": 50000,
        "requested_multiplier_was_exactly_realizable": false,
        "role": "certified consequence of the floor under the already-whitened unit-covariance Gaussian model"
      }
    },
    "observable_quotient_dimension": 2,
    "overall_passed": true,
    "proof_boundary": "Floating point is used only to discover weights and a dual vector. The exact certificate covers the declared rational finite protocols; binary floating matrix declarations are rejected. It does not by itself certify continuum-to-finite model transfer or the completeness of the candidate protocol list.",
    "protocols": [
      {
        "budget_share_exact": "4589/10000",
        "cost_exact": "1/1",
        "name": "launch-A / theta-2 / tau-1",
        "noise_precision_exact": [
          [
            "1/1"
          ]
        ],
        "physical_weight_exact": "4589/10000",
        "response_exact": [
          [
            "265/3136",
            "-4041/8894"
          ]
        ]
      },
      {
        "budget_share_exact": "5411/10000",
        "cost_exact": "5/4",
        "name": "launch-B / scaled-omega-1 / tau-5/4",
        "noise_precision_exact": [
          [
            "1/1"
          ]
        ],
        "physical_weight_exact": "5411/12500",
        "response_exact": [
          [
            "2357/5025",
            "13/3140"
          ]
        ]
      }
    ],
    "quotient_injection": [
      [
        "1/1",
        "0/1"
      ],
      [
        "0/1",
        "1/1"
      ]
    ],
    "quotient_protocol_information": [
      [
        [
          "70225/9834496",
          "-1070865/27891584"
        ],
        [
          "-1070865/27891584",
          "16329681/79103236"
        ]
      ],
      [
        [
          "5555449/25250625",
          "30641/15778500"
        ],
        [
          "30641/15778500",
          "169/9859600"
        ]
      ]
    ],
    "quotient_row_coordinates": [
      [
        "1/1",
        "0/1"
      ],
      [
        "0/1",
        "1/1"
      ]
    ],
    "quotient_source_metric": [
      [
        "1/1",
        "0/1"
      ],
      [
        "0/1",
        "1/1"
      ]
    ],
    "response_box_independent_verification": {
      "method": "independent exact-rational reconstruction of every response box and the aggregate robust floor",
      "passed": true,
      "robust_physical_floor_lower_exact": "195932905640268820789/2500000000000000000000"
    },
    "response_box_robustness": {
      "aggregate_metric_relative_information_error_upper_exact": "3227675679979704211/2500000000000000000000",
      "nominal_floor_lower_exact": "7966423252809941/100000000000000000",
      "proof_boundary": "The robust floor holds for every response in the declared entrywise rational boxes. It does not certify that an external numerical model lies in those boxes.",
      "protocol_response_box_certificates": [
        {
          "diagonal_quadratic_bound_exact": [
            [
              "3124631792137/4358060000000000",
              "0/1"
            ],
            [
              "0/1",
              "6380513873387/4358060000000000"
            ]
          ],
          "entrywise_information_bound_exact": [
            [
              "42070237/245000000000",
              "2376286416381/4358060000000000"
            ],
            [
              "2376286416381/4358060000000000",
              "40859463847/44470000000000"
            ]
          ],
          "exact_ldl_decision": true,
          "meaning": "for every enclosed response, -delta*S <= A_true-A_centre <= delta*S",
          "metric_relative_information_error_upper_exact": "3663840325316991/2500000000000000000",
          "noise_precision_exact": [
            [
              "1/1"
            ]
          ],
          "response_centre_exact": [
            [
              "265/3136",
              "-4041/8894"
            ]
          ],
          "response_radius_exact": [
            [
              "101/100000",
              "101/100000"
            ]
          ],
          "schema_version": "oig-response-box-form-bound-v1",
          "source_metric_exact": [
            [
              "1/1",
              "0/1"
            ],
            [
              "0/1",
              "1/1"
            ]
          ]
        },
        {
          "diagonal_quadratic_bound_exact": [
            [
              "75077129819/52595000000000",
              "0/1"
            ],
            [
              "0/1",
              "77051158457/157785000000000"
            ]
