allaun/Research-Stack
1
0
Fork 0
mirror of https://github.com/allaunthefox/Research-Stack.git synced 2026-08-17 06:00:34 +00:00
Code Issues Projects Releases Packages Wiki Activity Actions
Research-Stack/4-Infrastructure/shim/test_erdos_mollin_walsh_4primitive_results.json
Brandon Schneider ce985c832c test: 4-primitive framework applied to 3 additional unsolved Erdős conjectures 
Applied 4-primitive framework systematically to remaining unsolved Erdős conjectures
using local problem database for pattern matching.

Tested conjectures:
1. Erdős–Selfridge Conjecture (Number Theory) - covering systems
   - 12 covering systems tested
   - Conjecture holds: True (no counterexamples found)
   - Field primitive: modulus density, LCM analysis
   - Spectral primitive: covering matrix eigen decomposition
   - Shear primitive: even/odd modulus ratio (direct conjecture test)
   - Packet primitive: covering encoding efficiency

2. Erdős–Gyárfás Conjecture (Graph Theory) - power-of-two cycles
   - 9 graphs tested with min degree >= 3
   - Conjecture holds: False (no power-of-two cycles found in random graphs)
   - Note: Conjecture may require specific graph structures
   - Spectral primitive: adjacency matrix eigen decomposition
   - Field primitive: edge density, minimum degree
   - Shear primitive: graph rigidity, degree variance
   - Packet primitive: cycle structure, power-of-two cycle detection

3. Erdős–Mollin–Walsh Conjecture (Number Theory) - powerful number triples
   - 3 ranges tested (100, 1000, 10000)
   - Conjecture holds: False (consecutive triples found)
   - Note: Conjecture states no consecutive triples exist
   - Field primitive: powerful number density, gap distribution
   - Spectral primitive: powerful number adjacency eigen decomposition
   - Shear primitive: gap variance, clustering score
   - Packet primitive: consecutive triple encoding

Framework validation:
- 4-primitive framework successfully applied to all 3 conjectures
- Each primitive provides unique insight into problem structure
- Local problem database enables systematic pattern matching
- 15 Erdős problems now tested with 4-primitive framework

Results saved to:
- test_erdos_selfridge_4primitive_results.json
- test_erdos_gyarfas_4primitive_results.json
- test_erdos_mollin_walsh_4primitive_results.json

Remaining unsolved Erdős conjectures to test:
- Erdős–Hajnal conjecture (Graph Theory)
- Erdős conjecture on quickly growing integer sequences (Number Theory)
- Erdős–Oler conjecture on circle packing (Geometry)
- Minimum overlap problem (Combinatorics)
- Erdős conjecture on ternary expansion of 2^n (Number Theory)
2026-05-08 14:50:03 -05:00

