Recorded data and provenance for the associated calculation.
| title | Neutral Geometric Frames Realize the Hodge-Gated Quartic Dielectric |
|---|---|
| checks_passed | 543 |
| case_count | 22 |
| status | explicit positive retained geometric-mode action; master-field reduction and vacuum Cauchy problem remain open |
| conclusion | Squared middle-form volume gives a quartic dielectric on the active four-carrier. Exact Gram and gradient bounds reduce the entire neutral-frame fixed-flux energy to its balanced scalar branch. Amplitude imbalance, nonorthogonality and spatial frame twists cannot lower the tube minimum. All occupied-sector Hodge zeros and geometric-degree profile controls use the same law. |
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{
"title": "Neutral Geometric Frames Realize the Hodge-Gated Quartic Dielectric",
"sources": {
"experiments/neutral-frame-dielectric-checks.py": "78ff3b7ab1a1178a9d3c61ba5f76cc8bc8104369846a1de7ce08343200b22e62",
"experiments/hodge-flux-tube-checks.py": "24ef86b4855274c5c8ca74890aa5e6bfa5f15bcccf2bc6ae566c02d998adb38b"
},
"checks_passed": 543,
"case_count": 22,
"occupied_depths": [
2,
3,
4,
5,
6,
10
],
"status": "explicit positive retained geometric-mode action; master-field reduction and vacuum Cauchy problem remain open",
"conclusion": "Squared middle-form volume gives a quartic dielectric on the active four-carrier. Exact Gram and gradient bounds reduce the entire neutral-frame fixed-flux energy to its balanced scalar branch. Amplitude imbalance, nonorthogonality and spatial frame twists cannot lower the tube minimum. All occupied-sector Hodge zeros and geometric-degree profile controls use the same law.",
"cases": [
{
"case": "geometric-frame-depth-2",
"real_depth": 2,
"even_source_carrier": 2,
"frame_vectors": 1,
"balanced_dielectric_power": 2,
"hodge_coefficient": 0.0,
"balanced_coefficient_fraction": "1",
"pohozaev_gradient_coefficient": 4,
"pohozaev_mass_coefficient": 0,
"nongated_regular_finite_width_profile_possible": false,
"active_color_source_permitted": false,
"random_frame_comparisons": 12,
"largest_sampled_area_fraction": 1.0
},
{
"case": "geometric-frame-depth-3",
"real_depth": 3,
"even_source_carrier": 4,
"frame_vectors": 2,
"balanced_dielectric_power": 4,
"hodge_coefficient": 1.0,
"balanced_coefficient_fraction": "1/4",
"pohozaev_gradient_coefficient": 4,
"pohozaev_mass_coefficient": 4,
"nongated_regular_finite_width_profile_possible": true,
"active_color_source_permitted": true,
"random_frame_comparisons": 12,
"largest_sampled_area_fraction": 0.9458033041345619
},
{
"case": "geometric-frame-depth-4",
"real_depth": 4,
"even_source_carrier": 4,
"frame_vectors": 2,
"balanced_dielectric_power": 4,
"hodge_coefficient": 1.0,
"balanced_coefficient_fraction": "1/4",
"pohozaev_gradient_coefficient": 2,
"pohozaev_mass_coefficient": 6,
"nongated_regular_finite_width_profile_possible": true,
"active_color_source_permitted": true,
"random_frame_comparisons": 12,
"largest_sampled_area_fraction": 0.953272234119911
},
{
"case": "geometric-frame-depth-5",
"real_depth": 5,
"even_source_carrier": 6,
"frame_vectors": 3,
"balanced_dielectric_power": 6,
"hodge_coefficient": 0.0,
"balanced_coefficient_fraction": "1/27",
"pohozaev_gradient_coefficient": -4,
"pohozaev_mass_coefficient": 16,
"nongated_regular_finite_width_profile_possible": false,
"active_color_source_permitted": false,
"random_frame_comparisons": 12,
"largest_sampled_area_fraction": 0.7641702420801575
},
{
"case": "geometric-frame-depth-6",
"real_depth": 6,
"even_source_carrier": 6,
"frame_vectors": 3,
"balanced_dielectric_power": 6,
"hodge_coefficient": 0.0,
"balanced_coefficient_fraction": "1/27",
"pohozaev_gradient_coefficient": -8,
"pohozaev_mass_coefficient": 20,