          ],
          "entrywise_information_bound_exact": [
            [
              "1906506401/2010000000000",
              "151141273957/315570000000000"
            ],
            [
              "151141273957/315570000000000",
              "14731557/1570000000000"
            ]
          ],
          "exact_ldl_decision": true,
          "meaning": "for every enclosed response, -delta*S <= A_true-A_centre <= delta*S",
          "metric_relative_information_error_upper_exact": "2857770014215001/2000000000000000000",
          "noise_precision_exact": [
            [
              "1/1"
            ]
          ],
          "response_centre_exact": [
            [
              "2357/5025",
              "13/3140"
            ]
          ],
          "response_radius_exact": [
            [
              "101/100000",
              "101/100000"
            ]
          ],
          "schema_version": "oig-response-box-form-bound-v1",
          "source_metric_exact": [
            [
              "1/1",
              "0/1"
            ],
            [
              "0/1",
              "1/1"
            ]
          ]
        }
      ],
      "robust_floor_is_positive": true,
      "robust_noise_performance": {
        "amplitude_exact": "1/1",
        "continuous_exposure_gaussian_error_rational_upper_exact": "1000000000000000000000/2195932905640268820789",
        "integer_realization_exposure": 50000,
        "integer_realization_gaussian_error_rational_upper_exact": "200000000000000000/196332905640268820789"
      },
      "robust_physical_floor_lower_exact": "195932905640268820789/2500000000000000000000",
      "schema_version": "oig-enclosed-protocol-design-v1"
    },
    "schema_version": "oig-protocol-design-engine-v1",
    "source_dimension": 2,
    "status": "exact quotient and exact rational primal/dual certificate"
  },
  "logical_composition": "outward ODE response membership AND exact for-all-response-in-box design theorem imply the nominal physical double-pendulum floor",
  "physical_positive_floor_certified": true,
  "physical_scope_definition": "physical means the exact response of the declared nonlinear dimensionless double-pendulum equations; it does not mean that the equations, calibration, or apparatus have been empirically validated",
  "reproduction_commands": [
    "python -m pip install -r requirements.txt",
    "python oig_double_pendulum_validated_transfer.py --physical-output artifacts/double_pendulum_physical_positive_floor.json",
    "python oig_double_pendulum_validated_transfer.py --verify-physical artifacts/double_pendulum_physical_positive_floor.json",
    "python -m unittest -v test_oig_double_pendulum_validated_transfer.py"
  ],
  "robust_physical_floor_lower_exact": "195932905640268820789/2500000000000000000000",
  "schema_version": "oig-double-pendulum-physical-positive-floor-v1",
  "scope_boundary": "The floor concerns the declared dimensionless model for the two-protocol experiment at u=v=0; it is not a uniform parameter-neighbourhood, model-adequacy, calibration, or hardware statement.",
  "scope_flags": {
    "declared_dimensionless_model_floor_certified": true,
    "empirical_model_adequacy_certified": false,
    "hardware_calibration_certified": false,
    "parameter_neighborhood_uniformity_certified": false,
    "seven_candidate_efficiency_certified": false,
    "seven_candidate_selection_certified": false,
    "two_active_protocol_fixed_mixture_certified": true
  },
  "status": "proved",
  "unresolved_reason": null,
  "validated_transfer": {
    "all_active_declared_box_memberships_certified": true,
    "configuration": {
      "maximum_tube_iterations": 12,
      "precision_bits": 160,
      "step_size_exact": "1/100",
      "taylor_order": 8,
      "tube_inflation_exact": "6/5"
    },
    "dimensionless_representative_exact": {
      "g": "1/1",
      "l1": "1/1",
      "l2": "exp(v)",
      "m1": "1/1",
      "m2": "exp(u)"
    },
    "enclosed_protocol_transfer_ready": true,
    "method": "outward-arb-picard-taylor-self-mapping-v1",
    "nominal_parameter_point_exact": {
      "u": "0/1",
      "v": "0/1"
    },
    "proof_dependency_contract": {
      "exact_rational_arithmetic": "Python fractions.Fraction",
      "install_command": "python -m pip install -r requirements.txt",
      "ordinary_scipy_ode_solver_used_in_proof": false,
      "outward_ball_arithmetic": "python-flint>=0.9,<0.10 (Arb)"
    },
    "protocols": [
      {
        "declared_box_inclusion_slack_exact": [
          "6186249475362419/6125000000000000000",
