Recorded data and provenance for the associated calculation.
| title | Gradient Gauss Static Energy and Physical Retarded Response |
|---|---|
| checks_passed | 149 |
| case_count | 50 |
| conclusion | Positive ordinary gradient-electric Gauss law realizes the static k^-4 energy and ordinary continuity, but its minimally coupled physical field has a nonzero spacelike retarded response. Finite positive ordinary gauge channels have only simple infrared poles. These are scoped linear results, not a general confinement no-go. |
View complete data
{
"title": "Gradient Gauss Static Energy and Physical Retarded Response",
"sources": {
"experiments/gradient-gauss-causality-checks.py": "4f36b19294076a5c1b147ee65096f24b44ac78bc7f1814cf646f06ee0def543c",
"content/notes/topics/gauge-topology/biharmonic-color-window.md": "b89119a0a4b96f86b11ee9079ba07c833237b475a17c4423183d3756eb984ddc",
"content/notes/topics/gauge-topology/normal-color-kinetics.md": "33e558e6945c95c9bae48a734f6292c6ca9871b25caa969b5dd1d656f6f54098"
},
"checks_passed": 149,
"case_count": 50,
"occupied_depths": [
2,
3,
4,
5,
6,
10
],
"conclusion": "Positive ordinary gradient-electric Gauss law realizes the static k^-4 energy and ordinary continuity, but its minimally coupled physical field has a nonzero spacelike retarded response. Finite positive ordinary gauge channels have only simple infrared poles. These are scoped linear results, not a general confinement no-go.",
"cases": [
{
"case": "quadrupole-tail-depth-2",
"depth": 2,
"sphere_area": "2*pi",
"tail_at_r_4t": "-53*t**2/(2560*pi)",
"tail_relative_bracket": "159/160",
"infrared_energy_power": 1,
"isolated_cost_diverges": true,
"neutral_dipole_cost_finite": false,
"conditional_confinement_window": false
},
{
"case": "quadrupole-tail-depth-3",
"depth": 3,
"sphere_area": "4*pi",
"tail_at_r_4t": "-157*t/(61440*pi)",
"tail_relative_bracket": "157/160",
"infrared_energy_power": 2,
"isolated_cost_diverges": true,
"neutral_dipole_cost_finite": true,
"conditional_confinement_window": true
},
{
"case": "quadrupole-tail-depth-4",
"depth": 4,
"sphere_area": "2*pi**2",
"tail_at_r_4t": "-77/(61440*pi**2)",
"tail_relative_bracket": "77/80",
"infrared_energy_power": 3,
"isolated_cost_diverges": true,
"neutral_dipole_cost_finite": true,
"conditional_confinement_window": true
},
{
"case": "quadrupole-tail-depth-5",
"depth": 5,
"sphere_area": "8*pi**2/3",
"tail_at_r_4t": "-15/(65536*pi**2*t)",
"tail_relative_bracket": "15/16",
"infrared_energy_power": 4,
"isolated_cost_diverges": false,
"neutral_dipole_cost_finite": true,
"conditional_confinement_window": false
},
{
"case": "quadrupole-tail-depth-6",
"depth": 6,
"sphere_area": "pi**3",
"tail_at_r_4t": "-29/(196608*pi**3*t**2)",
"tail_relative_bracket": "29/32",
"infrared_energy_power": 5,
"isolated_cost_diverges": false,
"neutral_dipole_cost_finite": true,
"conditional_confinement_window": false
},
{
"case": "quadrupole-tail-depth-10",
"depth": 10,
"sphere_area": "pi**5/12",
"tail_at_r_4t": "-23/(4194304*pi**5*t**6)",
"tail_relative_bracket": "23/32",
"infrared_energy_power": 9,
"isolated_cost_diverges": false,
"neutral_dipole_cost_finite": true,
"conditional_confinement_window": false
},
{
"case": "gradient-electric-action",
"static_kernel": "1/(ell**2*k**4)",
"ordinary_current_conservation": true,
"positive_nonzero_k_energy": true,
"homogeneous_wave_speed_finite": true,
"gauge_invariant_source_response_has_spacelike_tail": true,
"zero_spatial_mode_requires_domain_choice": true
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 1.0,
"r": 4.0,
"damping": 0.5,
"numeric": 0.018286224141926245,
"closed": 0.018286224141933954,
