The astronomical sample and the proposed contribution

SPARC combines 3.6-micrometre stellar photometry with neutral-hydrogen and optical rotation curves for 175 disk galaxies. Its sample spans a wide range of stellar mass, surface brightness and gas fraction. The catalog paper and original rotation-curve studies establish the astronomical inputs; they do not independently validate the later MQ interpretation. [2] [3]

The MQ manuscript evaluates every catalog galaxy before subdivision by observational quality or galaxy property. “Complete catalog” therefore means all 175 SPARC objects. SPARC is not statistically complete or volume limited, so an average over these objects cannot automatically be interpreted as an unbiased average over all galaxies. [1] [2]

The empirical radial acceleration relation, or RAR, was published by McGaugh, Lelli and Schombert in 2016 using 2,693 points in 153 galaxies. Its functional shape predates this MQ construction. The proposed new contribution is a derivation of that response and a fixed acceleration scale from the MQ input chain. A SPARC comparison is consequently a test of the adopted normalization and source-to-velocity prediction, not a new discovery of the empirical shape. [1] [4] [5]

What enters the calculation

SPARC provides separate Newtonian velocity contributions from gas, disk stars and bulge stars. The adopted fiducial stellar mass-to-light ratios are 0.5 for the disk and 0.7 for the bulge, in solar units at 3.6 micrometres. They multiply the squared unit-normalization stellar templates. The gas contribution enters as its signed template value times its absolute value, preserving an outward contribution where the source geometry produces one. Simply squaring every gas value would change the source model. [1] [3]

These are adopted source assumptions, not additional MQ parameters fitted to the target rotation velocities. They still carry astrophysical uncertainty. Distance changes the physical radius and mass reconstruction; inclination changes the deprojected speed; stellar populations affect mass-to-light ratios; gas mapping and noncircular motions affect the dynamical reconstruction. Table 1 separates the fixed MQ ingredient from those observational choices. [1] [2]

Exact source-to-velocity relationships

Let $\mathrm{𝑅}$ be positive galactocentric radius in metres and ${\mathrm{𝑣}}_{\mathrm{New}}$ the Newtonian baryonic circular speed in metres per second obtained from the adopted source. Let ${\mathrm{𝑔}}_{\mathrm{bar}}\left(\mathrm{𝑅}\right)$ denote its inward baryonic acceleration in metres per second squared. Equation (1a) defines that acceleration on the positive circular-source branch. [1]

Let ${\mathrm{𝑎}}_{\mathrm{MQ}}$ be the positive MQ acceleration scale, fixed here at 1.20040492178 × 10⁻¹⁰ m s⁻². Let ${\mathrm{𝑔}}_{\mathrm{real}}\left(\mathrm{𝑅}\right)$ denote the realized gravitational acceleration in the same acceleration units and ${\mathrm{𝑣}}_{\mathrm{real}}$ the realized circular speed in metres per second. The proposed bounded-realization response gives Equations (2b) and (3c). The exponential argument is dimensionless. [1] [5]

These are the baryonic-acceleration and galactic-response equations in the current manuscript’s Sections 4.1–4.2. Their line labels here are local to this Brief. The symbol subscripts identify the canonical quantities; the numerical calculations use SI acceleration units and convert final velocities to kilometres per second for comparison. [1]

Bounded realization is MQ’s proposed connection between a baryonic gravitational source and its observable response through the Frames mapping in the System Frame. At large baryonic acceleration relative to the fixed scale, the denominator approaches unity and the Newtonian result is recovered. At low positive acceleration it decreases, enhancing the calculated response relative to the baryonic Newtonian prediction. The equations alone do not establish that nature implements this interpretation. [1] [5]

Equations (1a)–(3c) as written require a positive inward circular-source acceleration and positive radius. The manuscript notes two nonpositive realized velocities in one galaxy and excludes those two values from logarithmic diagnostics. This Brief does not invent a clipping or sign prescription for reproducing them. The exact implementation for such rows is part of the needed reproducibility record. [1]

The displayed fixed scale belongs to the reported calculation. A later calibration elsewhere in the research program is not silently substituted, and no scale is refitted here to reproduce the catalog mean. The upstream scale remains conditional on the stated MQ derivation and calibration premises; it is not a new independent acceleration measurement. [1]

What the pooled ratio actually measures

For each radial comparison, subtract the observed velocity from the predicted velocity to obtain the signed residual. The pooled signed mean weights each radius equally. Its near cancellation indicates a small net offset over that particular radial sample, while absolute and squared residuals measure the remaining departures. Table 2 gives the manuscript’s reported summaries, not a fresh reconstruction of its input sample. [1]

Dividing the two rounded mean velocities printed in Table 2 gives about 1.0001885, consistent with the more precisely reported 1.0001888 after rounding of the inputs. Subtracting the displayed means gives 0.0264 km s⁻¹, consistent with the more precise signed residual. The fractional displacement of the reported ratio from unity is +0.01888 percent. It is not an error bound for an individual prediction.

