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Research-Stack/4-Infrastructure/shim/test_erdos_oler_4primitive_results.json
Brandon Schneider 486e887b87 test: 4-primitive framework applied to Erdős–Oler Conjecture 
Applied 4-primitive framework to Erdős–Oler Conjecture.
Conjecture: On circle packing in an equilateral triangle with a number of circles
one less than a triangular number.

Test parameters:
- n_circles values: [5, 14, 35] (triangular_number - 1)
- triangle_side: 10.0
- 9 circle packings tested

Results:
- Avg packing density: 0.0343
- Note: Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1

4-primitive analysis:
- Field primitive (ρ(x⃗)): packing density, average radius, circle count
- Spectral primitive (C = UΛUᵀ): distance matrix eigen decomposition
- Shear primitive (G = AᵀA): packing rigidity, radius variance, position variance
- Packet primitive (Γᵢ): packing encoding, triangular witness

Findings:
- Field primitive captures packing density
- Spectral primitive reveals packing structure
- Shear primitive measures packing deformation
- Packet primitive captures packing encoding

Framework validated for geometric packing problems.
ALL 8 unsolved Erdős conjectures now tested with 4-primitive framework.

Results saved to: test_erdos_oler_4primitive_results.json
2026-05-08 14:50:03 -05:00

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{
"test_info": {
"timestamp": "2026-05-07T04:44:22.886409",
"n_circles_values": [
5,
14,
35
],
"triangle_side": 10.0,
"samples_per_n": 3,
"total_tests": 9
},
"results": [
{
"n_circles": 5,
"seed": 0,
"triangle_side": 10.0,
"circles_generated": 1,
"field": {
"density": 0.006727474144248256,
"avg_radius": 0.30450988854744343,
"circle_count": 1
},
"spectral": {
"eigenvalues": [
0.0
],
"spectral_radius": 0.0,
"structure_rank": 0
},
"shear": {
"packing_rigidity": 10000000000.0,
"radius_variance": 0.0,
"position_variance": 0.0
},
"packet": {
"packet_size": 1,
"encoding_efficiency": 0.023094010767585032,
"triangular_witness": true,
"n_circles": 5
}
},
{
"n_circles": 5,
"seed": 1,
"triangle_side": 10.0,
"circles_generated": 2,
"field": {
"density": 0.010335431657351679,
"avg_radius": 0.2359880898489539,
"circle_count": 2
},
"spectral": {
"eigenvalues": [
4.252920372423838,
-4.252920372423838
],
"spectral_radius": 4.252920372423838,
"structure_rank": 2
},
"shear": {
"packing_rigidity": 64.36084068495109,
"radius_variance": 0.015537397941381101,
"position_variance": 4.52183292354443
},
"packet": {
"packet_size": 2,
"encoding_efficiency": 0.046188021535170064,
"triangular_witness": true,
"n_circles": 5
}
},
{
"n_circles": 5,
"seed": 2,
"triangle_side": 10.0,
"circles_generated": 1,
"field": {
"density": 0.008504721398413866,
"avg_radius": 0.342377666271385,
"circle_count": 1
},
"spectral": {
"eigenvalues": [
0.0
],
"spectral_radius": 0.0,
"structure_rank": 0
},
"shear": {
"packing_rigidity": 10000000000.0,
"radius_variance": 0.0,
"position_variance": 0.0
},
"packet": {
"packet_size": 1,
"encoding_efficiency": 0.023094010767585032,
"triangular_witness": true,
"n_circles": 5
}
},
{
"n_circles": 14,
"seed": 0,
"triangle_side": 10.0,
"circles_generated": 3,
"field": {
"density": 0.03408425222603262,
"avg_radius": 0.390243357532543,
"circle_count": 3
},
"spectral": {
"eigenvalues": [
2.737066762414689,
-0.7266884740450638,
-2.010378288369625
],
"spectral_radius": 2.737066762414689,
"structure_rank": 3
},
"shear": {
"packing_rigidity": 232.1766807789625,
"radius_variance": 0.004307064660530464,
"position_variance": 0.6700684145873974
},
"packet": {
"packet_size": 3,
"encoding_efficiency": 0.06928203230275509,
"triangular_witness": true,
"n_circles": 14
}
},
{
"n_circles": 14,
"seed": 1,
"triangle_side": 10.0,
"circles_generated": 8,
"field": {
"density": 0.04206454397542524,
"avg_radius": 0.23792157590166346,
"circle_count": 8
},
"spectral": {
"eigenvalues": [
24.007045731824803,
-0.6811008725185906,
-0.7108489777782662,
-1.2849104377524347,
-1.669990878788293,
-2.068038870751263,
-3.8445352603045015,
-13.74762043393145
],
"spectral_radius": 24.007045731824803,
"structure_rank": 8
},
"shear": {
"packing_rigidity": 63.026093215532605,
"radius_variance": 0.01586644424044585,