"nongated_regular_finite_width_profile_possible": false,
"active_color_source_permitted": false,
"random_frame_comparisons": 12,
"largest_sampled_area_fraction": 0.7048127546329586
},
{
"case": "geometric-frame-depth-10",
"real_depth": 10,
"even_source_carrier": 10,
"frame_vectors": 5,
"balanced_dielectric_power": 10,
"hodge_coefficient": 0.0,
"balanced_coefficient_fraction": "1/3125",
"pohozaev_gradient_coefficient": -52,
"pohozaev_mass_coefficient": 72,
"nongated_regular_finite_width_profile_possible": false,
"active_color_source_permitted": false,
"random_frame_comparisons": 12,
"largest_sampled_area_fraction": 0.44307547865360264
},
{
"case": "frame-imbalance-n2-s0.1",
"real_depth": 3,
"relative_log_amplitude": 0.1,
"reference_tension": 2.7028439573935117,
"complete_energy": 2.7393649524492796,
"positive_excess": 0.03652099505576789
},
{
"case": "frame-imbalance-n2-s0.3",
"real_depth": 3,
"relative_log_amplitude": 0.3,
"reference_tension": 2.7028439573935117,
"complete_energy": 3.0680232076710006,
"positive_excess": 0.3651792502774889
},
{
"case": "frame-imbalance-n2-s0.7",
"real_depth": 3,
"relative_log_amplitude": 0.7,
"reference_tension": 2.7028439573935117,
"complete_energy": 5.970009486740711,
"positive_excess": 3.2671655293471993
},
{
"case": "frame-angle-n2-theta0.4",
"real_depth": 3,
"relative_angle": 0.4,
"area_fraction": 0.1516466453264173,
"complete_energy": 7.742996639248165,
"positive_excess": 5.040152681854654
},
{
"case": "frame-angle-n2-theta0.9",
"real_depth": 3,
"relative_angle": 0.9,
"area_fraction": 0.6136010473465435,
"complete_energy": 3.270192008072596,
"positive_excess": 0.5673480506790844
},
{
"case": "frame-angle-n2-theta1.2",
"real_depth": 3,
"relative_angle": 1.2,
"area_fraction": 0.8686968577706227,
"complete_energy": 2.839021843225628,
"positive_excess": 0.1361778858321161
},
{
"case": "frame-twist-n2-theta0.2",
"real_depth": 3,
"twist_amplitude": 0.2,
"positive_gradient_excess": 0.01514152915955072,
"rigid_common_rotations_remain_free": true
},
{
"case": "frame-twist-n2-theta0.6",
"real_depth": 3,
"twist_amplitude": 0.6,
"positive_gradient_excess": 0.13627376243595749,
"rigid_common_rotations_remain_free": true
},
{
"case": "frame-imbalance-n3-s0.1",
"real_depth": 4,
"relative_log_amplitude": 0.1,
"reference_tension": 3.9953743016380896,
"complete_energy": 4.049360046807625,
"positive_excess": 0.053985745169535804
},
{
"case": "frame-imbalance-n3-s0.3",
"real_depth": 4,
"relative_log_amplitude": 0.3,
"reference_tension": 3.9953743016380896,
"complete_energy": 4.535186371834373,
"positive_excess": 0.539812070196283
},
{
"case": "frame-imbalance-n3-s0.7",
"real_depth": 4,
"relative_log_amplitude": 0.7,
"reference_tension": 3.9953743016380896,
"complete_energy": 8.824935090567617,
"positive_excess": 4.829560788929528
},
{
"case": "frame-angle-n3-theta0.4",
"real_depth": 4,
"relative_angle": 0.4,
"area_fraction": 0.1516466453264173,
"complete_energy": 11.445784617202012,
"positive_excess": 7.450410315563922
},
{
"case": "frame-angle-n3-theta0.9",
"real_depth": 4,
"relative_angle": 0.9,
"area_fraction": 0.6136010473465435,
"complete_energy": 4.834034563759179,
"positive_excess": 0.8386602621210892
},
{
"case": "frame-angle-n3-theta1.2",
"real_depth": 4,
"relative_angle": 1.2,
"area_fraction": 0.8686968577706227,
"complete_energy": 4.1966739823021175,
"positive_excess": 0.20129968066402792
},
{
"case": "frame-twist-n3-theta0.2",
"real_depth": 4,
"twist_amplitude": 0.2,
"positive_gradient_excess": 0.012048232347417967,
"rigid_common_rotations_remain_free": true
},
{
"case": "frame-twist-n3-theta0.6",
"real_depth": 4,
"twist_amplitude": 0.6,
"positive_gradient_excess": 0.10843409112676371,
"rigid_common_rotations_remain_free": true
}
]
}