files / problems / simon-ionization-excess / compute / q2 / certs / beta3_explore.json

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{
  "not_a_lower_bound": true,
  "b3": 1.1184337992027067,
  "fmin_HPS": 0.8941074569749823,
  "apparent_inf_beta3": 0.9206549282371305,
  "apparent_inf_inv": 1.0861832911868505,
  "gap_vs_b3_percent": 2.8835419708208465,
  "power_law": {
    "best": {
      "I_over_D": 0.9206549282371305,
      "alpha": -1.9999609530952958,
      "n": 3.502345377447195,
      "p": 3.9999609530952958,
      "inv": 1.0861832911868505,
      "polished": true
    },
    "n_evals": 4002
  },
  "atomic": [
    {
      "k": 1,
      "radii": [
        1.0
      ],
      "masses": [
        1.0
      ],
      "I_over_D": 1.0
    },
    {
      "k": 2,
      "radii": [
        1.0,
        1.7807764549246516
      ],
      "masses": [
        0.6403882072864975,
        0.3596117927135025
      ],
      "I_over_D": 0.9432997272187643,
      "inv": 1.060108437588985
    },
    {
      "k": 3,
      "radii": [
        1.0,
        1.5000006798201382,
        2.25000073518568
      ],
      "masses": [
        0.46551750457052105,
        0.32758612594388314,
        0.20689636948559584
      ],
      "I_over_D": 0.9310344827586372,
      "inv": 1.074074074074055
    },
    {
      "k": 4,
      "radii": [
        1.0,
        1.375968291881089,
        1.854968958597191,
        2.552367446956357
      ],
      "masses": [
        0.36396554789946445,
        0.28329416255583223,
        0.2101434153673966,
        0.14259687417730668
      ],
      "I_over_D": 0.9265301853982271,
      "inv": 1.0792956514095602
    },
    {
      "k": 5,
      "radii": [
        1.0,
        1.3037095229014495,
        1.6616584979174427,
        2.1178879425747064,
        2.761141068266489
      ],
      "masses": [
        0.2981845986224017,
        0.24582445213823512,
        0.1966748656571877,
        0.1513275487021568,
        0.10798853488001874
      ],
      "I_over_D": 0.9244006294629274,
      "inv": 1.0817820413871801
    },
    {
      "k": 6,
      "radii": [
        1.0,
        1.2509485166627945,
        1.5417709429109967,
        1.8833334534910702,
        2.317996232261877,
        2.910379951332926
      ],
      "masses": [
        0.24895792377472611,
        0.216344339384039,
        0.18217225268302675,
        0.14743454099397063,
        0.11789791416729146,
        0.087193028996946
      ],
      "I_over_D": 0.9232316682648097,
      "inv": 1.0831517531016612
    }
  ],
  "piecewise": [
    {
      "n_bins": 12,
      "R": 8.0,
      "I_over_D": 0.9210407612936494,
      "inv": 1.0857282782962268,
      "heights": [
        3.5341778189535247e-06,
        0.01691193875713194,
        6.138048047482727e-05,
        0.0016045920626717926,
        0.17239211674720875,
        0.2652787883249596,
        0.1942606978989949,
        0.11440415766926838,
        0.11634168052295647,
        0.04601333469734734,
        0.0529028240296441,
        0.019824954631522822
      ]
    },
    {
      "n_bins": 16,
      "R": 12.0,
      "I_over_D": 0.9213954483383637,
      "inv": 1.0853103320657715,
      "heights": [
        0.0020589847932057813,
        0.012656479235630631,
        1.5458270638056498e-17,
        0.0014973079623951532,
        0.09156753488855816,
        0.2690654142214918,
        0.11886613857912534,
        0.21573718195115202,
        0.0675816580157523,
        0.07058796028598599,
        0.08731388471630142,
        0.02432689567156766,
        0.02888831466305815,
        0.009789450515836297,
        6.156002655151988e-05,
        1.2344733877241519e-06
      ]
    },
    {
      "n_bins": 20,
      "R": 16.0,
      "I_over_D": 0.922180354585748,
      "inv": 1.08438657907564,
      "heights": [
        0.0001237441314764108,
        9.080352645309473e-05,
        0.0021640631815854476,
        9.882092489290497e-05,
        0.005766743931206714,
        9.142185012056005e-05,
        3.656526940520967e-05,
        0.0032400814659065886,
        0.15722855082810497,
        0.12363567405238307,
        0.0465316584782352,
        0.331245678547003,
        0.006326998551995145,
        0.16104410024521937,
        0.05546964018241441,
        0.04929290913584779,
        0.002716760791236603,
        0.053647837198065935,
        0.0008179761921144152,
        0.00042997151633301516
      ]
    }
  ],
  "b_s": [
    {
      "s": 3.0,
      "t0": 0.5960716379833215,
      "b": 1.1184337992027067,
      "b_inv": 0.8941074569749823,
      "grid_max": 1.1184337988163346,
      "grid_max_t": 0.5960500000000001
    },
    {
      "s": 3.1,
      "t0": 0.6083230819154379,
      "b": 1.1135848284857235,
      "b_inv": 0.8980007399704084,
      "grid_max": 1.1135848280473404,
      "grid_max_t": 0.6083000000000001
    },
    {
      "s": 3.5,
      "t0": 0.6507648650218449,
      "b": 1.0976095248500195,
      "b_inv": 0.9110708110305828,
      "grid_max": 1.097609524667729,
      "grid_max_t": 0.65075
    },
    {
      "s": 4.0,
      "t0": 0.6925048425718423,
      "b": 1.0830249175076245,
      "b_inv": 0.9233397900957897,
      "grid_max": 1.08302491748782,
      "grid_max_t": 0.6925
    },
    {
      "s": 5.0,
      "t0": 0.7519263986940594,
      "b": 1.0639339187843844,
      "b_inv": 0.9399079983675742,
      "grid_max": 1.0639339182765568,
      "grid_max_t": 0.75195
    }
  ],
  "quadratic_minorant": {
    "t0": 0.5960716379833215,
    "fmin": 0.8941074569749823,
    "alpha_grid_min": 0.29803581899166076,
    "alpha_at": 0.0,
    "fpp_over_2_near_t0": 0.659712284800662
  },
  "note": "These I/D values are numerical upper bounds on \u03b2_3^{rad}. The ionization coefficient needs a lower bound on \u03b2_3."
}