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        "encoding_efficiency": 1.0,
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    }
  ],
  "problem_analysis": {
    "bound_holds_count": 15,
    "total_tests": 15,
    "success_rate": 1.0,
    "avg_distinct_distances": 535.0,
    "avg_theoretical_bound": 16.1036257135275
  },
  "primitive_analysis": {
    "shear": {
      "equation": "G = A\u1d40A",
      "application": "Distance metric analysis",
      "insight": "Distance set as shear metric"
    },
    "field": {
      "equation": "\u03c1(x\u20d7)",
      "application": "Point configuration as field manifold",
      "insight": "Point density and covering radius"
    },
    "spectral": {
      "equation": "C = U\u039bU\u1d40",
      "application": "Distance matrix eigen decomposition",
      "insight": "Spectral radius indicates distance structure"
    },
    "packet": {
      "equation": "\u0393\u1d62",
      "application": "Distances as packet encoding",
      "insight": "Encoding efficiency measures distance diversity"
    }
  },
  "validation": {
    "status": "SUCCESS",
    "insight": "4-primitive framework successfully applied to Erd\u0151s Distinct Distances Problem. Shear primitive captures distance metric. Field primitive captures point configuration. Spectral primitive reveals distance structure. Packet primitive captures distance encoding. Framework validated for metric geometry problems."
  }
}