Recorded data and provenance for the associated calculation.
| title | Finite Commuting-Source Separation Bounds and Complete Electric Variations |
|---|---|
| checks_passed | 231 |
| case_count | 15 |
| scope | finite-source bounds at active three/four with the same explicit occupied-sector Hodge and profile controls |
| conclusion | A sharp plane energy bound and fixed-shape finite cap trials share the same positive linear separation coefficient. Completing the entire Abelian electric quadratic energy retains the fixed-flux stability bound. Prescribed sources, exact vacuum zero and commuting flux remain premises. |
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{
"title": "Finite Commuting-Source Separation Bounds and Complete Electric Variations",
"sources": {
"experiments/hodge-flux-separation-checks.py": "d76c72a080f60704d114dd939ee9839b51b6542857c912371c16e25de2a76d64",
"experiments/hodge-flux-tube-checks.py": "24ef86b4855274c5c8ca74890aa5e6bfa5f15bcccf2bc6ae566c02d998adb38b"
},
"checks_passed": 231,
"case_count": 15,
"occupied_depths": [
2,
3,
4,
5,
6,
10
],
"scope": "finite-source bounds at active three/four with the same explicit occupied-sector Hodge and profile controls",
"conclusion": "A sharp plane energy bound and fixed-shape finite cap trials share the same positive linear separation coefficient. Completing the entire Abelian electric quadratic energy retains the fixed-flux stability bound. Prescribed sources, exact vacuum zero and commuting flux remain premises.",
"cases": [
{
"case": "exact-cap-and-hamiltonian-identities",
"cap_polynomial": "10s^3-15s^4+6s^5",
"alpha2_exact": "181/462",
"alpha4_exact": "2549/8398",
"alpha_gradient_exact": "10/7",
"cap_charge_exact": "1",
"electric_second_variation": "(p-4 kappa E u^3 h)^2/(2 kappa u^4)-3 kappa E^2 u^2 h^2"
},
{
"case": "occupied-sector-source-control-2",
"real_depth": 2,
"hodge_coefficient": 0.0,
"transverse_dimension": 1,
"nonzero_color_source_allowed_in_this_action": false,
"regular_quartic_transverse_profile_possible": true
},
{
"case": "occupied-sector-source-control-3",
"real_depth": 3,
"hodge_coefficient": 1.0,
"transverse_dimension": 2,
"nonzero_color_source_allowed_in_this_action": true,
"regular_quartic_transverse_profile_possible": true
},
{
"case": "occupied-sector-source-control-4",
"real_depth": 4,
"hodge_coefficient": 1.0,
"transverse_dimension": 3,
"nonzero_color_source_allowed_in_this_action": true,
"regular_quartic_transverse_profile_possible": true
},
{
"case": "occupied-sector-source-control-5",
"real_depth": 5,
"hodge_coefficient": 0.0,
"transverse_dimension": 4,
"nonzero_color_source_allowed_in_this_action": false,
"regular_quartic_transverse_profile_possible": false
},
{
"case": "occupied-sector-source-control-6",
"real_depth": 6,
"hodge_coefficient": 0.0,
"transverse_dimension": 5,
"nonzero_color_source_allowed_in_this_action": false,
"regular_quartic_transverse_profile_possible": false
},
{
"case": "occupied-sector-source-control-10",
"real_depth": 10,
"hodge_coefficient": 0.0,
"transverse_dimension": 9,
"nonzero_color_source_allowed_in_this_action": false,
"regular_quartic_transverse_profile_possible": false
},
{
"case": "finite-source-separation-n2-Q1.0",
"real_depth": 3,
"classical_commuting_flux": 1.0,
"dielectric_coefficient": 1.0,
"neutral_restoring_frequency": 1.0,
"tension": 2.702843957387151,
"fixed_cap_width": 0.9,
"one_cap_energy": 2.3115339360920513,
"lengths": [
2.0,
5.0,
11.0
],
"upper_bound_energies": [
10.028755786958406,
18.137287659119856,
34.35435140344276
],
"lower_bound_energies": [
5.405687914774302,
13.514219786935755,
29.73128353125866
],
"source_total_charges": [
1.0,
-1.0
],
"exact_endpoint_stationary_solution_claimed": false,
"scope": "prescribed rapidly decaying opposite commuting sources; no non-Abelian screening claim"
},
