files / problems / simon-ionization-excess / compute / q2 / certs / beta3_compact.json

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{
  "arxiv": "2504.18487v1",
  "status": "residue",
  "is_new_bound": false,
  "beats_1.1185_in_HPS_theorem": false,
  "reason": "Certified Q \u2265 compact_gamma for D-measures of aspect \u2264 R. HPS \u03b2_3 is an inf over all radial probabilities. The h(D_L,D_R) lift is withdrawn, so this does not replace b(3)<1.1185 in Theorem 2.2.",
  "compact_best_gamma": "0.9019242856410750002801111492108644464576",
  "compact_best_inv": "1.108740518378673185266124092640237288452",
  "compact_best_row": {
    "R": 4,
    "n": 18,
    "phi_target": 0.906,
    "slsqp_phi": 0.9104292161366282,
    "eig_min_M": -0.1200591542894005,
    "certified_phi": 0.906,
    "q_hi": "1.080059738892306169872930831288596912737",
    "P_max_hi": "0.03848915365043331298663978452326696210185",
    "err_P_hi": "0.004075714358925027253419861493017756048449",
    "compact_gamma": "0.9019242856410750002801111492108644464576",
    "compact_inv": "1.108740518378673185266124092640237288452",
    "beats_1.1185_on_aspect_le_R": true,
    "f_at_1_over_R": 0.9558823529411765,
    "separated_half_record_only": 0.9249949049580795,
    "faces": {
      "n": 18,
      "gamma_target": 0.906,
      "n_faces": 262143,
      "interior_critical": 56937,
      "singular_or_illconditioned": 0,
      "min_mMm": 0.01280935493138805,
      "min_mMm_safe": 0.01280935483138805,
      "min_phi": 0.9109739000464696,
      "min_phi_safe": 0.9109738999464696,
      "copositive": 1
    }
  },
  "configs": [
    {
      "R": 4,
      "n": 18,
      "phi_target": 0.906,
      "slsqp_phi": 0.9104292161366282,
      "eig_min_M": -0.1200591542894005,
      "certified_phi": 0.906,
      "q_hi": "1.080059738892306169872930831288596912737",
      "P_max_hi": "0.03848915365043331298663978452326696210185",
      "err_P_hi": "0.004075714358925027253419861493017756048449",
      "compact_gamma": "0.9019242856410750002801111492108644464576",
      "compact_inv": "1.108740518378673185266124092640237288452",
      "beats_1.1185_on_aspect_le_R": true,
      "f_at_1_over_R": 0.9558823529411765,
      "separated_half_record_only": 0.9249949049580795,
      "faces": {
        "n": 18,
        "gamma_target": 0.906,
        "n_faces": 262143,
        "interior_critical": 56937,
        "singular_or_illconditioned": 0,
        "min_mMm": 0.01280935493138805,
        "min_mMm_safe": 0.01280935483138805,
        "min_phi": 0.9109739000464696,
        "min_phi_safe": 0.9109738999464696,
        "copositive": 1
      }
    },
    {
      "R": 6,
      "n": 20,
      "phi_target": 0.9065,
      "slsqp_phi": 0.9091200307038747,
      "eig_min_M": -2.064538025527184,
      "certified_phi": 0.9065,
      "q_hi": "1.093723547726442383253407548345890653067",
      "P_max_hi": "0.04476405102680152421355974930124238818339",
      "err_P_hi": "0.004740179199329668840891479354261955420065",
      "compact_gamma": "0.9017598208006703036255775099418558420739",
      "compact_inv": "1.108942732791202080208775060810951593457",
      "beats_1.1185_on_aspect_le_R": true,
      "f_at_1_over_R": 0.9774774774774775,
      "separated_half_record_only": 0.9357924672262299,
      "faces": {
        "n": 20,
        "gamma_target": 0.9065,
        "n_faces": 1048575,
        "interior_critical": 49922,
        "singular_or_illconditioned": 0,
        "min_mMm": 0.007952095055300906,
        "min_mMm_safe": 0.007952094955300906,
        "min_phi": 0.909357805842676,
        "min_phi_safe": 0.909357805742676,
        "copositive": 1
      }
    },
    {
      "R": 8,
      "n": 20,
      "phi_target": 0.906,
      "slsqp_phi": 0.9077332752177658,
      "eig_min_M": -5.626508194781405,
      "certified_phi": 0.906,
      "q_hi": "1.109569472067845008489378357730636666456",
      "P_max_hi": "0.05193925752084508223389033495286470150499",
      "err_P_hi": "0.005499980061713562825939961735806120443292",
      "compact_gamma": "0.9005000199382864647075910489680760820628",
      "compact_inv": "1.110494145317767477261223778195233700751",
      "beats_1.1185_on_aspect_le_R": true,
      "f_at_1_over_R": 0.9865384615384616,
      "separated_half_record_only": 0.9403229592567219,
      "faces": {
        "n": 20,
        "gamma_target": 0.906,
        "n_faces": 1048575,
        "interior_critical": 22150,
        "singular_or_illconditioned": 0,
        "min_mMm": 0.005389159590705357,
        "min_mMm_safe": 0.0053891594907053576,
        "min_phi": 0.9078588916428175,
        "min_phi_safe": 0.9078588915428175,
        "copositive": 1
      }
    },
    {
      "R": 12,
      "n": 22,
      "phi_target": 0.9055,
      "slsqp_phi": 0.9069245919372396,
      "eig_min_M": -18.585978872027002,
      "certified_phi": 0.9055,
      "q_hi": "1.119576291157994294788606304264703060665",
      "P_max_hi": "0.05641518621283777005011851944016164545797",
      "err_P_hi": "0.005973947533307309786062677564920366542882",
      "compact_gamma": "0.8995260524666926617922278920310721986121",
      "compact_inv": "1.111696539814256966877569616043574522930",
      "beats_1.1185_on_aspect_le_R": true,
      "f_at_1_over_R": 0.9936781609195403,
      "separated_half_record_only": 0.9438928089472614,
      "faces": {
        "n": 22,
        "gamma_target": 0.9055,
        "n_faces": 4194303,
        "interior_critical": 17703,
        "singular_or_illconditioned": 5,
        "min_mMm": 0.00447959230442579,
        "min_mMm_safe": 0.004479592204425791,
        "min_phi": 0.907036714315739,
        "min_phi_safe": 0.907036714215739,
        "copositive": 1
      }
    }
  ],
  "P_max_formula": "(q-1)/(q+1)",
  "withdrawn": {
    "old_gamma_inv": "1.116823910980300911670485397084439427267",
    "why": "h(0,1)=1-(1/12)(1-fmin)\u22480.991 > power-law I/D\u22480.921 on a measure that sits in a fixed-window tail."
  },
  "HPS_fmin": "0.8941074569749822846692081215995227421700",
  "HPS_b3": "1.118433799202706636641762578621416983973"
}