files / problems / simon-ionization-excess / compute / q1 / certs / hps_replay.json
{
"arxiv": "2504.18487v1",
"urls_opened": [
"https://arxiv.org/abs/2504.18487",
"https://arxiv.org/html/2504.18487v1",
"https://arxiv.org/abs/1808.09017",
"https://arxiv.org/html/1808.09017"
],
"dps": 80,
"not_a_new_bound": true,
"note": "Replay of HPS Section 7 arithmetic. The printed 3.90 is a valid upper enclosure of a1 on the paper's interval [\u03b23^{-1}, 5/2], but the max is at x=5/2 (value < 3.90), not at the left endpoint. The paper's 3.893 is a valid enclosure of the left-endpoint formula only, not of the supremum on [\u03b23^{-1}, 5/2].",
"b2": {
"closed_form": "(sqrt(2)+1)/2",
"beta2": "2*(sqrt(2)-1)",
"value": "1.207106781186547524400844362104849039285",
"beta2_value": "0.8284271247461900976033774484193961571393",
"numerical_max": "1.207106781186547524400844362104849039285",
"t_star_closed": "sqrt(2)-1",
"t_star_numerical": "0.4142135623730950488016887242112509630654",
"t_star_closed_value": "0.4142135623730950488016887242096980785697",
"interval_enclosure": [
"1.207106781186547524400844362104849039285",
"1.207106781186547524400844362104849039285"
],
"printed_interval": [
"1.2071",
"1.2072"
],
"printed_interval_valid": true
},
"b3": {
"closed_form": "(2/3)*(1+sqrt(2))^{1/3} / ((1+sqrt(2))^{2/3}-1)",
"beta3": "(3/2)*((1+sqrt(2))^{2/3}-1)/(1+sqrt(2))^{1/3}",
"value": "1.118433799202706636641762578621416983973",
"beta3_value": "0.8941074569749822846692081215995227421700",
"numerical_max": "1.118433799202706636641762578621416983973",
"t_star_closed": "(1+sqrt(2))^{1/3} - (1+sqrt(2))^{-1/3}",
"t_star_numerical": "0.5960716379833215231128054144019321157153",
"t_star_closed_value": "0.5960716379833215231128054143996818281133",
"interval_enclosure": [
"1.118433799202706636641762578621416983973",
"1.118433799202706636641762578621416983973"
],
"printed_interval": [
"1.1184",
"1.1185"
],
"printed_interval_valid": true,
"1.1185_is_valid_upper": true
},
"C_p": {
"formula_A18": "(3\u221a\u03c0/4) * (4\u03c0)^{-p/3} / p^{1+p/2} * [(15\u221a\u03c0/8 * \u0393(3/p)/\u0393(7/2+3/p))^{p/2} * \u0393(3/p+1)/\u0393(3/p+7/2)] / [(\u221a\u03c0/4 * \u0393(3/p+1)/\u0393(3/p+5/2))^{1+5p/6}]",
"C1_A18": "0.4271110366576396277881780248054296960346",
"C1_inv_A18": "2.341311542369653845094875615140092825333",
"C1_closed_Lemma64": "0.4271110366576396277881780248054296960346",
"C1_closed_formula": "3^{5/3} 5^{5/6} (7/\u03c0)^{1/3} / (22 \u221a11)",
"C1_from_radial_quadrature": "0.4271110366576396277881780248054296960346",
"C1_inv_printed": "2.341...",
"C2_A18": "0.2037533601214680925922212957126387696621",
"C2_inv_sqrt_A18": "2.215376835236356794199019678595406726648",
"C2_inv_sqrt_closed": "2.215376835236356794199019678595406726648",
"C2_inv_sqrt_formula": "4 \u03c0^{2/3} / \u221a15",
"C2_from_radial_quadrature": "0.2037533601214680925922212957126387696621",
"C2_inv_sqrt_printed": "2.215..."
