Flagged rows
A flagged row stays listed in rank order but cannot be the best of its class.
Rules §10 says how a flag is lifted or upheld.
likely below exact, VMC (transformer BF), -0.52582
Finding. E/N = -0.52582 (total -33.65248) is printed identically, always without an error bar, in arXiv:2507.02644v1 Table S4 (p. 46), v2 Table S1 (p. 33) and Nature Communications 17, 7838 (2026) Supplementary Table 1 (p. 2, file 41467_2026_74028_MOESM1_ESM.pdf via Europe PMC PMC13439271); the main text (p. 8 / PMC full text) only says the NQS energies are 'consistent with the AFQMC results within twice the statistical errors', i.e. Qin's bar. No sigma exists anywhere: the journal Data Availability statement says 'the figures and tables shown contain all the available data' (no Source Data file; the Europe PMC supplementary bundle holds only the SI PDF, the peer-review file and figure images), and the released code (CodeOcean capsule 8740417, DOI 10.24433/CO.9301616.v1 = v2, tree 010f28f0) ships no logs or energies; its evaluation script hubbard/test.py (optimizer 'energy') writes only 'Energy: <mean> Variance: <var>' of the pooled local energies to result.txt, never a standard error, and the training log keeps loss/pmove/variance/lr per step. The capsule README documents exactly this run (L1 = L2 = 8, --fermions 64 -> 32 up + 32 down, --boundary1/2 pbc, U default 8, t = 1, hamiltonian.py H = -t sum c+c + U sum n_up n_dn with no shift), so instance, boundary, filling and coupling match. Against the stored VarBench AFQMC row -33.642(5) the NQS sits 0.01048 total = 1.64e-4 per site below, 2.10 sigma of that bar; against Qin, Shi & Zhang PRB 94, 085103 Table IV (8x8, U = 8, PBC) -33.68(3) it sits 0.02752 total = 4.3e-4 per site ABOVE, 0.92 sigma; the two exact rows differ by 0.038 = 1.25 sigma (combined), so they are consistent and their weighted mean -33.6430(49) puts the NQS 1.92 sigma below. The VarBench value is not in PRB 94, 085103: VarBench history shows it uploaded by Yiqi Yang on 2022-03-16 (commit b1b88c1) with no reference, 'TODO: ask Shiwei' on 2024-07-11 (b1a1a26), and the PRB link attached by Yiqi Yang in PR #23 on 2024-08-06 (3f160a4); its method string 'Metropolis, Trotter error extrapolated' is the VarBench-paper AFQMC (Wu et al., Science 386, 296 (2024), arXiv:2302.04919, Suppl. S3 E 2, co-authors Yiqi Yang and Shiwei Zhang), a separate computation of the same instance. Not established either way: a 2.1 sigma violation of a variational bound against one exact estimate, 0.9 sigma inside the other, with the NQS statistical error unknown. The flag stays on this row; the AFQMC row needs a reference correction, not a flag (see the correction entry on it). It holds no record while flagged (RULES.md 6.1).
How these numbers were read
Where each number below was read, and how it was checked.
-0.5262(5), AFQMC
Checked 2026-09-15 · arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription
Reported as. E = -33.68(3) (total), 8 x 8, U = 8, PBC
Table IV, "Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit." Row 8 x 8, column U = 8 PBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.
-0.52582, VMC (transformer BF)
Checked 2026-09-15 · arXiv PDF text extracted locally with pypdf in layout mode, no LLM transcription
Reported as. -0.52582 per site, no error bar
Supplementary Table S1 of arXiv:2507.02644, "Benchmark energy in pure Hubbard model at half-filling with PBC", column 8 x 8, row "NQS"; the AFQMC reference in the same table is -0.5262(5). Same method string as this paper's 16 x 4 row. No error bar. It sits 1.6e-4 per site below the stored AFQMC exact row -0.5256563(78) - see the defect flag. Read from the arXiv supplement; the Nat. Commun. version was not compared.
-0.52566(8), AFQMC
Checked 2026-09-29 · VarBench dataset v1.1.0 (Zenodo doi:10.5281/zenodo.13377360 = ref. [63] of the cited paper) downloaded and read; varbench-history (b1b88c1, b1a1a26, 3f160a4, 37648ad); arXiv:2302.04919 v2 and v1 text, Suppl. S3 E 1-2, for the method; arXiv:1605.09421 v1/v2 re-extracted with pypdf for the comparison
Reported as. -33.642 | 0.005 | (no variance) | 64 | 128 | AFQMC (Metropolis, Trotter error extrapolated), numerically exact (total energy, t = 1)
Record row (exact (stochastic), smallest sigma on the instance; metrics.json records[236]; is_record false in the group file). Instance, convention and class agree with the card: 8 x 8 PP, N_up = N_dn = 32, U = 8, Eq. S5 without shift (einf 128), sign-problem-free ground-state AFQMC with the Trotter error extrapolated, exact (stochastic) under RULES 4, which settles the M8 flag (bound_type_reason 'exact diagonalization|ex' is a classifier trace; the Qin row's wording would fit). The reference set by ruling 5 (2026-09-21) names the producer correctly; no article prints -33.642, and the dataset the paper cites as [63] holds it (still linked to PRB 94, 085103), so the reference could add 'VarBench dataset v1.1.0, doi:10.5281/zenodo.13377360'. Independent checks: Qin et al. Table IV -33.68(3) (1.25 sigma) and VarBench's own attractive-U run on the same lattice, -289.657(6) (Hubbard/square_64_P_32_-8, uploaded by the same author 253efd3, 2022-04-21, with AFQMCLAB inputs extrapolated to dt = 0 in the methods archive), which the half-filling identity E(U) = E(-U) + U N/2 maps to -33.657(6), 1.9 sigma from this row. The three exact estimates agree within 2 sigma. Against the -U estimate, Gu et al.'s sigma-less -33.65248 (flagged below-exact-suspected, 2.1 sigma below this row) sits 0.75 sigma above, and 0.9 sigma above Qin et al.: the flag rests on this one run. Not a duplicate: Qin et al.'s -33.68(3) on the instance is a separate computation.
-0.5245, 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. −0.5245
Table II, row '8 x 8', column 'U = 8', sub-column 'K = 2' | Same row as above.
-0.5241, 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. -0.5241
Table III, n=1, 8x8 row, BW2 column | Same instance.
-0.5229, 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. −0.5229
Table II, row '8 x 8', column 'U = 8', sub-column 'K = 1' | Same row (half filling, full P, U=8, 8x8) as HF entry.
-0.5188, 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. -0.5188
Table III, n=1, 8x8 row, BW1 column | Half filling, Nf=32, matches catalogue Nf=32,U=8.
-0.4743, 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. −0.4743
Table II, row '8 x 8', column 'U = 8', sub-column 'K = 0 (HF)' | Table II 8x8 row is half filling (Nf=32) under full periodic BC, U=8 column; consistent with instance identity.