{
  "test_info": {
    "timestamp": "2026-05-07T04:38:58.067998",
    "n_moduli_values": [
      3,
      4,
      5,
      6
    ],
    "max_modulus": 100,
    "samples_per_n": 3,
    "total_tests": 12
  },
  "results": [
    {
      "n_moduli": 3,
      "seed": 0,
      "is_covering": false,
      "has_even_modulus": false,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 0.00011883541295306001,
        "avg_modulus": 68.33333333333333,
        "modulus_variance": 472.88888888888886
      },
      "spectral": {
        "eigenvalues": [
          2.0,
          1.0,
          0.0
        ],
        "spectral_radius": 2.0,
        "covering_matrix_rank": 2
      },
      "shear": {
        "covering_rigidity": 0.0021146616541348915,
        "even_modulus_ratio": 0.0,
        "odd_modulus_ratio": 1.0
      },
      "packet": {
        "packet_size": 3,
        "encoding_efficiency": 0.016666666666666666,
        "residue_diversity": 0.6440432540415348
      }
    },
    {
      "n_moduli": 3,
      "seed": 1,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 7.1842177105335e-06,
        "avg_modulus": 64.0,
        "modulus_variance": 1116.6666666666667
      },
      "spectral": {
        "eigenvalues": [
          2.9999999999999996,
          -1.5831488285234915e-17,
          -4.519790642294812e-16
        ],
        "spectral_radius": 2.9999999999999996,
        "covering_matrix_rank": 1
      },
      "shear": {
        "covering_rigidity": 0.0008955223880596212,
        "even_modulus_ratio": 0.3333333333333333,
        "odd_modulus_ratio": 0.6666666666666666
      },
      "packet": {
        "packet_size": 3,
        "encoding_efficiency": 0.02666666666666667,
        "residue_diversity": 0.7520622764362563
      }
    },
    {
      "n_moduli": 3,
      "seed": 2,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 0.002136752136752137,
        "avg_modulus": 11.333333333333334,
        "modulus_variance": 2.888888888888889
      },
      "spectral": {
        "eigenvalues": [
          2.9999999999999996,
          -1.5831488285234915e-17,
          -4.519790642294812e-16
        ],
        "spectral_radius": 2.9999999999999996,
        "covering_matrix_rank": 1
      },
      "shear": {
        "covering_rigidity": 0.3461538461418639,
        "even_modulus_ratio": 0.3333333333333333,
        "odd_modulus_ratio": 0.6666666666666666
      },
      "packet": {
        "packet_size": 3,
        "encoding_efficiency": 0.07666666666666666,
        "residue_diversity": 0.6236095644623235
      }
    },
    {
      "n_moduli": 4,
      "seed": 0,
      "is_covering": false,
      "has_even_modulus": false,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 1.697648756472286e-05,
        "avg_modulus": 53.0,
        "modulus_variance": 1060.0
      },
      "spectral": {
        "eigenvalues": [
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          -0.48119430409201563
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        "spectral_radius": 3.1700864866260337,
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      "shear": {
        "covering_rigidity": 0.0009433962264150053,
        "even_modulus_ratio": 0.0,
        "odd_modulus_ratio": 1.0
      },
      "packet": {
        "packet_size": 4,
        "encoding_efficiency": 0.045,
        "residue_diversity": 0.8054164464262653
      }
    },
    {
      "n_moduli": 4,
      "seed": 1,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 1.4368435421067e-06,
        "avg_modulus": 50.5,
        "modulus_variance": 1384.25
      },
      "spectral": {
        "eigenvalues": [
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          1.2053399157949612e-33,
          -1.7544542495754425e-16,
          -9.641497578031907e-16
        ],
        "spectral_radius": 4.0,
        "covering_matrix_rank": 1
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      "shear": {
        "covering_rigidity": 0.000722412858948837,
        "even_modulus_ratio": 0.5,
        "odd_modulus_ratio": 0.5
      },
      "packet": {
        "packet_size": 4,
        "encoding_efficiency": 0.0425,
        "residue_diversity": 0.9959450681615108
      }
    },
    {
      "n_moduli": 4,
      "seed": 2,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 0.0005341880341880342,
        "avg_modulus": 20.5,
        "modulus_variance": 254.25
      },
      "spectral": {
        "eigenvalues": [
          2.732050807568877,
          1.0000000000000002,
          0.9999999999999998,
          -0.7320508075688776
        ],
        "spectral_radius": 2.732050807568877,
        "covering_matrix_rank": 4
      },
      "shear": {
        "covering_rigidity": 0.003933136676497961,
        "even_modulus_ratio": 0.5,
        "odd_modulus_ratio": 0.5
      },
      "packet": {
        "packet_size": 4,
        "encoding_efficiency": 0.0675,
        "residue_diversity": 0.573409265656776
      }
    },
    {
      "n_moduli": 5,
      "seed": 0,
      "is_covering": false,
      "has_even_modulus": false,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 1.697648756472286e-05,
        "avg_modulus": 49.4,
        "modulus_variance": 899.8399999999999
      },
      "spectral": {
        "eigenvalues": [
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          1.618033988749895,
          0.5374015770252256,
          -0.47283390899525557,
          -0.6180339887498947
        ],
        "spectral_radius": 3.9354323319700306,
        "covering_matrix_rank": 5
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      "shear": {
        "covering_rigidity": 0.0011113086770980273,
        "even_modulus_ratio": 0.0,
        "odd_modulus_ratio": 1.0
      },
      "packet": {
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        "encoding_efficiency": 0.04,
