What the independent publications actually say

The Newtonian gravitational constant, G, sets gravity’s strength. Precision experiments disagree more than their quoted uncertainties suggest. The 2026 NIST/BIPM paper opens by contrasting “new physics or unrecognized instrumental effects”; it explicitly calls new physics remote and instrumental systematics more plausible. MQ supplies a particular theoretical interpretation of that unresolved problem, developed by independent researcher Jody A. Geiger. [1] [2] [3]

Table 1 separates the external publications from the Institute’s interpretation. The review’s reference 154 is Geiger’s 2023 paper, Measurement quantization, International Journal of Geometric Methods in Modern Physics 20, 2350069; that citation is a public research trail rather than a statement that the review authors verified MQ. [2] [18]

The two external papers provide a reporting connection, not a joint endorsement: one leaves a remote new-physics possibility open; the other documents MQ among approaches to the discrepancy. The BIPM apparatus itself has a long experimental history, so successive campaigns cannot automatically be treated as wholly independent replications. [1] [2] [4] [5]

What MQ means, and where 34.9284 ppm comes from

MQ starts from the axiom measure is discrete and proposes a common logical origin for physical laws through count relationships. Its current architecture distinguishes deriving an equation’s form from selecting the universe, state and units in which that equation is evaluated. The Internal Frame is the discrete count-configuration domain; the System Frame is its observable relational realization under the Frames mapping. The underlying count domain is not assumed to be a microscopic material lattice in a pre-existing continuous space. [19]

For this comparison, the operative claim is narrower: a finite-count electromagnetic realization of the gravitational coupling differs slightly from its upper-count-limit realization. Testing that contrast addresses a specified MQ consequence; it does not by itself verify the framework’s proposed derivation of all physical laws. [3] [6] [7]

Let the dimensionless coefficient ${\mathrm{𝜃}}_{\mathrm{𝑠}\mathrm{𝑖}}$ specify the MQ calibration; let ${\mathrm{𝑛}}_{\mathrm{𝐿}}$ be the realized separation count, and ${\mathrm{𝑄}}_{\mathrm{𝐿}}$ the positive Diophantine realization residual. The calibrated count below is a positive real-valued realization; it is not a claim that this calibration value is itself an integer primitive count. The residual follows from the unit-side right-triangle relation. [6] [7]

In the source convention used here, upright G denotes the upper-count-limit coupling and italic G the finite-count electromagnetic coupling; both have SI units m³ kg⁻¹ s⁻². Their ratio removes the common dimensional normalization. The target below uses the finite-count value as its denominator; “ppm” means the resulting dimensionless fraction multiplied by one million. [3] [6]

Table 2 evaluates the current count relation; Table 3 preserves the historical edition used for the midpoint analysis. The dimensional normalization is upright G = c³t_f/m_f, where c is the speed of light and the source’s fundamental time and mass measures supply t_f and m_f. Multiplying this by twice the residual–count product gives the finite-count value. The normalization’s dimensions are (m s⁻¹)³ × s / kg = m³ kg⁻¹ s⁻². The complete realization and calibration chain is available in the linked preprint. [6] [7] [17]

The author identifies four inputs for the fundamental-measure/constant reconstruction: the fine-structure constant selects the dimensionless coefficient, the speed of light anchors realized length/time and SI units, and electron mass with Bohr radius supplies cross-dimensional normalization. These are declared universe/realization-selection roles, not four statistically independent observations or an assertion that measurement supplied the equation’s logical form. CMB temperature is separately an epoch selector and is not an extra input to the count-gap arithmetic above. The source-specific derivation must still establish each input’s actual role, and numerical uncertainty must retain shared CODATA dependence. The comparisons below treat the declared target as fixed without assigning an uncertainty for possible model discrepancy. [8] [17] [19]

^a All three numerical columns multiply 10⁻¹¹ m³ kg⁻¹ s⁻². The midpoint is the arithmetic mean of the two couplings. The historical difference is 2.331066 × 10⁻¹⁵ m³ kg⁻¹ s⁻². The central midpoint update is approximately −0.000674 ppm. Historical offsets below retain their frozen denominator; the current count calculation is identified separately. [6] [9]

The measured results and the denominator convention

In free-deflection operation, the torsion pendulum’s gravitationally driven motion is measured. In electrostatic-servo operation, electrical feedback supplies a balancing torque. Both modes still require geometry, optical readout and calibration. Associating free deflection with the MQ upper-count realization and servo with its finite-count realization is a retrospective MQ interpretation; it is not a statement made by the experimental collaboration. [1] [3]

