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Research-Stack/4-Infrastructure/shim/test_erdos_quickly_growing_sequences_4primitive_results.json
Brandon Schneider c1f5ff95c0 test: 4-primitive framework applied to Erdős quickly growing sequences 
Applied 4-primitive framework to Erdős conjecture on quickly growing integer sequences.
Conjecture: On integer sequences with rational reciprocal series (Sylvester's sequence).

Test parameters:
- n_terms values: [3, 4, 5, 6]
- Sequences tested: Sylvester's sequence + growth factors [2, 3, 4]
- 16 sequences tested

Results:
- Sylvester tests: 4
- Rational sum count: 0 (Sylvester's sequence converges to 1, but not exactly 1 for finite terms)
- Note: Sylvester's sequence has rational reciprocal sum (converges to 1)

4-primitive analysis:
- Field primitive (ρ(x⃗)): sequence density, reciprocal sum, growth rate
- Spectral primitive (C = UΛUᵀ): growth matrix eigen decomposition
- Shear primitive (G = AᵀA): sequence rigidity, growth variance, gap variance
- Packet primitive (Γᵢ): sequence encoding, convergence property

Findings:
- 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.
6 unsolved Erdős conjectures now tested with 4-primitive framework.

Results saved to: test_erdos_quickly_growing_sequences_4primitive_results.json
2026-05-08 14:50:03 -05:00

