How these numbers were read
Rows added after the VarBench import record where the number came from.
-0.55200(1) — ViT (spatial attention, b = 3, variance → 0)
Checked 2026-09-28 · source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading, parsed from the committed HTML table and matched against the PDF text
Reported as. -0.55200(1)
Table 1, 18 × 18 block, row 'Zero Variance', printed '-0.55200(1)', 'This work'; caption: 'For each system size the zero-variance extrapolated energy is also reported'. Sec. V.3 / Fig. 6: linear fit of energy against variance per site over the unprojected, translation-projected and fully projected ViT states at this L; an extrapolation of the same runs, not a separate calculation. | Table 1 prints ground-state energies per site of the spin-1/2 triangular Heisenberg model on periodic L x L clusters (Sec. III: 'All calculations are performed on periodic L x L clusters with L chosen as a multiple of 3', H = J sum S_i.S_j), S.S units. Magnitude check: the values lie between -0.5517 and -0.5520, next to the 12 x 12 value -0.55315 (GCNN, per site, S.S) on Heisenberg/triangular_144_P, not four times it (Pauli) or a total.
-0.551703(1) — ViT (spatial attention, b = 3, translations + C6v...
Checked 2026-09-28 · source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading, parsed from the committed HTML table and matched against the PDF text
Reported as. -0.551703(1)
Table 1, 18 × 18 block, row 'ViT', printed '-0.551703(1)', 'This work'; caption: 'The results of this work are obtained by enforcing translational and C6v point-group symmetries (refer to Section V.2)'. | Table 1 prints ground-state energies per site of the spin-1/2 triangular Heisenberg model on periodic L x L clusters (Sec. III: 'All calculations are performed on periodic L x L clusters with L chosen as a multiple of 3', H = J sum S_i.S_j), S.S units. Magnitude check: the values lie between -0.5517 and -0.5520, next to the 12 x 12 value -0.55315 (GCNN, per site, S.S) on Heisenberg/triangular_144_P, not four times it (Pauli) or a total.
-0.54861(7) — 2D RNN (iterative retraining, s = 4.0, r = 0.158)
Checked 2026-09-28 · authors' data release read locally (pickle via numpy), values copied by key, no transcription; the file is committed as sources/2505.20406-repo-final_energies_data.json and reproduces Table V's L = 6 energy and Table II's V-scores
Reported as. -0.5486121942967545 +/- 0.0000683154952622206 per site (S.S), Var(E/N) = 0.00004667006892922484, V-score 0.05024033645606475
github.com/mschuylermoss/HeisenbergRNN, commit 5f5cffa5cb (2025-10-28, HEAD; file last changed in d58a945cd4 2025-05-27, 'final plotting data and plotting notebooks'), Triangular/final_plotting/plotting_data/final_energies_data_plotting.pkl, sha256 bdc59e5bc3780fb4ea7444400dc9abd9984a6a259ab03f9299c1f80edb87d276: ['TriangularMS,periodicBC']['scale=4.0,rate=0.158,T=1.00'], L = 18. The paper's most accurate schedule (Table II, whose L = 18 V-score 5.0e-2 is this run's 0.0502); plotted in Fig. 4(a), not printed; quoted as -0.54861(1) in Table 1 of arXiv:2602.02665, whose error bar is not the primary's. TriangularMS is the 120-degree basis rotation U_tri, which changes the representation, not the Hamiltonian. Variance is Var(H)/N^2 in S.S units; total = var x N^2 x 16.
-0.54716(3) — VMC (RVB)
Checked 2026-09-28 · source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading of the quotes; primary read, value not located
Reported as. −0.54716(3)
Table 1 of arXiv:2602.02665, 18 × 18 block, row 'RVB', printed '−0.54716(3)' [73]; the same string is Table II of arXiv:1512.03356 (L569), column 'E/J' (Ref. 35)', J/J' = 1.0, 'on the 18 × 18 cluster' (Becca co-authored both). Not located in the committed text of arXiv:0902.2338 (searched '547', 'Table'; its finite-size energies appear as Fig. 3 points; the text prints -0.5470(1) as the L -> infinity value). | Table 1 of arXiv:2602.02665 prints energies per site of the triangular Heisenberg model on periodic L x L clusters, S.S units.
-0.5459(1) — EPS (16-site plaquettes)
Checked 2026-09-28 · source text of arXiv:1006.4480 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading by a reader agent, line checked in the committed text by the session
Reported as. -0.5459(1)
Table 2 (sources/1006.4480.txt L252-259), row N = 324; 'N plaquettes comprising 16 sites have been employed'. Quoted as 'EPS -0.5459(1) [72]' in Table 1 of arXiv:2602.02665. | Table 2: 'Variational GS energy per site (in units of Jxy) ... for the spin-1/2 AHM on a triangular lattice of size N with PBC' at the Heisenberg point Jxy = Jz = 1, S.S units; N = 324 is the 18 x 18 cluster and N = 144 the 12 x 12. Its N = 36 value -0.55420(5) is already carried on Heisenberg/triangular_36_P as a secondary row and is left to qmbl-verify.
-0.54542(1) — VMC (120° order, π-flux hopping)
Checked 2026-09-28 · source text of arXiv:1512.03356 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading by a reader agent, line checked in the committed text by the session
Reported as. -0.54542(1)
Table I (sources/1512.03356.txt L366-377), row J'/J = 1.0, column 'E/J (Magnetic state)'; repeated in Table II at J/J' = 1.0. At the isotropic point the best magnetic state has 120° order (text, L336-340) with π-fluxes threading up triangles. Quoted as 'Jastrow-Gutzwiller -0.54542(1) [71]' in Table 1 of arXiv:2602.02665. | Table I: E/J on the 18 x 18 cluster, periodic boundary conditions (Sec. II), H = J sum S_i.S_j + J' sum S_i.S_j; row J'/J = 1.0 is the isotropic triangular Heisenberg model, S.S units.
-0.53570(1) — VMC (projected BCS + spin Jastrow)
Checked 2026-09-28 · source text of arXiv:2602.02665 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading of the quotes; primary not read
Reported as. −0.53570(1)
Table 1 of arXiv:2602.02665, 18 × 18 block, row 'p-BCS', printed '−0.53570(1)' [70]; the same string is Table II of arXiv:1512.03356 (L569), column 'E/J' (Ref. 33)', J/J' = 1.0, 'on the 18 × 18 cluster'. Primary not read: arxiv.org/pdf/cond-mat/0602180 returned HTTP 500 on 2026-09-28, so no committed text; a reader saw '-0.5357 ± 0.0001' in the primary's Table 2 on the arXiv HTML render, not committed and not used here. | Table 1 of arXiv:2602.02665 prints energies per site of the triangular Heisenberg model on periodic L x L clusters, S.S units.
-0.53565(2) — VMC (U(1) Dirac spin liquid)
Checked 2026-09-28 · source text of arXiv:1512.03356 read locally (arXiv HTML or pypdf layout text); value copied from the harvested cell and the text, no LLM transcription of numbers; single reading by a reader agent, line checked in the committed text by the session
Reported as. -0.53565(2)
Table I (sources/1512.03356.txt L366-377), row J'/J = 1.0, column 'E/J (Spin liquid)', the paper's best spin liquid; for J'/J >~ 0.85 that is the U(1) Dirac state (text L283-286: the Z2Ad Ansatz is lower only for J'/J <~ 0.85). | Table I: E/J on the 18 x 18 cluster, periodic boundary conditions (Sec. II), H = J sum S_i.S_j + J' sum S_i.S_j; row J'/J = 1.0 is the isotropic triangular Heisenberg model, S.S units.