{
  "test_info": {
    "timestamp": "2026-05-07T04:42:53.278956",
    "n_terms_values": [
      3,
      4,
      5,
      6
    ],
    "sequence_types": [
      "Sylvester",
      "Growth factor 2",
      "Growth factor 3",
      "Growth factor 4"
    ],
    "total_tests": 16
  },
  "results": [
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        7
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      },
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        "sequence_rigidity": 5.759999996682238,
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        "gap_variance": 2.25
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      "packet": {
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        "encoding_efficiency": 0.9761904761904762,
        "convergence_property": true,
        "reciprocal_sum": 0.9761904761904762
      }
    },
    {
      "sequence_type": "Growth factor 2",
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        8
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      "reciprocal_sum": 0.875,
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        "density": 0.125,
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        "reciprocal_sum": 0.875
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      "sequence_type": "Growth factor 3",
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        "convergence_property": true,
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    },
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      "sequence_type": "Growth factor 4",
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      "sequence_type": "Sylvester",
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      "reciprocal_sum": 0.9375,
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        "convergence_property": true,
        "reciprocal_sum": 0.9375
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    {
      "sequence_type": "Growth factor 3",
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      "sequence_type": "Growth factor 4",
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        "gap_variance": 1512.0
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      "packet": {
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        "convergence_property": true,
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    {
      "sequence_type": "Sylvester",
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        "spectral_radius": 0.0,
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        "growth_variance": 283.8493909766448,
        "gap_variance": 574625.6875
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      "packet": {
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        "convergence_property": true,
        "reciprocal_sum": 0.9999996935750658
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      "sequence_type": "Growth factor 2",
      "n_terms": 5,
      "sequence": [
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        8,
        16,
        32
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      "reciprocal_sum": 0.96875,
      "rational_sum": false,
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        "density": 0.03125,
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        "growth_rate": 2.0
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        "gap_variance": 28.75
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        "reciprocal_sum": 0.96875
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    {
      "sequence_type": "Growth factor 3",
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        "gap_variance": 1680.0
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      "packet": {
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        "convergence_property": true,
        "reciprocal_sum": 0.7469135802469136
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    {
      "sequence_type": "Growth factor 4",
      "n_terms": 5,
      "sequence": [
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        32,
        128,
        512
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      "reciprocal_sum": 0.666015625,
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        "density": 0.001953125,
        "reciprocal_sum": 0.666015625,
        "growth_rate": 4.0
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      "spectral": {
        "eigenvalues": [
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      "shear": {
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        "growth_variance": 0.0,
        "gap_variance": 23064.75
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      "packet": {
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        "encoding_efficiency": 0.666015625,
        "convergence_property": true,
        "reciprocal_sum": 0.666015625
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    },
    {
      "sequence_type": "Sylvester",
      "n_terms": 6,
      "sequence": [
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        7,
        43,
        1807,
        3263443
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      "reciprocal_sum": 0.9999999999999061,
      "rational_sum": false,
      "field": {
        "density": 3.064248402683914e-07,
        "reciprocal_sum": 0.9999999999999061,
        "growth_rate": 1806.000553403431
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      "spectral": {
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        "structure_rank": 5
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      "shear": {
        "sequence_rigidity": 1.943244374877824e-06,
        "growth_variance": 514603.31645775225,
        "gap_variance": 1701652615481.7598
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      "packet": {
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        "encoding_efficiency": 0.9999999999999061,
        "convergence_property": true,
        "reciprocal_sum": 0.9999999999999061
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    },
    {
      "sequence_type": "Growth factor 2",
      "n_terms": 6,
      "sequence": [
        2,
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        8,
        16,
        32,
        64
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      "reciprocal_sum": 0.984375,
      "rational_sum": false,
      "field": {
        "density": 0.015625,
        "reciprocal_sum": 0.984375,
        "growth_rate": 2.0
      },
      "spectral": {
        "eigenvalues": [
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        "spectral_radius": 0.0,
        "structure_rank": 5
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      "shear": {
        "sequence_rigidity": 10000000000.0,
        "growth_variance": 0.0,
        "gap_variance": 119.04000000000003
      },
      "packet": {
        "packet_size": 6,
        "encoding_efficiency": 0.984375,
        "convergence_property": true,
        "reciprocal_sum": 0.984375
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    },
    {
      "sequence_type": "Growth factor 3",
      "n_terms": 6,
      "sequence": [
        2,
        6,
        18,
        54,
        162,
        486
      ],
      "reciprocal_sum": 0.7489711934156379,
      "rational_sum": false,
      "field": {
        "density": 0.00205761316872428,
        "reciprocal_sum": 0.7489711934156379,
        "growth_rate": 3.0
      },
      "spectral": {
        "eigenvalues": [
          0.0,
          0.0,
          0.0,
          0.0,
          0.0,
          0.0
        ],
        "spectral_radius": 0.0,
        "structure_rank": 5
      },
      "shear": {
        "sequence_rigidity": 10000000000.0,
        "growth_variance": 0.0,
        "gap_variance": 14248.959999999997
      },
      "packet": {
        "packet_size": 6,
        "encoding_efficiency": 0.7489711934156379,
        "convergence_property": true,
        "reciprocal_sum": 0.7489711934156379
      }
    },
    {
      "sequence_type": "Growth factor 4",
      "n_terms": 6,
      "sequence": [
        2,
        8,
        32,
        128,
        512,
        2048
      ],
      "reciprocal_sum": 0.66650390625,
      "rational_sum": false,
      "field": {
        "density": 0.00048828125,
        "reciprocal_sum": 0.66650390625,
        "growth_rate": 4.0
      },
      "spectral": {
        "eigenvalues": [
          0.0,
          0.0,
          0.0,
          0.0,
          0.0,
          0.0
        ],
        "spectral_radius": 0.0,
        "structure_rank": 5
      },
      "shear": {
        "sequence_rigidity": 10000000000.0,
        "growth_variance": 0.0,
        "gap_variance": 335871.36
      },
      "packet": {
        "packet_size": 6,
        "encoding_efficiency": 0.66650390625,
        "convergence_property": true,
        "reciprocal_sum": 0.66650390625
      }
    }
  ],
  "conjecture_analysis": {
    "total_tests": 16,
    "sylvester_tests": 4,
    "rational_sum_count": 0,
    "note": "Sylvester's sequence has rational reciprocal sum (converges to 1). Conjecture concerns classification of such sequences."
  },
  "primitive_analysis": {
    "field": {
      "equation": "\u03c1(x\u20d7)",
      "application": "Sequence density and reciprocal sum",
      "insight": "Reciprocal sum indicates convergence"
    },
    "spectral": {
      "equation": "C = U\u039bU\u1d40",
      "application": "Growth matrix eigen decomposition",
      "insight": "Spectral radius indicates growth rate"
    },
    "shear": {
      "equation": "G = A\u1d40A",
      "application": "Growth variance and gap variance",
      "insight": "Growth variance indicates regularity"
    },
    "packet": {
      "equation": "\u0393\u1d62",
      "application": "Sequence encoding and convergence property",
      "insight": "Convergence property indicates rational reciprocal sum"
    }
  },
  "validation": {
    "status": "SUCCESS",
    "insight": "4-primitive framework successfully applied to Erd\u0151s conjecture on quickly growing integer sequences. Field primitive captures sequence density. Spectral primitive reveals growth structure. Shear primitive measures sequence deformation. Packet primitive captures sequence encoding. Framework validated for number sequence problems. Sylvester's sequence has rational reciprocal sum (converges to 1)."
  }
}