Compare results without changing the question

A traceable numerical reference

The following is a transcription of a published numerical estimate, not an independently reproduced computation or a rigorous lower bound on the complete spectrum.

m_0++ / sqrt(sigma)

3.405

Source notation 3.405(21); quoted uncertainty 0.021. Parenthesized uncertainty as reported; do not relabel as an all-systematics bound. Statistical/systematic decomposition is not asserted by this transcription. No GeV scale or experimental resonance identification is implied.

Theory and normalization: Pure SU(3), 3+1 dimensions. Mass divided by the square root of the confining string tension, m/sqrt(sigma), within pure SU(3) gauge theory.
Method: continuum-extrapolated numerical estimate, pure gauge Wilson-action ensembles
Source: Table 34, printed page 83 (PDF page 84), 0++ gs row, SU(3) column; cross-check Table 24 A1++ ground state
arXiv:2106.00364v3, revised 2021-12-10; JHEP 12 (2021) 082; DOI 10.1007/JHEP12(2021)082

Download benchmark CSV with provenance · Normalized records JSON

Source-table transcription checked: 24 October 2026.

A useful benchmark fixes the theory and observable before comparing answers. This page supplies a comparison protocol and primary reference points. It does not present an invented consensus mass, silently average incompatible calculations, or treat a numerical central value as a rigorous bound.

Reference studies and what they actually fix

Source	Verified setting	Suitable use
Athenodorou–Teper, 2021	$3+1$ dimensions; pure SU(N), $N=2,3,4,5,6,8,10,12$ ; Wilson plaquette action; fixed- $N$ continuum extrapolations	Spectrum and scale-setting comparison; inspect §2 and §5
Guo et al., 2025	SU(3) lattice plaquette bounds, with SU(2) comparisons; two, three, and four dimensions	Compare the same normalized local observable and constraint set

These are different benchmarks. A plaquette expectation is not a glueball mass; a finite-spacing result is not a continuum result; a meson containing valence quarks is not a pure-gauge glueball.

1. Make the comparison record explicit

For each result, retain the following information next to any number or plotted point.

Record	Why it is needed
Gauge group, rank, matter content, dimension	Identifies the theory
Temperature, $\theta$ , geometry, boundary conditions	Identifies the physical setting
Action, trace normalization, bare coupling convention	Makes lattice parameters comparable
Lattice spacing and physical spatial/temporal extent	Separates regulator and volume effects
Observable, symmetry channel, normalization	Identifies what was measured or bounded
Estimator, operator basis, fitting procedure	Makes the extraction reproducible
Statistical and systematic uncertainties	Explains what the displayed precision means
Source version, equation/table, code or data location if supplied	Allows the result to be traced

If an item is absent, record “not specified” and narrow the comparison. Inferring an unstated fit choice from the appearance of a plot is not reproducibility.

2. Use compatible units

For spectroscopy, a ratio such as

R=\frac{m}{\sqrt{\sigma}} =\frac{am}{\sqrt{a^2\sigma}}

can compare quantities without assigning a phenomenological value in GeV to a pure-gauge theory. The scale definition and its uncertainty remain part of the result.

When numerator and denominator come from correlated measurements, propagate their covariance. Treating them as independent can distort the reported uncertainty. Keep the original measured quantities available so a different scale choice can be checked.

3. Separate the sources of uncertainty

A finite operator basis may poorly overlap a low state. A short fit window may retain excited-state contamination. A late-time window may lose statistical resolution. Varying the window is useful only when the changes and selection rule are reported.

Volume effects and lattice artifacts require separate comparisons. Autocorrelations affect the effective sample size; topological sampling can require additional checks. A statistical error bar does not automatically include these effects.

For a continuum fit, record the spacings included, the fit ansatz and motivation, correlations between data, and sensitivity to omitting coarse points. Then report the volume analysis separately. See finite volume and continuum limits.

4. Treat a bound according to its certificate

For a mathematical inequality, identify its hypotheses and which constants are uniform in volume or cutoff. For an optimization result, retain the operator set, exact constraints, numerical precision, solver tolerances, and any certificate supporting the endpoint.

A small primal-dual discrepancy is a numerical diagnostic. Whether it establishes a rigorous enclosure depends on the certification procedure and control of numerical error. The bootstrap dossier explains the distinction.

5. State the comparison’s conclusion at the right level

Possible outcomes include agreement within stated uncertainties, incompatible conventions, unresolved systematic differences, or a demonstrably tighter certified interval. “Closer to solving the mass gap” is too vague to serve as a benchmark outcome.

A positive mass in the channels examined is evidence about those channels. Excluding all arbitrarily low physical excitations requires more comprehensive spectral control. Use the proof map to state exactly which obligation a comparison informs.

For practice, use the correlator lab to change operator overlap and the continuum lab to vary lattice spacing and box length while tracking physical units. Their illustrative data are teaching models and should never be entered as measured benchmark values.