"undamped_limit": 0.019894367886486918,
"quadrature_error": 2.3980443085718728e-09,
"outside_wave_cone": true
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 1.0,
"r": 4.0,
"damping": 0.2,
"numeric": 0.019247999011165275,
"closed": 0.019247999011633047,
"undamped_limit": 0.019894367886486918,
"quadrature_error": 1.21504949510511e-08,
"outside_wave_cone": true
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 1.0,
"r": 4.0,
"damping": 0.05,
"numeric": 0.019732635227773486,
"closed": 0.019732635214406137,
"undamped_limit": 0.019894367886486918,
"quadrature_error": 2.18848432258633e-08,
"outside_wave_cone": true
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 0.5,
"r": 3.0,
"damping": 0.5,
"numeric": 0.011855962520459368,
"closed": 0.011855962520463111,
"undamped_limit": 0.013262911924324612,
"quadrature_error": 3.6958380352229553e-09,
"outside_wave_cone": true
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 0.5,
"r": 3.0,
"damping": 0.2,
"numeric": 0.01269559553965035,
"closed": 0.012695595539785475,
"undamped_limit": 0.013262911924324612,
"quadrature_error": 5.288015995288931e-09,
"outside_wave_cone": true
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 0.5,
"r": 3.0,
"damping": 0.05,
"numeric": 0.013120876689200452,
"closed": 0.013120876685033478,
"undamped_limit": 0.013262911924324612,
"quadrature_error": 1.0768529484395426e-08,
"outside_wave_cone": true
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 2.0,
"r": 0.7,
"damping": 0.5,
"numeric": 0.0666786778501255,
"closed": 0.06667867785012932,
"undamped_limit": 0.07957747154594767,
"quadrature_error": 5.64931715429008e-10,
"outside_wave_cone": false
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 2.0,
"r": 0.7,
"damping": 0.2,
"numeric": 0.07430961365818115,
"closed": 0.07430961365818393,
"undamped_limit": 0.07957747154594767,
"quadrature_error": 1.0099315641051044e-08,
"outside_wave_cone": false
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 2.0,
"r": 0.7,
"damping": 0.05,
"numeric": 0.0782554142443273,
"closed": 0.07825541425139138,
"undamped_limit": 0.07957747154594767,
"quadrature_error": 1.525190297812798e-08,
"outside_wave_cone": false
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 1.0,
"r": 1.0,
"damping": 0.5,
"numeric": 0.04922517439269182,
"closed": 0.04922517439269186,
"undamped_limit": 0.07957747154594767,
"quadrature_error": 1.4737411284153968e-09,
"outside_wave_cone": false
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 1.0,
"r": 1.0,
"damping": 0.2,
"numeric": 0.0628379617892564,
"closed": 0.06283796178926714,
"undamped_limit": 0.07957747154594767,
"quadrature_error": 9.407668506533048e-10,
"outside_wave_cone": false
},
{
"case": "three-dimensional-retarded-Fourier",
"t": 1.0,
"r": 1.0,
"damping": 0.05,
"numeric": 0.07363880436802249,
"closed": 0.07363880443896412,
"undamped_limit": 0.07957747154594767,
"quadrature_error": 1.9230052985944644e-08,
"outside_wave_cone": false
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 1,
"wave_number": 0.001,
"response": 44521.71717924774,
"spectral_response": 44521.717179247746,
"k_squared_response": 0.04452171717924774,
"upper_bound": 0.04452171717924774,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 1,
"wave_number": 0.03,
"response": 49.468574643608605,
"spectral_response": 49.468574643608605,
"k_squared_response": 0.044521717179247744,
"upper_bound": 0.04452171717924774,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 1,
"wave_number": 0.2,
"response": 1.1130429294811932,
"spectral_response": 1.1130429294811934,
"k_squared_response": 0.04452171717924774,