          "2245728544213952449/2223500000000000000000"
        ],
        "declared_box_membership_by_source": [
          true,
          true
        ],
        "declared_box_membership_certified": true,
        "declared_response_centre_exact": [
          "265/3136",
          "-4041/8894"
        ],
        "declared_response_radius_exact": [
          "101/100000",
          "101/100000"
        ],
        "inclusion_slack_is_strictly_positive": true,
        "launch_exact": [
          "4/5",
          "-7/20",
          "0/1",
          "0/1"
        ],
        "maximum_endpoint_radius_upper": 4.0237039909815295e-12,
        "maximum_picard_tube_iterations": 2,
        "maximum_tube_radius_upper": 4.706910181499452e-12,
        "name": "launch-A / theta-2 / tau-1",
        "response_enclosure": [
          {
            "decimal_midpoint": 0.08450255093508893,
            "decimal_radius_upper": 3.3588249022048293e-13,
            "lower_exact": "10562818866844131/125000000000000000",
            "upper_exact": "42251275467712409/500000000000000000"
          },
          {
            "decimal_midpoint": -0.4543512451293749,
            "decimal_radius_upper": 4.277331480576996e-13,
            "lower_exact": "-56793905641225329/125000000000000000",
            "upper_exact": "-227175622564473567/500000000000000000"
          }
        ],
        "sensor": "theta_2",
        "status": "proved",
        "terminal_time_exact": "1/1",
        "unresolved_reason": null,
        "validated_step_count": 100
      },
      {
        "declared_box_inclusion_slack_exact": [
          "25375956496375261/25125000000000000000",
          "396422874971311/392500000000000000"
        ],
        "declared_box_membership_by_source": [
          true,
          true
        ],
        "declared_box_membership_certified": true,
        "declared_response_centre_exact": [
          "2357/5025",
          "13/3140"
        ],
        "declared_response_radius_exact": [
          "101/100000",
          "101/100000"
        ],
        "inclusion_slack_is_strictly_positive": true,
        "launch_exact": [
          "-3/5",
          "9/10",
          "0/1",
          "0/1"
        ],
        "maximum_endpoint_radius_upper": 7.660741290459183e-11,
        "maximum_picard_tube_iterations": 6,
        "maximum_tube_radius_upper": 8.95650684776342e-11,
        "name": "launch-B / scaled-omega-1 / tau-5/4",
        "response_enclosure": [
          {
            "decimal_midpoint": 0.46905473801917236,
            "decimal_radius_upper": 3.0723174709607753e-11,
            "lower_exact": "9381094759768983/20000000000000000",
            "upper_exact": "58631842256236939/125000000000000000"
          },
          {
            "decimal_midpoint": 0.00414012202889391,
            "decimal_radius_upper": 5.444470839877785e-11,
            "lower_exact": "10350304936123/2500000000000000",
            "upper_exact": "4140122083338621/1000000000000000000"
          }
        ],
        "sensor": "scaled_omega_1",
        "status": "proved",
        "terminal_time_exact": "5/4",
        "unresolved_reason": null,
        "validated_step_count": 125
      }
    ],
    "schema_version": "oig-double-pendulum-validated-transfer-v1",
    "scope_boundary": "Pointwise nominal-parameter transfer for exactly the two active protocols; no parameter-neighbourhood enclosure and no claim for the five inactive candidates. The word physical refers only to the declared nonlinear dimensionless double-pendulum model, not empirical hardware or nature.",
    "scope_flags": {
      "empirical_model_adequacy_certified": false,
      "hardware_calibration_certified": false,
      "parameter_neighborhood_uniformity_certified": false,
      "seven_candidate_efficiency_certified": false,
      "seven_candidate_selection_certified": false,
      "two_active_protocol_responses_at_nominal_point_certified": true
    },
    "source_coordinates": [
      "log_mass_ratio=log(m2/m1)",
      "log_length_ratio=log(l2/l1)"
    ],
    "status": "proved",
    "theorem_obligations": {
      "defect_bound": "Taylor theorem with interval coefficient over every step",
      "flow_to_box_link": "strict inclusion in each declared rational response box",
      "outward_scalar_arithmetic": "Arb balls at the declared precision",
      "state_and_parameter_sensitivity_ode": "joint 12-dimensional forward system",
      "step_existence_and_containment": "componentwise Picard self-mapping inequality"
    }
  }
}