What is being compared

A Galactic rotation curve gives the circular speed associated with the inferred gravitational field at different distances from the Galactic center. It is not a direct record of every star following a circular orbit. Eilers and collaborators inferred the circular curve through a Jeans-equation analysis of more than 23,000 luminous red giants, assuming an axisymmetric potential. Their study excludes the barred inner 5 kpc, where that assumption is inappropriate. [2]

The MQ comparison uses 38 tabulated circular-velocity values from 5.27 to 24.82 kpc, with asymmetric statistical uncertainties and radius-dependent systematic fractions reproduced by Jiao and collaborators. These measurements supply the target. The McMillan density model supplies the baryonic source through a separate published construction. Distinct source and target provenance does not imply that every Galactic assumption is statistically independent. [1] [2] [3] [4]

McMillan’s complete model includes a dark halo as well as ordinary matter. The MQ calculation retains only the bulge, thin stellar disk, thick stellar disk, neutral atomic hydrogen disk and molecular-gas disk. Their axisymmetric Newtonian mid-plane radial forces are integrated at the 38 target radii and summed. The Navarro–Frenk–White halo force is excluded before the MQ input is formed. This is a specific use of McMillan’s baryonic reconstruction, not a claim that his original mass model omitted dark matter. [1] [3]

No density normalization, MQ acceleration scale or halo profile is re-estimated from the Eilers velocity residuals in the principal prediction. The adopted astrophysical model nonetheless contains published fitted quantities, scale lengths and source assumptions. “No galaxy-specific MQ fit” is therefore more precise than “parameter-free.” Figure 1 shows the source and target roles. [1]

The fixed source-to-response law

Let $\mathrm{𝑅}$ denote galactocentric radius in metres, ${\mathrm{𝑣}}_{\mathrm{New}}\left(\mathrm{𝑅}\right)$ the positive Newtonian baryonic circular speed in metres per second, and ${\mathrm{𝑔}}_{\mathrm{bar}}\left(\mathrm{𝑅}\right)$ the corresponding positive baryonic acceleration in metres per second squared. Equation (1a) converts the source speed into acceleration. These are Brief equation labels, distinct from the manuscript’s numbering. [1]

Let ${\mathrm{𝑎}}_{\mathrm{MQ}}$ denote the positive MQ characteristic acceleration scale and ${\mathrm{𝑔}}_{\mathrm{real}}\left(\mathrm{𝑅}\right)$ the realized radial acceleration, both in metres per second squared. The current galactic construction gives Equation (2b) for a positive baryonic source. The accelerations here are evaluated at the same radius. [1]

Let ${\mathrm{𝑣}}_{\mathrm{real}}\left(\mathrm{𝑅}\right)$ denote the predicted positive circular speed in metres per second. Using the circular relation between acceleration, squared speed and radius gives Equation (3c). [1]

Within MQ, bounded realization describes the count-constrained Frames mapping into measurable gravitational response. The source derivation assigns the square-root depth to the positive quadratic System Frame norm and the exponential fraction to subdivision consistency. These are theoretical ingredients of the proposed law. Agreement with a rotation curve tests the resulting response; it does not by itself establish that this underlying ontology is implemented in nature. [1]

The denominator approaches unity at high baryonic acceleration, recovering the Newtonian source response. At low acceleration it produces a larger realized response for the same baryonic source. The functional form matches the empirical radial acceleration relation reported by McGaugh, Lelli and Schombert in 2016. That functional shape is acknowledged prior art. The claimed MQ contribution is its proposed derivation and upstream normalization, not a new discovery of that empirical curve. [1] [6]

The reported Milky Way statistics use the frozen nominal acceleration 1.20040492178 × 10⁻¹⁰ m s⁻². In MQ its normalization inherits non-galactic inputs through the frame-resolved cosmological construction. It is not fitted from these target velocities. A later overview distinguishes a revised shared-input calibration from the nominal value used for the reported galaxy analyses; the statistics here are not relabeled as a rerun under that revision. [1] [8]

How much the residuals change

The signed velocity residual is the MQ prediction minus the observed circular speed. Root-mean-square residual is obtained by squaring those differences, averaging across the retained radii and taking the square root. It gives larger discrepancies more weight than a mean absolute residual. It is a speed statistic, not a percentage uncertainty or a likelihood. [1]

Table 1 and Figure 2 give the same source-reported RMS values for the complete sample and its radial subsets. The percentage reduction is calculated by subtracting the MQ-to-Newtonian RMS ratio from unity and multiplying by 100. Using the displayed complete-sample values yields 87.112%. This arithmetic checks the stated reduction without reproducing the component-force integration or the 38 raw residuals.

