⚛️ general relativity

Covariance scaling and a dual-mode torque diagnostic in measurements of the gravitational constant

This paper analyzes the CODATA-2018 archive of gravitational constant measurements using a proportional covariance-scale model to demonstrate that the recent NIST torsion-balance result is statistically consistent with historical data once extra dispersion is applied, while also identifying a common absolute torque offset in NIST's dual-mode measurements that challenges the assumption of independent Type-A uncertainties.

Original authors: Vineet Kumar

Published 2026-10-06

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CC BY 4.0

Original authors: Vineet Kumar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Gravity is the most familiar force in our daily lives, yet it remains the most mysterious in the precise language of physics. For over two centuries, scientists have tried to pin down a single number that describes how strongly the Earth pulls on objects, a value known as the gravitational constant. While we use this number to calculate the orbits of planets and the tides of oceans, laboratory experiments designed to measure it have stubbornly refused to agree with one another. Different teams, using different machines in different labs, keep getting slightly different answers. The spread between these results is far wider than the scientists' own error bars suggest should be possible. This disagreement has become a quiet crisis in metrology, the science of measurement, because it implies that either the experiments are missing something fundamental, or the way scientists report their uncertainty is incomplete.

A recent analysis by Vineet Kumar takes a fresh look at this puzzle, not by building a new machine, but by re-examining the history of how these measurements have been recorded. The study focuses on a specific set of sixteen measurements collected by an international committee known as CODATA, which serves as the standard reference for fundamental constants. The core of the investigation is a statistical audit: checking whether the reported uncertainties in these past experiments truly reflect the reality of the data, or if they are simply too optimistic. The researcher also looked at a very recent, high-profile experiment conducted at the National Institute of Standards and Technology (NIST) in 2026. This new experiment is particularly interesting because it uses a device that shares a direct family history with one of the older measurements in the archive, allowing for a unique test of consistency.

The analysis begins by treating the sixteen historical measurements as a single group. When the researcher applied a statistical model that assumes the reported errors might be systematically too small, the data revealed a clear pattern. The spread of the results was so large that the reported uncertainties had to be multiplied by a factor of nearly three to make the numbers fit together logically. This suggests that the historical record contains a hidden layer of variation that the original reports did not capture. To ensure this finding wasn't just an artifact of one specific experiment, the researcher removed the measurement that shared the most direct equipment lineage with the new NIST study. Even without that specific data point, the remaining fifteen measurements still required the same large adjustment to their uncertainties. The conclusion is robust: the historical archive, as it stands, underestimates the true variability of these measurements.

With this historical context established, the study turned to the 2026 NIST result. This experiment measured gravity in four different configurations, using two types of test masses and two different operating modes. Because these four measurements were taken with the same equipment, they are not independent; they share the same potential sources of error. When the researcher compared these four new values against the historical average using the original, unadjusted uncertainties, the new result looked like a statistical outlier, a result so unlikely it should almost never happen. However, once the researcher applied the same "inflation factor" of nearly three that was derived from the historical data, the picture changed completely. The NIST result, which had seemed strange, became perfectly consistent with the rest of the archive. The new measurement was not an error; it was simply part of a larger, more variable reality that the old uncertainty estimates had failed to describe.

The paper goes further to investigate a specific physical mechanism that might explain why the two operating modes of the NIST machine gave slightly different answers. The researchers looked at the difference in torque, or twisting force, measured when the machine was free to move versus when it was held in place by an electronic servo. They tested two competing ideas: either the difference was caused by a tiny, fixed error in the machine's calibration that affected all measurements equally, or it was a proportional error that grew larger as the force being measured grew larger. The data strongly favored the first idea. The four different configurations all showed a nearly identical fixed difference of about 1.918 piconewton-meters. This specific value is small, but it is consistent across different materials and setups, pointing toward a constant offset rather than a scaling mistake.

This finding offers a concrete diagnostic tool for future experiments. If the error is indeed a fixed offset, then changing the size of the gravitational force in a future experiment should change the percentage error in a predictable way. If the error were proportional instead, the percentage would stay the same. The study suggests that a future experiment could deliberately vary the distance between the masses to test this directly. While the paper does not claim to have solved the mystery of the gravitational constant or identified the exact physical cause of the offset—whether it is a thermal effect, an electrical glitch, or something else—it provides a clear path forward. It confirms that the disagreement in the field is real and quantifiable, and it offers a method to check if new measurements are truly independent or if they are simply repeating the same hidden biases. The work does not propose a new value for gravity, but it clarifies the landscape of uncertainty, showing that the numbers are not as scattered as they seem, provided one accounts for the hidden scale of the errors.

Technical Summary: Covariance Scaling and a Dual-Mode Torque Diagnostic in Measurements of the Gravitational Constant

Problem Statement
The Newtonian gravitational constant, $G$ , remains the least precisely determined fundamental constant. Despite two centuries of experimentation, independent laboratory determinations disagree by margins significantly larger than their individual reported uncertainties suggest. The CODATA-2018 adjustment addressed this discordance by applying a uniform multiplicative uncertainty expansion (a factor of 3.9) rather than treating the reported covariance structure as adequate. This paper investigates whether the reported covariance structure of the sixteen determinations in the CODATA-2018 archive is sufficient under a proportional covariance-scale model. Furthermore, it evaluates the statistical consistency of a later, four-configuration NIST measurement (2026) with this historical archive, given that the NIST apparatus shares a lineage with the BIPM Mark II balance (recorded as BIPM-14 in the archive). Finally, the paper examines the physical nature of discrepancies in dual-mode torsion-balance measurements by analyzing matched free-minus-servo torque differences.

