files / problems / simon-ionization-excess / compute / q4 / certs / smallz.json
{
"not_a_new_bound": true,
"is_new_bound": false,
"status": "residue",
"record": [
"Lieb Phys. Rev. A 29 (1984): Nc < 2Z+1",
"Nam arXiv:1009.2367v3: Nc < 1.22 Z + 3 Z^{1/3}",
"HPS arXiv:2504.18487v1 Prop. 2.4\u20132.5 (printed remainders)",
"q1 remainder tightening 2.953 / 3.892 / 3.9781",
"q3 dent (Z>=4): N_c < 1.1118 Z + 3.966 Z^{1/3} via beta3 >= 0.899526"
],
"constants": {
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"b3": 1.1184337992027067,
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"q3_Z4_remainder": 3.966
},
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],
"q3_integer_exclusion_at_Z_ge_4": {
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"by_Z": [
{
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{
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{
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}
]
},
"finite_Z_try": {
"nam_lemma1_at_lieb_edge_and_Z2_candidates": [
{
"Z": 2,
"N": 4,
"alpha_lower": 0.5590169943749475,
"lhs_lower": 1.6770509831248424,
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"contradiction": false,
"margin": 0.8626653745443456
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{
"Z": 3,
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{
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{
"Z": 6,
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{
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{
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],
"he4_hylleraas_product": {
"separated_He_plus_2H": {
"trial": "He_Hylleraas_at_infinity_plus_2H_1s",
"Z": 2,
"N": 4,
"energy_upper": -3.8911170212765955,
"hylleraas_core_exact": -2.8911170212765955,
"hydrogenic_1s_exact": -0.5
},
"compact_attempt": {
"trial": "Hylleraas_1s2_times_exp(-zeta(r3+r4))^2_outer_1s_only",
"best_energy_if_repulsion_ignored": -5.738768755651595,
"best_zeta": 1.6853750000000003,
"valid_variational_upper": false,
"reason": "core\u2013outer repulsion omitted; energy too negative"
},
"q1_three_electron_hydrogenic_1s2_2s_upper": -7.289057803205487,
"separated_upper_above_N3_upper": true,
"proves_N4_nonbinding": false,
"note": "Separated He+2H gives a legal upper bound E(4,2) <= -54353/18800 - 1. Comparing to the q1 N=3 hydrogenic upper does not prove E(4,2) >= E(3,2). Variational uppers alone cannot prove non-binding."
}
},
"N0_minus_Z": {
"can_move_this_session": false,
"reason": "No verified integer envelope beats Lieb at Z=2..6. The q3 leading dent is Z>=4 and does not tighten integer caps. A leading coefficient >1 cannot prove a Z-independent excess bound on N0(Z)-Z."
},
"punch": "Best published integer bound at Z=2..6 remains Lieb Nc<=2Z. The q3 simplified envelope 1.1118 Z + 3.966 Z^{1/3} beats the printed HPS leading at Z>=4 as reals but excludes no extra integer N beyond Lieb/Nam/HPS printed. Nam Lemma 1 at the Lieb edge and a four-electron Hylleraas product trial for Z=2, N=4 do not yield a dent. N0(Z)-Z bounded cannot move here."
}