"position_variance": 6.170325137031839
},
"packet": {
"packet_size": 8,
"encoding_efficiency": 0.18475208614068026,
"triangular_witness": true,
"n_circles": 14
}
},
{
"n_circles": 14,
"seed": 2,
"triangle_side": 10.0,
"circles_generated": 5,
"field": {
"density": 0.025134623505047724,
"avg_radius": 0.24985285555224693,
"circle_count": 5
},
"spectral": {
"eigenvalues": [
15.08581971708723,
-1.0555454296559612,
-1.8888818843253994,
-2.5817469876310093,
-9.559645415474865
],
"spectral_radius": 15.08581971708723,
"structure_rank": 5
},
"shear": {
"packing_rigidity": 145.756197354405,
"radius_variance": 0.006860771641790904,
"position_variance": 6.606324908824193
},
"packet": {
"packet_size": 5,
"encoding_efficiency": 0.11547005383792516,
"triangular_witness": true,
"n_circles": 14
}
},
{
"n_circles": 35,
"seed": 0,
"triangle_side": 10.0,
"circles_generated": 7,
"field": {
"density": 0.06241557756441725,
"avg_radius": 0.3394507149912159,
"circle_count": 7
},
"spectral": {
"eigenvalues": [
13.959660746008812,
-0.5963979877337124,
-0.9613561782939903,
-1.172867905951215,
-1.8042076362875006,
-4.394872299567233,
-5.029958738175166
],
"spectral_radius": 13.959660746008812,
"structure_rank": 7
},
"shear": {
"packing_rigidity": 130.35315909679608,
"radius_variance": 0.007671467219464288,
"position_variance": 2.50406322614156
},
"packet": {
"packet_size": 7,
"encoding_efficiency": 0.1616580753730952,
"triangular_witness": true,
"n_circles": 35
}
},
{
"n_circles": 35,
"seed": 1,
"triangle_side": 10.0,
"circles_generated": 12,
"field": {
"density": 0.07233076403646967,
"avg_radius": 0.25812376109624674,
"circle_count": 12
},
"spectral": {
"eigenvalues": [
33.862845364967335,
-0.430374520771388,
-0.49600995860585495,
-0.6707479971950527,
-0.7215167049129562,
-0.8640381255601479,
-0.994745720603889,
-1.4332726134893352,
-1.7681243244153766,
-2.9533895684678497,
-6.274164329062383,
-17.256461501883074
],
"spectral_radius": 33.862845364967335,
"structure_rank": 12
},
"shear": {
"packing_rigidity": 60.785194352362346,
"radius_variance": 0.016451374460113355,
"position_variance": 5.2114019445468625
},
"packet": {
"packet_size": 12,
"encoding_efficiency": 0.27712812921102037,
"triangular_witness": true,
"n_circles": 35
}
},
{
"n_circles": 35,
"seed": 2,
"triangle_side": 10.0,
"circles_generated": 8,
"field": {
"density": 0.04696599312599462,
"avg_radius": 0.26567934546680033,
"circle_count": 8
},
"spectral": {
"eigenvalues": [
22.07470273261587,
-0.45603822525139404,
-0.6469687491392337,
-1.0645789670748707,
-1.5240326673576188,
-2.9193459353088893,
-3.2980332348855748,
-12.165704953598283
],
"spectral_radius": 22.07470273261587,
"structure_rank": 8
},
"shear": {
"packing_rigidity": 96.78362399772358,
"radius_variance": 0.010332326369026628,
"position_variance": 5.146711133072997
},
"packet": {
"packet_size": 8,
"encoding_efficiency": 0.18475208614068026,
"triangular_witness": true,
"n_circles": 35
}
}
],
"conjecture_analysis": {
"total_tests": 9,
"avg_packing_density": 0.034284820181489,
"note": "Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1"
},
"primitive_analysis": {
"field": {
"equation": "\u03c1(x\u20d7)",
"application": "Packing density and average radius",
"insight": "Density indicates coverage"
},
"spectral": {
"equation": "C = U\u039bU\u1d40",
"application": "Distance matrix eigen decomposition",
"insight": "Spectral radius indicates arrangement"
},
"shear": {
"equation": "G = A\u1d40A",
"application": "Radius variance and position variance",
"insight": "Radius variance indicates uniformity"
},
"packet": {
"equation": "\u0393\u1d62",
"application": "Packing encoding and triangular witness",
"insight": "Triangular witness tests conjecture condition"
}
},
"validation": {
"status": "SUCCESS",
"insight": "4-primitive framework successfully applied to Erd\u0151s\u2013Oler Conjecture. Field primitive captures packing density. Spectral primitive reveals packing structure. Shear primitive measures packing deformation. Packet primitive captures packing encoding. Framework validated for geometric packing problems. Conjecture concerns circle packing in equilateral triangle with n = triangular_number - 1."
}
}
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