{
  "seed": 2742,
  "dimension": 8,
  "beta": 2.0,
  "kappa": 1.5,
  "algebra_check": {
    "max_projector_tangent_or_idempotence_residual": 4.996003610813204e-16
  },
  "prediction_checks": {
    "P1_intrinsic_constraint": {
      "prediction": "post-retraction intermediate |norm-1| stays at floating-point precision, independent of h",
      "h_values": [
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        0.02,
        0.01,
        0.005
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      "observed_max_each": [
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        3.3306690738754696e-16,
        3.3306690738754696e-16
      ],
      "observed_max_over_sweep": 3.3306690738754696e-16
    },
    "P2_ambient_local_scaling": {
      "prediction": "one-step unconstrained radial norm violation scales as h^0.5",
      "h_values": [
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        0.02,
        0.01,
        0.005
      ],
      "median_violation": [
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      "q95_violation": [
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      "observed_loglog_exponent": 0.5700982569303314
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    "P3_fixed_state_control_cost": {
      "prediction": "beta/4 E||u||^2 scales exactly as amplitude^2",
      "amplitudes": [
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        0.5,
        1.0,
        2.0,
        4.0
      ],
      "costs": [
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        0.10953079588961495,
        0.4381231835584598,
        1.7524927342338392,
        7.009970936935357
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      "observed_loglog_exponent": 2.0,
      "intrinsic_endpoint_alignment_by_amplitude": [
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        0.5762669444184964,
        0.6068380879001029,
        0.6569455105838411,
        0.7410485078624091
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  },
  "endpoint_comparison_equal_steps": {
    "intrinsic": {
      "max_violation": 3.3306690738754696e-16,
      "median_violation": 2.220446049250313e-16,
      "mean_final_norm_error": 5.0515147620444624e-17,
      "mean_alignment": 0.6095356317860299,
      "mean_control_energy": 0.09991351576752919,
      "q95_violation": 2.220446049250313e-16
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    "projected_each": {
      "max_violation": 3.3306690738754696e-16,
      "median_violation": 2.220446049250313e-16,
      "mean_final_norm_error": 5.0515147620444624e-17,
      "mean_alignment": 0.6095356317860299,
      "mean_control_energy": 0.09991351576752919,
      "q95_violation": 2.220446049250313e-16
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    "projected_final": {
      "max_violation": 1.2964174194605937,
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      "mean_final_norm_error": 3.874678355941796e-17,
      "mean_alignment": 0.6873637767586465,
      "mean_control_energy": 0.21509418408407457,
      "q95_violation": 0.9736673453575195
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    "ambient": {
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      "mean_alignment": 0.9739198921624161,
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      "q95_violation": 0.9736673453575195
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  }
}