Every published energy
Grouped by what the number is. The exact energy comes first where there is one; in every other group the lowest unflagged energy is its best value, and each row's distance is to that best (rules §6).
Numerically exact (1)
Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established. An exact energy is shown first, before the best of each other class. QMC is exact only within its error bar, which it must state, and reads exact (stochastic); an exact diagonalization outranks it. For a diagonalization resolved by symmetry sector only the ground state's sector is listed; the lowest energies of the other sectors are the instance's spectrum, in its JSON, and are not shown as the exact energy.
Strict variational upper bounds (3)
Energies that a variational state reached (VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE). Each is an upper bound on the ground-state energy.
How these numbers were read
Where each number below was read, and how it was checked.
-0.5168588, ED
Checked 2026-09-16 · ed_full.py (numba matrix-free Lanczos, S^z = 0, no spatial symmetry, 601,080,390 states, residual 6.9e-10) and ed_perm.py (fully symmetric sector of the 384-element space group x spin inversion, 789,438 states, residual 6.3e-10), an AMD EPYC 9654 server, 2026-09-16
Reported as. -66.15792523719963 (ed_full.py) | -66.15792523721132 (ed_perm.py)
Pauli total; E/N = -0.51685879 in S.S units. The ground state is the unique fully symmetric singlet; the lowest S^z = 1 level -63.40912320391 gives the triplet gap (E1 - E0)/4 = 0.68720, the 0.6872 printed for this cluster in arXiv:2010.03563 Table I and arXiv:2101.08787 Table II, whose E/N = -0.5168 is this energy truncated. Replaces VarBench's exact row -66.1514 (-0.5168078 per site), 6.5e-3 above it (RULES.md 11, removals.mjs).
-0.5168, DMRG
Checked 2026-09-15 · source text of arXiv:2010.03563 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. -0.5168
Table I, row 'Cluster 32': GS energy -0.5168, Singlet gap 0.0318, Triplet gap 0.6872. Text: 'Our extrapolated (χ→∞) cluster energies and gaps are summarized in Table I.' | Value -0.5168 matches instance exact_per_site=-0.5168078125 to 4 significant figures (both essentially the same S·S-convention per-site energy).
-0.5162(1), VMC (PP)
Checked 2026-10-15 · published PDF Phys. Rev. X 11, 041021 (2021) (open access), sources/journal/prx11-041021.pdf, sha256 e65c26b3, pypdf layout (ð1Þ = (1) in the font); arXiv:2101.08787v2 compared (qmbl/sources/2101.08787.txt Table II identical)
Reported as. -0.5162(1) per site (S.S, J = 1)
H = j1 sum h_lambda(i,j) + j2 sum ..., h_lambda = lambda/2 (S+S- + S-S+) + SzSz (Eqs. 1-2), Heisenberg point lambda = 1, j2 = 0; Table II: 'The ground-state energy per site is listed in the first line of every cell'; 'We consider pyrochlore clusters with periodic boundary conditions'. Geometry 4 x L^3 = 4L^3 sites. Stored Pauli total = 4 x N x E/N. Own computation (mVMC / NQS CNN columns). The paper prints no energy variance: the stored energy_variance is the VarBench upload's only. Published 29 October 2021, peer reviewed. 0.0843 above the exact -66.15792523719963 (Table II ED column prints -0.5168). No duplicate.
-0.5128(1), VMC (CNN)
Checked 2026-10-16 · source text of arXiv:2101.08787 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. -0.5128(1)
Appendix H Table II, geometry 4x2^3, column NQS CNN, first line of cell; sources/2101.08787.txt line 1124; version read: arXiv text, Appendix H Table II (sources/2101.08787.txt) | Spin-1/2 operators S = sigma/2, J = 1; Table II caption: first line of each cell is the energy per site (S.S units). Cluster 4 x L^3 is L^3 FCC primitive cells of 4 sites (4 x 2^3 = VarBench's pyrochlore-2x2x2_32_P).