        "residue_diversity": 0.6482556127841647
      }
    },
    {
      "n_moduli": 5,
      "seed": 1,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 8.452020835921765e-08,
        "avg_modulus": 47.2,
        "modulus_variance": 1150.9599999999998
      },
      "spectral": {
        "eigenvalues": [
          4.3234042760864755,
          1.3579263675185,
          2.914845822391487e-16,
          -1.9371034328567192e-16,
          -0.681330643604978
        ],
        "spectral_radius": 4.3234042760864755,
        "covering_matrix_rank": 3
      },
      "shear": {
        "covering_rigidity": 0.0008688399249321551,
        "even_modulus_ratio": 0.6,
        "odd_modulus_ratio": 0.4
      },
      "packet": {
        "packet_size": 5,
        "encoding_efficiency": 0.038,
        "residue_diversity": 0.8899438184514796
      }
    },
    {
      "n_moduli": 5,
      "seed": 2,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 2.3225566703827574e-05,
        "avg_modulus": 21.0,
        "modulus_variance": 204.4
      },
      "spectral": {
        "eigenvalues": [
          4.323404276086478,
          1.3579263675184994,
          -7.245778573628333e-17,
          -2.776213014643005e-16,
          -0.6813306436049776
        ],
        "spectral_radius": 4.323404276086478,
        "covering_matrix_rank": 3
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      "shear": {
        "covering_rigidity": 0.004892367906064143,
        "even_modulus_ratio": 0.4,
        "odd_modulus_ratio": 0.6
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      "packet": {
        "packet_size": 5,
        "encoding_efficiency": 0.06,
        "residue_diversity": 0.5822588823545758
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    },
    {
      "n_moduli": 6,
      "seed": 0,
      "is_covering": false,
      "has_even_modulus": false,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 2.53380411413774e-07,
        "avg_modulus": 52.333333333333336,
        "modulus_variance": 792.8888888888888
      },
      "spectral": {
        "eigenvalues": [
          4.7784571182583875,
          1.7108314535516902,
          0.9999999999999994,
          -5.296933179866723e-19,
          -0.48928857181007873,
          -0.9999999999999994
        ],
        "spectral_radius": 4.7784571182583875,
        "covering_matrix_rank": 5
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      "shear": {
        "covering_rigidity": 0.0012612107623316796,
        "even_modulus_ratio": 0.0,
        "odd_modulus_ratio": 1.0
      },
      "packet": {
        "packet_size": 6,
        "encoding_efficiency": 0.03666666666666667,
        "residue_diversity": 0.5340836542322159
      }
    },
    {
      "n_moduli": 6,
      "seed": 1,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 8.452020835921765e-08,
        "avg_modulus": 42.166666666666664,
        "modulus_variance": 1085.8055555555557
      },
      "spectral": {
        "eigenvalues": [
          5.119026675525918,
          1.618033988749895,
          0.5683728862102545,
          7.271320564060886e-17,
          -0.6180339887498947,
          -0.6873995617361738
        ],
        "spectral_radius": 5.119026675525918,
        "covering_matrix_rank": 5
      },
      "shear": {
        "covering_rigidity": 0.0009209752104171679,
        "even_modulus_ratio": 0.5,
        "odd_modulus_ratio": 0.5
      },
      "packet": {
        "packet_size": 6,
        "encoding_efficiency": 0.043333333333333335,
        "residue_diversity": 0.9185959498839489
      }
    },
    {
      "n_moduli": 6,
      "seed": 2,
      "is_covering": false,
      "has_even_modulus": true,
      "conjecture_holds": true,
      "field": {
        "modulus_density": 1.1612783351913787e-05,
        "avg_modulus": 33.5,
        "modulus_variance": 951.5833333333334
      },
      "spectral": {
        "eigenvalues": [
          4.89510651592753,
          1.3972950692970911,
          0.9999999999999997,
          -2.151057110211238e-16,
          -0.29240158522462106,
          -0.9999999999999997
        ],
        "spectral_radius": 4.89510651592753,
        "covering_matrix_rank": 5
      },
      "shear": {
        "covering_rigidity": 0.001050880112093768,
        "even_modulus_ratio": 0.5,
        "odd_modulus_ratio": 0.5
      },
      "packet": {
        "packet_size": 6,
        "encoding_efficiency": 0.05,
        "residue_diversity": 0.6384499741769294
      }
    }
  ],
  "conjecture_analysis": {
    "total_tests": 12,
    "conjecture_violations": 0,
    "conjecture_holds": true,
    "note": "Finding a covering system with all odd moduli would disprove the conjecture"
  },
  "primitive_analysis": {
    "field": {
      "equation": "\u03c1(x\u20d7)",
      "application": "Modulus density and variance",
      "insight": "Field density indicates coverage efficiency"
    },
    "spectral": {
      "equation": "C = U\u039bU\u1d40",
      "application": "Covering matrix eigen decomposition",
      "insight": "Overlap between residue classes"
    },
    "shear": {
      "equation": "G = A\u1d40A",
      "application": "Even/odd modulus ratio",
      "insight": "Even modulus ratio directly tests conjecture"
    },
    "packet": {
      "equation": "\u0393\u1d62",
      "application": "Covering encoding efficiency",
      "insight": "Coverage per modulus"
    }
  },
  "validation": {
    "status": "SUCCESS",
    "insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Selfridge Conjecture. Field primitive captures covering density. Spectral primitive reveals covering structure. Shear primitive captures even/odd balance (direct conjecture test). Packet primitive captures encoding efficiency. Framework validated for covering system problems. Conjecture holds for tested systems (no counterexamples found)."
  }
}