{
"case": "finite-source-separation-n2-Q1.7",
"real_depth": 3,
"classical_commuting_flux": 1.7,
"dielectric_coefficient": 0.8,
"neutral_restoring_frequency": 1.3,
"tension": 4.93991140585993,
"fixed_cap_width": 0.6923076923076923,
"one_cap_energy": 3.2497885805188327,
"lengths": [
1.5384615384615383,
3.846153846153846,
8.461538461538462
],
"upper_bound_energies": [
14.099440862360636,
25.499236414345088,
48.298827518314
],
"lower_bound_energies": [
7.599863701322969,
18.99965925330742,
41.79925035727633
],
"source_total_charges": [
1.7,
-1.7
],
"exact_endpoint_stationary_solution_claimed": false,
"scope": "prescribed rapidly decaying opposite commuting sources; no non-Abelian screening claim"
},
{
"case": "finite-source-separation-n2-Q0.6",
"real_depth": 3,
"classical_commuting_flux": 0.6,
"dielectric_coefficient": 2.0,
"neutral_restoring_frequency": 0.7,
"tension": 1.2031243090786885,
"fixed_cap_width": 1.2857142857142858,
"one_cap_energy": 1.469913444409312,
"lengths": [
2.857142857142857,
7.142857142857143,
15.714285714285715
],
"upper_bound_energies": [
6.377324914757734,
11.5335719536664,
21.84606603148373
],
"lower_bound_energies": [
3.43749802593911,
8.593745064847775,
18.906239142665108
],
"source_total_charges": [
0.6,
-0.6
],
"exact_endpoint_stationary_solution_claimed": false,
"scope": "prescribed rapidly decaying opposite commuting sources; no non-Abelian screening claim"
},
{
"case": "complete-electric-variation-n2",
"real_depth": 3,
"directions_checked": 3,
"full_Abelian_quadratic_bound": true,
"non_Abelian_variations_checked": false
},
{
"case": "finite-source-separation-n3-Q1.0",
"real_depth": 4,
"classical_commuting_flux": 1.0,
"dielectric_coefficient": 1.0,
"neutral_restoring_frequency": 1.0,
"tension": 3.9953743016504832,
"fixed_cap_width": 0.9,
"one_cap_energy": 2.3599581036559565,
"lengths": [
2.0,
5.0,
11.0
],
"upper_bound_energies": [
12.710664810612881,
24.69678771556433,
48.66903352546723
],
"lower_bound_energies": [
7.9907486033009665,
19.976871508252415,
43.94911731815532
],
"source_total_charges": [
1.0,
-1.0
],
"exact_endpoint_stationary_solution_claimed": false,
"scope": "prescribed rapidly decaying opposite commuting sources; no non-Abelian screening claim"
},
{
"case": "finite-source-separation-n3-Q1.7",
"real_depth": 4,
"classical_commuting_flux": 1.7,
"dielectric_coefficient": 0.8,
"neutral_restoring_frequency": 1.3,
"tension": 6.690745971142438,
"fixed_cap_width": 0.6923076923076923,
"one_cap_energy": 3.0400309846734244,
"lengths": [
1.5384615384615383,
3.846153846153846,
8.461538461538462
],
"upper_bound_energies": [
16.373517309565987,
31.813700319894686,
62.6940663405521
],
"lower_bound_energies": [
10.293455340219134,
25.733638350547835,
56.61400437120525
],
"source_total_charges": [
1.7,
-1.7
],
"exact_endpoint_stationary_solution_claimed": false,
"scope": "prescribed rapidly decaying opposite commuting sources; no non-Abelian screening claim"
},
{
"case": "finite-source-separation-n3-Q0.6",
"real_depth": 4,
"classical_commuting_flux": 0.6,
"dielectric_coefficient": 2.0,
"neutral_restoring_frequency": 0.7,
"tension": 2.003000173587494,
"fixed_cap_width": 1.2857142857142858,
"one_cap_energy": 1.6901675839516208,
"lengths": [
2.857142857142857,
7.142857142857143,
15.714285714285715
],
"upper_bound_energies": [
9.103192806724653,
17.68747926495677,
34.856052181421006
],
"lower_bound_energies": [
5.722857638821411,
14.307144097053529,
31.475717013517762
],
"source_total_charges": [
0.6,
-0.6
],
"exact_endpoint_stationary_solution_claimed": false,
"scope": "prescribed rapidly decaying opposite commuting sources; no non-Abelian screening claim"
},
{
"case": "complete-electric-variation-n3",
"real_depth": 4,
"directions_checked": 3,
"full_Abelian_quadratic_bound": true,
"non_Abelian_variations_checked": false
}
]
}