What is being combined

The four observations are differences between free-deflection and electrostatic-servo determinations within BIPM Mark I, corrected BIPM Mark II and NIST’s copper and sapphire configurations. Both methods infer the gravitational interaction. Free deflection uses the angular response and torsional calibration; servo operation uses a balancing electrostatic torque. Different calibration chains can expose systematic errors, but active electronics alone does not prove the MQ realization assignment. [2] [3] [4]

A mode contrast is defined here as free deflection divided by servo, minus one, multiplied by one million. Its units are parts per million, abbreviated ppm; the servo value is the denominator. A combined gravitational-constant result, an absolute difference from MQ and a within-apparatus mode contrast are distinct quantities. MSL absolute measurements and the NIST consensus are not added as extra direct pairs. [1] [4]

The four primary pairs and their uncertainty inputs

Table 1 gives the explicit central values and rounded relative standard uncertainties used in this reproduction. All central values multiply 10⁻¹¹ m³ kg⁻¹ s⁻². Relative uncertainties are in ppm, and the final column is the dimensionless within-pair correlation. Mark II uses the corrected 2014 record rather than the uncorrected 2013 result. NIST values and correlations come from the published paper’s Tables 16–18. [1] [2] [3] [4]

For each ratio, first-order propagation multiplies the free-to-servo ratio by the square root of the two squared relative uncertainties minus twice their correlation times their product. Retaining this covariance is essential. The negative BIPM correlations increase the contrast uncertainty relative to independent-input propagation; the positive within-pair NIST correlations reduce it. These are the stated primary budgets, not statistical-only errors. [3] [4] [10]

The resulting contrasts are 17.98 ± 97.66 ppm for Mark I, 106.36 ± 103.87 ppm for Mark II, 56.79 ± 29.43 ppm for NIST copper and 149.72 ± 92.01 ppm for NIST sapphire. Each uncertainty is one standard uncertainty. All central values are positive, but none individually excludes zero at two standard uncertainties under these budgets. Their signs alone do not quantify independence or establish a common physical origin.

Covariance between the NIST contrasts

The NIST measurements share correlated inputs across materials. In the order copper servo, copper free, sapphire servo and sapphire free, the six off-diagonal correlations are 0.42, 0.38, 0.12, 0.23, 0.23 and 0.25. Propagating the four-input covariance through the two ratios gives a copper–sapphire contrast covariance of 462.3295 ppm² and correlation 0.170759. Treating the contrast error bars as independent would omit that contribution. [4]

Table 2 shows the complete contrast covariance used here, rounded for display. Computation retains the additional digits implied by Table 1. Covariance has units ppm², unlike a dimensionless correlation matrix. Zeros between campaigns are an explicit synthesis assumption, not evidence that all historical systematic influences were independently measured and found absent. The nonzero NIST off-diagonal entries are retained. [1] [3] [4]

For an executable reconstruction, form the covariance of the four NIST absolute values from their central values, relative standard uncertainties and correlations. Differentiate each contrast with respect to its free and servo values, then multiply the resulting derivative rows through that covariance. Equivalently, the NIST cross-term is the product of the two free-to-servo ratios times four relative-error products: free/free plus servo/servo, minus free/servo and servo/free. This reproduces Table 2 without assuming independent materials.

How the common-gap fit is obtained

The generalized least-squares fit assumes that all four contrasts estimate one common mean with the covariance held fixed. Solve the linear system consisting of the covariance matrix times an unnormalized weight vector equal to a column of ones. Divide the resulting weights by their sum. The weighted sum of the four contrasts is the fitted common gap; the square root of the reciprocal unnormalized-weight sum is its standard uncertainty.

Table 3 reports the calculated contrasts and normalized linear weights. These weights are not probabilities of a physical cause. Copper contributes 81.56 percent of the estimator’s weight. The narrower copper budget therefore strongly influences the common mean; describing this as four equally strong replications would obscure the actual information balance.

This calculation gives 60.9932 ± 26.9309 ppm, or 61.0 ± 26.9 ppm at appropriate summary precision. The public source quotes 60.994 ± 26.930 ppm. The final-digit difference is immaterial at the precision of the historical inputs, but the two numerical records are identified rather than silently equated. The fitted center is 2.265 standard uncertainties above zero. The residual chi-square is 1.412 for three degrees of freedom under the specified fixed-covariance common-gap model. [1]

Figure 1 shows the individual intervals and the common fit. The fit is derived from the four pairs; it must not be counted as a fifth independent observation. A low residual chi-square under this model does not show that omitted mode-dependent effects have been bounded.

The fixed MQ target and its additional apparatus assignment

Let ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ denote the dimensionless MQ invariant coefficient and ${\mathrm{𝑛}}_{\mathrm{𝐿}}$ the positive-real electromagnetic-demarcation length-count parameter. This calibrated continuation is distinct from primitive integer cardinality. The current central coefficient used here is 3.26239030504. Equation (1a) is the source count constraint; these Brief labels differ from the source’s equation numbers. [1] [6]

Let ${\mathrm{𝑄}}_{\mathrm{𝐿}}$ be the dimensionless Diophantine realization residual. Its positive-count definition is Eq. (2b). Evaluation uses the algebraically equal reciprocal of the root plus the count to avoid subtractive cancellation. [6]

Upright $\mathrm{G}$ denotes the upper-count gravitational coupling, and italic $\mathrm{𝐺}$ the finite-count electromagnetic-demarcation coupling. Both have units m³ kg⁻¹ s⁻². The source relates them through Eq. (3c). [1] [6] [7]

Dividing Eq. (3c) gives the exact fractional target in Eq. (4d). Scaling this dimensionless expression by one million gives 34.9284 ppm after rounding, with count 84.6005456716. No fit to the four contrasts is used to obtain that target.