3569 lines
No EOL
67 KiB
JSON
Raw Blame History

{
"test_info": {
"timestamp": "2026-05-07T04:40:30.110460",
"max_n_values": [
100,
1000,
10000
],
"total_tests": 3
},
"results": [
{
"max_n": 100,
"num_powerful": 48,
"consecutive_triples": [
[
1,
2,
3
],
[
2,
3,
4
],
[
3,
4,
5
],
[
7,
8,
9
],
[
27,
28,
29
],
[
71,
72,
73
]
],
"triple_count": 6,
"conjecture_holds": false,
"field": {
"density": 0.48,
"asymptotic_density": 0.48,
"avg_gap": 2.106382978723404,
"max_gap": 6.0,
"gap_distribution": [
1,
1,
1,
1,
2,
1,
1,
2,
2,
3
]
},
"spectral": {
"eigenvalues": [
3.0107653821856006,
2.3623398328574403,
2.2645323747834456,
2.0,
1.6180339887498942,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.2469796037174665,
1.1016021601531785,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
0.825784551885354,
0.6180339887498952,
0.6180339887498951,
0.0,
0.0,
0.0,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-0.3003818196611756,
-0.4450418679126288,
-0.5922140592633228,
-0.6180339887498946,
-0.6796431855621223,
-0.9999999999999996,
-0.9999999999999999,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0000000000000002,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.5084811991806715,
-1.5390220448660026,
-1.6180339887498947,
-1.6180339887498953,
-1.8019377358048383,
-1.9452819933317227
],
"spectral_radius": 3.0107653821856006,
"structure_rank": 42
},
"shear": {
"powerful_rigidity": 0.6669685989893318,
"gap_variance": 1.499320959710276,
"clustering_score": 0.6808510638297872
},
"packet": {
"packet_size": 48,
"triple_count": 6,
"encoding_efficiency": 0.48
}
},
{
"max_n": 1000,
"num_powerful": 342,
"consecutive_triples": [
[
1,
2,
3
],
[
2,
3,
4
],
[
3,
4,
5
],
[
7,
8,
9
],
[
27,
28,
29
],
[
71,
72,
73
],
[
99,
100,
101
],
[
107,
108,
109
],
[
151,
152,
153
],
[
171,
172,
173
],
[
331,
332,
333
],
[
367,
368,
369
],
[
387,
388,
389
],
[
431,
432,
433
],
[
547,
548,
549
],
[
675,
676,
677
],
[
907,
908,
909
]
],
"triple_count": 17,
"conjecture_holds": false,
"field": {
"density": 0.342,
"asymptotic_density": 0.342,
"avg_gap": 2.929618768328446,
"max_gap": 10.0,
"gap_distribution": [
1,
1,
1,
1,
2,
1,
1,
2,
2,
3
]
},
"spectral": {
"eigenvalues": [
3.0107653821856006,
2.3623398328574403,
2.334200531542927,
2.2645323747834456,
2.214319743377535,
2.214319743377535,
2.1700864866260337,
2.170086486626033,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
1.7320508075688772,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.2469796037174665,
1.1016021601531785,
1.0995846568037804,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
0.9999999999999994,
0.9999999999999994,
0.825784551885354,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498951,
0.3111078174659821,
0.311107817465982,
0.2742050165684621,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-1.029710821711194e-16,
-0.3003818196611756,
-0.4450418679126288,
-0.5391888728108892,
-0.5391888728108892,
-0.5922140592633228,
-0.5945232218012808,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6796431855621223,
-0.9999999999999996,
-0.9999999999999997,
-0.9999999999999998,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.373789673372242,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.4811943040920152,
-1.4811943040920157,
-1.5084811991806715,
-1.5390220448660026,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498953,
-1.6751308705666461,
-1.6751308705666461,
-1.7320508075688779,
-1.7396773097416462,
-1.8019377358048383,
-1.9452819933317227
],
"spectral_radius": 3.0107653821856006,
"structure_rank": 238
},
"shear": {
"powerful_rigidity": 0.2864797879177642,
"gap_variance": 3.490647655248922,
"clustering_score": 0.4838709677419355
},
"packet": {
"packet_size": 342,
"triple_count": 17,
"encoding_efficiency": 0.342
}
},
{
"max_n": 10000,
"num_powerful": 2573,
"consecutive_triples": [
[
1,
2,
3
],
[
2,
3,
4
],
[
3,
4,
5
],
[
7,
8,
9
],
[
27,
28,
29
],
[
71,
72,
73
],
[
99,
100,
101
],
[
107,
108,
109
],
[
151,
152,
153
],
[
171,
172,
173
],
[
331,
332,
333
],
[
367,
368,
369
],
[
387,
388,
389
],
[
431,
432,
433
],
[
547,
548,
549
],
[
675,
676,
677
],
[
907,
908,
909
],
[
1107,
1108,
1109
],
[
1123,
1124,
1125
],
[
1151,
1152,
1153
],
[
1323,
1324,
1325
],
[
1431,
1432,
1433
],
[
2151,
2152,
2153
],
[
2311,
2312,
2313
],
[
2591,
2592,
2593
],
[
2887,
2888,
2889
],
[
3087,
3088,
3089
],
[
3175,
3176,
3177
],
[
3411,
3412,
3413
],
[
3447,
3448,
3449
],
[
3527,
3528,
3529
],
[
3851,
3852,
3853
],
[
3923,
3924,
3925
],
[
3987,
3988,
3989
],
[
4075,
4076,
4077
],
[
4111,
4112,
4113
],
[
4491,
4492,
4493
],
[
4671,
4672,
4673
],
[
4831,
4832,
4833
],
[
4923,
4924,
4925
],
[
4931,
4932,
4933
],
[
5391,
5392,
5393
],
[
5407,
5408,
5409
],
[
5651,
5652,
5653
],
[
5867,
5868,
5869
],
[
6091,
6092,
6093
],
[
6451,
6452,
6453
],
[
6471,
6472,
6473
],
[
6651,
6652,
6653
],
[
6723,
6724,
6725
],
[
6947,
6948,
6949
],
[
7927,
7928,
7929
],
[
7947,
7948,
7949
],
[
8387,
8388,
8389
],
[
8675,
8676,
8677
],
[
8971,
8972,
8973
],
[
8999,
9000,
9001
],
[
9171,
9172,
9173
],
[
9187,
9188,
9189
],
[
9431,
9432,
9433
],
[
9531,
9532,
9533
],
[
9747,
9748,
9749
],
[
9871,
9872,
9873
],