Across the complete sample the source also reports a mean absolute residual of 7.4312 km/s, a maximum absolute residual of 25.6269 km/s and a mean absolute relative error of 3.54277%. The rounded signed mean is −3.0744 km/s. Positive and negative departures can offset in that mean while leaving a much larger RMS. None of these quantities can be substituted for the covariance-dependent goodness-of-fit statistic. [1]

The inner 30 points extend to 19.71 kpc and have a negative signed mean; the outer eight run from 20.27 to 24.82 kpc and have a positive signed mean. The outer RMS is larger, but its observational uncertainties are also much larger. Accordingly, an easier statistical acceptance in the outer sample does not establish a more accurate outer prediction. The two subsets also do not constitute two independently observed galaxies. [1]

Why the covariance treatment changes the conclusion

Covariance describes how errors at different radii move together. A source normalization change can shift a whole predicted profile coherently, rather than independently perturbing each point. A diagonal treatment retains variances but discards cross-radius covariance. Table 2 separates the diagonal result from the specific propagated source calculation; the central frozen prediction is the same. [1]

For the observational diagonal, the MQ analysis combines each adopted systematic fraction, converted to a speed uncertainty, with the appropriate asymmetric statistical uncertainty in quadrature. This pointwise construction omits a measured full observational covariance. Its very small right-tail probability formally rejects the complete sample under that incomplete error model. Reporting only the later 0.400 result would conceal this dependence. [1] [2] [4]

For the source propagation, McMillan’s published posterior standard deviations and eight-parameter correlation matrix are used to construct a three-parameter conditional covariance. Only the thin-disk, thick-disk and bulge density normalizations are varied. Disk scale lengths and the other parameters are held at their adopted best-fit values. Conditional covariance describes uncertainty with those quantities fixed; it is not the larger marginal uncertainty obtained by allowing every correlated parameter to vary. [1] [3]

The reported Monte Carlo calculation uses 200,000 correlated source draws. For each draw, those three density normalizations rescale their component forces, the baryonic speed is recomputed, and the same frozen MQ law is applied. The resulting source standard deviations range from 2.09 to 3.17 km/s across the 38 radii. The correlated source contribution is added to the adopted observational diagonal for the generalized residual test. Neither an MQ parameter nor a covariance amplitude is fitted from the velocity residuals. [1]

The generalized statistic weights the residual vector using the inverse of that combined covariance. A correlated shift can have a different statistical cost from many independent deviations, so a modest marginal source standard deviation can strongly affect the result. The reported value 39.57 divided by 38 gives 1.041, consistent with the reduced statistic in Table 2. The source reports a right-tail probability of 0.400, which means non-rejection under this model. It is not the probability that MQ is true. [1]

Under the same source-covariance procedure, the inner subset has generalized chi-square 30.85 for 30 degrees of freedom and right-tail probability 0.423; the outer subset has 3.356 for eight degrees of freedom and probability 0.910. Separate subset statistics need not sum to the complete generalized statistic when cross-subset correlations enter the full covariance. Their separate probabilities must not be multiplied into a global confirmation. [1]

What remains outside that error model

The reported propagation does not jointly vary disk scale lengths, gas-disk parameters and all other source quantities, and it does not supply the full Eilers point-to-point covariance. The full McMillan Markov chain Monte Carlo source chain is publicly archived on Zenodo and offers a route to broader propagation. Availability of that chain does not mean it was used for the reported 200,000-draw conditional calculation. [1] [7]

The preprint additionally explores hypothetical observational correlation structures. They are sensitivity tests, not measurements of the missing Eilers covariance. Oman and Riley independently show that neighboring Galactic rotation-curve points can be correlated and that covariance can change inferred masses. Their fitted kernel concerns another circular-velocity dataset and cannot simply be transferred to Eilers as measured uncertainty. It motivates the missing-data requirement without validating MQ. [1] [5]

The earlier McGaugh-based source comparison is retained in the preprint only as a sensitivity analysis because that Galactic source was conditioned on an empirical radial acceleration relation. It gave reported MQ and Newtonian RMS values of 11.08 and 58.57 km/s. Those figures describe another source construction and must not be mixed with the principal McMillan pair. The present test’s source provenance is the reason for distinguishing the two. [1]

Chronology, reproducibility and the scope of the story

Table 3 distinguishes publication and source dates from the retrospective archive label. Neither September 8 nor September 27 is assigned here as a first discovery or journal-publication date. This Brief was prepared October 26, 2026.

A useful replication package would contain all 38 radii and target speeds, asymmetric statistical errors, adopted systematic fractions, five component forces, density-law conventions, force-integration tolerances, halo-exclusion check, nominal acceleration, source-normalization covariance construction, random-draw procedure and combined residual covariance. It should also preserve the full and subset residual vectors and specify the rules for asymmetric errors. The preprint states that derived numerical products needed to reproduce its results are available from the author on reasonable request. [1]

The result is a fixed-law application within the MQ program using external source and kinematic studies. Those study teams did not execute an independent MQ replication. The displayed reductions and summary arithmetic were checked for this Brief, but the raw 38-point integration, Monte Carlo covariance and right-tail probabilities were not independently regenerated here. A complete joint source-and-observational covariance calculation remains the principal precision extension. [1]

The reported RMS improvement supports further testing of the galactic response law under controlled source choices. It does not establish an advantage over every fitted halo model, measure a universal percentage accuracy, or resolve the lensing, cluster-merger and cosmological requirements of a comprehensive dark-matter alternative. Those remain separate physical tests. [1] [9] [10]