^a Coefficients multiply 10⁻¹¹ m³ kg⁻¹ s⁻². Relative uncertainties are one-standard uncertainties. Corrected Mark II values replace the preliminary result affected by a known servo error. Inputs and their provenance are tabulated in the MQ technical reanalysis; the original experimental publications remain the measurement sources. [1] [3] [4] [5]

For context, the review reports MSL results of (6.6742 ± 0.0007) × 10⁻¹¹ in 1999 and (6.67387 ± 0.00027) × 10⁻¹¹ in 2003, in m³ kg⁻¹ s⁻². NIST’s 2026 consensus is (6.67387 ± 0.00038) × 10⁻¹¹ in the same units, with 57-ppm relative uncertainty. The matching rounded central values do not make these one result, nor are successive MSL publications automatically independent inputs. [1] [2]

The contrast uses the servo value as denominator. Let F and E denote free and servo coefficients. If their quoted relative standard uncertainties are r_F and r_E in ppm and their correlation is ρ, first-order propagation of the full quotient gives:

Worked copper example. Substitution gives 10⁶ × (6.674021 / 6.673642 − 1) = 56.7906 ppm. Its uncertainty is (6.674021 / 6.673642) × √(30.3² + 23.2² − 2 × 0.42 × 30.3 × 23.2) = 29.4269 ppm. The distance from zero is 1.930 standard uncertainties; from the MQ target, 0.743. The two-sided Gaussian zero-gap p-value is approximately 0.0536. Compatibility with the target is not a detection of its mechanism. [1] [3] [10]

The frozen book calculation quotes 56.7889 ± 29.4252 ppm for copper; reconstructing the explicitly declared quotient from published rounded coefficients gives 56.7906 ± 29.4269 ppm. Both round to the release’s 56.79 ± 29.43 ppm. This brief uses the reproducible rounded-input reconstruction throughout the plots and fit. [3] [9]

Figure 1 puts all four Table 4 contrasts on one axis; the broad BIPM bars and copper’s stronger precision are visible without treating the fitted row as extra evidence.

How the four-pair fit handles shared uncertainty

The NIST paper’s correlation matrix, in order copper servo, copper free, sapphire servo, sapphire free, has off-diagonal entries 0.42, 0.38, 0.12, 0.23, 0.23 and 0.25. Multiplying these correlations by the corresponding absolute standard uncertainties constructs the four-result covariance. Propagating it through both quotient derivatives, ∂δ/∂F = 10⁶/E and ∂δ/∂E = −10⁶F/E², gives the two-mode covariance block below. The reconstruction assumes zero covariance between the two BIPM campaigns and between those campaigns and NIST; that assumption is not evidence that every inherited apparatus error is independent. [1] [3]

Table 5 gives the resulting contrast covariance; Table 6 reports the fitted weights and reconstructed contrasts. Equation labels in this brief identify its own calculation steps, rather than reproducing numbering from the original papers.

For contrast vector d and covariance V, generalized least squares (GLS) estimates one common gap μ. The vector 1 has four unit entries. This is a model of a shared contrast; it does not establish that all four instruments realized exactly the same physical mode effect.

^a The copper pair supplies about 81.56% of the estimator weight. The aggregate is strongly influenced by that pair, rather than four equally precise replications.

The fit is 60.9932 ± 26.9309 ppm, or 60.99 ± 26.93 ppm. The earlier frozen summary 60.994 ± 26.930 ppm differs only in final reconstruction digits. The residual χ² is 1.412 for three degrees of freedom. Zero lies 2.265 standard uncertainties below the fitted mean; the MQ target lies 0.968 below it. A two-sided Gaussian zero-gap test gives p ≈ 0.0235; a one-sided test gives half that value only if the positive direction was specified in advance. [3] [9]

Interpreting the 8.14-to-1 likelihood ratio

With the same fixed covariance for each candidate gap, χ²(μ) = (d − μ1)ᵀV⁻¹(d − μ1), and relative likelihood is exp[−(χ²(μ) − χ² at the fitted mean)/2]. Both fixed comparisons have no fitted gap parameter. Their ratio quantifies support inside this particular Gaussian model; it is neither a posterior probability nor odds that MQ is true.

Table 7 reports the numerical model comparisons; Figure 2 shows why both the MQ target and zero remain plausible locations on the broad likelihood curve.

