Rules

Version 0.1 (draft, 2026-09-11). These rules exist so that accepting or rejecting a row is a matter of citing a clause, not of anyone's judgment about anyone's work. Where a rule was written in response to a concrete case, the case is named.

1. Scope#

Track 1, the only track open, covers ground-state energies of lattice Hamiltonians computed by any method. A row is a claim about one (Hamiltonian instance, method) pair.

Out of scope for now: real-time dynamics, finite temperature, continuum and molecular systems. Each needs an accuracy metric that track 1's does not provide, and no track opens until track 1 is maintained.

2. Instances#

An instance is a fully specified Hamiltonian: model, lattice, site count, boundary conditions, and every coupling. Rows are comparable only within an instance.

Instance identity is part of the claim. A submitter asserting a record must state the symmetry sector and particle number; if a paper's number turns out to be from a different sector, the row moves rather than competes.

3. Required fields#

A missing field never excludes a row. Only §6 and §7 govern what can hold a record.

4. bound_type#

Every row declares exactly one. This is the field that keeps the table honest, because energies from different classes are not comparable and mixing them manufactures false records.

• variational: a strict variational upper bound on the ground-state energy in the stated sector. VMC, DMRG and MPS at finite bond dimension, PEPS, NQS, VQE and other parameterised circuits, and p-step Lanczos applied to a variational state.
• projected: variational only within a constraint, or otherwise biased. Fixed-node and constrained-path methods, GFMC on a trial state, constraint-release schemes. Still an upper bound in most cases, but node- or constraint-dependent, so two projected energies from different trial states are not cleanly comparable to each other or to variational ones.
• extrapolated: the reported number is an extrapolation, not an achieved energy: zero-variance, bond-dimension, or Trotter-error extrapolation. Not a bound. No ansatz ever reached it.
• exact: numerically exact for this instance: exact diagonalization, exact solution, sign-problem-free QMC where that is established.

Worked cases that fixed the boundaries:

• "exact grad" is not an exact energy. 21 rows read VQE + symm. circuit (64 pars., exact grad, statevector). That describes how the gradient was computed. They are variational. Rule order in scripts/bound_type.mjs is load-bearing for this reason.
• p-step Lanczos stays variational. A Lanczos step applied to a variational state is a variational improvement, not a projection.
• Zero-variance extrapolation is its own class. Chen & Heyl report both a variational −0.4976921(4) and an extrapolated −0.497715(9) on 10x10 J1-J2. Ranking them in one column would award the record to a number no wave function achieved.
• A method name is not a guarantee. QMC (continuous-time expansion) is numerically exact for impurity models, but on t-V lattice instances its rows sit above exact diagonalization and 4.7e-2 above DMRG. It is null pending per-model classification.
• When in doubt, null. Bare QMC and AFQMC strings are unresolved: sign-problem-free (exact) and constrained-path (projected) are different classes and the string does not say.

5. Units and conventions#

Stored conventions follow VarBench and are asserted per instance, not per row:

• Spin models are written with Pauli matrices and stored as totals. Papers quote E/N in the S.S convention: E/N = energy / (4 * n_sites).
• Hubbard and t-V are stored as totals; papers quote per site.
• dof is the spin count for spin models, N_up + N_dn for Hubbard, and the particle count for spinless t-V.
• einf is 0 for traceless spin Hamiltonians, U * N_up * N_dn / n_sites for the plain Hubbard model. Hopping terms are traceless, so a t' variant keeps the same einf; extra density-density terms do not.

Any submitted number is converted by the loader, never by hand.

6. Records and ties#

The record for an instance is the lowest eligible variational energy. projected, extrapolated and exact rows are displayed alongside but never hold it.

• Rows whose error bars overlap at 2 sigma share the rank.
• A sampled energy must state its sigma to be eligible. sigma must also state how it was estimated. An error bar that ignores autocorrelation understates by around an order of magnitude, which is enough to manufacture a record. Report the blocking or binning analysis, or the raw chain. A sampled row without sigma is listed and is eligible for nothing.
• A deterministic energy needs no sigma. DMRG at a stated bond dimension, exact diagonalization, statevector circuits and tensor-network contractions carry no statistical error, so there is no error bar to withhold; their convergence control is the bond dimension or truncation error carried in method. 144 of the 399 variational rows are of this kind. Requiring sigma of them would vacate more than half the table's records over a field that cannot exist. scripts/units.mjs draws the line by explicit deterministic markers, and anything not so marked counts as sampled.
• Case: the 4x4 J1-J2 VQE rows. exact grads & metric, statevector and 2^14 samples/grad appear as separate rows on the same instance. The first is deterministic and eligible without a sigma; the second is sampled, reports none, and is not. On four instances the sampled variant sat lower and would have taken the record with no error bar to check it against.
• A row without energy_variance is completely fine. The V-score renders n/a and nothing is inferred or reconstructed. Most of the literature stops at the energy, and a missing V-score is never held against a row; it only means the row cannot be compared across instances.
• A row with no error metric we could find is marked, not penalised. Where neither a sigma nor an energy_variance turned up in the source that was read, there is currently no way to judge how converged the number is, and a reader comparing two energies should see that rather than infer it from two blank cells. The marker is o in scripts/records.mjs and a circle in the README table, derived at display time from the two absent fields.

