{
  "test_info": {
    "timestamp": "2026-05-07T04:44:22.886409",
    "n_circles_values": [
      5,
      14,
      35
    ],
    "triangle_side": 10.0,
    "samples_per_n": 3,
    "total_tests": 9
  },
  "results": [
    {
      "n_circles": 5,
      "seed": 0,
      "triangle_side": 10.0,
      "circles_generated": 1,
      "field": {
        "density": 0.006727474144248256,
        "avg_radius": 0.30450988854744343,
        "circle_count": 1
      },
      "spectral": {
        "eigenvalues": [
          0.0
        ],
        "spectral_radius": 0.0,
        "structure_rank": 0
      },
      "shear": {
        "packing_rigidity": 10000000000.0,
        "radius_variance": 0.0,
        "position_variance": 0.0
      },
      "packet": {
        "packet_size": 1,
        "encoding_efficiency": 0.023094010767585032,
        "triangular_witness": true,
        "n_circles": 5
      }
    },
    {
      "n_circles": 5,
      "seed": 1,
      "triangle_side": 10.0,
      "circles_generated": 2,
      "field": {
        "density": 0.010335431657351679,
        "avg_radius": 0.2359880898489539,
        "circle_count": 2
      },
      "spectral": {
        "eigenvalues": [
          4.252920372423838,
          -4.252920372423838
        ],
        "spectral_radius": 4.252920372423838,
        "structure_rank": 2
      },
      "shear": {
        "packing_rigidity": 64.36084068495109,
        "radius_variance": 0.015537397941381101,
        "position_variance": 4.52183292354443
      },
      "packet": {
        "packet_size": 2,
        "encoding_efficiency": 0.046188021535170064,
        "triangular_witness": true,
        "n_circles": 5
      }
    },
    {
      "n_circles": 5,
      "seed": 2,
      "triangle_side": 10.0,
      "circles_generated": 1,
      "field": {
        "density": 0.008504721398413866,
        "avg_radius": 0.342377666271385,
        "circle_count": 1
      },
      "spectral": {
        "eigenvalues": [
          0.0
        ],
        "spectral_radius": 0.0,
        "structure_rank": 0
      },
      "shear": {
        "packing_rigidity": 10000000000.0,
        "radius_variance": 0.0,
        "position_variance": 0.0
      },
      "packet": {
        "packet_size": 1,
        "encoding_efficiency": 0.023094010767585032,
        "triangular_witness": true,
        "n_circles": 5
      }
    },
    {
      "n_circles": 14,
      "seed": 0,
      "triangle_side": 10.0,
      "circles_generated": 3,
      "field": {
        "density": 0.03408425222603262,
        "avg_radius": 0.390243357532543,
        "circle_count": 3
      },
      "spectral": {
        "eigenvalues": [
          2.737066762414689,
          -0.7266884740450638,
          -2.010378288369625
        ],
        "spectral_radius": 2.737066762414689,
        "structure_rank": 3
      },
      "shear": {
        "packing_rigidity": 232.1766807789625,
        "radius_variance": 0.004307064660530464,
        "position_variance": 0.6700684145873974
      },
      "packet": {
        "packet_size": 3,
        "encoding_efficiency": 0.06928203230275509,
        "triangular_witness": true,
        "n_circles": 14
      }
    },
    {
      "n_circles": 14,
      "seed": 1,
      "triangle_side": 10.0,
      "circles_generated": 8,
      "field": {
        "density": 0.04206454397542524,
        "avg_radius": 0.23792157590166346,
        "circle_count": 8
      },
      "spectral": {
        "eigenvalues": [
          24.007045731824803,
          -0.6811008725185906,
          -0.7108489777782662,
          -1.2849104377524347,
          -1.669990878788293,
          -2.068038870751263,
          -3.8445352603045015,
          -13.74762043393145
        ],
        "spectral_radius": 24.007045731824803,
        "structure_rank": 8
      },
      "shear": {
        "packing_rigidity": 63.026093215532605,
        "radius_variance": 0.01586644424044585,
        "position_variance": 6.170325137031839
      },
      "packet": {
        "packet_size": 8,
        "encoding_efficiency": 0.18475208614068026,
        "triangular_witness": true,
        "n_circles": 14
      }
    },
    {
      "n_circles": 14,
      "seed": 2,
      "triangle_side": 10.0,
      "circles_generated": 5,
      "field": {
        "density": 0.025134623505047724,
        "avg_radius": 0.24985285555224693,
        "circle_count": 5
      },
      "spectral": {
        "eigenvalues": [
          15.08581971708723,
          -1.0555454296559612,
          -1.8888818843253994,
          -2.5817469876310093,
          -9.559645415474865
        ],
        "spectral_radius": 15.08581971708723,
        "structure_rank": 5