},
"kappa": {
"formula": "\u221a5 * (2/(9\u03c0\u00b2) * 1.456)^{1/3}",
"LT_factor": "1.456",
"LT_source": "Frank\u2013Hundertmark\u2013Jex\u2013Nam arXiv:1808.09017 Theorem 1",
"value": "0.7156511510061795178815821763078492748336",
"interval_enclosure": [
"0.7156511510061795178815821763078492748336",
"0.7156511510061795178815821763078492748336"
]
},
"prop_2_4": {
"lambda_formula": "(3/8) C1^{-1} \u03ba",
"lambda": "0.6283358625603360818460567972501173627912",
"lambda_printed": "0.6284",
"a_formula": "\u03b22^{-1} \u03bb (N/Z)^{-2/3} + \u03b22^{-1} (9/2 \u03b22)^{1/3} (N/Z)^{1/3}",
"a_at_N/Z=1": "2.630161544855095512489556755871369394617",
"a_at_N/Z=5/2": "2.952038354343948425034033355200872450699",
"a_at_N/Z=5/2_interval": [
"2.952038354343948425034033355200872450699",
"2.952038354343948425034033355200872450699"
],
"grid_max_on_[1,5/2]": "2.952038354343948425034033355200872450699",
"grid_max_at": "2.500000000000000000000000000000000000000",
"max_is_at_right_endpoint": true,
"printed_2.953_valid": true,
"printed_2.96_valid": true
},
"prop_2_5": {
"c_formula": "\u03ba * C2^{-1/2}",
"c": "1.585436982049326057582984553635934279879",
"c_interval": [
"1.585436982049326057582984553635934279879",
"1.585436982049326057582984553635934279879"
],
"c_printed_upper": "1.5855",
"c_lt_1.5855": true,
"lambda": "0.7469007910928607847912470023397632451945",
"lambda_formula": "(5/12)^{1/3}",
"a1_left_endpoint_formula": "3 (3/10)^{1/3} \u03b23^{-1} + c \u03b23^{-1/3}",
"a1_left_endpoint": "3.891855917536407862423677607764617923966",
"a1_left_endpoint_interval": [
"3.891855917536407862423677607764617923966",
"3.891855917536407862423677607764617923966"
],
"a1_at_5/2": "3.899495157762337995751372241432359472817",
"a1_at_5/2_interval": [
"3.899495157762337995751372241432359472817",
"3.899495157762337995751372241432359472817"
],
"a1_at_9/4": "3.868190406136808397861494672915143824966",
"a1_at_9/4_interval": [
"3.868190406136808397861494672915143824966",
"3.868190406136808397861494672915143824966"
],
"a1_critical_point_is_a_minimum": true,
"a1_critical_x": "1.638906086431086191436146167164845039263",
"a1_at_critical": "3.826876761905421574375018707315347997183",
"grid_max_on_[b3,5/2]": "3.899495157762337995751372241432359472817",
"grid_max_at": "2.500000000000000000000000000000000000000",
"HPS_says_sup_at_x=beta3^{-1}": true,
"independent_sup_is_at": "x=5/2",
"printed_3.893_valid_on_HPS_interval_[b3,5/2]": false,
"printed_3.893_valid_on_left_endpoint_only": true,
"printed_3.90_valid": true,
"a2_formula": "\u03b23^{-1}/84",
"a2": "0.01331468808574650757906860212644544028539",
"a3_formula": "(c/5) (5/12)^{2/3} \u03b23^{-1/3}",
"a3": "0.1836150378723424123136965427940548227968",
"a4_formula": "c \u03b23^{-1/3}/84",
"a4": "0.01959174529191402018517833493881986367727",
"printed_a2_0.0134_valid": true,
"printed_a3_0.184_valid": true,
"printed_a4_0.0196_valid": true
},
"printed_enclosures": {
"b2_in_(1.2071,1.2072)": true,
"b3_in_(1.1184,1.1185)": true,
"1.1185_is_valid_upper_on_b3": true,
"2.96_is_valid_upper_on_a_s2": true,
"2.953_is_valid_upper_on_a_s2": true,
"c_lt_1.5855": true,
"3.90_is_valid_upper_on_a1_over_[b3,5/2]": true,
"3.893_is_valid_upper_on_a1_over_[b3,5/2]": false,
"3.893_is_valid_upper_on_left_endpoint_formula": true,
"HPS_claim_sup_of_a1_at_left_on_[b3,5/2]": false,
"a2_le_0.0134": true,
"a3_le_0.184": true,
"a4_le_0.0196": true,
"sup_of_a1_on_[b3,5/2]_is_at": "x=5/2"
}
}