The marker describes our search, not the authors. The figure may be in a supplement, a companion paper, or the group's own records; what the row records is that we did not find it. It is therefore an open question addressed to whoever can close it - the row stays in the table and in rank order, and closing it takes one message. It is deliberately not a defect (11), which asserts a suspected error and withholds the record: missing data is not an error, and by 3 it excludes nothing. exact rows are not marked, since an exact diagonalization has no error to report and nothing is missing.

6.1 A flagged row cannot hold a record#

A row carrying a defect is displayed in rank order, in place, with its flag - and is skipped when the record is assigned. Suspicion is enough to withhold a record; it is never enough to hide or delete a row (11). Section 10 is how a flag gets lifted or upheld.

This exists because the opposite happened on first implementation: 5 of 7 flagged rows held their instance record, including the three TFIsing RBM (alpha = 1) rows a rerun had already shown to be minima of an optimization trace. Ranking purely on energy promotes precisely the numbers that are wrong, because being wrong downward is what makes a number look like a record.

7. Aggregate#

The headline is records held: the number of instances where a group holds the record. There is no cross-instance score. The V-score is shown where energy_variance is present and is never ranked on.

8. Provenance#

• arXiv preprint minimum. Rows flag whether the source is peer reviewed.
• Preprint-only sources older than 12 months are flagged stale-preprint.
• A row whose number is quoted from a different paper than the one that produced it is marked secondary and names both.
• A cited number that cannot be found in the cited source is not admissible. Case: arXiv 2604.25775 reports "the previous best result −0.766073(6) from NNBF [24]", citing Loehr & Clark arXiv 2510.26906. That value appears nowhere in the only public version of that paper, whose best 4x16 energy is −0.76560(1); its lower −0.768337(2) is a 4x8 number. The −0.76560(1) row is admissible, the −0.766073(6) row is not.

8.1 Venue type#

"Peer reviewed" is not one thing. A number in Phys. Rev. B has been refereed by people who check physics numerics; the same number in a machine-learning conference proceedings has been refereed for machine-learning contribution. Both are peer reviewed; only one is evidence about the energy. Rows therefore record the venue, and a physics-numerics venue outranks an ML venue when the two disagree.

Case: arXiv:2607.00398 (conference proceedings) claims -0.49782(3) on 10x10 J1-J2, described in its own abstract as "statistically consistent with the variational state of the art". It is 1.3e-4 below the best variational energy, i.e. an unclaimed record, and on 8x8 it beats the field by 1.1e-3 while reporting a variance 6x-14x worse - the tension in 9.3. Both rows are listed and flagged energy-variance-inconsistent. Note what is not claimed: there is no exact reference at either size, so nothing here is proof of error, only grounds for objection.

8.2 baseline: collected versus computed#

VarBench did two different things, and the imported rows must say which. A row whose reference cites a paper is one VarBench collected from the literature. A row whose only reference is a run script in varbench/methods is one VarBench computed itself, and carries baseline: true. Of the 578 imported rows, 369 are computed and 209 collected.

The distinction matters because the two are different kinds of claim. A collected row is somebody's published result, defended in a paper. A computed row is a reference calculation - the same ansatz run across the instance set so the V-score would have something to measure against. Jastrow baseline is named as one. Nobody ever claimed state of the art for a plain RBM (alpha = 1); it is the floor, not a contender.

The evidence that these are reference runs rather than results is in what they report. Rows VarBench computed carry energy_variance 76% of the time and sigma 46%; rows citing a paper carry variance 35% and sigma 90%. That inversion is the V-score's fingerprint: the metric needs Var(E), almost no paper publishes it, so the benchmark had to produce it. The same ansatz then recurs across dozens of instances - exact diagonalization on 74, RBM (alpha = 1) on 44, Jastrow baseline on 44.

A baseline row is a full citizen of the table: it ranks, it can hold a record, and it is cited to the VarBench paper. The flag changes nothing about eligibility. It exists so that "record held by RBM (alpha = 1)" reads as *nobody has published a better number for this instance*, rather than as a defended claim - because those are the instances worth attacking, and one of those readings finds them and the other does not.

9. Validation#

scripts/validate.mjs runs on every change and checks:

1. dof and einf against the instance definition (§5).
2. The V-score recomputed from the row's own energy, variance, dof and einf.
3. Energy against the row's own variance. Where a row reports both, the V-score must be consistent with the energy: a state further from an eigenstate cannot also be lower in energy. This is the check that works where no exact reference exists at all - ED for frustrated 2D models reaches roughly 6x6, so 8x8 and 10x10 J1-J2 have no exact row and never will. Being lower than an extrapolated ground state proves nothing on its own, since the extrapolation may carry systematic error; being lower than the field while reporting a worse variance is grounds for objection.