Figure 1 compares signed cancellation with absolute error in the same velocity units. The small teal point does not make the amber and navy bars disappear. A ratio near unity can coexist with systematic errors in groups of galaxies and appreciable radial scatter.

Galaxies are clusters of radial measurements

Individual galaxies contribute different numbers of radii. Measurements within a galaxy also share distance, inclination and source reconstruction. The 3,366 comparisons are therefore not 3,366 independent galaxies. The manuscript uses a galaxy-cluster bootstrap to assess the stability of pooled summaries under catalog resampling. It generates 10,000 catalogs by sampling 175 galaxies with replacement and retaining each selected galaxy’s full radial aggregate. [1]

Table 3 records the source’s 95-percent percentile intervals. These are intervals for catalog summary estimators under galaxy resampling. They are not pointwise prediction bands and do not propagate all uncertainty in distance, inclination or stellar populations. They also do not make SPARC a random volume-limited population sample. [1]

Equal galaxy weight exposes a different bias

First averaging residuals within each galaxy, then averaging those 175 means equally, gives the source-reported bias of +3.3195 km s⁻¹. The median galaxy mean is +3.7614 km s⁻¹. Of the 175 galaxy means, 112 are positive and 63 negative. The exact two-sided binomial sign-test probability is 0.0002622, reproduced from those reported sign counts. It tests equal sign probabilities under that null model; it is not a probability that MQ is true or false. [1]

This does not contradict the near-zero point-weighted mean. A densely sampled galaxy has greater influence when each radius has equal weight and the same influence as every other galaxy when each galaxy has equal weight. The differing results show that the distribution of residuals and sampling density matter. Figure 2 presents those weighting choices without treating them as interchangeable measurements.

The highest-quality subset answers a narrower question

SPARC’s independently assigned highest-quality class contains 99 galaxies in the reported MQ analysis. For that subset, the manuscript first averages the logarithm of the predicted-to-observed velocity ratio within each galaxy, averages those galaxy offsets with equal weight, and exponentiates the result. The resulting geometric velocity ratio is 1.0042, with a reported 95-percent galaxy-bootstrap interval of 0.9763–1.0325. [1] [2]

Table 4 makes explicit why 1.0001888 and 1.0042 are not a before-and-after ranking of the same statistic. The source reports no statistically significant marginal velocity dependence in the highest-quality subset, while conditional analyses retain residual structure. Quality is correlated with galaxy properties and dynamical regime. The subset result does not prove that all lower-quality residuals are observational errors. [1] [6]

Uncertainty and independence of the evidence

The standardized rotation-velocity error includes formal fitting uncertainty and asymmetry between approaching and receding sides, but does not include the full inclination uncertainty. A common radial covariance matrix is not supplied for all 175 heterogeneous curves. A joint likelihood for all radial points therefore requires additional, explicit correlation assumptions. Random-error diagnostics and cluster-bootstrap summaries cannot be combined into a global confirmation significance without such a model. [1] [2]

The data are independently produced astronomical observations. The MQ calculation and its residual summaries are the author’s analysis of those data. Since the earlier empirical RAR was developed using SPARC, agreement with that functional shape on SPARC is not an independent test of discovering the shape. Separate source constructions and out-of-SPARC samples can test transfer of the frozen prediction, but their own selection and source uncertainties still require evaluation. [4] [8]

This galaxy calculation also does not by itself resolve lensing, galaxy-cluster dynamics or cosmological evidence commonly associated with dark matter. Its stated scope is the rotation-curve response to adopted baryonic sources. Extending the interpretation requires separately specified predictions and tests. [7]

Reproducibility record and version limits

For this Brief, the public Rotmod_LTG.zip download was inspected on October 26, 2026. Counting its non-comment numerical rows yields 3,391 across 175 profiles, while the manuscript reports 3,366 comparisons. The 25-row difference is unresolved here. It cannot be assigned to a quality cut, an archive change or a physical exclusion without the author’s exact input and row manifest. The published MQ summary numbers and bootstrap intervals in this Brief remain source-reported, not independently reproduced from that download. [1] [9]

The arithmetic linking the reported rounded means, ratio and percentage, and the exact sign-test calculation from 112 positive and 63 negative galaxy means, were checked separately. Those checks verify the stated summary arithmetic, not the original predictions or the bootstrap intervals. An exploratory reconstruction using a guessed prescription for nonpositive sources would not repair the missing manifest and is not offered as independent validation.

A reproducible package should identify the downloaded files and their versions, all retained radial rows, source normalization and signed-gas handling, unit conversions, nonpositive-source implementation, per-row predictions, quality membership, statistic definitions, and the bootstrap seed and resampling procedure. Those records would allow another researcher to locate a discrepancy without changing the frozen law to improve agreement.

September 27, 2026 is the release’s recorded source/audit snapshot. The public manuscript identifies a September 8 completed-source milestone and now includes later revisions; these current qualifications are not assigned retrospectively to an uninspected September snapshot. This Brief was prepared October 26, 2026. No journal-publication date or original discovery date is inferred from those milestones. [1]