Recorded data and provenance for the associated calculation.
| generated_at | 2026-09-23T17:16:49.576024+00:00 |
|---|---|
| status | diagnostic_pass |
| checks | 32 |
| check_meaning | one completed candidate-equation comparison per maintained mode and strength |
| purpose | All current candidate equations in one comparison run under shared mode assignments |
View complete data
{
"generated_at": "2026-09-23T17:16:49.576024+00:00",
"status": "diagnostic_pass",
"checks": 32,
"check_meaning": "one completed candidate-equation comparison per maintained mode and strength",
"purpose": "All current candidate equations in one comparison run under shared mode assignments",
"normalization": {
"response": "unit wave norm for the displayed algebraic source controls",
"linked_radial": "unit wave norm; its a*v^2 values belong only to the linked finite-carrier action",
"whole_depth_current": "unit phase charge for the reported common a*v^2 threshold; wave norm N=J/(2*omega)"
},
"equations": {
"response": "K_ij=sum_even_ell s_iell*s_jell*d/((ell-1)*(ell+d)); bias Kc=-diag(K); fixed-norm frequency shift=-gamma^2*N*K_ii",
"linked_radial": "H=D/R+gK; p=-R*omega_R/omega; G_R=2*N*p*omega/R^2; a*v^2>N*p*omega/2",
"whole_depth_current": "E1=<psi,H0(g)psi>-int g(kappa)*|j|; g(kappa)=g0*kappa^4/(1+kappa^4); G_ij=d_i d_j E1; F_i=d_i E1; a*v^2>J*F^T*G^-1*F on each tested metric subspace",
"positive_wave_action": "L+=||psi_dot||^2-||delta W/delta psi*||^2+positive collective kinetics-a*xi^2/2; J=2*omega*N",
"finite_chain_identity": "For B with diagonal n+j and positive n-independent subdiagonal b_j, omega/(n)_d=1/||adj(B)||_2; the positive adjugate is entrywise increasing in n, so omega/S(n,d) strictly decreases at finite b_j.",
"finite_chain_strong_gap": "omega/(n)_d=(prod b_j)^-1*[1-(n^2/b_1^2+(n+d-1)^2/b_(d-1)^2)/2+O(b^-4)] when all b_j grow at a fixed positive ratio.",
"fractional_current": "For k<0, f_d(k)=g*k+g*(d-1)^(-p)*|k|^p, p=1+2/d; G=f_d'(k), C=f_d(k)-G*k, omega=epsilon_n(G)+C. The power and coefficient cancel the leading high-index finite-gap drift without particle-specific parameters.",
"exact_depth_two_current": "Put P=n(n+1), mu=(1+g^2)^(-1/2), G^2=g^2-4P, k=G*(mu-(1+G^2)^(-1/2))/2. Define f(k(G))=mu*(g^2+G^2)/4-1/(2*mu)+1/(2*sqrt(1+G^2)); then f'(k)=G and omega=epsilon_n(G)+f-G*k=mu*P on the convex branch.",
"exact_depth_two_radius": "For physical radius R, let h(R)=g0*R/c/(1+(R/R0)^4) and omega_n(R)=c*n*(n+1)/(R*sqrt(1+h(R)^2)); every retained index has the same stationary-radius equation 2*gamma^2*x^2*(x^4-1)=(1+x^4)^3, x=R/R0, gamma=g0*R0/c."
},
"comparison_scope": [
"The three routes are separate actions and their Hessian entries are never added.",
"Strengths 1000 and 10000 are dimensionless controls, not physical fits.",
"Current Hessian rows show the uniform-radius and shared-q plane; pointwise coefficients test bare spinor positivity for all smooth conformal directions, and depth-two form rows include nine zonal directions.",
"The listed current a*v^2 values are lower bounds from finite tested metric subspaces, not sufficient thresholds for all metric perturbations.",
"The background potential, local causal completion, tensor metric response, complete wave stability, and physical exact mass law remain unproved.",
"The fractional-current calculation tests a new inverse fixed-geometry law only; its metric Hessian and full spatial stability have not been computed.",
"The depth-two exact law applies only on its finite invertible current branch and has no derived origin in the master action.",
"The exact depth-two radius control is an on-branch inverse trial; it does not establish angular stationarity, full coupled stability, or global confinement."