Methodology
The analysis employs a proportional covariance-scale model where the reported covariance matrix $S$ is scaled by a dimensionless factor $\lambda^2$ . The model assumes the vector of historical determinations $y$ follows a normal distribution $y | \mu, \lambda \sim N(\mu \mathbf{1}, \lambda^2 S)$ .

Historical Covariance Audit: The author estimates the maximum-likelihood scale factor $\hat{\lambda}_{MLE}$ for two configurations: the complete 16-determination archive and a 15-determination subset excluding BIPM-14 (to avoid direct apparatus overlap with the test measurement). The consistency of the archive is assessed using the residual quadratic $Q$ and compared against the classical Birge ratio. Model selection is performed using the Bayesian Information Criterion (BIC) to compare the proportional scale against additive floor and power-law alternatives.
Predictive Consistency Check: The 2026 NIST result is treated as a single, correlated four-component vector. A predictive statistic is constructed by combining the NIST reported covariance with the uncertainty of the historical mean. The discrepancy is evaluated under two conventions:
- Uniform Scaling: The inferred historical scale $\hat{\lambda}_{MLE}$ is applied to both the historical mean uncertainty and the NIST internal covariance.
- Mean-Only Scaling: The scale is applied only to the historical mean uncertainty, leaving the NIST internal covariance unscaled.
  Significance is assessed using finite-sample $F$ distributions and diagnostic $\chi^2$ tests.
Dual-Mode Torque Diagnostic: Matched free-minus-servo torque differences ( $d_i$ $d_{i}$ ) from four NIST configurations are analyzed. Two competing one-parameter models are fitted:
- Common Absolute Offset: $d_i = \Delta N + \epsilon_i$ (constant torque bias).
- Fractional Scale: $d_i = f N_{free,i} + \epsilon_i$ (constant fractional error).
  The models are compared via the difference in their chi-square statistics ( $\Delta\chi^2$ ), noting that they are not nested.

Key Results

Historical Inadequacy: The reported covariance of the CODATA-2018 archive is inadequate under a proportional-scale audit. Even after excluding the lineage-linked BIPM-14 entry, the required inflation factor is $\hat{\lambda}_{MLE} = 2.967$ (with $Q = 132.088$ for $N=15$ , $p \approx 2.6 \times 10^{-21}$ ). The proportional scale model is preferred over additive floor or power-law alternatives based on BIC.
NIST Consistency: Evaluated against the unscaled historical covariance, the 2026 NIST four-configuration vector is highly unusual ( $Q_0 = 42.625$ , $p \approx 1.24 \times 10^{-8}$ ). However, when the historically inferred extra dispersion ( $\hat{\lambda}_{MLE} = 2.967$ ) is applied uniformly, the result becomes statistically unremarkable ( $Q_\lambda = 4.840$ , finite-sample $F$ $p = 0.382$ ). Under a mean-only scaling convention, the vector remains discrepant ( $p \approx 2.4 \times 10^{-7}$ ).
Torque Model Preference: The four matched torque differences are consistent with a single common absolute offset of $\hat{\Delta N} = (1.918 \pm 0.215)$ pN m ( $\chi^2_{abs}/\nu = 0.364$ , $p=0.779$ ). In contrast, a fractional-scale model yields a poor fit ( $\chi^2_{frac}/\nu = 4.045$ , $p=0.0069$ ). The contrast $\Delta\chi^2 = 11.041$ strongly favors the absolute offset model over the fractional alternative under the stated independent-Type-A treatment.
Physical Interpretation: The preference for an absolute offset implies that the discrepancy between free and servo modes corresponds to a fractional shift in $G$ that is inversely proportional to the gravitational signal torque ( $N_{grav}$ ). This is consistent with the observation that the sapphire configuration (lower $N_{grav}$ ) exhibits a larger fractional discrepancy than the copper configuration.

Significance and Claims
The paper does not propose a revised value for $G$ nor does it identify a specific physical mechanism for the observed dispersion. Instead, it provides:

A Covariance Audit: A likelihood-based quantification of the under-reporting of uncertainty in the historical $G$ archive, confirming that a proportional inflation factor of nearly three is necessary to achieve statistical consistency.
A Lineage-Qualified Consistency Check: A demonstration that the 2026 NIST result is not an outlier when viewed through the lens of the historically inferred dispersion, provided a uniform scaling convention is adopted.
An Experimentally Testable Diagnostic: A characterization of the dual-mode torque discrepancy as a common absolute offset rather than a fractional error. The paper suggests that a future experiment systematically varying the source-mass separation (and thus $N_{grav}$ ) could definitively discriminate between the absolute-offset and fractional-scale hypotheses.

The author emphasizes that their conclusions are descriptive of the statistical structure of the existing data and do not establish the physical origin of the systematic errors. The exclusion of BIPM-14 mitigates but does not eliminate shared apparatus lineage, and a truly independent redetermination using an unrelated technique remains the decisive test for these conclusions.

Technical Summary: Covariance Scaling and a Dual-Mode Torque Diagnostic in Measurements of the Gravitational Constant

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