files / problems / simon-ionization-excess / compute / q1 / certs / hps_tight.json

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{
  "status": "dent",
  "reason": "interval-certified improvements of the printed HPS remainders 2.96, 3.90, and 4, using the same Section 7 chain",
  "arxiv": "2504.18487v1",
  "theorem_unchanged": true,
  "did_not_beat_1.1185": true,
  "LT_factor": "1.456",
  "LT_source": "Frank\u2013Hundertmark\u2013Jex\u2013Nam arXiv:1808.09017 Theorem 1",
  "later_LT_lead_not_used": {
    "arxiv": "2403.04347",
    "claimed": "L_{1,1,1}/L^{cl} <= 1.44655",
    "why_not_used": "not independently replayed"
  },
  "tightenings": {
    "1_exact_b3_in_a1": true,
    "2_Lieb_N/Z_lt_2+1/Z_for_Z_ge_4": true,
    "3_actual_a1_function": true,
    "4_optimize_lambda": {
      "best_lambda": "0.7465500000000000000000000000000000000000",
      "best_max_a1": "3.891856413153097230002275417045379336481",
      "hps_lambda": "0.7469007910928607847912470023397632451945",
      "hps_max_a1": "3.891855917536407862423677607764617923966",
      "moved_a1": false
    },
    "5_interval_enclosures": true
  },
  "why_lambda_did_not_move_a1": "For Z>=4 the allowed x-interval is [b(3), 9/4]. a1(x) has a minimum inside and the larger endpoint is x=b(3). At that point Z = N \u03b23, so the crude replacement Z < N \u03b23 is equality, and a1 = (\u03bb^{-1}+6\u03bb\u00b2/5) \u03b23^{-1} + c \u03b23^{-1/3}, which HPS already minimises by \u03bb=(5/12)^{1/3}.",
  "HPS_5/2_interval_replay": {
    "a1_at_left": [
      "3.891855917536407862423677607764617923966",
      "3.891855917536407862423677607764617923966"
    ],
    "a1_at_5/2": [
      "3.899495157762337995751372241432359472817",
      "3.899495157762337995751372241432359472817"
    ],
    "sup_is_at": "x=5/2",
    "3.893_valid_on_5/2_interval": false,
    "3.90_valid_on_5/2_interval": true,
    "note": "HPS claimed the sup on [\u03b23^{-1}, 5/2] is at the left. Independently the sup is at 5/2 and is < 3.90 but not < 3.893. Prop. 2.5 is only stated for Z>=4, where Lieb gives 9/4 not 5/2."
  },
  "intermediates": {
    "b2_interval": [
      "1.207106781186547524400844362104849039285",
      "1.207106781186547524400844362104849039285"
    ],
    "b3_interval": [
      "1.118433799202706636641762578621416983973",
      "1.118433799202706636641762578621416983973"
    ],
    "kappa_interval": [
      "0.7156511510061795178815821763078492748336",
      "0.7156511510061795178815821763078492748336"
    ],
    "C1_inv_interval": [
      "2.341311542369653845094875615140092825333",
      "2.341311542369653845094875615140092825333"
    ],
    "C2_inv_sqrt_interval": [
      "2.215376835236356794199019678595406726648",
      "2.215376835236356794199019678595406726648"
    ],
    "c_interval": [
      "1.585436982049326057582984553635934279879",
      "1.585436982049326057582984553635934279879"
    ],
    "lambda_s2_interval": [
      "0.6283358625603360818460567972501173627912",
      "0.6283358625603360818460567972501173627912"
    ],
    "lambda_s3": "0.7469007910928607847912470023397632451945",
    "a_s2_at_5/2": [
      "2.952038354343948425034033355200872450699",
      "2.952038354343948425034033355200872450699"
    ],
    "a1_at_b3": [
      "3.891855917536407862423677607764617923966",
      "3.891855917536407862423677607764617923966"
    ],
    "a1_exact_x_at_b3": [
      "3.891855917536407862423677607764617923966",
      "3.891855917536407862423677607764617923966"
    ],
    "a1_at_9/4": [
      "3.868190406136808397861494672915143824966",
      "3.868190406136808397861494672915143824966"
    ],
    "a1_at_2": [
      "3.843380631427884734180282404371191047149",
      "3.843380631427884734180282404371191047149"
    ],
    "a2": [
      "0.01331468808574650757906860212644544028539",
      "0.01331468808574650757906860212644544028539"
    ],
    "a3": [
      "0.1836150378723424123136965427940548227968",
      "0.1836150378723424123136965427940548227968"
    ],
    "a4": [
      "0.01959174529191402018517833493881986367727",
      "0.01959174529191402018517833493881986367727"
    ],
    "Z4_remainder_coeff": [
      "3.978009257824328899981904022527038575515",
      "3.978009257824328899981904022527038575515"
    ],
    "r_max_on_contradiction_set": [
      "0.4705180144405142455387763635073145689990",
      "0.4705180144405142455387763635073145689990"
    ],
    "r_max_lt_0.5": true
  },
  "enclosures": {
    "a_s2_lt_2.953": true,
    "a1_lt_3.892": true,
    "a1_lt_3.893": true,
    "a2_lt_0.01332": true,
    "a2_lt_0.0134": true,
    "a3_lt_0.18362": true,
    "a3_lt_0.184": true,
    "a4_lt_0.01960": true,
    "Z4_coeff_lt_3.9781": true,
    "Z4_coeff_lt_3.979": true
  },
  "inequalities": [
    {
      "beats": "HPS Prop. 2.4 printed remainder 2.96",
      "range": "Z >= 2",
      "inequality": "N_c < b(2) Z + 2.953 Z^{1/3}",
      "b2": "(sqrt(2)+1)/2"
    },
    {
      "beats": "HPS Prop. 2.5 printed remainder 3.90",
      "range": "Z >= 4",
      "inequality": "N < b(3) Z + 3.892 Z^{1/3} + 0.0134 + 0.184 Z^{-1/3} + 0.0196 Z^{-2/3}",
      "b3": "(2/3)(1+sqrt(2))^{1/3}/((1+sqrt(2))^{2/3}-1)",
      "note": "same extras as the printed (2.9); only a1 is tightened. Uses Lieb N/Z < 2+1/Z <= 9/4 so the a1 max is at x=b(3)."
    },
    {
      "beats": "HPS simplified printed remainder 4",
      "range": "Z >= 4",
      "inequality": "N_c < b(3) Z + 3.9781 Z^{1/3}"
    },
    {
      "beats": "HPS simplified printed remainder 4 (with 1.1185 kept)",
      "range": "Z >= 4",
      "inequality": "N_c < 1.1185 Z + 3.9781 Z^{1/3}",
      "note": "1.1185 is still only an enclosure of b(3), not a new leading coefficient"
    }
  ]
}