],
"sources": {
"experiments/joint-candidate-equations.py": "3f1ebcdbee9018e7509974d16086c2e70d8f8c28210922b1aee9090af59f3464",
"experiments/results/flux-moment-20260915T033331_031324Z.json": "6ed720878d217cc502ef9d7ba94e70b3d5b0afd10e08e1a7d3410336878f21e5"
},
"response_depth_summaries": [
{
"depth": 2,
"unbiased_response_unit_spread": "1.52670739761425149811797417707",
"common_bias_next_mode": 96,
"common_bias_next_residual": "0.00000111384682091369598343901997867",
"one_scale_size_curvature": "no finite stationary size"
},
{
"depth": 3,
"unbiased_response_unit_spread": "560.99045825018205064048546714",
"common_bias_next_mode": 18,
"common_bias_next_residual": "0.0000243434165573608154403295120663",
"one_scale_size_curvature": "flat at equality"
},
{
"depth": 4,
"unbiased_response_unit_spread": "46197.9775183491321099073459152",
"common_bias_next_mode": 73,
"common_bias_next_residual": "0.000446067882280155034157241105503",
"one_scale_size_curvature": "negative"
},
{
"depth": 5,
"unbiased_response_unit_spread": "28.3618456039244013278354074498",
"common_bias_next_mode": 23,
"common_bias_next_residual": "0.000121131941950498495500393189928",
"one_scale_size_curvature": "negative"
},
{
"depth": 6,
"unbiased_response_unit_spread": "175.578403851341661033508338772",
"common_bias_next_mode": 36,
"common_bias_next_residual": "-0.000702411175544740834562977331403",
"one_scale_size_curvature": "negative"
},
{
"depth": 10,
"unbiased_response_unit_spread": null,
"common_bias_next_mode": 24,
"common_bias_next_residual": "0.00361044022890621980587624078647",
"one_scale_size_curvature": "negative"
}
],
"depth_summaries": [
{
"depth": 2,
"strength": 1000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "17.7367210583195493711627932498",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "16.4999371238239938008973695105",
"whole_depth_current_tested_metric_basis": "nine zonal form directions through scalar degree 16",
"linked_frequency_unit_spread": "1.01315247993163137785188609996",
"current_frequency_unit_spread": "1.01207752477367018961729172588",
"fractional_current_frequency_unit_spread": "1.00000009702100611796993234748",
"exact_depth_two_frequency_unit_spread": "1",
"exact_depth_two_radius_control": {
"radius_over_reference": "1.00000100000250001350007837551",
"log_radius_curvature_of_frequency_unit": "3.99997599999999975999942399371",
"common_frequency_unit_times_reference_radius_over_speed": "0.00399999200001599995200009599957",
"scope": "positive on-branch local radius curvature under g(R)=g0/(1+(R/R0)^4); full field-metric stability and global binding untested"
},
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 3,
"strength": 1000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "0.0191888348940945224763154897119",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.054709747896032404068655807805",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00070250544374772516675875522",
"current_frequency_unit_spread": "1.00128164427240484712910983944",
"fractional_current_frequency_unit_spread": "1.00000207776444294260898004723",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 4,
"strength": 1000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "0.0953834984877321018226226101433",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.89677191023405256053990797227",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00804107268321030410618277529",
"current_frequency_unit_spread": "1.02200248352782185889964239323",
"fractional_current_frequency_unit_spread": "1.00049206165787873956674047804",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 5,
"strength": 1000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "0.0000198224606980954164603576928005",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.000724852128331527592770833212654",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00048883213512751488849517045",
"current_frequency_unit_spread": "1.00173199726907502515871328796",
"fractional_current_frequency_unit_spread": "1.00000438736756902535773861018",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 6,
"strength": 1000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "0.00000434789670734456319114904360168",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.000667252356433764137050301812337",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00110345292441715869813378958",
"current_frequency_unit_spread": "1.00469047158687740939342347701",
"fractional_current_frequency_unit_spread": "1.00002496963947629105007308681",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 10,
"strength": 1000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "1.74539319982332225822155368221e-14",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.00000000178279414117739837793730651728",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": null,
"current_frequency_unit_spread": null,
"fractional_current_frequency_unit_spread": null,
