files / problems / simon-ionization-excess / compute / q2 / certs / s_gt_3.json
{
"arxiv": "2504.18487v1",
"urls_opened": [
"https://arxiv.org/abs/2504.18487",
"https://arxiv.org/html/2504.18487v1"
],
"dps": 80,
"definition": "I_s(\u03bd) = \u222c (|x|^s+|y|^s)/(2|x-y|) d\u03bd(x) d\u03bd(y), \u03bd orthogonal to radial functions",
"verdict": {
"I_s_goes_negative_for_s_gt_3": true,
"on_which_examples": [
"dipole_two_shell_s31",
"dipole_two_shell_s35",
"dipole_two_shell_s4"
],
"I_3_stays_nonnegative_on_all_examples": true,
"certified_path_to_s_gt_3": false,
"reason": "Lemma 4.3 needs I_s(\u03bd)\u22650 for every \u03bd \u22a5 radial. Two-shell \u2113=1 measures give I_s<0 for every s>3 (interval enclosure of the closed quadratic form; s=4 is the exact rational \u22121025/2048). Hardy is applied to u=|x|^{s/2} V, not to V. Q_H(V) stays near 4.5 and does not control the sign of I_s. For the s=3.1 dipole, Q_H(u) < (3.1)\u00b2/4 already, so there is no uniform gap past 9/4. Lemma 5.9 (\u03b1_N vs \u03b2, 2\u2264s\u22644) does not replace radialization. A 3D search is only an upper bound on \u03b2_s.",
"b4": "1.083024917507624435206881756796524210931",
"b4_interval": [
"1.083024917507624435206881756796524210931",
"1.083024917507624435206881756796524210931"
],
"b4_lt_1.1185": true,
"cannot_use_b4_in_Theorem_2_2": true
},
"b_s": {
"3.00000": {
"s": "3.000000000000000000000000000000000000000",
"t0": "0.5960716379833215231128054143996818281133",
"b": "1.118433799202706636641762578621416983973",
"b_inv": "0.8941074569749822846692081215995227421700",
"interval_b": [
"1.118433799202706636641762578621416983973",
"1.118433799202706636641762578621416983973"
],
"closed_form": "(s-1)/(s t0) with t0^s + s t0 + 1 - s = 0"
},
"3.10000": {
"s": "3.100000000000000000000000000000000000000",
"t0": "0.6083230819154380588217237554522006962168",
"b": "1.113584828485723269305085172419647046252",
"b_inv": "0.8980007399704085630225445913818200753677",
"interval_b": [
"1.113584828485723269305085172419647046252",
"1.113584828485723269305085172419647046252"
],
"closed_form": "(s-1)/(s t0) with t0^s + s t0 + 1 - s = 0"
},
"3.50000": {
"s": "3.500000000000000000000000000000000000000",
"t0": "0.6507648650218449293031083758351068797090",
"b": "1.097609524850019497044479594554830684084",
"b_inv": "0.9110708110305829010243517261691496315925",
"interval_b": [
"1.097609524850019497044479594554830684084",
"1.097609524850019497044479594554830684084"
],
"closed_form": "(s-1)/(s t0) with t0^s + s t0 + 1 - s = 0"
},
"4.00000": {
"s": "4.000000000000000000000000000000000000000",
"t0": "0.6925048425718423433201934034003832859846",
"b": "1.083024917507624435206881756796524210931",
"b_inv": "0.9233397900957897910935912045338443813128",
"interval_b": [
"1.083024917507624435206881756796524210931",
"1.083024917507624435206881756796524210931"
],
"closed_form": "(s-1)/(s t0) with t0^s + s t0 + 1 - s = 0"
}
},
"s3_dipole_discriminant": {
"identity": "(t^4+t)^2 - 4 t^2 = t^2 (t^3-1)(t^3+3)",
"sign_on_(0,1)": "<= 0, =0 only at endpoints of the closed interval",
"conclusion": "Q_s=3,\u2113=1 (\u03b1,\u03b2) \u2265 0 for all t\u2208(0,1] and all real \u03b1,\u03b2"
},
"exact_negative_Q": {
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"Q": "-1025/2048",
"Q_value": "-0.5004882812500000000000000000000000000000",
"sign": "negative"
},
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"Q": "1/sqrt(2) - 1 - 1/(65536 sqrt(2))",
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}
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},
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{
"name": "dipole_two_shell_s3_inner_heavy",
