files / problems / simon-ionization-excess / compute / q13 / certs / leading.json

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{
  "arxiv": "2504.18487v1",
  "beta3_lower": "0.9086061090539742",
  "source": "q13 compact R=10 n=37 target=0.9119, lifted by mass-opt dichotomy",
  "row": "certs/raise_R10_n37_t0p9119.json",
  "R_split": 10,
  "b_interval": [
    "1.100586921037966191361489740098386991442",
    "1.100586921037966191361489740098386991442"
  ],
  "b_hi": "1.100586921037966191361489740098386991442",
  "printed_leading": "1.1006",
  "a1_at_left": [
    "3.847213547185210982221867667241172117546",
    "3.847213547185210982221867667241172117546"
  ],
  "a1_at_9/4": [
    "3.821466832189506466727989903205785790237",
    "3.821466832189506466727989903205785790237"
  ],
  "a1_max_hi_on_interval": "3.847213547185210982221867667241172117546",
  "a1_printed": "3.848",
  "a2": [
    "0.01310222525045197846858916357259984513621",
    "0.01310222525045197846858916357259984513621"
  ],
  "a3": [
    "0.1826331464844667572856636334383632803897",
    "0.1826331464844667572856636334383632803897"
  ],
  "a4": [
    "0.01948697736986093536304421962317788751862",
    "0.01948697736986093536304421962317788751862"
  ],
  "Z4_remainder_coeff": [
    "3.932817188437856703715530365867203964984",
    "3.932817188437856703715530365867203964984"
  ],
  "Z4_printed": "3.933",
  "extras_printed": [
    "0.01320",
    "0.1827",
    "0.019490"
  ],
  "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_1.1010": true,
    "beats_1.1013": true,
    "beats_1.1017": true,
    "beats_1.1020": true,
    "beats_1.1021": true,
    "beats_1.1026": true,
    "beats_1.1035": true,
    "beats_1.108741_class_number": true
  },
  "inequalities": [
    {
      "beats": "q11 leading 1.1010, q10 1.1013, q9 1.1017, q8 1.1020, q7 1.1021, q6 1.1026, q5 1.1035, q4 1.1057, q3 1.1118, and HPS printed 1.1185",
      "range": "Z >= 4",
      "inequality": "N < 1.1006 Z + 3.848 Z^{1/3} + 0.01320 + 0.1827 Z^{-1/3} + 0.019490 Z^{-2/3}"
    },
    {
      "beats": "q11 simplified 1.1010 Z + 3.934 Z^{1/3}",
      "range": "Z >= 4",
      "inequality": "N_c < 1.1006 Z + 3.933 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."
}