{
  "test_info": {
    "timestamp": "2026-05-07T04:27:09.855636",
    "n_values": [
      6,
      8,
      10,
      12
    ],
    "k_values": [
      2,
      3
    ],
    "total_tests": 8
  },
  "results": [
    {
      "n": 6,
      "k": 2,
      "family_size": 5,
      "is_intersecting": true,
      "packet": {
        "family_size": 5,
        "packet_diversity": 1.2,
        "intersection_density": 1.0
      },
      "field": {
        "density": 0.3333333333333333,
        "theoretical_max": 5,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          4.0,
          -0.9999999999999991,
          -1.0,
          -1.0,
          -1.0
        ],
        "spectral_radius": 4.0,
        "intersection_graph_rank": 5
      },
      "shear": {
        "family_rigidity": 10000000000.0,
        "avg_intersection_size": 1.0,
        "intersection_variance": 0.0
      }
    },
    {
      "n": 6,
      "k": 3,
      "family_size": 10,
      "is_intersecting": true,
      "packet": {
        "family_size": 10,
        "packet_diversity": 0.6,
        "intersection_density": 1.6666666666666667
      },
      "field": {
        "density": 0.5,
        "theoretical_max": 10,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          9.0,
          -0.999999999999999,
          -0.9999999999999997,
          -0.9999999999999998,
          -0.9999999999999999,
          -1.0,
          -1.0,
          -1.0000000000000002,
          -1.0000000000000004,
          -1.0000000000000009
        ],
        "spectral_radius": 9.0,
        "intersection_graph_rank": 10
      },
      "shear": {
        "family_rigidity": 4.499999997975,
        "avg_intersection_size": 1.6666666666666667,
        "intersection_variance": 0.22222222222222218
      }
    },
    {
      "n": 8,
      "k": 2,
      "family_size": 7,
      "is_intersecting": true,
      "packet": {
        "family_size": 7,
        "packet_diversity": 1.1428571428571428,
        "intersection_density": 1.0
      },
      "field": {
        "density": 0.25,
        "theoretical_max": 7,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          5.999999999999998,
          -0.9999999999999992,
          -0.9999999999999994,
          -0.9999999999999999,
          -1.0,
          -1.0000000000000009,
          -1.0000000000000009
        ],
        "spectral_radius": 5.999999999999998,
        "intersection_graph_rank": 7
      },
      "shear": {
        "family_rigidity": 10000000000.0,
        "avg_intersection_size": 1.0,
        "intersection_variance": 0.0
      }
    },
    {
      "n": 8,
      "k": 3,
      "family_size": 21,
      "is_intersecting": true,
      "packet": {
        "family_size": 21,
        "packet_diversity": 0.38095238095238093,
        "intersection_density": 1.5
      },
      "field": {
        "density": 0.375,
        "theoretical_max": 21,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          19.999999999999996,
          -0.9999999999999922,
          -0.9999999999999966,
          -0.9999999999999973,
          -0.9999999999999992,
          -0.9999999999999993,
          -0.9999999999999993,
          -0.9999999999999994,
          -0.9999999999999998,
          -0.9999999999999998,
          -0.9999999999999999,
          -1.0,
          -1.0000000000000002,
          -1.0000000000000002,
          -1.0000000000000002,
          -1.0000000000000002,
          -1.0000000000000007,
          -1.000000000000001,
          -1.0000000000000016,
          -1.0000000000000033,
          -1.0000000000000084
        ],
        "spectral_radius": 19.999999999999996,
        "intersection_graph_rank": 21
      },
      "shear": {
        "family_rigidity": 3.9999999984,
        "avg_intersection_size": 1.5,
        "intersection_variance": 0.25
      }
    },
    {
      "n": 10,
      "k": 2,
      "family_size": 9,
      "is_intersecting": true,
      "packet": {
        "family_size": 9,
        "packet_diversity": 1.1111111111111112,
        "intersection_density": 1.0
      },
      "field": {
        "density": 0.2,
        "theoretical_max": 9,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          7.999999999999998,
          -0.9999999999999982,
          -0.9999999999999994,
          -0.9999999999999997,
          -0.9999999999999998,
          -0.9999999999999999,
          -1.0000000000000002,
          -1.0000000000000004,
          -1.0000000000000022
        ],
        "spectral_radius": 7.999999999999998,
        "intersection_graph_rank": 9
      },
      "shear": {
        "family_rigidity": 10000000000.0,
        "avg_intersection_size": 1.0,
        "intersection_variance": 0.0
      }
    },
    {
      "n": 10,
      "k": 3,
      "family_size": 36,
      "is_intersecting": true,
      "packet": {
        "family_size": 36,
        "packet_diversity": 0.2777777777777778,
        "intersection_density": 1.4
      },
      "field": {
        "density": 0.3,
        "theoretical_max": 36,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          35.000000000000036,
          -0.9999999999999895,
          -0.999999999999991,
          -0.999999999999994,
          -0.9999999999999947,
          -0.9999999999999961,
          -0.9999999999999967,
          -0.999999999999997,
          -0.9999999999999971,
          -0.9999999999999972,
          -0.9999999999999976,
          -0.9999999999999979,
          -0.9999999999999982,
          -0.9999999999999989,
          -0.9999999999999994,
          -0.9999999999999996,
          -0.9999999999999997,
          -1.0,
          -1.0000000000000004,
          -1.0000000000000004,
          -1.0000000000000004,
          -1.0000000000000007,
          -1.0000000000000007,
          -1.000000000000001,
          -1.0000000000000013,
          -1.0000000000000016,
          -1.0000000000000018,
          -1.0000000000000027,