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},
{
"name": "dipole_two_shell_s31",
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{
"name": "dipole_two_shell_s35",
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{
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"orthogonal_to_radial": true
},
{
"name": "quad_two_shell_s4_opposite",
"s": "4",
"ell": 2,
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},
{
"name": "quad_two_shell_s35_opposite",
"s": "3.5",
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{
"name": "quad_two_shell_s3_opposite",
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{
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{
"name": "dipole_two_shell_s4_same_sign",
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"note": "\u03bd = \u03b3_\u03bc \u2212 \u03b3_{\u2212\u03bc}, same |center|",
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{
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"note": "\u03bd = \u03b3_\u03bc \u2212 \u03b3_{\u2212\u03bc}, same |center|",
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{
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"note": "\u03bd = \u03b3_\u03bc \u2212 \u03b3_{\u2212\u03bc}, same |center|",
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},
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"note": "Unregularized \u222b |x|^{s-2} V\u00b2 d\u00b3x diverges for a dipole tail when s\u22653. I_s itself stays finite (compact support). HPS regularize the weight; the informal IMS+Hardy bound is not available without that cutoff."
},
{
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"note": "Unregularized \u222b |x|^{s-2} V\u00b2 d\u00b3x diverges for a dipole tail when s\u22653. I_s itself stays finite (compact support). HPS regularize the weight; the informal IMS+Hardy bound is not available without that cutoff."
}
],
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],
"pointwise_minorant_half_b_inv": "0.4555354055152914505121758630845748157963",
"note": "two_shell_modulated and nam-like are upper bounds on \u03b2_s. They are not lower bounds. The pointwise minorant is a genuine lower bound but only (1/2) b(s)^{-1}."
},
"4.000": {
"b_inv": "0.9233397900957897910935912045338443813128",
"two_shell_modulated_best_ratio": "0.9721168836640964058483967471453250747904",
"at_t_w": [
"0.6666666666666666666666666666666666666667",
"0.6500000000000000000000000000000000000000"
],
"beats_b_inv": false,
"radial_nam_like_best": "0.9497680167605484937783067962186920530397",
"radial_nam_like_at_p_nhi": [
"1.200000000000000000000000000000000000000",
"4.000000000000000000000000000000000000000"
],
"pointwise_minorant_half_b_inv": "0.4616698950478948955467956022669221906564",
"note": "two_shell_modulated and nam-like are upper bounds on \u03b2_s. They are not lower bounds. The pointwise minorant is a genuine lower bound but only (1/2) b(s)^{-1}."
}
},
"pairwise_check_s4_dipole": {
"s": 4,
"t": "1/8",
"alpha": 16,
"beta": -1,
"n_fibonacci": 180,
"pairwise_I_s": -0.09638642199793518,
"closed_Q": "-0.5004882812500000000000000000000000000000",
"same_sign": true,
"note": "Fibonacci-sphere pairwise sum omits the same-shell diagonal. The missing self-energy is positive, so a negative pairwise sum is even stronger evidence that I_s < 0."
},
"pairwise_one_shell_s4": 0.08228344966815743,
"improved_Hardy": {
"citation": "Ekholm\u2013Frank, Comm. Math. Phys. 264 (2006), Lemma 2.4; HPS (4.15)",
"c_H_radial": "1/4 = ((d-2)/2)^2",
"c_H_ell_ge_1": "9/4 = d^2/4",
"c_H_pure_ell": "\u2113(\u2113+1)+1/4, so s \u2264 2\u2113+1 is the IMS+Hardy range",
"dipole_ell_1_threshold": 3,
"quadrupole_ell_2_threshold": 5
}
}