below exact energy, VMC (RBM), -1.238172(2)
Finding. -12.381718(22) sits 2.28e-4 (10.4 sigma) below the exact energy -12.38148999965 (VarBench ED via NetKet and DMRG rows; Lanczos on the 1024-state space here agrees to 1e-13). Same provenance as chain_10_P_1: the number entered VarBench in commit 6e7a021 (Dian Wu, 2023-01-09, 'Update Ising Jastrow and RBM baselines', no code), replacing -12.381447(21), var 0.00047990793 (77a95d7, 2022-12-12; 4.3e-5 = 2.0 sigma ABOVE exact) with a value 2.7e-4 (12 sigma) lower at a variance that changed by only 3% (0.00049440652). The linked script scripts/TFIsing/chain_10_O_1/vmc_rbm.sh -> programs/vmc_netket/vmc.py was written in July 2024 (methods 1347303, link added in dataset 5047bfd with the value unchanged) and does not produce the stored number: the bit-faithful rerun (NetKet 3.13.0, jax 0.4.30, CPU, seed 123) prints energy -12.381393 +- 0.000027 [var 0.000753, R_hat 1.0005], 9.7e-5 ABOVE exact and 3.2e-4 (12 sigma) above the stored value; full summation of the final parameters gives -12.381410828 (above exact) and five repeated 2^20-sample estimates (-12.381428, -12.381419, -12.381402, -12.381353, -12.381416; sigma 2.6e-5, tau_corr <= 0.04, R_hat 1.0005) scatter around it within 2.2 sigma (checks/tfising-rbm-varbench-rerun/ising_open_1d_L10_h=1.eval.json). The published code (persistent chains, float64 matrix elements, between-chain error estimate; see the chain_10_P_1 finding) has no mechanism for a below-exact value; the stored sigma = sqrt(0.00049440652/2^20) = 2.17e-5 marks the number as a 2^20-sample final estimate, so it is neither a trace point (per-step sigma 9e-4 in the rerun) nor a 10-sigma fluctuation, but a biased estimate from an unpublished January 2023 run. Earlier 'minimum of the optimization trace' diagnosis withdrawn.
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.
below exact energy, VMC (LRU), -1.2381549(7)
Finding. E/N = -1.2381549(7) (arXiv:2605.13807v1, Table I, row N = 10, column e_1DLRU(N); the only arXiv version, submitted 2026-05-13, PDF byte-identical to the cached copy) against e_exact(10) = -1.2381490 printed in the same row: 5.90e-6 per site, 8.4 sigma, below an exact value the paper itself derives from [1 - csc(pi/(2(2N+1)))]/N and that its released code recomputes with the same formula (PSR-NQS/lru_1d/run_training.py line 22-23, exact_obc_critical_energy_density) for the same Hamiltonian H = -sum sz sz - h sum sx, J = -1, h = 1, open chain (run_training.py 282, 318-319; vmc_utils.py 12-13, 28). Stored total -12.381549 +/- 0.000007 = 10 x per-site value, matches the instance exact row -12.38148999965 to 7 digits. The released code shows what the number is: run_training.py 189-230 evaluates the model state at the END of training with 1000 x 1024 = 1,024,000 fresh autoregressive samples and writes energy_mean and energy_error_of_mean = sqrt(Var/1024000) to final_stats.csv; the Table I caption says the same. A 1024-sample training-step bar would be sqrt(1000) = 31.6x larger (2.2e-5 per site, printed as (2) at five decimals), so the printed (7) is the final-estimate standard error, not a per-step one. The lowest_energy checkpoint (run_training.py 52-58, 141-160, UCB rule after half the steps) is saved but never restored; the reported energy is not a minimum over the optimization trace, and the mechanism established for the VarBench RBM (alpha = 1) row on this instance does not apply here (that dip would be ~4 sigma_step = 9e-5 per site, 15x deeper than observed, with a 30x wider bar). Samples are exact autoregressive draws (wvfn_1d.py 116-151, log_softmax conditionals, categorical sampling), so no autocorrelation correction is missing either. With a normalized causal ansatz (checked: the parallel path shifts inputs by one site and skip-adds the unshifted input, ssm_1d.py 108-112, ssm_blocks.py 50-55, identical to the sampler loop at wvfn_1d.py 134-149) and consistent psi evaluations, the estimator is unbiased and 8.4 sigma below E0 has probability ~1e-17: the printed value cannot come from the released procedure unless the two psi evaluations it mixes disagree. They are different code paths: the denominator psi(sigma) is the sampler sequential recurrence (run_training.py 200-201, model.log_psis(samples, clp)), the N numerators psi(sigma^i) are a parallel associative-scan forward pass (vmc_utils.py 22-24, parallel=True). A mismatch delta between them enters as exp(delta) and biases the field term downward (Jensen); an rms mismatch of ~4e-3 in log psi would reproduce the N = 10 dip. The repository sanity check never tests this: benchmark_trainstep_vs_L_1D_lru.py 280-292 passes clp into both sides of the sampled-input comparison, as its own comment admits. Not run here (out of scope for this pass); no results, logs or CSVs were ever committed (27 commits, 1D code in one commit 8335644 of 2026-05-12). Whatever the mechanism, the same column has two more sub-exact entries (N = 24: -1.57e-5, 15.7 sigma; N = 32: -5.92e-5, 29.6 sigma) and two entries within 1 sigma of exact (N = 8, N = 128), so the printed bars understate the true uncertainty of the column by an order of magnitude; the paper and the script both print the relative error unsigned (rel_err = abs(E - exact)/abs(exact), run_training.py 221), which is why the sign flips went unnoticed and Table I text reads the non-monotonic error as optimization noise. The flag stands; the row holds no record. Corrections: the RBM-rerun diagnosis attached to this row through SHARED["below-exact"] in scripts/defects.mjs is about a different row and must come off (separate correction entry).