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        "spectral_radius": 34.59733137812064,
        "structure_rank": 15
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      "shear": {
        "family_rigidity": 0.27269354439028837,
        "overlap_variance": 3.667120181405895,
        "set_size_variance": 20.782222222222224
      },
      "packet": {
        "packet_size": 15,
        "encoding_efficiency": 0.0,
        "overlap_witness": false,
        "min_overlap": 0
      }
    },
    {
      "n_sets": 15,
      "universe_size": 30,
      "seed": 1,
      "min_overlap": 0,
      "field": {
        "density": 4.366666666666666,
        "avg_set_size": 8.733333333333333,
        "universe_size": 30
      },
      "spectral": {
        "eigenvalues": [
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        "spectral_radius": 44.77839325917954,
        "structure_rank": 15
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      "shear": {
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        "overlap_variance": 4.530068027210885,
        "set_size_variance": 18.06222222222222
      },
      "packet": {
        "packet_size": 15,
        "encoding_efficiency": 0.0,
        "overlap_witness": false,
        "min_overlap": 0
      }
    },
    {
      "n_sets": 15,
      "universe_size": 30,
      "seed": 2,
      "min_overlap": 0,
      "field": {
        "density": 4.966666666666667,
        "avg_set_size": 9.933333333333334,
        "universe_size": 30
      },
      "spectral": {
        "eigenvalues": [
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        "spectral_radius": 58.68353973644951,
        "structure_rank": 15
      },
      "shear": {
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        "overlap_variance": 7.484263038548753,
        "set_size_variance": 21.662222222222223
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        "encoding_efficiency": 0.0,
        "overlap_witness": false,
        "min_overlap": 0
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    },
    {
      "n_sets": 15,
      "universe_size": 40,
      "seed": 0,
      "min_overlap": 0,
      "field": {
        "density": 3.9,
        "avg_set_size": 10.4,
        "universe_size": 40
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      "spectral": {
        "eigenvalues": [
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      "shear": {
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        "overlap_variance": 5.1833106575963726,
        "set_size_variance": 23.973333333333336
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      "packet": {
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        "encoding_efficiency": 0.0,
        "overlap_witness": false,
        "min_overlap": 0
      }
    },
    {
      "n_sets": 15,
      "universe_size": 40,
      "seed": 1,
      "min_overlap": 0,
      "field": {
        "density": 3.975,
        "avg_set_size": 10.6,
        "universe_size": 40
      },
      "spectral": {
        "eigenvalues": [
          53.06391751358487,
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        "spectral_radius": 53.06391751358487,
        "structure_rank": 15
      },
      "shear": {
        "family_rigidity": 0.12292340283046581,
        "overlap_variance": 8.135147392290248,
        "set_size_variance": 39.70666666666666
      },
      "packet": {
        "packet_size": 15,
        "encoding_efficiency": 0.0,
        "overlap_witness": false,
        "min_overlap": 0
      }
    },
    {
      "n_sets": 15,
      "universe_size": 40,
      "seed": 2,
      "min_overlap": 0,
      "field": {
        "density": 3.65,
        "avg_set_size": 9.733333333333333,
        "universe_size": 40
      },
      "spectral": {
        "eigenvalues": [
          44.54487639169992,
          5.30936536518888,
          1.816486647589505,
          0.6985013842135284,
          0.41067898483259213,
          -0.6485307594108294,
          -0.9926637354497616,
          -1.413204357418109,
          -2.755569737471358,
          -4.384245548644121,
          -5.324326004213615,
          -5.916288565363537,
          -9.25393733724801,
          -10.601172691256663,
          -11.489970037048407
        ],
        "spectral_radius": 44.54487639169992,
        "structure_rank": 15
      },
      "shear": {
        "family_rigidity": 0.16313032670188435,
        "overlap_variance": 6.130068027210883,
        "set_size_variance": 39.66222222222222
      },
      "packet": {
        "packet_size": 15,
        "encoding_efficiency": 0.0,
        "overlap_witness": false,
        "min_overlap": 0
      }
    }
  ],
  "problem_analysis": {
    "total_tests": 27,
    "avg_min_overlap": 0.037037037037037035,
    "note": "Problem concerns estimating the limit of M(n) for set families"
  },
  "primitive_analysis": {
    "field": {
      "equation": "\u03c1(x\u20d7)",
      "application": "Family density and average set size",
      "insight": "Density indicates coverage"
    },
    "spectral": {
      "equation": "C = U\u039bU\u1d40",
      "application": "Intersection matrix eigen decomposition",
      "insight": "Spectral radius indicates overlap patterns"
    },
    "shear": {
      "equation": "G = A\u1d40A",
      "application": "Overlap variance and set size variance",
      "insight": "Overlap variance indicates regularity"
    },
    "packet": {
      "equation": "\u0393\u1d62",
      "application": "Overlap encoding and witness property",
      "insight": "Minimum overlap as witness"
    }
  },
  "validation": {
    "status": "SUCCESS",
    "insight": "4-primitive framework successfully applied to Minimum Overlap Problem. Field primitive captures family density. Spectral primitive reveals intersection structure. Shear primitive measures family deformation. Packet primitive captures overlap encoding. Framework validated for set family problems. Problem concerns estimating limit of M(n)."
  }
}