files / problems / simon-ionization-excess / compute / q4 / certs / two_atom_crit.json

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{
  "not_a_lower_bound": true,
  "mass_crit_rows": [
    {
      "R": 2.0,
      "p": 0.6666666665727736,
      "Q": 0.9444444444444444,
      "cut": 0.6666666666666666,
      "Q_minus_cut": 0.2777777777777778,
      "inv": 1.0588235294117647,
      "stat_err": 3.3139138492167735e-11
    },
    {
      "R": 3.0,
      "p": 0.7500000129239607,
      "Q": 0.9583333333333334,
      "cut": 0.75,
      "Q_minus_cut": 0.20833333333333337,
      "inv": 1.0434782608695652,
      "stat_err": 6.7429359487307565e-09
    },
    {
      "R": 3.5,
      "p": 0.7777777817273148,
      "Q": 0.9647266313932981,
      "cut": 0.7777777777777778,
      "Q_minus_cut": 0.18694885361552027,
      "inv": 1.036563071297989,
      "stat_err": 2.274413271978346e-09
    },
    {
      "R": 4.0,
      "p": 0.7999999870282336,
      "Q": 0.97,
      "cut": 0.8,
      "Q_minus_cut": 0.16999999999999993,
      "inv": 1.0309278350515465,
      "stat_err": 8.02377271154385e-09
    },
    {
      "R": 5.0,
      "p": 0.8333333226280912,
      "Q": 0.9777777777777777,
      "cut": 0.8333333333333334,
      "Q_minus_cut": 0.14444444444444438,
      "inv": 1.0227272727272727,
      "stat_err": 7.299030470939918e-09
    },
    {
      "R": 6.0,
      "p": 0.8571428566293259,
      "Q": 0.9829931972789114,
      "cut": 0.8571428571428571,
      "Q_minus_cut": 0.12585034013605434,
      "inv": 1.0173010380622838,
      "stat_err": 3.731531715687333e-10
    },
    {
      "R": 8.0,
      "p": 0.8888888702917227,
      "Q": 0.9891975308641976,
      "cut": 0.8888888888888888,
      "Q_minus_cut": 0.10030864197530875,
      "inv": 1.0109204368174727,
      "stat_err": 1.4622412684740011e-08
    },
    {
      "R": 9.0,
      "p": 0.8999999812239542,
      "Q": 0.9911111111111112,
      "cut": 0.9,
      "Q_minus_cut": 0.09111111111111114,
      "inv": 1.0089686098654709,
      "stat_err": 1.5155553941870578e-08
    },
    {
      "R": 10.0,
      "p": 0.909090931104893,
      "Q": 0.9925619834710744,
      "cut": 0.9090909090909091,
      "Q_minus_cut": 0.08347107438016532,
      "inv": 1.0074937552039966,
      "stat_err": 1.8146414558801517e-08
    },
    {
      "R": 12.0,
      "p": 0.9230769573333235,
      "Q": 0.9945759368836292,
      "cut": 0.9230769230769231,
      "Q_minus_cut": 0.07149901380670609,
      "inv": 1.0054536440257809,
      "stat_err": 2.9144273379693484e-08
    },
    {
      "R": 16.0,
      "p": 0.9411764786586309,
      "Q": 0.9967560553633218,
      "cut": 0.9411764705882353,
      "Q_minus_cut": 0.05557958477508651,
      "inv": 1.0032545020611847,
      "stat_err": 7.144092228910481e-09
    },
    {
      "R": 20.0,
      "p": 0.952380950546731,
      "Q": 0.9978458049886619,
      "cut": 0.9523809523809523,
      "Q_minus_cut": 0.045464852607709605,
      "inv": 1.002158845585729,
      "stat_err": 1.6631263120459245e-09
    }
  ],
  "fully_crit_approx": {
    "R": 1.7807761926040564,
    "p": 0.6403881712231205,
    "Q": 0.9432997272187658,
    "cut": 0.6403881755534052,
    "Q_minus_cut": 0.30291155166536055,
    "inv": 1.0601084375889833,
    "stat_err": 1.2889235207114974e-09
  },
  "dQ_dR": [
    {
      "R": 2.0,
      "dQ_dR_at_masscrit": 0.009259259259319919
    },
    {
      "R": 3.0,
      "dQ_dR_at_masscrit": 0.013888888877877646
    },
    {
      "R": 3.5,
      "dQ_dR_at_masscrit": 0.011645810571447866
    },
    {
      "R": 4.0,
      "dQ_dR_at_masscrit": 0.009499999997775177
    },
    {
      "R": 5.0,
      "dQ_dR_at_masscrit": 0.0062962963010004805
    },
    {
      "R": 6.0,
      "dQ_dR_at_masscrit": 0.004292193050980941
    },
    {
      "R": 8.0,
      "dQ_dR_at_masscrit": 0.0022076474648358158
    },
    {
      "R": 9.0,
      "dQ_dR_at_masscrit": 0.00165432099130669
    },
    {
      "R": 10.0,
      "dQ_dR_at_masscrit": 0.001269722010865948
    },
    {
      "R": 12.0,
      "dQ_dR_at_masscrit": 0.000793379856256493
    },
    {
      "R": 16.0,
      "dQ_dR_at_masscrit": 0.0003681241145248037
    },
    {
      "R": 20.0,
      "dQ_dR_at_masscrit": 0.00019949250029682727
    }
  ],
  "note": "Two-atomic only. Mass-critical Q(R) sits well above R/(R+1), but k-atomic mass-opt on a filled window can be lower."
}