files / problems / simon-lieb-thirring / compute / q3 / certs / ef_ratio.json

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{
  "EF_kappa_proved": 1.0,
  "kappa_needed_to_beat_CCR": 1.35,
  "one_bound_state_kappa": 2.4674011002723395,
  "min_empirical": 2.717301004834492,
  "records": [
    {
      "family": "hermite",
      "n_fun": 1,
      "T": 0.49937552050797895,
      "I3": 0.183776298473932,
      "kappa": 2.717301004834492,
      "L_over_Lcl_if_K_eq_kappa": 1.1003234549066563
    },
    {
      "family": "hermite",
      "n_fun": 2,
      "T": 1.9962541634134268,
      "I3": 0.6534268390184256,
      "kappa": 3.055053824253974,
      "L_over_Lcl_if_K_eq_kappa": 1.0377191046948484
    },
    {
      "family": "hermite",
      "n_fun": 3,
      "T": 4.4881421709103435,
      "I3": 1.4225646807796963,
      "kappa": 3.1549652761310147,
      "L_over_Lcl_if_K_eq_kappa": 1.021155670827616
    },
    {
      "family": "hermite",
      "n_fun": 5,
      "T": 12.446992024574357,
      "I3": 3.8673817150361534,
      "kappa": 3.2184544846403917,
      "L_over_Lcl_if_K_eq_kappa": 1.0110335315519638
    },
    {
      "family": "hermite",
      "n_fun": 8,
      "T": 31.78573180167499,
      "I3": 9.808469923456315,
      "kappa": 3.240641205990905,
      "L_over_Lcl_if_K_eq_kappa": 1.007566618125378
    },
    {
      "family": "finite-well-eigs",
      "n_fun": 1,
      "n_bound_available": 4,
      "T": 0.3682149849831864,
      "I3": 0.10917412930045024,
      "kappa": 3.3727311345882
    },
    {
      "family": "finite-well-eigs",
      "n_fun": 2,
      "n_bound_available": 4,
      "T": 1.8052396794650654,
      "I3": 0.565438527743727,
      "kappa": 3.1926364951969664
    },
    {
      "family": "finite-well-eigs",
      "n_fun": 3,
      "n_bound_available": 4,
      "T": 4.862706971891784,
      "I3": 1.5596158894527237,
      "kappa": 3.1178875547351153
    },
    {
      "family": "finite-well-eigs",
      "n_fun": 4,
      "n_bound_available": 4,
      "T": 9.47883767712177,
      "I3": 3.0444037080639204,
      "kappa": 3.1135284890156076
    }
  ],
  "beats_CCR": false,
  "note": "Empirical kappa sits near 2.5\u20134, above the 1.35 threshold, but this is not a proof that kappa\u22651.35 for every orthonormal system. The EF CS step still only gives kappa\u22651."
}