The book also reports a roughly 5.4-to-1 comparison after introducing additional heterogeneity. Its precise extra-variance model and fitting inputs are not supplied in the materials reviewed here, so that figure is not regenerated or used as this brief’s main result. NIST’s published consensus separately allows configuration-specific additional “dark uncertainty”—an uncertainty treatment, not dark matter. Adding plausible unknown offsets can weaken discrimination between fixed gaps. [1] [9]

The primary NIST uncertainty budgets contain substantial conventional terms: autocollimator nonlinearity reaches 85.7 ppm for sapphire free deflection, while mass integration contributes 16.2 ppm to each configuration. The paper investigates unknown torque offsets, including possible resistive-heating effects. A remaining mode difference is not evidence that these effects have been eliminated. [1]

The historical ±3.5-ppm graphic, with the full uncertainty

The useful historical comparison is the midpoint between the two MQ coupling realizations, not an independently observed 35-ppm split. Chapter 7 of The Rules Beneath Reality and the Institute’s experimental page give two aggregate historical treatments: a base analysis and an expanded analysis including MSL. Their central offsets are +3.04 and −2.65 ppm. A ±3.5-ppm display window contains both centers; it is not their measured uncertainty, and the values are not exactly ±2.5 ppm. [9] [13]

Use the frozen midpoint M₀ = (6.6738448362 + 6.6740779428)/2 = 6.6739613895, in coefficient units of 10⁻¹¹ m³ kg⁻¹ s⁻². For a reported historical midpoint M with standard uncertainty u, calculate its fractional offset, uncertainty in ppm and standardized residual as follows:

^a Coefficients multiply 10⁻¹¹ m³ kg⁻¹ s⁻². The p-values describe compatibility with a fixed midpoint under a Gaussian approximation; they are not probabilities that MQ is correct.

For the base treatment, the subtraction is 6.67398168 − 6.6739613895 = 0.0000202905; dividing by the frozen midpoint and multiplying by one million gives +3.040248 ppm. For the expanded treatment, 6.67394370 − 6.6739613895 = −0.0000176895 gives −2.650525 ppm. At those small standardized residuals, fixing the center to MQ retains approximately 99.72% and 99.70% of each Gaussian curve’s maximum likelihood. That high compatibility reflects their broad uncertainties and does not measure a theory’s chance of being true.

What can be reconstructed from the historical record

The MQ interpretation separates gravitational results by their physical measurement route, including electromagnetic versus macroscopic realization, then compares the midpoint of the resulting group estimates with MQ. If the two group means are μ_E and μ_M with covariance C, their midpoint and variance are:

A reproducible population fit would supply each experiment’s value, inclusion rule, group assignment, revised/corrected status and full covariance. Given a membership design matrix A, individual-value vector g and their covariance Σ, the group estimator is (AᵀΣ⁻¹A)⁻¹AᵀΣ⁻¹g and its covariance is (AᵀΣ⁻¹A)⁻¹. The reviewed book and public summary supply the aggregate midpoints above, but not that complete per-experiment classification and covariance file. Thus Table 8 reproduces the downstream midpoint arithmetic; it does not independently rebuild the original historical population analysis. That underlying file should be requested before describing the grouping as independently replicated. [9] [13]

A midpoint match alone cannot establish two distinct populations or determine their separation. Absolute coupling values also remain a constraint: the corrected BIPM Mark II combined value, (6.67554 ± 0.00016) × 10⁻¹¹ m³ kg⁻¹ s⁻², lies about 9.14 quoted standard uncertainties above the current MQ upper-count central value, ignoring any MQ model-discrepancy allowance. Agreement of a mode contrast does not resolve that absolute-value tension. [3] [5] [6]

The human record: a dated message, then a later mode interpretation

Geiger’s preserved message to NIST contacts has an embedded sent date of September 26, 2021, 3:11 p.m. It describes different electromagnetic and macroscopic measurement routes and asks about the physical interpretation of the 2018 CODATA gravitational constant. The screenshot’s date governs this reproduction; a September 21 date appearing in a secondary narrative is not substituted for it. [9] [14]

Table 9 distinguishes the correspondence, theoretical publication, external citation and experimental publication. Figure 4 preserves the supplied message image with a selected accessible transcription below it.

Figure 4 · Open the supplied 2021 email facsimile

“Most simply, you are selecting experiments that use either an electromagnetic (quantum) or macroscopic approach to the measures of G and ħ.”