"upper_bound": 0.04452171717924774,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 1,
"wave_number": 1.0,
"response": 0.04452171717924774,
"spectral_response": 0.044521717179247744,
"k_squared_response": 0.04452171717924774,
"upper_bound": 0.04452171717924774,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 1,
"wave_number": 10.0,
"response": 0.00044521717179247736,
"spectral_response": 0.00044521717179247746,
"k_squared_response": 0.04452171717924774,
"upper_bound": 0.04452171717924774,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 1,
"wave_number": 100.0,
"response": 4.452171717924774e-06,
"spectral_response": 4.452171717924775e-06,
"k_squared_response": 0.04452171717924774,
"upper_bound": 0.04452171717924774,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 2,
"wave_number": 0.001,
"response": 69078.62871971204,
"spectral_response": 69078.62868731003,
"k_squared_response": 0.06907862871971204,
"upper_bound": 0.2477486136784356,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 2,
"wave_number": 0.03,
"response": 76.79863970372253,
"spectral_response": 76.79863970372382,
"k_squared_response": 0.06911877573335028,
"upper_bound": 0.2477486136784356,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 2,
"wave_number": 0.2,
"response": 1.7711898553613303,
"spectral_response": 1.7711898553613397,
"k_squared_response": 0.07084759421445323,
"upper_bound": 0.2477486136784356,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 2,
"wave_number": 1.0,
"response": 0.10481258995153823,
"spectral_response": 0.10481258995153814,
"k_squared_response": 0.10481258995153823,
"upper_bound": 0.2477486136784356,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 2,
"wave_number": 10.0,
"response": 0.0024087668946079633,
"spectral_response": 0.0024087668946079625,
"k_squared_response": 0.24087668946079632,
"upper_bound": 0.2477486136784356,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 2,
"wave_number": 100.0,
"response": 2.4767717424234656e-05,
"spectral_response": 2.4767717424234652e-05,
"k_squared_response": 0.24767717424234656,
"upper_bound": 0.2477486136784356,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 3,
"wave_number": 0.001,
"response": 211448.93995107635,
"spectral_response": 211448.9399970211,
"k_squared_response": 0.21144893995107633,
"upper_bound": 0.589699540512747,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 3,
"wave_number": 0.03,
"response": 235.03770299568464,
"spectral_response": 235.03770299569968,
"k_squared_response": 0.21153393269611617,
"upper_bound": 0.589699540512747,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 3,
"wave_number": 0.2,
"response": 5.37984754553216,
"spectral_response": 5.379847545532138,
"k_squared_response": 0.21519390182128642,
"upper_bound": 0.589699540512747,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 3,
"wave_number": 1.0,
"response": 0.2870989844500461,
"spectral_response": 0.2870989844500461,
"k_squared_response": 0.2870989844500461,
"upper_bound": 0.589699540512747,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 3,
"wave_number": 10.0,
"response": 0.005751514368558865,
"spectral_response": 0.0057515143685588615,
"k_squared_response": 0.5751514368558865,
"upper_bound": 0.589699540512747,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 3,
"wave_number": 100.0,
"response": 5.895483007306285e-05,
"spectral_response": 5.8954830073062835e-05,
"k_squared_response": 0.5895483007306285,
"upper_bound": 0.589699540512747,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 8,
"wave_number": 0.001,
"response": 2250209.7071148125,
"spectral_response": 2250209.7110282118,
