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

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{
  "is_new_bound": false,
  "status": "residue",
  "certified_finite_Z_dent": null,
  "hydrogen_uniqueness_not_claimed": true,
  "best_published_integer_bound": {
    "Z=2": {
      "Nc_le": 4,
      "source": "Lieb Nc < 2Z+1",
      "unsettled_N": [
        2,
        3,
        4
      ],
      "N0_ge": 2,
      "comment": "Zhislin binds N=2; Lieb excludes N>=5; N=3,4 open"
    },
    "Z=3": {
      "Nc_le": 6,
      "source": "Lieb Nc < 2Z+1",
      "unsettled_N": [
        3,
        4,
        5,
        6
      ],
      "N0_ge": 3
    },
    "Z=4": {
      "Nc_le": 8,
      "source": "Lieb Nc < 2Z+1",
      "unsettled_N": [
        4,
        5,
        6,
        7,
        8
      ],
      "N0_ge": 4
    },
    "Z=5": {
      "Nc_le": 10,
      "source": "Lieb Nc < 2Z+1",
      "unsettled_N": [
        5,
        6,
        7,
        8,
        9,
        10
      ],
      "N0_ge": 5
    }
  },
  "tried": {
    "published_envelopes": "Lieb, Nam, HPS s=2 printed and q1, HPS s=3 printed and q1. At Z=2,3,4,5 Lieb is strictly best. Nam/HPS exclude no extra integer.",
    "lieb_weights": "\u03c6 = |x|^s, |x|/(1+\u03bb|x|), 1-e^{-\u03b1|x|}, |x|e^{-\u03bc|x|}, min(|x|,\u03c1), compact C^1 cutoff. Search upper on inf R_N(\u03c6) is \u2264 2 for every family at N=3 and N=4, so none of these weights can exclude those N at Z=2.",
    "nam_hps_plug_in": "Nam Lemma 1 + Prop. 1 and HPS (7.10) at N=3,4 and Z=2..5: no contradiction from published alpha lowers. Prop. 1 remainder destroys the bound at N=4.",
    "alpha_search": "Numerical upper on alpha_4,2 \u2248 0.6123724356957944. The regular tetrahedron gives the exact upper sqrt(6)/4, so alpha_4,2 * 3 < 2 by 54<64. Need > 2/3 if kinetic is dropped. Pair geometry cannot exclude N=4 at Z=2.",
    "tetrahedron": "alpha_4,2 <= sqrt(6)/4, hence alpha_4,2 * 3 < 2. The s=2 pair geometry cannot exclude N=4 at Z=2 even with the kinetic remainder dropped.",
    "temple_IH": "3e Slater trials have \u03bc above the helium Hylleraas upper, so Temple vs E_1 \u2265 E(2,2) does not apply. IH minorants 1/|x-y|\u22650 and 1/(|x|+|y|) sit well below E(2,2).",
    "not_used": "HF/DFT tables are not bounds and were not used."
  },
  "flags": {
    "any_nam_lemma1_contradiction": false,
    "any_hps710_contradiction": false,
    "any_weight_certified_R_gt_2": false,
    "temple_applies": false,
    "families_have_R_leq_2_config": {
      "(3, 'power')": true,
      "(3, 'screened_rational')": true,
      "(3, 'one_minus_exp')": true,
      "(3, 'screened_exp')": true,
      "(3, 'min_r')": true,
      "(3, 'compact_cutoff')": true,
      "(4, 'power')": true,
      "(4, 'screened_rational')": true,
      "(4, 'one_minus_exp')": true,
      "(4, 'screened_exp')": true,
      "(4, 'min_r')": true,
      "(4, 'compact_cutoff')": true
    }
  },
  "replay": "problems/simon-ionization-excess/compute/q2/run_all.sh",
  "note": "Residue: nothing here certifies Nc(2)<4 or Nc(2)<3, nor any unique N0(Z) for Z>1. Best published integer bounds remain Lieb: Nc(2)\u22644, Nc(3)\u22646, Nc(4)\u22648."
}