files / problems / simon-ionization-excess / compute / q2 / certs / aspect_try.json

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{
  "status": "residue",
  "is_new_bound": false,
  "beats_1.1185_in_HPS_theorem": false,
  "reason": "No lift of the aspect-\u22644 compact \u03b3 to every radial probability. Geometric t0-chains stay above compact \u03b3 in this scan but that is a search, not a lower bound. The withdrawn h(D_L,D_R) lift remains withdrawn.",
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      "equal_z_inv": 1.0150886427571022
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      "k": 20,
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      "equal_z_Q": 0.9858788566361459,
      "equal_z_aspect": 18592.1365177416,
      "equal_m_below_compact": false,
      "equal_z_below_compact": false,
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      "equal_z_inv": 1.0143234062367823
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    {
      "k": 21,
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      "equal_m_aspect": 31191.110821249677,
      "equal_z_Q": 0.9865512914587895,
      "equal_z_aspect": 31191.110821249677,
      "equal_m_below_compact": false,
      "equal_z_below_compact": false,
      "equal_m_inv": 1.0136320418995397,
      "equal_z_inv": 1.013632041899539
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    {
      "k": 22,
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      "equal_z_Q": 0.9871625961865519,
      "equal_z_aspect": 52327.7888657444,
      "equal_m_below_compact": false,
      "equal_z_below_compact": false,
      "equal_m_inv": 1.0130043458524867,
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    {
      "k": 23,
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      "equal_z_Q": 0.9877207441048192,
      "equal_z_aspect": 87787.7515574874,
      "equal_m_below_compact": false,
      "equal_z_below_compact": false,
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      "equal_z_inv": 1.0124319105055444
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    {
      "k": 24,
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      "equal_z_Q": 0.9882323797408114,
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      "equal_m_below_compact": false,
      "equal_z_below_compact": false,
      "equal_m_inv": 1.0119077460933594,
      "equal_z_inv": 1.011907746093358
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  ],
  "t0_chain_min_equal_m_Q": 0.9379263179805126,
  "t0_chain_min_equal_z_Q": 0.9379263179805125,
  "any_chain_below_compact": false,
  "below_compact": [],
  "k_atomic_opt": [
    {
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      "aspect": 1.7807749372140407,
      "radii": [
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      "masses": [
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      "equilibrium_max_rel_err": 3.219377509945267e-07,
      "inv": 1.0601084375869274
    },
    {
      "k": 3,
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      "masses": [
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      "equilibrium_max_rel_err": 3.4597826242739903e-07,
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    {
      "k": 4,
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      "equilibrium_max_rel_err": 1.1491619443368968e-07,
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    {
      "k": 5,
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      "masses": [
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    {
      "k": 6,
      "Q": 0.9232307844581592,
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      "masses": [
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  "truncation_power_alpha_m2": [
    {
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      "Q_increased": false,
      "n": 4.0,
      "alpha": -2.0
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    {
      "Q_full": 0.9267008090743243,
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      "aspect_full": 5.780158745650013,
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      "n": 6.0,
      "alpha": -2.0
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    {
      "Q_full": 0.932973630620418,
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      "Q_decreased": true,
      "n": 8.0,
      "alpha": -2.0
    },
    {
      "Q_full": 0.943113367795106,
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      "Q_increased": false,
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      "n": 12.0,
      "alpha": -2.0
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    {
      "Q_full": 0.9558718866815704,
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      "Q_increased": false,
      "Q_decreased": true,
      "n": 20.0,
      "alpha": -2.0
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    {
      "Q_full": 0.9705980888510506,
      "Q_trunc": 0.921975822736133,
      "aspect_full": 37.04108858610879,
      "truncated": true,
      "Q_increased": false,
      "Q_decreased": true,
      "n": 40.0,
      "alpha": -2.0
    },
    {
      "Q_full": 0.981394856899672,
      "Q_trunc": 0.9220948373287527,
      "aspect_full": 73.02007571379706,
      "truncated": true,
      "Q_increased": false,
      "Q_decreased": true,
      "n": 80.0,
      "alpha": -2.0
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  ],
  "truncation_t0_chain16": {
    "Q_full": 0.9823486292221693,
    "Q_trunc": 0.9379263179805126,
    "aspect_full": 2347.0542676558634,
    "truncated": true,
    "Q_increased": false,
    "Q_decreased": true
  },
  "truncation_can_increase_Q": false,
  "random_atomic_min_Q": 0.9304279244788176,
  "random_below_compact": [],
  "note_on_equilibrium": "For a probability-constrained critical point, V(r)=(Q/2)(r\u00b2+D) on the support. A one-point measure satisfies it with Q=1. A discrete k-atomic local min is only approximately critical (Nelder\u2013Mead). The identity is not used as a bound."
}