below exact energy, VMC (RBM), -1.278523(5)
Finding. -12.785231(48) sits 3.25e-4 (6.8 sigma) below the exact free-fermion energy -12.784906443 (recomputed here; VarBench's Exact Solution and DMRG rows agree to 4e-15). The value is not reproducible from the cited source: the linked script programs/vmc_netket/vmc.py was first committed to varbench/methods on 2024-07-23 (1347303) and linked from the dataset on 2024-07-30 (5047bfd, link only, value unchanged), eighteen months after the number, which entered the dataset in commit 6e7a021 (Dian Wu, 2023-01-09, 'Update Ising Jastrow and RBM baselines', no code, no message body). The script also uses nk.operator.IsingJax (NetKet 3.9, 24 Jul 2023) and, for larger RBMs, nkx.driver.VMC_SRt (NetKet 3.10, 7 Nov 2023; CHANGELOG.md of the v3.13.0 clone), neither of which existed in January 2023, so it is a later reimplementation of the baseline, not the program that ran. That commit moved the row from -12.784707(48), var 0.0024197807 (77a95d7, 2022-12-12; 2.0e-4 = 4.2 sigma ABOVE exact) to the stored value with an almost unchanged variance (0.0024565159): a state of the same quality cannot have an energy 5.2e-4 (11 sigma) lower. A bit-faithful rerun of the linked script (checks/tfising-rbm-varbench-rerun/run.sh: NetKet 3.13.0, jax 0.4.30, CPU, seed 123, methods ed31bb0) prints energy -12.784382 +- 0.000065 [var 0.004806, R_hat 1.0005], i.e. 5.2e-4 ABOVE exact and 8.5e-4 (13 sigma) above the stored number; full summation of the same final parameters over all 1024 basis states gives -12.784376149 (above exact, as the variational principle requires) and five repeated 2^20-sample estimates of those parameters (-12.784403, -12.784375, -12.784276, -12.784286, -12.784475; sigma 6.7e-5, tau_corr <= 0.003, R_hat 1.0005) scatter around it within 1.5 sigma (checks/tfising-rbm-varbench-rerun/ising_peri_1d_L10_h=1.eval.json). By construction the script's printed energy is vstate.expect(H) with 2^20 samples from 1024 persistent Metropolis chains (NetKet 3.13.0: MetropolisSampler reset_chains=False, MCState.n_samples setter only changes chain_length; IsingJax mels are float64, only the amplitude ratios are float32; error_of_mean = sqrt(Var(chain means)/1024), the between-chain estimate, which equals sqrt(Var/2^20) only because tau_corr ~ 0), so the code as published has no mechanism that yields a value below the exact energy beyond its own error bar. The stored sigma equals sqrt(0.0024565159/2^20) = 4.84e-5 to two digits, which identifies the published number as a 2^20-sample final estimate, not a training-trace point (per-step sigma with 1024 samples is 2e-3 in the rerun log): the earlier diagnosis 'minimum of the optimization trace' (checks/results-tfising-rbm.json, complex RBM, netket 3.22) is withdrawn. What the January 2023 run did to obtain an estimate 3e-4 below E0 with a 5e-5 error bar cannot be determined from any published source (the VarBench paper arXiv:2302.04919 does not describe the baseline protocol; the methods repository has no output logs, only the 2024 scripts). The row stays as published with this flag; it is a 2023 number whose code was never published and whose only cited code produces a different, variationally consistent value.
Diagnosis. The three values were entered in VarBench commit 6e7a021 (Dian Wu, 2023-01-09, 'Update Ising Jastrow and RBM baselines', no code, no log), replacing uploads that sat above the exact energy: chain_10_P_1 -12.784707(48) and chain_10_O_1 -12.381447(21) of 2022-12-12 (77a95d7; 4.2 and 2.0 sigma above), while chain_32_P_0.5 was below exact in both its uploads (2022-07 and 2023-01). The linked program, programs/vmc_netket in varbench/methods, was written in July 2024 (IsingJax and VMC_SRt did not exist in January 2023) and the links were attached on 2024-07-30 (5047bfd) without regenerating a number. Run as pinned (NetKet 3.13.0, jax 0.4.30, seed 123, CPU, 2026-09-19) the script gives -12.784382(65), -12.381393(27) and -34.03197(10): all above the exact energies, with full summation of the same parameters at -12.784376 and -12.381411 and five repeated 2^20-sample estimates scattering around them (tau_corr ~ 0, R_hat 1.0005). The stored sigmas equal sqrt(stored variance / 2^20), so the stored numbers are final 2^20-sample estimates of a better-converged 2023 state (variance about half of the rerun's) whose bias below the exact energy no published source explains.
Ruled out. Not a minimum of the optimisation trace: the program prints a separate final estimate, the per-step noise (2.2e-3, 8.9e-4, 3.1e-3 at 1024 samples) is 40x the stored bars, and the trace minima lie about 1e-2 below the exact energy, not 2-3e-4. Not an unthermalised restart of the sampler: the chains persist into the final estimate, and a fresh restart biases the estimate upward (seen on the 32-site evaluation). Not float32 local energies: the local energy is accumulated in float64 with float32 amplitude ratios. The 2026-09-11 check (checks/tfising_rbm_check.py: complex RBM, netket 3.22.4, 2000 steps) ran a different program and is withdrawn as a diagnosis.
Evidence. VarBench dataset commits 77a95d7, 6e7a021, 5047bfd; varbench/methods programs/vmc_netket/vmc.py (1347303, c686621, bba4595); checks/tfising-rbm-varbench-rerun/ (run.sh, run.log, *.eval.json, trace_summary.json); the first rerun, checks/tfising_rbm_check.py, for the withdrawn diagnosis.