files / problems / simon-ionization-excess / compute / q9 / certs / leading.json

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{
  "arxiv": "2504.18487v1",
  "beta3_lower": "0.9077156846635223",
  "source": "q9 compact R=10 n=34 target=0.9113, lifted by mass-opt dichotomy",
  "row": "certs/raise_R10_n34_t0p9113.json",
  "R_split": 10,
  "b_interval": [
    "1.101666542614261673164542435977138547969",
    "1.101666542614261673164542435977138547969"
  ],
  "b_hi": "1.101666542614261673164542435977138547969",
  "printed_leading": "1.1017",
  "a1_at_left": [
    "3.849916816462881110086962578403136156356",
    "3.849916816462881110086962578403136156356"
  ],
  "a1_at_9/4": [
    "3.824297931403986100996742715628395389119",
    "3.824297931403986100996742715628395389119"
  ],
  "a1_max_hi_on_interval": "3.849916816462881110086962578403136156356",
  "a1_printed": "3.850",
  "a2": [
    "0.01311507788826501991862550519020403033297",
    "0.01311507788826501991862550519020403033297"
  ],
  "a3": [
    "0.1826928450095448204656923963053306034147",
    "0.1826928450095448204656923963053306034147"
  ],
  "a4": [
    "0.01949334720923348998844437494820652295758",
    "0.01949334720923348998844437494820652295758"
  ],
  "Z4_remainder_coeff": [
    "3.935553838205211099959529654902135127557",
    "3.935553838205211099959529654902135127557"
  ],
  "Z4_printed": "3.936",
  "extras_printed": [
    "0.01320",
    "0.1827",
    "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_1.1020": true,
    "beats_1.1021": true,
    "beats_1.1026": true,
    "beats_1.1035": true,
    "beats_1.108741_class_number": true
  },
  "inequalities": [
    {
      "beats": "q8 leading 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 < 1.1017 Z + 3.850 Z^{1/3} + 0.01320 + 0.1827 Z^{-1/3} + 0.019500 Z^{-2/3}"
    },
    {
      "beats": "q8 simplified 1.1020 Z + 3.937 Z^{1/3}",
      "range": "Z >= 4",
      "inequality": "N_c < 1.1017 Z + 3.936 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."
}