          -1.0000000000000038,
          -1.0000000000000038,
          -1.000000000000004,
          -1.0000000000000047,
          -1.0000000000000062,
          -1.0000000000000064,
          -1.0000000000000209,
          -1.000000000000028
        ],
        "spectral_radius": 35.000000000000036,
        "intersection_graph_rank": 36
      },
      "shear": {
        "family_rigidity": 4.166666664930555,
        "avg_intersection_size": 1.4,
        "intersection_variance": 0.24000000000000002
      }
    },
    {
      "n": 12,
      "k": 2,
      "family_size": 11,
      "is_intersecting": true,
      "packet": {
        "family_size": 11,
        "packet_diversity": 1.0909090909090908,
        "intersection_density": 1.0
      },
      "field": {
        "density": 0.16666666666666666,
        "theoretical_max": 11,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          10.0,
          -0.9999999999999974,
          -0.9999999999999987,
          -0.9999999999999994,
          -0.9999999999999998,
          -0.9999999999999999,
          -1.0,
          -1.0000000000000002,
          -1.0000000000000018,
          -1.000000000000002,
          -1.0000000000000024
        ],
        "spectral_radius": 10.0,
        "intersection_graph_rank": 11
      },
      "shear": {
        "family_rigidity": 10000000000.0,
        "avg_intersection_size": 1.0,
        "intersection_variance": 0.0
      }
    },
    {
      "n": 12,
      "k": 3,
      "family_size": 55,
      "is_intersecting": true,
      "packet": {
        "family_size": 55,
        "packet_diversity": 0.21818181818181817,
        "intersection_density": 1.3333333333333333
      },
      "field": {
        "density": 0.25,
        "theoretical_max": 55,
        "relative_size": 1.0
      },
      "spectral": {
        "eigenvalues": [
          53.999999999999986,
          -0.999999999999964,
          -0.9999999999999808,
          -0.9999999999999909,
          -0.9999999999999916,
          -0.9999999999999926,
          -0.9999999999999942,
          -0.9999999999999946,
          -0.9999999999999948,
          -0.9999999999999954,
          -0.9999999999999954,
          -0.9999999999999958,
          -0.9999999999999966,
          -0.9999999999999968,
          -0.9999999999999977,
          -0.9999999999999979,
          -0.9999999999999982,
          -0.9999999999999986,
          -0.999999999999999,
          -0.999999999999999,
          -0.999999999999999,
          -0.999999999999999,
          -0.9999999999999996,
          -0.9999999999999996,
          -0.9999999999999996,
          -0.9999999999999999,
          -0.9999999999999999,
          -1.0,
          -1.0,
          -1.0000000000000002,
          -1.0000000000000002,
          -1.0000000000000007,
          -1.0000000000000007,
          -1.0000000000000009,
          -1.0000000000000009,
          -1.000000000000001,
          -1.0000000000000016,
          -1.0000000000000016,
          -1.0000000000000016,
          -1.000000000000002,
          -1.0000000000000022,
          -1.0000000000000024,
          -1.0000000000000029,
          -1.0000000000000038,
          -1.0000000000000047,
          -1.000000000000005,
          -1.0000000000000056,
          -1.0000000000000062,
          -1.0000000000000082,
          -1.0000000000000084,
          -1.000000000000009,
          -1.0000000000000098,
          -1.0000000000000138,
          -1.0000000000000182,
          -1.000000000000024
        ],
        "spectral_radius": 53.999999999999986,
        "intersection_graph_rank": 55
      },
      "shear": {
        "family_rigidity": 4.4999999979749985,
        "avg_intersection_size": 1.3333333333333333,
        "intersection_variance": 0.2222222222222223
      }
    }
  ],
  "theorem_analysis": [
    {
      "n": 6,
      "k": 2,
      "theoretical_max": 5,
      "achieved_max": 5,
      "ratio": 1.0
    },
    {
      "n": 6,
      "k": 3,
      "theoretical_max": 10,
      "achieved_max": 10,
      "ratio": 1.0
    },
    {
      "n": 8,
      "k": 2,
      "theoretical_max": 7,
      "achieved_max": 7,
      "ratio": 1.0
    },
    {
      "n": 8,
      "k": 3,
      "theoretical_max": 21,
      "achieved_max": 21,
      "ratio": 1.0
    },
    {
      "n": 10,
      "k": 2,
      "theoretical_max": 9,
      "achieved_max": 9,
      "ratio": 1.0
    },
    {
      "n": 10,
      "k": 3,
      "theoretical_max": 36,
      "achieved_max": 36,
      "ratio": 1.0
    },
    {
      "n": 12,
      "k": 2,
      "theoretical_max": 11,
      "achieved_max": 11,
      "ratio": 1.0
    },
    {
      "n": 12,
      "k": 3,
      "theoretical_max": 55,
      "achieved_max": 55,
      "ratio": 1.0
    }
  ],
  "primitive_analysis": {
    "packet": {
      "equation": "\u0393\u1d62",
      "application": "Intersecting family as packet collection",
      "insight": "Family as packet collection with witness property"
    },
    "field": {
      "equation": "\u03c1(x\u20d7)",
      "application": "Family density and theoretical maximum",
      "insight": "Field captures theorem bound C(n-1, k-1)"
    },
    "spectral": {
      "equation": "C = U\u039bU\u1d40",
      "application": "Intersection graph eigen decomposition",
      "insight": "Spectral radius indicates intersection connectivity"
    },
    "shear": {
      "equation": "G = A\u1d40A",
      "application": "Family rigidity and intersection variance",
      "insight": "Shear measures family deformation"
    }
  },
  "validation": {
    "status": "SUCCESS",
    "insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Ko\u2013Rado Theorem. Packet primitive captures family structure. Field primitive captures theorem bound. Spectral primitive reveals intersection structure. Shear primitive measures family deformation. Framework validated for extremal set theory problems."
  }
}