Hubbard rectangular 4x16, U = 8, n = 0.875

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. −0.76748

Sec III, 'Variance Extrapolation' paragraph, Fig.1 | same instance as above; paper explicitly notes this extrapolated value is 'not necessarily variational' and uses it as a ground-truth reference

Checked 2026-10-22 · source text of arXiv:2507.02644 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. -0.7673

main text, Scalability section: 'zero-variance extrapolation on a 16 × 4 pure Hubbard model ... The extrapolated ground state energies are —0.7673'; dl/gu-natcommun.txt line 150 (the printed dash is an em dash); not variational | Energy per site in units of t. PBC per the Supplementary Table 5 caption ('Raw data for the energies of Hubbard model under PBC'); U = 8 and hole doping 1/8 per the main text (Fig. 4 caption: 'U = 8 and hole doping δ = 1/8 under PBC'; the t' = -0.2 study: 'Lx × 8 with Lx ranging from 8 to 32, with t' = -0.2 and δ = 1/8'). H has -t' sum_<<ij>>, so t2 = t'. The 16 x 4 and 16 x 16 values at t' = 0 equal the arXiv text's -0.76298 and -0.75630 to the printed digits. No error bar printed. Version read: journal Supplementary Information (41467_2026_74028_MOESM1_ESM.pdf); the arXiv text has no raw-data table.

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.76647 (+/- 0.00002) per site

Sec. III.3.1: "a GFMC calculation with the ACE trial wave-function gives an even lower energy of -0.76647(2)". Fixed-node: upper bound but node-dependent.

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.76623 (+/- 0.00001) per site

Sec. III.3.1: "reach a new state-of-the-art energy of -0.76623(1)".

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.7656 (+/- 0.00001) per site

Sec. III: "further projected at essentially no cost to the fully symmetric state to reach an energy of -0.76560(1)". This is the best 4x16 number in arXiv:2510.26906v1.

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.76486 (+/- 0.00002) per site

Same section, before the free projection.

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.76464 (+/- 0.00001) per site

Sec. III.3.1: "achieves a ground-state energy of -0.76464(1)".

Checked 2026-09-15 · source text of arXiv:2507.01856 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. -0.7642

S III.2 p3: 'The extrapolated energy for PBC shown in Fig.(2)(c) is -0.7642, which matches the state-of-the-art neural Pfaffian result [11]', where Fig.2(c) is explicitly the 4x16-PBC lattice | 4x16=64-site lattice, n=0.875 (Nf=28 per spin), U=8, t'=0, PBC; matches Hubbard/rectangular-4x16_64_P_28_8.

Checked 2026-09-13 · arXiv HTML parsed locally, no LLM transcription

Reported as. -0.76413 (+/- 0.00003) per site

Sec. III: "The energy we obtain in the 16x4 lattice is E/N = -0.76413, outperforming a recent result E/N = -0.76298 produced by Transformer-NNBF". Instance identity confirmed by that comparison value; the number is independently quoted as the HFPS result by both arXiv:2604.25775 and arXiv:2510.26906. No error bar in Sec. III; Table IV of the same paper (raw data of Fig. 7, L x 4 PBC lattices, 1/8 doping, U = 8) gives -0.76413(3) and sigma^2/N_site = 0.031283, added 2026-09-15. Published as PNAS 123, e2535288123 (2026); read from the arXiv version, the journal version not compared.

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; single reading

Reported as. −0.76340(2)

Sec III, 'Optimizing multi-determinants' paragraph | same instance as above

Checked 2026-09-10 · source text/table parsed locally (arXiv HTML or pypdf), no LLM transcription

Reported as. -0.76298 per site

Supp. Sec. 4.1: "our NQS yields a ground state energy of -0.76298" on 16x4 PBC. Instance identity confirmed: same passage quotes HFDS -0.753(2), matching the VarBench row -0.753000(47). No error bar given.

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; single reading

Reported as. −0.76258(1)

Sec III, 'Increasing neurons' paragraph | 4x16=64 sites (P=PBC); U=8, t=1; hole doping 1/8 => Nf=28 per spin

Checked 2026-09-15 · source text of arXiv:2507.10705 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. −0.76256

Sec III.3, 'we find a fairly similar energy...when we initialize without a pairing field' | same instance as above

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. -0.7618

Table III, n=0.875 block, 4x16 row, BW2 column | same as BW1 row above

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. −0.7611

Table IV, row '4 x 16', column 'RHB' | Same row; paper's own best ansatz for this instance.

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. −0.7597

Table IV, row '4 x 16', column 'K = 2' | Same row as above.

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. -0.7597

Table III, n=0.875 block, 4x16 row, BW1 column | 64 sites, Nf=28 (n=2*28/64=0.875), P boundary (fully periodic, per Table III caption); U=8 confirmed by section context (only U studied for this filling in the paper)

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. −0.7585

Table IV, row '4 x 16', column 'K = 1' | Same row as above.

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. -0.753(2) (energy per site, units of t)

Energy verified; sigma differs (ambiguous entry). PBC-PBC = P. 4 x 16, n = 7/8 = 28 up + 28 down at zero magnetization, U = 8, t = 1, Eq. 1 with no shift. N~ = 16 and alpha = 6 for L = 16 ('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 (many lower rows, e.g. ACE + full symmetry projection -49.03872). arXiv:2308.11823 Table III quotes the same number as 'Ref.2 ... from HFSD' (-0.753, 4 x 16); no QMBL row carries that quote. Upload 8df3719 (2022-07-16): -48.192 | 0.003 | 0.44(1); energy_variance upload-only. 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). No duplicate.

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; single reading

Reported as. -0.75066(2)

Appendix E: 'the highest energy wavefunction we study, n_h=128 with energy per site of -0.75066(2)' | Same 64-site instance; deliberately under-converged baseline.

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. −0.746

Table IV, row '4 x 16', column 'NNB' | Same row/instance as above; quoted comparison column.

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. −0.6035

Table IV, row '4 x 16', column 'K = 0' | Table IV 'under periodic boundary conditions' (full P), nh=0.125, U=8; row '4x16' (64 sites, Nf=28) matches this instance exactly.