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": null
},
{
"depth": 2,
"strength": 10000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "1.90883776047025799371681387129",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "1.82215728008407484850295077194",
"whole_depth_current_tested_metric_basis": "nine zonal form directions through scalar degree 16",
"linked_frequency_unit_spread": "1.00014326115294860386484288028",
"current_frequency_unit_spread": "1.00013061121423429966625917586",
"fractional_current_frequency_unit_spread": "1.00000000000785882470211163309",
"exact_depth_two_frequency_unit_spread": "1",
"exact_depth_two_radius_control": {
"radius_over_reference": "1.0000000100000002500000135",
"log_radius_curvature_of_frequency_unit": "3.99999975999999999999975999999",
"common_frequency_unit_times_reference_radius_over_speed": "0.0003999999920000001599999952",
"scope": "positive on-branch local radius curvature under g(R)=g0/(1+(R/R0)^4); full field-metric stability and global binding untested"
},
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 3,
"strength": 10000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "0.000192114212129299380144714648466",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.000548741494202082953461629966085",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00000702008337194470755093789",
"current_frequency_unit_spread": "1.0000128001648596054451445525",
"fractional_current_frequency_unit_spread": "1.00000000107398134829850278038",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 4,
"strength": 10000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "0.0000965310530947897287110809338984",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.000934237089593066886115957370151",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00007987578767187919481784775",
"current_frequency_unit_spread": "1.00021532651912226086210466747",
"fractional_current_frequency_unit_spread": "1.00000004635983841261293036951",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 5,
"strength": 10000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "0.00000000198403247044893972944233450197",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.0000000727577948935955023813650305656",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00000488602753101474718278041",
"current_frequency_unit_spread": "1.00001728039883142074017996492",
"fractional_current_frequency_unit_spread": "1.00000000025266722047945222585",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 6,
"strength": 10000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "4.35550084596553062232361381615e-11",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "0.00000000672865289745143545714536876092",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": "1.00001102464072810782624856074",
"current_frequency_unit_spread": "1.00004664263250875322341926221",
"fractional_current_frequency_unit_spread": "1.00000000204096139988507729868",
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": true
},
{
"depth": 10,
"strength": 10000,
"linked_radial_common_av_squared_must_exceed_per_unit_wave_norm": "1.74635088676926502171673326929e-23",
"whole_depth_current_tested_metric_av_squared_lower_bound_per_unit_phase_charge": "1.78966548618756833262699116165e-18",
"whole_depth_current_tested_metric_basis": "uniform radius and shared q shape; bare positivity holds for all smooth conformal directions",
"linked_frequency_unit_spread": null,
"current_frequency_unit_spread": null,
"fractional_current_frequency_unit_spread": null,
"exact_depth_two_frequency_unit_spread": null,
"exact_depth_two_radius_control": null,
"finite_chain_unit_strictly_decreases_with_index": null
}
],
"records": [
{
"depth": 2,
"index": 76,
"strength": 1000,
"simplex_count": 2926,
"response": {
"response_norm": "0.697490123554727113901019237765",
"response_per_simplex": "0.000238376665603119314388591673877",
"biased_amplitude_curvature": "-2.78996049421890845560407695106"
},
"linked_radial": {
"frequency": "11.9585903437687946193781543657",
"frequency_per_simplex": "0.00408700968686561675303422910654",
"radius_exponent": "1.95228683244675076784221443477",
"bare_radial_curvature_per_unit_wave_norm": "46.693196925529360812627245741",
"background_threshold_av_squared_per_unit_wave_norm": "11.6732992313823402031568114353",
"field_radius_mixed_block_at_reference": "0",
"inverse_adjugate_norm": "0.00204350484343280837651711455327"
},
"whole_depth_current": {
"frequency": "11.4421323311893256945191180982",
"frequency_per_simplex": "0.00391050318905991992293886469522",
"canonical_phase_charge_at_unit_wave_norm": "22.8842646623786513890382361964",
"metric_force_per_unit_wave_norm": [
"-0.5008128675367741715969388211",
"-21.9530611902206457280805801613"
],
"bare_conformal_hessian_per_unit_wave_norm": [
[
"22.3834517948418772174412973753",
"22.2390608236594323962215296186"
