files / problems / simon-ionization-excess / compute / q5 / certs / leading.json

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{
  "arxiv": "2504.18487v1",
  "beta3_lower": "0.9062382172832278",
  "source": "q5 compact R=10 n=30 target=0.9103, lifted by mass-opt dichotomy",
  "row": "certs/raise_R10_n30_t0p9103.json",
  "R_split": 10,
  "b_interval": [
    "1.103462622662125834846772916963824281788",
    "1.103462622662125834846772916963824281788"
  ],
  "b_hi": "1.103462622662125834846772916963824281788",
  "printed_leading": "1.1035",
  "a1_at_left": [
    "3.854413255310619783235097560758835153791",
    "3.854413255310619783235097560758835153791"
  ],
  "a1_at_9/4": [
    "3.829006477723663152886311327103666454330",
    "3.829006477723663152886311327103666454330"
  ],
  "a1_max_hi_on_interval": "3.854413255310619783235097560758835153791",
  "a1_printed": "3.855",
  "a2": [
    "0.01313645979359673612912824901147409859271",
    "0.01313645979359673612912824901147409859271"
  ],
  "a3": [
    "0.1827920743140776952827075237138994736643",
    "0.1827920743140776952827075237138994736643"
  ],
  "a4": [
    "0.01950393498724137550236234513670132971735",
    "0.01950393498724137550236234513670132971735"
  ],
  "Z4_remainder_coeff": [
    "3.940105772929359278400521551694153112844",
    "3.940105772929359278400521551694153112844"
  ],
  "Z4_printed": "3.941",
  "extras_printed": [
    "0.01320",
    "0.1828",
    "0.019510"
  ],
  "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_1.108741_class_number": true
  },
  "inequalities": [
    {
      "beats": "q4 leading 1.1057, q3 1.1118, and HPS printed 1.1185",
      "range": "Z >= 4",
      "inequality": "N < 1.1035 Z + 3.855 Z^{1/3} + 0.01320 + 0.1828 Z^{-1/3} + 0.019510 Z^{-2/3}"
    },
    {
      "beats": "q4 simplified 1.1057 Z + 3.946 Z^{1/3}",
      "range": "Z >= 4",
      "inequality": "N_c < 1.1035 Z + 3.941 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 Not the withdrawn 1.1168.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."
}