"k_squared_response": 2.2502097071148124,
"upper_bound": 2.778415999123257,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 8,
"wave_number": 0.03,
"response": 2500.348284120526,
"spectral_response": 2500.3482841245504,
"k_squared_response": 2.2503134557084734,
"upper_bound": 2.778415999123257,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 8,
"wave_number": 0.2,
"response": 56.36954900784704,
"spectral_response": 56.369549007847226,
"k_squared_response": 2.254781960313882,
"upper_bound": 2.778415999123257,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 8,
"wave_number": 1.0,
"response": 2.343060791731941,
"spectral_response": 2.343060791731944,
"k_squared_response": 2.343060791731941,
"upper_bound": 2.778415999123257,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 8,
"wave_number": 10.0,
"response": 0.027435175105165084,
"spectral_response": 0.02743517510516509,
"k_squared_response": 2.7435175105165084,
"upper_bound": 2.778415999123257,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "positive-ordinary-gauge-spectral-bound",
"channels": 8,
"wave_number": 100.0,
"response": 0.0002777998721485137,
"spectral_response": 0.0002777998721485138,
"k_squared_response": 2.7779987214851367,
"upper_bound": 2.778415999123257,
"applies_to_depths": [
2,
3,
4,
5,
6,
10
]
},
{
"case": "covariant-box-factor-control",
"partial_fractions": "(1/z - 1/(z+m^2))/m^2",
"residues_opposite_signs": true,
"static_with_lower_term_has_simple_IR_pole": true,
"scope": "ordinary finite quadratic canonical realization"
},
{
"case": "neutral-determinant-mobility-depth-2",
"depth": 2,
"dipole_squared": 6.666666666666666,
"translated_dipole_squared": 6.666666666666666,
"dilated_dipole_squared": 26.666666666666664,
"translation_allowed_by_Abelian_dipole_conservation": true,
"unassisted_dilation_preserves_dipole": false,
"non_Abelian_completion_not_inferred": true
},
{
"case": "neutral-determinant-mobility-depth-3",
"depth": 3,
"dipole_squared": 6.666666666666666,
"translated_dipole_squared": 6.666666666666666,
"dilated_dipole_squared": 26.666666666666664,
"translation_allowed_by_Abelian_dipole_conservation": true,
"unassisted_dilation_preserves_dipole": false,
"non_Abelian_completion_not_inferred": true
},
{
"case": "neutral-determinant-mobility-depth-4",
"depth": 4,
"dipole_squared": 6.666666666666666,
"translated_dipole_squared": 6.666666666666666,
"dilated_dipole_squared": 26.666666666666664,
"translation_allowed_by_Abelian_dipole_conservation": true,
"unassisted_dilation_preserves_dipole": false,
"non_Abelian_completion_not_inferred": true
},
{
"case": "neutral-determinant-mobility-depth-5",
"depth": 5,
"dipole_squared": 6.666666666666666,
"translated_dipole_squared": 6.666666666666666,
"dilated_dipole_squared": 26.666666666666664,
"translation_allowed_by_Abelian_dipole_conservation": true,
"unassisted_dilation_preserves_dipole": false,
"non_Abelian_completion_not_inferred": true
},
{
"case": "neutral-determinant-mobility-depth-6",
"depth": 6,
"dipole_squared": 6.666666666666666,
"translated_dipole_squared": 6.666666666666666,
"dilated_dipole_squared": 26.666666666666664,
"translation_allowed_by_Abelian_dipole_conservation": true,
"unassisted_dilation_preserves_dipole": false,
"non_Abelian_completion_not_inferred": true
},
{
"case": "neutral-determinant-mobility-depth-10",
"depth": 10,
"dipole_squared": 6.666666666666666,
"translated_dipole_squared": 6.666666666666664,
"dilated_dipole_squared": 26.666666666666664,
"translation_allowed_by_Abelian_dipole_conservation": true,
"unassisted_dilation_preserves_dipole": false,
"non_Abelian_completion_not_inferred": true
}
]
}