This selected transcription makes the message’s general distinction accessible without opening an attachment. The screenshot’s calculation table includes expressions using both ℏ and G as inputs; it should not be described as proving that all those tabulated values were derived independently of G. The correspondence also does not demonstrate that the later exact mode assignment was preregistered, that a recipient replied, or that NIST endorsed or acted on it. Source versions and the original export can be requested through the Institute’s media contact. [14]

The proposed experiment: a controlled comparison in one shared apparatus

The Institute proposes a contactless beam experiment that keeps the apparatus and persistent magnetic support circuit shared while alternating calibration, free-mode and electrostatic-mode states. The current public top graphic identifies Project 2.39, October 30, 2026, retaining the geometry/method snapshot identified as 2.37. Its lower page text also contains older baseline material. The specifications below come from the dated current graphic; earlier design numbers are not combined with them. [13]

Figure 5 uses the Institute’s current concept illustration; Figure 6 redraws the functional arrangement and Table 10 collects the declared dimensions and schedule. Table 11 preserves the current 14-entry comparison ledger.

The proposed reference source-reversal torque is 260.791678 nN m. Applying the design’s frozen, rounded 34.93-ppm contrast conditionally gives 9.1094533 pN m. This engineering normalization is not an independently derived apparatus response to MQ. Using the current unrounded 34.928385-ppm target instead gives approximately 9.1090323 pN m; the ledger below retains the design’s frozen denominator so its ppm values remain reproducible. [13]

The current design ledger consolidates overlapping terms into 14 comparison entries. Its largest entry combines optical, mechanical and thermal behavior; the next is source/self-gravity modeling. Their stated root-sum-square total is summarized below. These are unqualified comparison estimates that may overestimate or underestimate real uncertainty, not achieved instrument performance. [13]

^a Here 1 aN m = 10⁻¹⁸ N m, 1 pN m = 10⁻¹² N m, and “ppm of gap” uses the approximately 9.10945-pN m conditional gap as denominator. These ppm values do not use the full source torque or the gravitational constant as denominator. Displayed rounded entries need not reproduce every last digit of the retained design total.

The two leading entries account for approximately 92.29% and 7.41% of the retained variance. A 5-ppm-of-gap benchmark corresponds to about 45.55 aN m; the listed 48.30-aN m total is 5.30 ppm of gap. Neither number is an experimental uncertainty already demonstrated. Establishing that budget requires measurements of drift, alignment, thermal response, source reversal, optical phase, support history and covariance between retained terms. Root-sum-square addition is justified only to the extent that the retained grouping removes double counting and residual cross-correlations are negligible or included. [13]

Table 12 gives a different precision benchmark: the standard uncertainty of the measured mode contrast, in ppm relative to the servo coupling. For an unbiased Gaussian contrast with the full systematic budget included, nominal separation from zero is the target divided by that standard uncertainty. These thresholds do not use “ppm of gap,” and they do not establish that the proposed apparatus can reach them. [3] [9]

^a A sensitivity calculation under the stated model, not a forecast of an achieved significance, a posterior probability or a substitute for conventional-offset testing.

What would make this a discriminating MQ test?

Before examining the relevant contrast, the experiment would need a fixed mode-to-realization map, signed target, torque estimator, material/configuration choices, inclusion rules and correlated uncertainty model. Calibration must separate real torque from optical-path change, local elastic strain and feedback effects. A universal extra rescaling of every apparatus length is not an established conversion model and can give the wrong sign. The required prediction is for the actual instrument estimator, with existing laboratory electrical calibrations counted once. [3] [6] [13]

An independently reviewed implementation could then compare a fixed MQ contrast with zero and with explicit conventional-offset models, repeat source reversals and configuration changes, and publish raw time series with calibration and covariance records. The proposed experiment has not yet supplied such a result. It provides a prospective route to a stronger test than a retrospective comparison of historical means. [13]

Source trail and useful follow-up reading

For the experimental story, start with NIST’s abstract, motivation, uncertainty tables and consensus construction; then read the review’s Section 5.3 rather than relying on the citation alone. For the MQ claim, follow the count relation, denominator convention and apparatus assignment separately. The existing mode-gap evidence brief provides the input and covariance tables; the gravity derivation page links the current normalization; and the planned experiment page carries the evolving design snapshot. [1] [2] [3] [6] [13]

The publication index lists the eight published MQ papers and their journal links. The historical-comparison page explains the Institute’s contribution catalog and scoring method. Those catalog scores are the Institute’s authored assessment, not an independently calibrated measure of theory validity or a substitute for the experimental comparisons here. The correspondence brief expands the documentary timeline. [14] [15] [16]

For an interview, the unresolved technical questions are concrete: Which physical estimator makes the mode assignment valid? Can the original historical grouping file be released? How robust is the contrast to configuration-specific unknown offsets? Which experiment parameters and analysis rules will be fixed prospectively? Answers to those questions would materially strengthen a story without changing the published measurements.