files / problems / simon-ionization-excess / compute / q4 / certs / leading.json

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{
  "arxiv": "2504.18487v1",
  "beta3_lower": "0.9044140893458346",
  "source": "q4 compact R=10 n=26 target=0.9091, lifted by mass-opt dichotomy",
  "row": "certs/raise_R10_n26_t0p9091.json",
  "R_split": 10,
  "b_interval": [
    "1.105688214922992823332048674155594982768",
    "1.105688214922992823332048674155594982768"
  ],
  "b_hi": "1.105688214922992823332048674155594982768",
  "printed_leading": "1.1057",
  "a1_at_left": [
    "3.859983628776694008616046452564538695087",
    "3.859983628776694008616046452564538695087"
  ],
  "a1_at_9/4": [
    "3.834838724784706004917969729046135916540",
    "3.834838724784706004917969729046135916540"
  ],
  "a1_max_hi_on_interval": "3.859983628776694008616046452564538695087",
  "a1_printed": "3.860",
  "a2": [
    "0.01316295493955943837300057945423327360438",
    "0.01316295493955943837300057945423327360438"
  ],
  "a3": [
    "0.1829148839183991026859093638567910576153",
    "0.1829148839183991026859093638567910576153"
  ],
  "a4": [
    "0.01951703878590158413149404495757330830421",
    "0.01951703878590158413149404495757330830421"
  ],
  "Z4_remainder_coeff": [
    "3.945744850264930678911309873235527742144",
    "3.945744850264930678911309873235527742144"
  ],
  "Z4_printed": "3.946",
  "extras_printed": [
    "0.01320",
    "0.1830",
    "0.019520"
  ],
  "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_1.108741_class_number": true
  },
  "inequalities": [
    {
      "beats": "q3 leading 1.1118 and HPS printed 1.1185",
      "range": "Z >= 4",
      "inequality": "N < 1.1057 Z + 3.860 Z^{1/3} + 0.01320 + 0.1830 Z^{-1/3} + 0.019520 Z^{-2/3}"
    },
    {
      "beats": "q3 simplified 1.1118 Z + 3.966 Z^{1/3}",
      "range": "Z >= 4",
      "inequality": "N_c < 1.1057 Z + 3.946 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 Not the withdrawn 1.1168.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."
}