],
[
"22.2390608236594323962215296186",
"64.1964705307373836879937132198"
]
],
"background_threshold_av_squared_per_unit_phase_charge": "10.945363815293031555991965858",
"background_threshold_av_squared_per_unit_wave_norm": "250.476602375188294423827373563",
"spinor_pointwise_positive_coefficients": null,
"depth_two_form_zonal_control": {
"maximum_scalar_degree": 16,
"minimum_bare_hessian_eigenvalue": "2.81571851970735441654538495675e-13",
"background_threshold_av_squared_per_unit_phase_charge": "10.9539576886332583245281363264"
},
"field_metric_mixed_block_at_reference": "0",
"inverse_adjugate_norm": "0.00195525159452995996146943234761"
},
"fractional_current": {
"frequency": "11.7039748085512977281189245426",
"frequency_per_simplex": "0.00399999139048233005062164201728",
"effective_edge_over_reference_gap": "0.952033946645767089796203076987",
"normalized_current": "-0.0239830266771164551018984615065",
"scalar_offset": "-0.28759278429763977698001033655",
"fixed_point_slope": "0.906369286614458219324178996767",
"power": "2.0",
"coefficient": "1.0",
"scope": "fixed-geometry frequency and component fixed point; spatial support and full stability untested"
},
"exact_depth_two_current": {
"frequency": "11.7039765920702237659208192771",
"frequency_per_simplex": "0.00399999200002399992000027999899",
"effective_edge_over_reference_gap": "0.952033612851983288628954665467",
"normalized_current": "-0.0239830423037989543514712843647",
"source_curvature": "1000.00663557736221671676739691",
"allowed_product_upper_bound": "62484.5009658776579187091921832",
"scope": "exact on the stated finite convex branch of an inverse current law; its master-field and metric origin are unproved"
}
},
{
"depth": 2,
"index": 81,
"strength": 1000,
"simplex_count": 3321,
"response": {
"response_norm": "0.701827090046728237623487723781",
"response_per_simplex": "0.000211330048192330092629776490148",
"biased_amplitude_curvature": "-2.80730836018691295049395089512"
},
"linked_radial": {
"frequency": "13.5295264732792859511661242643",
"frequency_per_simplex": "0.00407393148849120323732795069687",
"radius_exponent": "1.946355760611160568304935145",
"bare_radial_curvature_per_unit_wave_norm": "52.6665435792166747760905960404",
"background_threshold_av_squared_per_unit_wave_norm": "13.1666358948041686940226490101",
"field_radius_mixed_block_at_reference": "0",
"inverse_adjugate_norm": "0.00203696574424560161866397534844"
},
"whole_depth_current": {
"frequency": "12.9486395693527433256010623698",
"frequency_per_simplex": "0.00389901823828748669846463787106",
"canonical_phase_charge_at_unit_wave_norm": "25.8972791387054866512021247396",
"metric_force_per_unit_wave_norm": [
"-0.637690678834767573581857954715",
"-24.8044609614454257553673667846"
],
"bare_conformal_hessian_per_unit_wave_norm": [
[
"25.2595884598707190776202667849",
"25.1065254789972912775168565299"
],
[
"25.1065254789972912775168565299",
"72.4370036946887688404869393689"
]
],
"background_threshold_av_squared_per_unit_phase_charge": "12.3199617095861122910695381551",
"background_threshold_av_squared_per_unit_wave_norm": "319.053487371314808979474844179",
"spinor_pointwise_positive_coefficients": null,
"depth_two_form_zonal_control": {
"maximum_scalar_degree": 16,
"minimum_bare_hessian_eigenvalue": "1.34712309705180730979423076651e-13",
"background_threshold_av_squared_per_unit_phase_charge": "12.3284703887303778875723989741"
},
"field_metric_mixed_block_at_reference": "0",
"inverse_adjugate_norm": "0.00194950911914374334923231893553"
},
"fractional_current": {
"frequency": "13.2839711077137968631486080771",
"frequency_per_simplex": "0.00399999130012460007923776214306",
"effective_edge_over_reference_gap": "0.945372272839295636967464818503",
"normalized_current": "-0.0273138635803521815162675907483",
"scalar_offset": "-0.373023571843044646090792540383",
"fixed_point_slope": "0.893729488922429432602520416154",
"power": "2.0",
"coefficient": "1.0",
"scope": "fixed-geometry frequency and component fixed point; spatial support and full stability untested"
},
"exact_depth_two_current": {
"frequency": "13.2839734320797037343209298767",
"frequency_per_simplex": "0.00399999200002399992000027999899",
"effective_edge_over_reference_gap": "0.945371884498370302575884011329",
"normalized_current": "-0.0273138842149578484860140165885",
"source_curvature": "1000.00673426581858735293085328",
"allowed_product_upper_bound": "62484.5009658776579187091921832",
"scope": "exact on the stated finite convex branch of an inverse current law; its master-field and metric origin are unproved"
}
},
{
"depth": 2,
"index": 95,
"strength": 1000,
"simplex_count": 4560,
"response": {
"response_norm": "0.71198816279324300540738418584",
"response_per_simplex": "0.000156137754998518202940215830228",
"biased_amplitude_curvature": "-2.84795265117297202162953674336"
},
"linked_radial": {
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}
]
}