{
  "title": "An Exact-Zero Dielectric Leaves a Vacuum-History Ambiguity",
  "sources": {
    "experiments/vacuum-connection-checks.py": "95e318a98b676905fd9146002f73918ae0b87d13ba6d564ce690c0248ce4cf7b",
    "experiments/hodge-gated-gauss-checks.py": "d24406f940aea7e81ee65cb9596a57a3dc88e0f5d03c07106db93cc948ff1faa"
  },
  "checks_passed": 472,
  "dependency_checks_passed": 18,
  "case_count": 16,
  "occupied_depths": [
    2,
    3,
    4,
    5,
    6,
    10
  ],
  "scope": "exact local homogeneous driven classical solutions with smooth raw vacuum connection histories; energy-density controls, not finite-total-energy particles",
  "conclusion": "In active sectors, different smooth zero-energy vacuum connection histories sharing identical initial raw data yield different responses to the same later neutral drive. A positive floor rejects those histories but removes the exact source constraints. A geometric vacuum-flatness or connection-reduction rule requires derivation. Inactive sectors keep their zero source and neutral harmonic controls.",
  "cases": [
    {
      "case": "vacuum-response-depth-2-B-0",
      "real_depth": 2,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 0.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-2-B-1",
      "real_depth": 2,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 1.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-2-B-3",
      "real_depth": 2,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 3.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-3-B-0",
      "real_depth": 3,
      "hodge_coefficient_active": true,
      "settled_magnetic_curvature": 0.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-3-B-1",
      "real_depth": 3,
      "hodge_coefficient_active": true,
      "settled_magnetic_curvature": 1.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.3833787434220946,
      "maximum_energy_work_error": 3.209932319947484e-14,
      "maximum_independent_scalar_PDE_error": 5.468203634340085e-08,
      "post_probe_complete_energy_density": 0.07251764543751306,
      "unforced_common_initial_velocity_period": 4.9173822305399515,
      "unforced_turning_amplitude": -0.17205289820856973,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-3-B-3",
      "real_depth": 3,
      "hodge_coefficient_active": true,
      "settled_magnetic_curvature": 3.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.22686144342728434,
      "maximum_energy_work_error": 4.835021272242557e-14,
      "maximum_independent_scalar_PDE_error": 5.487742082976865e-08,
      "post_probe_complete_energy_density": 0.07246051405469903,
      "unforced_common_initial_velocity_period": 3.111923495369455,
      "unforced_turning_amplitude": -0.26193038339572744,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-4-B-0",
      "real_depth": 4,
      "hodge_coefficient_active": true,
      "settled_magnetic_curvature": 0.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-4-B-1",
      "real_depth": 4,
      "hodge_coefficient_active": true,
      "settled_magnetic_curvature": 1.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.3833787434220946,
      "maximum_energy_work_error": 3.209932319947484e-14,
      "maximum_independent_scalar_PDE_error": 5.468203634340085e-08,
      "post_probe_complete_energy_density": 0.07251764543751306,
      "unforced_common_initial_velocity_period": 4.9173822305399515,
      "unforced_turning_amplitude": -0.17205289820856973,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-4-B-3",
      "real_depth": 4,
      "hodge_coefficient_active": true,
      "settled_magnetic_curvature": 3.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.22686144342728434,
      "maximum_energy_work_error": 4.835021272242557e-14,
      "maximum_independent_scalar_PDE_error": 5.487742082976865e-08,
      "post_probe_complete_energy_density": 0.07246051405469903,
      "unforced_common_initial_velocity_period": 3.111923495369455,
      "unforced_turning_amplitude": -0.26193038339572744,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-5-B-0",
      "real_depth": 5,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 0.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-5-B-1",
      "real_depth": 5,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 1.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-5-B-3",
      "real_depth": 5,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 3.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-6-B-0",
      "real_depth": 6,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 0.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-6-B-1",
      "real_depth": 6,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 1.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-6-B-3",
      "real_depth": 6,
      "hodge_coefficient_active": false,
      "settled_magnetic_curvature": 3.0,
      "initial_raw_connection_and_curvature": "0",
      "initial_neutral_amplitude_and_velocity": "0",
      "identical_neutral_probe": "smooth pulse supported in 2<t<3",
      "response_at_t4": 0.4128976872814609,
      "maximum_energy_work_error": 3.0059288391726113e-14,
      "maximum_independent_scalar_PDE_error": 5.4657472436936416e-08,
      "post_probe_complete_energy_density": 0.07252479676807998,
      "unforced_common_initial_velocity_period": 6.981317007977318,
      "unforced_turning_amplitude": -0.0035577020538688294,
      "local_homogeneous_control": true,
      "finite_total_energy_global_solution": false
    },
    {
      "case": "vacuum-response-depth-10",
      "real_depth": 10,
      "independent_normal_connection_exists": false,
      "neutral_clock_period": 6.981317007977318
    }
  ]
}