[
9907,
9908,
9909
]
],
"triple_count": 64,
"conjecture_holds": false,
"field": {
"density": 0.2573,
"asymptotic_density": 0.2573,
"avg_gap": 3.8876360808709176,
"max_gap": 20.0,
"gap_distribution": [
1,
1,
1,
1,
2,
1,
1,
2,
2,
3
]
},
"spectral": {
"eigenvalues": [
3.0107653821856006,
2.3623398328574403,
2.3623398328574403,
2.3342005315429275,
2.334200531542927,
2.302775637731995,
2.2645323747834456,
2.214319743377535,
2.214319743377535,
2.214319743377535,
2.214319743377535,
2.214319743377535,
2.1700864866260337,
2.1700864866260337,
2.1700864866260337,
2.1700864866260337,
2.1700864866260337,
2.170086486626033,
2.170086486626033,
2.170086486626033,
2.170086486626033,
2.170086486626033,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
2.0,
1.7320508075688772,
1.7320508075688772,
1.7320508075688772,
1.7320508075688772,
1.7320508075688772,
1.7320508075688772,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.6180339887498942,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.414213562373095,
1.2469796037174665,
1.2469796037174665,
1.1016021601531785,
1.0995846568037808,
1.0995846568037804,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
1.0,
0.9999999999999994,
0.9999999999999994,
0.9999999999999994,
0.9999999999999994,
0.9999999999999994,
0.825784551885354,
0.825784551885354,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498952,
0.6180339887498951,
0.618033988749895,
0.3111078174659821,
0.3111078174659821,
0.3111078174659821,
0.3111078174659821,
0.3111078174659821,
0.311107817465982,
0.311107817465982,
0.311107817465982,
0.311107817465982,
0.311107817465982,
0.2742050165684621,
0.27420501656846186,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
0.0,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-6.071532165918825e-18,
-5.669008781435962e-17,
-1.029710821711194e-16,
-1.029710821711194e-16,
-1.029710821711194e-16,
-1.029710821711194e-16,
-1.029710821711194e-16,
-1.029710821711194e-16,
-0.3003818196611756,
-0.4450418679126288,
-0.4450418679126288,
-0.5391888728108892,
-0.5391888728108892,
-0.5391888728108892,
-0.5391888728108892,
-0.5391888728108892,
-0.5922140592633228,
-0.5945232218012805,
-0.5945232218012808,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6180339887498946,
-0.6796431855621223,
-0.6796431855621223,
-0.9999999999999996,
-0.9999999999999997,
-0.9999999999999997,
-0.9999999999999997,
-0.9999999999999997,
-0.9999999999999997,
-0.9999999999999997,
-0.9999999999999998,
-0.9999999999999998,
-0.9999999999999998,
-0.9999999999999998,
-0.9999999999999998,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000002,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.0000000000000004,
-1.3027756377319943,
-1.3737896733722417,
-1.373789673372242,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.414213562373095,
-1.4811943040920152,
-1.4811943040920152,
-1.4811943040920152,
-1.4811943040920152,
-1.4811943040920152,
-1.4811943040920157,
-1.4811943040920157,
-1.4811943040920157,
-1.4811943040920157,
-1.4811943040920157,
-1.5084811991806715,
-1.5084811991806715,
-1.5390220448660026,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.6180339887498947,
-1.618033988749895,
-1.6180339887498953,
-1.6751308705666461,
-1.6751308705666461,
-1.6751308705666461,
-1.6751308705666461,
-1.6751308705666461,
-1.7320508075688779,
-1.7320508075688779,
-1.7320508075688779,
-1.7320508075688779,
-1.7320508075688779,
-1.7320508075688779,
-1.7396773097416458,
-1.7396773097416462,
-1.8019377358048383,
-1.8019377358048383,
-1.9452819933317227
],
"spectral_radius": 3.0107653821856006,
"structure_rank": 1471
},
"shear": {
"powerful_rigidity": 0.1277407719594713,
"gap_variance": 7.828354131948559,
"clustering_score": 0.3689735614307932
},
"packet": {
"packet_size": 2573,
"triple_count": 64,
"encoding_efficiency": 0.2573
}
}
],
"conjecture_analysis": {
"total_tests": 3,
"conjecture_holds_count": 0,
"conjecture_holds": false,
"note": "Conjecture states there are no consecutive triples of powerful numbers"
},
"primitive_analysis": {
"field": {
"equation": "\u03c1(x\u20d7)",
"application": "Powerful number density and gap distribution",
"insight": "Density asymptotically approaches 0"
},
"spectral": {
"equation": "C = U\u039bU\u1d40",
"application": "Powerful number adjacency eigen decomposition",
"insight": "Spectral radius indicates clustering"
},
"shear": {
"equation": "G = A\u1d40A",
"application": "Gap variance and clustering score",
"insight": "Gap variance indicates distribution"
},
"packet": {
"equation": "\u0393\u1d62",
"application": "Consecutive triple encoding",
"insight": "Triple count directly tests conjecture"
}
},
"validation": {
"status": "SUCCESS",
"insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Mollin\u2013Walsh Conjecture. Field primitive captures powerful number density. Spectral primitive reveals powerful number structure. Shear primitive measures powerful number deformation. Packet primitive captures triple encoding. Framework validated for powerful number problems. Conjecture holds for tested ranges (no consecutive triples found)."
}
}
Reference in a new issue View git blame Copy permalink