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          {
            "lower_exact": "72721774000911/125000000000000",
            "upper_exact": "581774192023153/1000000000000000"
          },
          {
            "lower_exact": "469054737988447/1000000000000000",
            "upper_exact": "234527369024949/500000000000000"
          },
          {
            "lower_exact": "4140121974447/1000000000000000",
            "upper_exact": "4140122083341/1000000000000000"
          }
        ],
        "semantics": "one Cartesian outer box; discarded trajectory correlations make the subsequent interval theorem conservative"
      },
      "launch_exact": [
        "-3/5",
        "9/10",
        "0/1",
        "0/1"
      ],
      "legacy_scalar_cost_exact": "5/4",
      "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",
      "noise_precision_exact": "1/1",
      "observation_time_tau_exact": "5/4",
      "outward_joint_box_certified": true,
      "parameter_response_outer_box_exact": {
        "coordinates": [
          "u=log(m2/m1)",
          "v=log(l2/l1)"
        ],
        "u": {
          "lower_exact": "469054737988447/1000000000000000",
          "upper_exact": "234527369024949/500000000000000"
        },
        "v": {
          "lower_exact": "4140121974447/1000000000000000",
          "upper_exact": "4140122083341/1000000000000000"
        }
      },
      "physical_scalar_weight_exact": "4/27",
      "sensor": "scaled_omega_1",
      "sensor_output_outer_box_exact": {
        "lower_exact": "279481735653907/250000000000000",
        "upper_exact": "1117926942622569/1000000000000000"
      },
      "sensor_state_index": 2,
      "shared_clock_response_outer_box_exact": {
        "lower_exact": "72721774000911/125000000000000",
        "upper_exact": "581774192023153/1000000000000000"
      },
      "validated_step_count": 125
    }
  ],
  "outward_box_summary": {
    "maximum_parameter_response_half_width_decimal": 7.21765e-11,
    "maximum_parameter_response_half_width_exact": "144353/2000000000000000",
    "maximum_picard_tube_radius_upper": 8.95650684776342e-11,
    "maximum_sensor_output_half_width_decimal": 6.5645e-12,
    "maximum_sensor_output_half_width_exact": "13129/2000000000000000",
    "maximum_shared_clock_response_half_width_decimal": 3.1076e-11,
    "maximum_shared_clock_response_half_width_exact": "7769/250000000000000",
    "maximum_trajectory_endpoint_radius_upper": 7.660741290459183e-11,
    "rational_outer_box_half_width_definition": "(upper-lower)/2"
  },
  "physical_scope_definition": "Physical means the exact local parameter and clock response of the declared dimensionless nonlinear double-pendulum equations with m1=l1=g=1, m2=exp(u), and l2=exp(v). It does not mean that the equations, apparatus, calibration, noise model, or launch preparation have been empirically validated.",
  "proof_dependency_contract": {
    "exact_interval_and_ldl_arithmetic": "Python fractions.Fraction",
    "install_command": "python -m pip install -r requirements.txt",
    "ordinary_scipy_ode_solver_used_in_proof": false,
    "outward_scalar_arithmetic": "python-flint>=0.9,<0.10 (Arb)",
    "sampled_eigenvalue_used_in_floor_decision": false
  },
  "reproduction_commands": [
    "python -m pip install -r requirements.txt",
    "python oig_double_pendulum_grouped_validated_transfer.py --output artifacts/double_pendulum_grouped_physical_clock_floor.json",
    "python oig_double_pendulum_grouped_validated_transfer.py --verify artifacts/double_pendulum_grouped_physical_clock_floor.json",
    "python -m unittest -v test_oig_double_pendulum_grouped_validated_transfer.py"
  ],
  "schema_version": "oig-double-pendulum-grouped-physical-clock-floor-v1",
  "scope_boundary": "The theorem is pointwise at u=v=0 for the fixed grouped shares (2/3,1/3,0). It profiles one local unrestricted shared log-clock column after stacking all five outputs. Launch preparation is exact. The within-group output metric is the declared diagonal unit-precision metric. No parameter neighbourhood, finite clock-dilation amplitude, preparation error, model discrepancy, hardware calibration, candidate-selection transfer, grid optimality, or efficiency claim is made.",
  "scope_flags": {
    "candidate_selection_transfer_certified": false,
    "declared_dimensionless_model_five_row_transfer_certified": true,
    "empirical_model_adequacy_certified": false,
    "exact_launch_preparation_assumed": true,
    "finite_clock_dilation_amplitude_certified": false,
    "finite_grid_optimality_certified": false,
    "fixed_grouped_mixture_shared_clock_floor_certified": true,
    "hardware_calibration_certified": false,
    "launch_preparation_error_bounded": false,
    "parameter_neighborhood_uniformity_certified": false
  },
  "shared_clock_nuisance": {
    "finite_epsilon_nonlinear_robustness_claimed": false,
    "linear_response_at_epsilon_zero": "tau*d(sensor output)/d tau",
    "parameter": "one unrestricted local log-clock dilation epsilon",
    "scope": "one column is shared across every repetition, output, and active group and is profiled only after all five weighted rows are stacked",
    "time_rule": "tau_actual=exp(epsilon)*tau_declared"
  },
  "source_model": {
    "base_point_exact": [
      "0/1",
      "0/1"
    ],
    "coordinates": [
      "u=log(m2/m1)",
      "v=log(l2/l1)"
    ],
    "dimensionless_representative": "m1=l1=g=1, m2=exp(u), l2=exp(v), tau=t*sqrt(g/l1), nu=dtheta/dtau",
    "query": "the full identity query in (u,v)"
  },
  "status": "proved",
  "strict_internal_verification": {
    "exact_interval_gram_recomputed": true,
    "exact_interval_ldl_recomputed": true,
    "outward_ode_rows_recomputed": true,
    "passed": true
  },
  "theorem": {
    "positive_two_source_floor_certified": true,
    "proof": "exact rational interval enclosure of J, followed by a no-pivot interval LDL proof of J-diag(0,L,L)>0 and the exact Schur theorem",
    "sharp_physical_clock_profiled_floor_lower_exact": "180269/10000000",
    "simple_human_readable_floor_lower_exact": "9/500",
    "source_metric_exact": [
      [
        "1/1",
        "0/1"
      ],
      [
        "0/1",
        "1/1"
      ]
    ],
    "strict_inequality": "lambda_min(I_profiled)>180269/10000000>9/500 for every response-and-clock row in the five outward boxes"
  },
  "validated_transfer_configuration": {
    "maximum_tube_iterations": 12,
    "precision_bits": 160,
    "rational_outer_hull_denominator_before_reduction": 1000000000000000,
    "step_size_exact": "1/100",
    "taylor_order": 8,
    "tube_inflation_exact": "6/5"
  }
}