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{
"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": [
{
"sequence_type": "Sylvester",
"n_terms": 3,
"sequence": [
2,
3,
7
],
"reciprocal_sum": 0.9761904761904762,
"rational_sum": false,
"field": {
"density": 0.14285714285714285,
"reciprocal_sum": 0.9761904761904762,
"growth_rate": 2.3333333333333335
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 2
},
"shear": {
"sequence_rigidity": 5.759999996682238,
"growth_variance": 0.17361111111111116,
"gap_variance": 2.25
},
"packet": {
"packet_size": 3,
"encoding_efficiency": 0.9761904761904762,
"convergence_property": true,
"reciprocal_sum": 0.9761904761904762
}
},
{
"sequence_type": "Growth factor 2",
"n_terms": 3,
"sequence": [
2,
4,
8
],
"reciprocal_sum": 0.875,
"rational_sum": false,
"field": {
"density": 0.125,
"reciprocal_sum": 0.875,
"growth_rate": 2.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 2
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 1.0
},
"packet": {
"packet_size": 3,
"encoding_efficiency": 0.875,
"convergence_property": true,
"reciprocal_sum": 0.875
}
},
{
"sequence_type": "Growth factor 3",
"n_terms": 3,
"sequence": [
2,
6,
18
],
"reciprocal_sum": 0.7222222222222222,
"rational_sum": false,
"field": {
"density": 0.05555555555555555,
"reciprocal_sum": 0.7222222222222222,
"growth_rate": 3.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 2
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 16.0
},
"packet": {
"packet_size": 3,
"encoding_efficiency": 0.7222222222222222,
"convergence_property": true,
"reciprocal_sum": 0.7222222222222222
}
},
{
"sequence_type": "Growth factor 4",
"n_terms": 3,
"sequence": [
2,
8,
32
],
"reciprocal_sum": 0.65625,
"rational_sum": false,
"field": {
"density": 0.03125,
"reciprocal_sum": 0.65625,
"growth_rate": 4.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 2
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 81.0
},
"packet": {
"packet_size": 3,
"encoding_efficiency": 0.65625,
"convergence_property": true,
"reciprocal_sum": 0.65625
}
},
{
"sequence_type": "Sylvester",
"n_terms": 4,
"sequence": [
2,
3,
7,
43
],
"reciprocal_sum": 0.9994462901439646,
"rational_sum": false,
"field": {
"density": 0.023255813953488372,
"reciprocal_sum": 0.9994462901439646,
"growth_rate": 6.142857142857143
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 3
},
"shear": {
"sequence_rigidity": 0.2448111025383398,
"growth_variance": 4.084782060972538,
"gap_variance": 250.8888888888889
},
"packet": {
"packet_size": 4,
"encoding_efficiency": 0.9994462901439646,
"convergence_property": true,
"reciprocal_sum": 0.9994462901439646
}
},
{
"sequence_type": "Growth factor 2",
"n_terms": 4,
"sequence": [
2,
4,
8,
16
],
"reciprocal_sum": 0.9375,
"rational_sum": false,
"field": {
"density": 0.0625,
"reciprocal_sum": 0.9375,
"growth_rate": 2.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 3
},
"shear": {
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"growth_variance": 0.0,
"gap_variance": 6.222222222222221
},
"packet": {
"packet_size": 4,
"encoding_efficiency": 0.9375,
"convergence_property": true,
"reciprocal_sum": 0.9375
}
},
{
"sequence_type": "Growth factor 3",
"n_terms": 4,
"sequence": [
2,
6,
18,
54
],
"reciprocal_sum": 0.7407407407407407,
"rational_sum": false,
"field": {
"density": 0.018518518518518517,
"reciprocal_sum": 0.7407407407407407,
"growth_rate": 3.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 3
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 184.8888888888889
},
"packet": {
"packet_size": 4,
"encoding_efficiency": 0.7407407407407407,
"convergence_property": true,
"reciprocal_sum": 0.7407407407407407
}
},
{
"sequence_type": "Growth factor 4",
"n_terms": 4,
"sequence": [
2,
8,
32,
128
],
"reciprocal_sum": 0.6640625,
"rational_sum": false,
"field": {
"density": 0.0078125,
"reciprocal_sum": 0.6640625,
"growth_rate": 4.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 3
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 1512.0
},
"packet": {
"packet_size": 4,
"encoding_efficiency": 0.6640625,
"convergence_property": true,
"reciprocal_sum": 0.6640625
}
},
{
"sequence_type": "Sylvester",
"n_terms": 5,
"sequence": [
2,
3,
7,
43,
1807
],
"reciprocal_sum": 0.9999996935750658,
"rational_sum": false,
"field": {
"density": 0.0005534034311012728,
"reciprocal_sum": 0.9999996935750658,
"growth_rate": 42.02325581395349
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 4
},
"shear": {
"sequence_rigidity": 0.0035229950522667423,
"growth_variance": 283.8493909766448,
"gap_variance": 574625.6875
},
"packet": {
"packet_size": 5,
"encoding_efficiency": 0.9999996935750658,
"convergence_property": true,
"reciprocal_sum": 0.9999996935750658
}
},
{
"sequence_type": "Growth factor 2",
"n_terms": 5,
"sequence": [
2,
4,
8,
16,
32
],
"reciprocal_sum": 0.96875,
"rational_sum": false,
"field": {
"density": 0.03125,
"reciprocal_sum": 0.96875,
"growth_rate": 2.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 4
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 28.75
},
"packet": {
"packet_size": 5,
"encoding_efficiency": 0.96875,
"convergence_property": true,
"reciprocal_sum": 0.96875
}
},
{
"sequence_type": "Growth factor 3",
"n_terms": 5,
"sequence": [
2,
6,
18,
54,
162
],
"reciprocal_sum": 0.7469135802469136,
"rational_sum": false,
"field": {
"density": 0.006172839506172839,
"reciprocal_sum": 0.7469135802469136,
"growth_rate": 3.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 4
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 1680.0
},
"packet": {
"packet_size": 5,
"encoding_efficiency": 0.7469135802469136,
"convergence_property": true,
"reciprocal_sum": 0.7469135802469136
}
},
{
"sequence_type": "Growth factor 4",
"n_terms": 5,
"sequence": [
2,
8,
32,
128,
512
],
"reciprocal_sum": 0.666015625,
"rational_sum": false,
"field": {
"density": 0.001953125,
"reciprocal_sum": 0.666015625,
"growth_rate": 4.0
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 4
},
"shear": {
"sequence_rigidity": 10000000000.0,
"growth_variance": 0.0,
"gap_variance": 23064.75
},
"packet": {
"packet_size": 5,
"encoding_efficiency": 0.666015625,
"convergence_property": true,
"reciprocal_sum": 0.666015625
}
},
{
"sequence_type": "Sylvester",
"n_terms": 6,
"sequence": [
2,
3,
7,
43,
1807,
3263443
],
"reciprocal_sum": 0.9999999999999061,
"rational_sum": false,
"field": {
"density": 3.064248402683914e-07,
"reciprocal_sum": 0.9999999999999061,
"growth_rate": 1806.000553403431
},
"spectral": {
"eigenvalues": [
0.0,
0.0,
0.0,
0.0,
0.0,
0.0
],
"spectral_radius": 0.0,
"structure_rank": 5
},
"shear": {
"sequence_rigidity": 1.943244374877824e-06,
"growth_variance": 514603.31645775225,
"gap_variance": 1701652615481.7598
},
"packet": {
"packet_size": 6,
"encoding_efficiency": 0.9999999999999061,
"convergence_property": true,
"reciprocal_sum": 0.9999999999999061
}
},
{
"sequence_type": "Growth factor 2",
"n_terms": 6,
"sequence": [
2,
4,
8,
16,
32,
64
],
"reciprocal_sum": 0.984375,
"rational_sum": false,
"field": {
"density": 0.015625,
"reciprocal_sum": 0.984375,
"growth_rate": 2.0
},
"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": 119.04000000000003
},
"packet": {
"packet_size": 6,
"encoding_efficiency": 0.984375,
"convergence_property": true,
"reciprocal_sum": 0.984375
}
},
{
"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)."
}
}
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