files / problems / simon-ionization-excess / compute / q7 / certs / leading.json

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{
  "arxiv": "2504.18487v1",
  "beta3_lower": "0.9074071558364936",
  "source": "q7 compact R=10 n=33 target=0.9111, lifted by mass-opt dichotomy",
  "row": "certs/raise_R10_n33_t0p9111.json",
  "R_split": 10,
  "b_interval": [
    "1.102041121857970863838942578552020678384",
    "1.102041121857970863838942578552020678384"
  ],
  "b_hi": "1.102041121857970863838942578552020678384",
  "printed_leading": "1.1021",
  "a1_at_left": [
    "3.850854645415426269283283027292022579266",
    "3.850854645415426269283283027292022579266"
  ],
  "a1_at_9/4": [
    "3.825280053151238278034060772197551540266",
    "3.825280053151238278034060772197551540266"
  ],
  "a1_max_hi_on_interval": "3.850854645415426269283283027292022579266",
  "a1_printed": "3.851",
  "a2": [
    "0.01311953716497584361713026879228596045695",
    "0.01311953716497584361713026879228596045695"
  ],
  "a3": [
    "0.1827135485499046086099358681875839578990",
    "0.1827135485499046086099358681875839578990"
  ],
  "a4": [
    "0.01949555627932965409366346615232611522689",
    "0.01949555627932965409366346615232611522689"
  ],
  "Z4_remainder_coeff": [
    "3.936503244799014585001426179718284537655",
    "3.936503244799014585001426179718284537655"
  ],
  "Z4_printed": "3.937",
  "extras_printed": [
    "0.01320",
    "0.1828",
    "0.019500"
  ],
  "r_max": [
    "0.4705180144405142455387763635073145689990",
    "0.4705180144405142455387763635073145689990"
  ],
  "checks": {
    "leading_lt_printed": true,
    "a1_lt_printed": true,
    "Z4_coeff_lt_printed": true,
    "extras_ok": true,
    "r_max_lt_0.5": true,
    "beats_1.1185": true,
    "beats_1.1118": true,
    "beats_1.1057": true,
    "beats_1.1026": true,
    "beats_1.1035": true,
    "beats_1.108741_class_number": true
  },
  "inequalities": [
    {
      "beats": "q6 leading 1.1026, q5 1.1035, q4 1.1057, q3 1.1118, and HPS printed 1.1185",
      "range": "Z >= 4",
      "inequality": "N < 1.1021 Z + 3.851 Z^{1/3} + 0.01320 + 0.1828 Z^{-1/3} + 0.019500 Z^{-2/3}"
    },
    {
      "beats": "q6 simplified 1.1026 Z + 3.938 Z^{1/3}",
      "range": "Z >= 4",
      "inequality": "N_c < 1.1021 Z + 3.937 Z^{1/3}"
    }
  ],
  "LT_factor": "1.456",
  "note": "\u03b2_3 \u2265 compact \u03b3 on every radial probability, by the aspect-10 compact certificate plus the mass-stationary cut Q>10/11. Not the withdrawn 1.1168. The aspect-\u22644 number 1.1087 is not used as a class-only quote; the unrestricted leading is the compact \u03b3 at the split R."
}