{
  "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,
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      "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,
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      "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
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    {
      "case": "three-dimensional-retarded-Fourier",
      "t": 2.0,
      "r": 0.7,
      "damping": 0.05,
      "numeric": 0.0782554142443273,
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      "quadrature_error": 1.525190297812798e-08,
      "outside_wave_cone": false
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      "case": "three-dimensional-retarded-Fourier",
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      "quadrature_error": 9.407668506533048e-10,
      "outside_wave_cone": false
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      "case": "three-dimensional-retarded-Fourier",
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      "quadrature_error": 1.9230052985944644e-08,
      "outside_wave_cone": false
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      "spectral_response": 49.468574643608605,
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      "case": "positive-ordinary-gauge-spectral-bound",
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      "response": 1.1130429294811932,
      "spectral_response": 1.1130429294811934,
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      "case": "positive-ordinary-gauge-spectral-bound",
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      "case": "positive-ordinary-gauge-spectral-bound",
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    },
    {
      "case": "positive-ordinary-gauge-spectral-bound",
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      "wave_number": 10.0,
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    },
    {
      "case": "positive-ordinary-gauge-spectral-bound",
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      "wave_number": 100.0,
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      "applies_to_depths": [
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    },
    {
      "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,
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      "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,
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      "translated_dipole_squared": 6.666666666666666,
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      "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
    }
  ]
}