How these numbers were read
Where each number below was read, and how it was checked.
-0.768416, VMC (NNBF)
Checked 2026-09-15 · source text of arXiv:2510.26906 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; adversarial re-read: confirmed
Reported as. −0.768416
Sec III.2, '4x8 Lattice' | same instance as above
-0.768337(2), VMC (NNBF)
Checked 2026-09-15 · source text of arXiv:2510.26906 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; adversarial re-read: confirmed
Reported as. −0.768337(2)
Sec III.2, '4x8 Lattice' | 4x8=32 sites (P=PBC); U=8, hole doping 1/8 => Nf=14 per spin; catalogue best_per_site for this instance is -0.7669, so this value is a new lower (better) variational energy
-0.76690(1), VMC (HFPS)
Checked 2026-09-15 · arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription
Reported as. -0.76690(1) per site, sigma^2/N_site = 0.016320
Table IV of arXiv:2507.10705, "Raw data of Fig. 7 in the L x 4 lattice with 1/8 doping and U = 8", row 8 x 4; Sec. III states Fig. 7 shows "L x 4 PBC lattices". 32 sites, 28 electrons, U = 8, periodic: this instance. The same table's 16 x 4 value -0.76413(3) is the HFPS row on rectangular-4x16_64_P_28_8, and its 4 x 4 value -0.74177(5) sits 3e-5 above the ED energy -0.7418 quoted by arXiv:2606.00924. New record, 0.47% below the VarBench HFDS row -0.7633125 and below the RHB -0.7641(3) of arXiv:2606.00924 on the same lattice. Variance: sigma^2/N_site stored as a total (x 32). Its V-score, 2.7e-3, is ~25x the VarBench HFDS row's 1.1e-4 at a higher energy; the HFDS variance is the likelier outlier, since on 4x16 the same VarBench ansatz has ten times the per-site variance and this paper puts HFDS 3e-4 above exact already at 4x4. Read from the arXiv version; the PNAS version was not compared (pnas.org refuses this sandbox).
-0.7641, VMC (residual HB)
Checked 2026-09-15 · arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription
Reported as. -0.7641 per site, table-wide sampling error ~0.0003
Table IV of arXiv:2606.00924, "Energies per site for the Hubbard model at nh = 0.125, U = 8 for different systems under periodic boundary conditions. The energy sampling errors are around 0.0003", row 4 x 8, column RHB (Ndet = 5, Nneuron = 100, the setting adopted for all large-scale runs after Table III). Instance identity: the same row's HFDS column, -0.7633, is the VarBench HFDS row -0.7633125. sigma is the caption's table-wide "around 0.0003". Layout mode shifts this row one column left in sweep-tables.tsv; read here from the printed table. Internal check: the same ansatz gives -0.7410 on 4 x 4 against ED -0.7418 (Table III), above exact as it must be.
-0.76331(6), 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. -0.7633(7) (energy per site, units of t)
Energy verified; sigma differs (ambiguous entry). PBC-PBC = P. 4 x 8, n = 7/8 = 14 up + 14 down at zero magnetization, U = 8, t = 1, Eq. 1 with no shift. N~ = 16 and alpha = 14 for L = 8 ('A good trade-off ... is achieved by the addition of N~ = 16 hidden fermions for all system sizes. We also consider a two-layer fully connected neural network of hidden-unit density α = {60, 14, 6} for the L = {4, 8, 16} sizes', main Sec. 3 B, p. 6), lambda = 8 soft mean-field constraint on phi_v ('To guide the wave-function ansatz toward the λ = 8 stripe order, we add a soft mean-field constraint', p. 6): method string and detail right. Variational VMC bound, own computation. Not the record (MLP-NNBF -24.586784). Upload 93b5ac0 (2022-07-16): -24.426 | 0.002 | 0.021(2); energy_variance upload-only. Duplicate check: arXiv:2308.11823 Table III quotes this value as 'Ref.2 (HFSD)' -0.7633 and prints the same -0.7633 in its own BW2 column (QMBL row 'BW2 (p=1)' -24.4256, sweep-allresults); that BW2 cell is the other paper's own number and moved across its versions (v1 -0.7635, v3/v4 -0.7637, v5/v6 -0.7633), so it is not a duplicate of this row; its coincidence with the adjacent Ref.2 cell is noted in the report for whoever reads arXiv:2308.11823. M6/M8 settled from the source: a sampled VMC energy ('General expectation values and gradients of the objective function are computed using Markov chain Monte Carlo sampling', main text Sec. 2 C, p. 5) whose stored sigma is the uploader's Monte Carlo error; units.mjs counts the row deterministic only because 'Soft mean-field constraint' matches /mean[- ]field/ (units.mjs line 20), which changes nothing while sigma is stored (isSampled is consulted only when sigma is null). 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).
-0.7633, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2308.11823 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. -0.7633
Table III, n=0.875, 4x8 row, BW2 column | Same instance.
-0.7617, VMC (HB)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. −0.7617
Table IV, row '4 x 8', column 'K = 2' | Same row as above.
-0.7599, VMC (HB)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. −0.7599
Table IV, row '4 x 8', column 'K = 1' | Same row as above.
-0.7591, VMC (tensor BF)
Checked 2026-09-15 · source text of arXiv:2308.11823 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. -0.7591
Table III, n=0.875, 4x8 row, BW1 column | n=0.875, Nf=0.875*32/2=14, matches catalogue Nf=14,U=8, boundary PBC.
-0.75500(4), VMC (NNBF)
Checked 2026-09-15 · source text of arXiv:2606.00924 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. −0.755
Table IV, row '4 x 8', column 'NNB' | Same row/instance as above; quoted comparison column.
-0.6146, HF
Checked 2026-09-15 · source text of arXiv:2606.00924 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. −0.6146
Table IV, row '4 x 8', column 'K = 0' | Table IV states 'under periodic boundary conditions' (full P in both directions) at nh=0.125, U=8; row '4x8' (32 sites, Nf=14) matches this instance's lattice/boundary/filling/U exactly.