How these numbers were read
Where each number below was read, and how it was checked.
-1.1061269, ED
Checked 2026-09-16 · recomputed: Lanczos ED over the full fixed-(N_up, N_dn) space by two independent codes (checks/hubbard-u-labels/)
Reported as. -17.698030018702013 (ed_check.py) | -17.69803001870263 (ed_lib.mjs)
Reproduced at the grid U the instance now carries, not at the U of VarBench's name (relabels.mjs).
-1.105848(3), VMC (HFDS)
Checked 2026-09-29 · PNAS 119(32), e2122059119 (2022), SI Appendix PDF (PMC9371695; sources/pnas.2122059119.sapp.PMC9371695.pdf, sha256 c6e62d9c...), pypdf layout mode, re-extracted 2026-09-29 (byte-identical to the earlier extraction); same cell read in arXiv:2111.10420v2 (18 Jun 2022) Appendix H (sources/2111.10420v2.pdf, pypdf layout)
Reported as. -17.69357(4) (total energy, units of t)
Record row (best variational bound; the instance record is the ED row, 0.00446 below). 4 x 4 torus, n = 1/2 = 4 + 4 at zero magnetization ('In all cases we focus on the zero magnetization and fixed visible- and hidden-particle subspaces', p. 5), t = 1, U n_up n_dn with no shift (Eq. 1). U '3.6' is the grid value 3.59381366 (the paper's own ED in this row is the ground state there). N~ = 8 (main Fig. 3 caption, 'Average physical site occupation is n = 1/2 and N~ = 8'), single hidden layer, alpha = 32: method string and detail right. Variational VMC bound, own computation. Stored sigma equals the printed bar. energy_variance 0.033 is the upload's (7f646d3: -17.69357 | 0.00004 | 0.033(2)); not printed as a number. Venue PNAS, peer reviewed ('Edited by Angel Rubio ... received December 6, 2021; accepted June 28, 2022'; 'This article is a PNAS Direct Submission'); the imported row carries no peer_reviewed field (loader-level, see report). No duplicate (the only other row is the ED).
-1.10581(5), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.6929(8)
Appendix H Table I, U=3.6, column HFDS alpha=16; sources/2111.10420.txt line 1177; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29 | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.10578(4), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.6925(6)
Appendix H Table I, U=3.6, column HFDS alpha=8; sources/2111.10420.txt line 1177; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#65: sigma 0.000608 -> 0.0006 (The table prints totals, so the bar is 0.0006 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 1.3%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.10561(3), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.6897(4)
Appendix H Table I, U=3.6, column HFDS alpha=4; sources/2111.10420.txt line 1177; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29 | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.105581(6), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.6893(1)
Appendix H Table I, U=3.6, column HFDS alpha=2; sources/2111.10420.txt line 1177; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#63: sigma 0.0001008 -> 0.0001 (The table prints totals, so the bar is 0.0001 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.105239(6), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.68383(9)
Appendix H Table I, U=3.6, column HFDS alpha=1; sources/2111.10420.txt line 1177; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#62: sigma 0.0000896 -> 0.00009 (The table prints totals, so the bar is 0.00009 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.4%. Same fix as the trailing-float-noise ruling: store sigma_per_) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.103532(4), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.65651(7)
Appendix H Table I, U=3.6, column HFDS alpha=0.5; sources/2111.10420.txt line 1177; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#61: sigma 0.0000704 -> 0.00007 (The table prints totals, so the bar is 0.00007 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.6%. Same fix as the trailing-float-noise ruling: store sigma_per_) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.102726(6), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.64362(9)
Appendix H Table II, U=3.6, row Unrestricted hid. spin(RBM); sources/2111.10420.txt line 1202; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#75: sigma 0.0000896 -> 0.00009 (The table prints totals, so the bar is 0.00009 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.4%. Same fix as the trailing-float-noise ruling: store sigma_per_) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.101150(6), VMC (RBM)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.6184(1)
Appendix H Table II, U=3.6, row USD (RBM); sources/2111.10420.txt line 1197; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#71: sigma 0.0001008 -> 0.0001 (The table prints totals, so the bar is 0.0001 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.098930(5), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.58288(8)
Appendix H Table II, U=3.6, row Unrestricted hidden spin; sources/2111.10420.txt line 1199; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29 | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.098918(6), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.58269(9)
Appendix H Table II, U=3.6, row Gutzwiller inspired CF; sources/2111.10420.txt line 1198; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#72: sigma 0.0000896 -> 0.00009 (The table prints totals, so the bar is 0.00009 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.4%. Same fix as the trailing-float-noise ruling: store sigma_per_) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.09700(6), VMC (SJ)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.552(1)
Appendix H Table II, U=3.6, row USD (Gutzwiller); sources/2111.10420.txt line 1195; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#69: sigma 0.001008 -> 0.001 (The table prints totals, so the bar is 0.001 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 0.8%. Same fix as the trailing-float-noise ruling: store sigma_per_si) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.09679(2), VMC (SJ)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -17.5487(3)
Appendix H Table II, U=3.6, row USD (Jastrow); sources/2111.10420.txt line 1196; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#70: sigma 0.000304 -> 0.0003 (The table prints totals, so the bar is 0.0003 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 1.3%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.05434(1), VMC (HFDS)
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -16.8694(2)
Appendix H Table II, U=3.6, row Jastrow inspired CF; sources/2111.10420.txt line 1200; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29; qmbl-verify 2026-10-19 SP2-arxiv-2111-10420#74: sigma 0.000208 -> 0.0002 (The table prints totals, so the bar is 0.0002 as printed; the round trip through a rounded per-site sigma inflates or shrinks it by 4.0%. Same fix as the trailing-float-noise ruling: store sigma_per_s) | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).
-1.053(5), HF
Checked 2026-10-16 · source text of arXiv:2111.10420 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading
Reported as. -16.85(8)
Appendix H Table II, U=3.6, row USD; sources/2111.10420.txt line 1194; version read: arXiv:2111.10420v2 text, Appendix H (sources/2111.10420.txt); the PNAS SI (Cloudflare 403 on 2026-10-16) was read for this paper's earlier rows on 2026-09-29 | Eq. (1): H = -sum t_ij (c+c + h.c.) + U sum n_up n_dn, t = 1. Tables I-III print totals on the 4x4 torus (ED -17.698 on 16 sites); Tables IV and V print 'energy per site'. The rounded U of the tables (0.6, 2.15, 3.6, 7.75) are the grid 10^(k/9): QMBL's ED at U = 0.59948425 gives -19.47476333 and at 2.15443469 (n = 1/2) -18.40532088, the printed ED values, against -19.47434179 at U = 0.6 and -18.40534078 at U = 2.1544 (checks/hubbard-u-labels/results-2026-10-16/, run_fermions.mjs).