It is not proof, because the inference only runs one way. A small variance means close to an eigenstate, not to the ground state. A state pinned near a competing, excited configuration has little variance and a large energy error, and a better state below it will then report the worse variance. Case (2026-09-15): VarBench's HFDS rows on the four 4xL Hubbard instances at 1/8 doping are each pinned to lambda = 8 stripe order, among near-degenerate stripe states. Their variance per unit energy error is 0.18-1.6, against 6-80 for every other Hubbard row with an exact or DMRG reference - including on the open-boundary instances, where extrapolated DMRG is an independent check. HFPS sits 0.47% below HFDS on 4x8 with 25x its variance, and is the better state. Two VarBench DMRG rows on 4x4 (U = 3.5981, 8 and 10 electrons) show the same signature from having converged to an excited state. The check stands; what it establishes is a question to the authors, which is how the HQT case in 8.1 is carried.

1. The variational principle: no variational row may sit below an exact row in the same instance. Sector-resolved ED rows are excluded, since an unconstrained state may legitimately sit below the lowest state of one sector. A violation counts only past 3 sigma, or past a relative 1e-8 when no sigma is given.

Six issues survive these checks on the imported VarBench data. They are carried as known defects attached to the row (§11), not silently corrected.

Resolved case, 2026-09-11: the three TFIsing RBM (alpha = 1) rows. They sit 3.8 to 10.4 sigma below an exact solution, which no variational state can be. Rerun on three seeds each (netket 3.22.4, alpha=1 complex RBM, 2000 SR steps): evaluating the trained parameters by full summation over all 1024 basis states, with zero Monte-Carlo error, puts every converged energy above the exact value as the variational principle requires, while every training trace dips 1.5e-3 to 3.5e-3 below it. Each published value lies between the two. The conclusion is that the published number is a minimum of the optimization trace, not a converged measurement. An underestimated error bar is ruled out: tau_corr <= 0.05 and R_hat = 1.0000 across all six runs. Evidence in checks/.

This is why §6 requires a stated sigma and how it was estimated, and why §10 admits "violates the variational principle against a known exact reference" as grounds for objection. A leaderboard that ranked on energy alone, with no exact-reference check, would have carried all three as records.

The case none of the above would catch: kagome, 108 sites (2026-09-14). Ðurić et al., Phys. Rev. X 15, 011047 (2025), report a spinon pair-density-wave ground state at E₀ = −48.18 on a 108-site cluster, 1.78% below the DMRG benchmark of −47.33964 they compare against. A Comment (arXiv:2605.28861) shows the single-spin-flip update used at that size does not conserve total magnetisation. As the network concentrates on the physical S^z_tot = 0 sector the acceptance rate collapses to exactly zero beyond 5000 iterations, so the chains freeze and the average is taken over a non-representative set of configurations. Under the magnetisation-preserving exchange update the same architecture converges to ≈ −45.6, and re-evaluating the spin-flip-optimised parameters ergodically gives ≈ −42.6; the published number lies below all of them.

Three things make this the sharpest case in this document:

1. No check on the number itself could have caught it. DMRG at finite bond dimension is an upper bound too, so an energy below it is not evidence of error; ordinarily it is just a better state. There is no exact reference at 108 sites. The refutation had to come from the sampler, which is why §10 now admits non-ergodic sampling as its own ground.
2. The instance ends with no record at all. Its only variational row is flagged, and the DMRG row is extrapolated: Depenbrock et al. state their energies are "extrapolated in the truncation error of single-site DMRG", which lies below any energy an MPS achieved. "No eligible record" is the correct answer here, and a table ranking on energy alone would instead have printed a refuted number as the record.
3. DMRG is not one bound_type. At a stated bond dimension it is a strict variational bound; extrapolated in the truncation error it is not. The same method name maps to two different classes depending on one sentence in the methods section.

10. Pending, confirmed, objections#

A new row is pending for 30 days and confirmed if no objection is unresolved. Anyone may object by opening an issue against the row. An objection must be specific and technical. Admissible grounds:

• wrong symmetry sector, particle number, or boundary conditions;
• wrong unit convention (§5);
• bound_type misdeclared (§4);
• sigma estimated without autocorrelation correction (§6);
• the energy violates the variational principle against a known exact reference (§9);
• the sampling is non-ergodic: the Monte Carlo update rule is incompatible with a conservation law or symmetry of the Hamiltonian, so the reported average is taken over a non-representative set of configurations (§9, the kagome case);
• the number cannot be located in the cited source (§8).

Not admissible: that a method is uninteresting, that a comparison is unflattering, or that a competing row ought to have been cited.

The submitter may correct or withdraw. A confirmed row is overturned only by an objection that meets the same bar. Rulings cite a clause.

11. Corrections#

Errors found by the maintainers are corrected in public with the reason recorded on the row. Rows are never silently deleted; superseded ones are marked and kept.