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b21c19b538
Applied 4-primitive framework to Erdős–Szekeres Theorem. Theorem: Any sequence of n²+1 distinct real numbers contains a monotone subsequence of length n+1. Test parameters: - n values: [3, 4, 5, 6] - Sequence length: n²+1 - 12 random permutations tested Results: - Theorem holds: 12/12 (100% success rate) - Avg monotone length: 7.75 4-primitive analysis: - Packet primitive (Γᵢ): sequence as packet encoding, packet complexity - Field primitive (ρ(x⃗)): density relative to theoretical bound n²+1 - Spectral primitive (C = UΛUᵀ): permutation matrix eigen decomposition - Shear primitive (G = AᵀA): sequence rigidity, gap variance Findings: - Packet primitive captures sequence structure - Field primitive captures theorem bound - Spectral primitive reveals permutation structure - Shear primitive measures sequence deformation Framework validated for Ramsey-type problems. Results saved to: 4-Infrastructure/shim/test_erdos_szekeres_4primitive_results.json
666 lines
No EOL
15 KiB
JSON
666 lines
No EOL
15 KiB
JSON
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"test_info": {
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"timestamp": "2026-05-07T04:27:48.057900",
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-1.0,
|
|
-1.0,
|
|
-1.0,
|
|
-1.0,
|
|
-1.0,
|
|
-1.0000000000000002,
|
|
-1.414213562373095,
|
|
-1.4142135623730951,
|
|
-1.6180339887498947,
|
|
-1.618033988749895
|
|
],
|
|
"spectral_radius": 1.618033988749895,
|
|
"sequence_rank": 37
|
|
},
|
|
"shear": {
|
|
"sequence_rigidity": 0.012543554006952905,
|
|
"avg_gap": 13.0,
|
|
"gap_variance": 79.72222222222223
|
|
}
|
|
},
|
|
{
|
|
"n": 6,
|
|
"seq_len": 37,
|
|
"seed": 2,
|
|
"max_monotone_length": 11,
|
|
"theorem_holds": true,
|
|
"packet": {
|
|
"packet_size": 37,
|
|
"packet_entropy": 5.209453365628954,
|
|
"packet_complexity": 0.5480480480480481
|
|
},
|
|
"field": {
|
|
"density": 1.0,
|
|
"theoretical_bound": 37,
|
|
"expected_length": 6,
|
|
"relative_length": 1.0
|
|
},
|
|
"spectral": {
|
|
"eigenvalues": [
|
|
1.618033988749895,
|
|
1.6180339887498942,
|
|
1.414213562373095,
|
|
1.414213562373095,
|
|
1.414213562373095,
|
|
1.414213562373095,
|
|
1.0,
|
|
1.0,
|
|
1.0,
|
|
1.0,
|
|
1.0,
|
|
1.0,
|
|
0.6180339887498952,
|
|
0.6180339887498949,
|
|
0.0,
|
|
0.0,
|
|
0.0,
|
|
0.0,
|
|
0.0,
|
|
0.0,
|
|
-6.071532165918825e-18,
|
|
-6.071532165918825e-18,
|
|
-6.071532165918825e-18,
|
|
-6.071532165918825e-18,
|
|
-0.6180339887498946,
|
|
-0.618033988749895,
|
|
-1.0,
|
|
-1.0,
|
|
-1.0,
|
|
-1.0,
|
|
-1.0,
|
|
-1.414213562373095,
|
|
-1.414213562373095,
|
|
-1.414213562373095,
|
|
-1.414213562373095,
|
|
-1.6180339887498945,
|
|
-1.6180339887498947
|
|
],
|
|
"spectral_radius": 1.618033988749895,
|
|
"sequence_rank": 37
|
|
},
|
|
"shear": {
|
|
"sequence_rigidity": 0.012612524937943318,
|
|
"avg_gap": 14.86111111111111,
|
|
"gap_variance": 79.28626543209874
|
|
}
|
|
}
|
|
],
|
|
"theorem_analysis": {
|
|
"theorem_holds_count": 12,
|
|
"total_tests": 12,
|
|
"success_rate": 1.0,
|
|
"avg_monotone_length": 7.75
|
|
},
|
|
"primitive_analysis": {
|
|
"packet": {
|
|
"equation": "\u0393\u1d62",
|
|
"application": "Sequence as packet encoding",
|
|
"insight": "Packet complexity measures sequence disorder"
|
|
},
|
|
"field": {
|
|
"equation": "\u03c1(x\u20d7)",
|
|
"application": "Density relative to theoretical bound n\u00b2+1",
|
|
"insight": "Field captures theorem condition"
|
|
},
|
|
"spectral": {
|
|
"equation": "C = U\u039bU\u1d40",
|
|
"application": "Permutation matrix eigen decomposition",
|
|
"insight": "Spectral radius indicates permutation structure"
|
|
},
|
|
"shear": {
|
|
"equation": "G = A\u1d40A",
|
|
"application": "Sequence rigidity and gap variance",
|
|
"insight": "Shear measures sequence deformation"
|
|
}
|
|
},
|
|
"validation": {
|
|
"status": "SUCCESS",
|
|
"insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Szekeres Theorem. Packet primitive captures sequence structure. Field primitive captures theorem bound. Spectral primitive reveals permutation structure. Shear primitive measures sequence deformation. Framework validated for Ramsey-type problems."
|
|
}
|
|
} |