Associating free deflection with the upper-count realization and servo with the electromagnetic-demarcation realization is an additional physical interpretation. MQ’s numerical construction predates the 2026 NIST paper; this apparatus-specific association was retrospective. Current calibration digits do not establish an exact historical preregistered prediction. The equations also do not justify applying one universal rescaling to every already realized laboratory length or electrical calibration. [1] [8]

Three model comparisons answer different questions

Under the Gaussian model with covariance fixed, the relative likelihood of any candidate common mean can be reconstructed from its distance to the fitted center. Square that difference, divide by twice the squared common-fit standard uncertainty, change the sign and exponentiate. Normalize the result to one at the freely fitted center. All residual components orthogonal to the common mean cancel in these likelihood ratios; this is why the scalar reconstruction agrees with the full four-contrast calculation.

Table 4 gives 8.136 for fixed MQ divided by exact zero, reported as 8.14 to 1. The freely fitted model gives 1.597 times the fixed target’s raw likelihood, reported as 1.60. The fitted center is 0.968 standard uncertainties above the MQ target. This broad compatibility does not establish that the true gap equals the target or that other positive offsets are excluded. Figure 2 plots the same likelihood, rather than a posterior density. [1]

Akaike’s information criterion adds twice the fitted-parameter count to minus twice the maximized log likelihood. The free-gap model adds one parameter, giving a limited relative-criterion weight of 1.70 to 1 in favor of the fixed target over the free gap. Schwarz’s criterion uses the fitted-parameter count times the logarithm of sample size; treating the four contrasts as the nominal sample size gives 1.25 to 1. These are illustrative complexity comparisons within the stated likelihood, not calibrated odds that MQ is physically true. The small correlated dataset particularly limits an asymptotic Schwarz interpretation. [11] [12]

The public MQ source reports that a broader heterogeneity treatment reduces fixed MQ versus zero to about 5.4 to 1. That profile is retained as a reported sensitivity result and was not independently regenerated for this Brief. A reproducible extension needs the exact additional variance structure, allowed range, fit procedure and likelihood-normalization treatment. It must not be represented as the same calculation as Tables 1–4. [1]

Historical exclusions and absolute-value tension

An early preliminary discrepancy of roughly 300 ppm associated with servo operation is not counted as another mature pair. A known apparatus problem is evidence that method-dependent bias can occur; it is not a successful independent prediction. Corrected Mark II values and its revised correlation must replace the uncorrected historical combination in a present-day calculation. [1] [3]

The corrected Mark II combined constant is 6.67554 ± 0.00016, in units of 10⁻¹¹ m³ kg⁻¹ s⁻². The current MQ upper-count coefficient is 6.6740779383 in the same units. Subtracting and dividing by the published BIPM standard uncertainty gives 9.14. This is an absolute-value tension before allowing any unquantified MQ model discrepancy. A favorable sign for the within-pair contrast does not remove it. [1] [3]

The combined NIST result, 6.67387 ± 0.00038 in the same units, is a consensus with a different uncertainty construction. Its full analysis allows additional unidentified configuration effects beyond the primary budgets used here. The term dark uncertainty in that paper refers to unexplained measurement variation, not dark matter. The present common-gap uncertainty is therefore not a full uncertainty bound on every possible apparatus bias. [4]

What a discriminating experiment must resolve

A common positive mode gap can be compatible with conventional offsets, a physical contribution or a mixture. Four positive central contrasts do not establish four independent replications of an MQ mechanism. This fit holds a common mean and stated covariance fixed; the actual instruments differ in material, calibration and known sensitivities. Assumptions about unmodeled campaign effects are part of the inference, not empirical findings supplied by the fit.

A prospective test must fix the target, mode assignment, normalization and analysis before evaluating new results. It needs independent checks of angular calibration, electrical torque transfer, pressure and thermal effects, suspension response and source reversals. The experimental estimator must be derived for the actual apparatus without applying a correction twice. Better precision by itself cannot separate a physical mode contribution from an unrecognized mode-specific bias. [4] [8] [9]

Dates, authorship and reproduction scope

The historical experimental records are Mark I in 2001, the corrected Mark II account in 2014 following its 2013 report, and NIST’s paper published April 16, 2026. September 27, 2026 is the archive source-audit snapshot. This Brief was prepared October 26, 2026. The experiments belong to their respective collaborations; the combined likelihood and MQ apparatus interpretation are subsequent analyses by the MQ research program. [1] [2] [3] [4]

This Brief reproduces the four primary-budget contrasts, their stated covariance synthesis, fit weights, current numerical target and conditional model comparisons. It does not reprocess torque data, refit the full NIST consensus or reproduce the reported heterogeneity profile. For follow-up reporting, request those separate analyses and the provenance of each covariance assumption, rather than treating a short likelihood ratio as a complete evidence assessment.