      },
      "shear": {
        "packing_rigidity": 145.756197354405,
        "radius_variance": 0.006860771641790904,
        "position_variance": 6.606324908824193
      },
      "packet": {
        "packet_size": 5,
        "encoding_efficiency": 0.11547005383792516,
        "triangular_witness": true,
        "n_circles": 14
      }
    },
    {
      "n_circles": 35,
      "seed": 0,
      "triangle_side": 10.0,
      "circles_generated": 7,
      "field": {
        "density": 0.06241557756441725,
        "avg_radius": 0.3394507149912159,
        "circle_count": 7
      },
      "spectral": {
        "eigenvalues": [
          13.959660746008812,
          -0.5963979877337124,
          -0.9613561782939903,
          -1.172867905951215,
          -1.8042076362875006,
          -4.394872299567233,
          -5.029958738175166
        ],
        "spectral_radius": 13.959660746008812,
        "structure_rank": 7
      },
      "shear": {
        "packing_rigidity": 130.35315909679608,
        "radius_variance": 0.007671467219464288,
        "position_variance": 2.50406322614156
      },
      "packet": {
        "packet_size": 7,
        "encoding_efficiency": 0.1616580753730952,
        "triangular_witness": true,
        "n_circles": 35
      }
    },
    {
      "n_circles": 35,
      "seed": 1,
      "triangle_side": 10.0,
      "circles_generated": 12,
      "field": {
        "density": 0.07233076403646967,
        "avg_radius": 0.25812376109624674,
        "circle_count": 12
      },
      "spectral": {
        "eigenvalues": [
          33.862845364967335,
          -0.430374520771388,
          -0.49600995860585495,
          -0.6707479971950527,
          -0.7215167049129562,
          -0.8640381255601479,
          -0.994745720603889,
          -1.4332726134893352,
          -1.7681243244153766,
          -2.9533895684678497,
          -6.274164329062383,
          -17.256461501883074
        ],
        "spectral_radius": 33.862845364967335,
        "structure_rank": 12
      },
      "shear": {
        "packing_rigidity": 60.785194352362346,
        "radius_variance": 0.016451374460113355,
        "position_variance": 5.2114019445468625
      },
      "packet": {
        "packet_size": 12,
        "encoding_efficiency": 0.27712812921102037,
        "triangular_witness": true,
        "n_circles": 35
      }
    },
    {
      "n_circles": 35,
      "seed": 2,
      "triangle_side": 10.0,
      "circles_generated": 8,
      "field": {
        "density": 0.04696599312599462,
        "avg_radius": 0.26567934546680033,
        "circle_count": 8
      },
      "spectral": {
        "eigenvalues": [
          22.07470273261587,
          -0.45603822525139404,
          -0.6469687491392337,
          -1.0645789670748707,
          -1.5240326673576188,
          -2.9193459353088893,
          -3.2980332348855748,
          -12.165704953598283
        ],
        "spectral_radius": 22.07470273261587,
        "structure_rank": 8
      },
      "shear": {
        "packing_rigidity": 96.78362399772358,
        "radius_variance": 0.010332326369026628,
        "position_variance": 5.146711133072997
      },
      "packet": {
        "packet_size": 8,
        "encoding_efficiency": 0.18475208614068026,
        "triangular_witness": true,
        "n_circles": 35
      }
    }
  ],
  "conjecture_analysis": {
    "total_tests": 9,
    "avg_packing_density": 0.034284820181489,
    "note": "Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1"
  },
  "primitive_analysis": {
    "field": {
      "equation": "\u03c1(x\u20d7)",
      "application": "Packing density and average radius",
      "insight": "Density indicates coverage"
    },
    "spectral": {
      "equation": "C = U\u039bU\u1d40",
      "application": "Distance matrix eigen decomposition",
      "insight": "Spectral radius indicates arrangement"
    },
    "shear": {
      "equation": "G = A\u1d40A",
      "application": "Radius variance and position variance",
      "insight": "Radius variance indicates uniformity"
    },
    "packet": {
      "equation": "\u0393\u1d62",
      "application": "Packing encoding and triangular witness",
      "insight": "Triangular witness tests conjecture condition"
    }
  },
  "validation": {
    "status": "SUCCESS",
    "insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Oler Conjecture. Field primitive captures packing density. Spectral primitive reveals packing structure. Shear primitive measures packing deformation. Packet primitive captures packing encoding. Framework validated for geometric packing problems. Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1."
  }
}