{
  "passed": true,
  "ambient_dimension": 128,
  "sparse_dimensions": 16,
  "sample_count_per_distribution": 20000,
  "seed": 20261111,
  "outer_radius": 1,
  "controls": {
    "uniform_d_ball": {
      "active_dimensions": 128,
      "max_norm": 0.9999994375833048,
      "mean_squared_norm": 0.9846031601593415,
      "radial_dimension": 128
    },
    "uniform_k_ball": {
      "active_dimensions": 16,
      "max_norm": 0.999996278718437,
      "mean_squared_norm": 0.8894302831319619,
      "radial_dimension": 16
    },
    "sparse_same_radial": {
      "active_dimensions": 16,
      "max_norm": 0.9999995857902936,
      "mean_squared_norm": 0.9847321079586993,
      "radial_dimension": 128
    },
    "sparse_sphere": {
      "active_dimensions": 16,
      "max_norm": 1.0000000000000002,
      "mean_squared_norm": 1.0,
      "radial_dimension": null
    }
  },
  "elapsed_seconds": 3.858549994882196,
  "limitations": [
    "synthetic distributions, not trained-network parameters",
    "same outer radius, not same mean squared norm",
    "changing angular support alone leaves the chosen radial law unchanged",
    "fixed-radius shell is a deliberately structured counterexample"
  ]
}