{
  "title": "An Actual Nested Spatial Flag Generates a Conditional Rank-Three Normal Color Geometry",
  "sources": {
    "experiments/nested-normal-curvature-checks.py": "a77678aea22b4191f1f5180c8095cc03701e96c2a8c1b7db7e3d1ada0d872d2f",
    "experiments/hodge-gated-gauss-checks.py": "d24406f940aea7e81ee65cb9596a57a3dc88e0f5d03c07106db93cc948ff1faa"
  },
  "checks_passed": 689,
  "dependency_checks_passed": 174,
  "case_count": 13,
  "occupied_depths": [
    2,
    3,
    4,
    5,
    6,
    10
  ],
  "scope": "one exact nested ordinary spatial flag and induced normal connections; quadratic and higher-jet controls, not a derived force action",
  "cases": [
    {
      "case": "nested-normal-curvature-depth-2",
      "real_depth": 2,
      "even_source_carrier": 2,
      "intrinsic_real_normal_rank": 8,
      "intrinsic_curvature_lie_dimension": 1,
      "even_shadow_real_normal_rank": 8,
      "complex_normal_rank": 4,
      "even_shadow_curvature_lie_dimension": 1,
      "quadratic_first_normal_real_rank": 2,
      "symmetric_complex_second_fundamental_dimension": 1,
      "full_second_fundamental_normal_isomorphism": false,
      "odd_shadow_equals_intrinsic_normal_connection": true,
      "independent_gauge_action_or_force_derived": false
    },
    {
      "case": "nested-normal-curvature-depth-3",
      "real_depth": 3,
      "even_source_carrier": 4,
      "intrinsic_real_normal_rank": 7,
      "intrinsic_curvature_lie_dimension": 10,
      "even_shadow_real_normal_rank": 6,
      "complex_normal_rank": 3,
      "even_shadow_curvature_lie_dimension": 9,
      "quadratic_first_normal_real_rank": 6,
      "symmetric_complex_second_fundamental_dimension": 3,
      "full_second_fundamental_normal_isomorphism": true,
      "odd_shadow_equals_intrinsic_normal_connection": false,
      "independent_gauge_action_or_force_derived": false
    },
    {
      "case": "nested-normal-curvature-depth-4",
      "real_depth": 4,
      "even_source_carrier": 4,
      "intrinsic_real_normal_rank": 6,
      "intrinsic_curvature_lie_dimension": 9,
      "even_shadow_real_normal_rank": 6,
      "complex_normal_rank": 3,
      "even_shadow_curvature_lie_dimension": 9,
      "quadratic_first_normal_real_rank": 6,
      "symmetric_complex_second_fundamental_dimension": 3,
      "full_second_fundamental_normal_isomorphism": true,
      "odd_shadow_equals_intrinsic_normal_connection": true,
      "independent_gauge_action_or_force_derived": false
    },
    {
      "case": "nested-normal-curvature-depth-5",
      "real_depth": 5,
      "even_source_carrier": 6,
      "intrinsic_real_normal_rank": 5,
      "intrinsic_curvature_lie_dimension": 10,
      "even_shadow_real_normal_rank": 4,
      "complex_normal_rank": 2,
      "even_shadow_curvature_lie_dimension": 4,
      "quadratic_first_normal_real_rank": 4,
      "symmetric_complex_second_fundamental_dimension": 6,
      "full_second_fundamental_normal_isomorphism": false,
      "odd_shadow_equals_intrinsic_normal_connection": false,
      "independent_gauge_action_or_force_derived": false
    },
    {
      "case": "nested-normal-curvature-depth-6",
      "real_depth": 6,
      "even_source_carrier": 6,
      "intrinsic_real_normal_rank": 4,
      "intrinsic_curvature_lie_dimension": 4,
      "even_shadow_real_normal_rank": 4,
      "complex_normal_rank": 2,
      "even_shadow_curvature_lie_dimension": 4,
      "quadratic_first_normal_real_rank": 4,
      "symmetric_complex_second_fundamental_dimension": 6,
      "full_second_fundamental_normal_isomorphism": false,
      "odd_shadow_equals_intrinsic_normal_connection": true,
      "independent_gauge_action_or_force_derived": false
    },
    {
      "case": "nested-normal-curvature-depth-10",
      "real_depth": 10,
      "even_source_carrier": 10,
      "intrinsic_real_normal_rank": 0,
      "intrinsic_curvature_lie_dimension": 0,
      "even_shadow_real_normal_rank": 0,
      "complex_normal_rank": 0,
      "even_shadow_curvature_lie_dimension": 0,
      "quadratic_first_normal_real_rank": 0,
      "symmetric_complex_second_fundamental_dimension": 15,
      "full_second_fundamental_normal_isomorphism": false,
      "odd_shadow_equals_intrinsic_normal_connection": true,
      "independent_gauge_action_or_force_derived": false
    },
    {
      "case": "actual-complex-graph-diagonal-1-x-0",
      "complex_base_rank": 2,
      "complex_normal_rank": 3,
      "diagonal_second_derivative": 1.0,
      "position_x1": 0.0,
      "pointwise_curvature_lie_dimension": 9,
      "traceless_curvature_lie_dimension": 8
    },
    {
      "case": "actual-complex-graph-diagonal-1-x-0.4",
      "complex_base_rank": 2,
      "complex_normal_rank": 3,
      "diagonal_second_derivative": 1.0,
      "position_x1": 0.4,
      "pointwise_curvature_lie_dimension": 9,
      "traceless_curvature_lie_dimension": 8
    },
    {
      "case": "actual-complex-graph-diagonal-1.41421-x-0",
      "complex_base_rank": 2,
      "complex_normal_rank": 3,
      "diagonal_second_derivative": 1.4142135623730951,
      "position_x1": 0.0,
      "pointwise_curvature_lie_dimension": 4,
      "traceless_curvature_lie_dimension": 3
    },
    {
      "case": "actual-complex-graph-diagonal-1.41421-x-0.4",
      "complex_base_rank": 2,
      "complex_normal_rank": 3,
      "diagonal_second_derivative": 1.4142135623730951,
      "position_x1": 0.4,
      "pointwise_curvature_lie_dimension": 9,
      "traceless_curvature_lie_dimension": 8
    },
    {
      "case": "quadratic-curve-control",
      "real_depth": 2,
      "complex_normal_rank": 4,
      "normal_graph_degrees": [
        2
      ],
      "transported_curvature_lie_dimension": 1,
      "traceless_transport_lie_dimension": 1,
      "pointwise_only_curvature_lie_dimension": 1,
      "quadratic_selection_is_universal": false
    },
    {
      "case": "higher-normal-jet-curve",
      "real_depth": 2,
      "complex_normal_rank": 4,
      "normal_graph_degrees": [
        2,
        3,
        4,
        5
      ],
      "transported_curvature_lie_dimension": 16,
      "traceless_transport_lie_dimension": 15,
      "pointwise_only_curvature_lie_dimension": 1,
      "quadratic_selection_is_universal": false
    },
    {
      "case": "exact-rational-curvature-and-higher-jet-certificate",
      "all_occupied_intrinsic_and_shadow_dimensions_exact": true,
      "covariant_curvature_jet_maximum_order": 6,
      "depth_two_exact_normal_holonomy_dimension": 16,
      "depth_two_exact_traceless_holonomy_dimension": 15,
      "four_base_parallel_second_fundamental_obstruction_squared": 1536,
      "higher_jet_control_embeds_in_one_common_nested_flag": true
    }
  ]
}