The physical problem and the proposed contribution

A rotation curve traces orbital speed as a function of radius. The gravitational field calculated from the reconstructed stars and gas can be smaller than the acceleration inferred from that motion. MQ’s galactic construction retains the baryonic source and changes the relation between its Newtonian gravitational availability and the realized response. The calculation discussed here is a stationary circular-motion prescription, rather than a complete dynamical theory for every galactic environment. [1] [7]

McGaugh, Lelli and Schombert reported the empirical RAR in 2016, using 2,693 points in 153 rotationally supported galaxies. It connects acceleration inferred from rotation curves with that predicted from the baryonic distribution. Their published fitting function has the same functional form as the later MQ law. That observational relation and its authorship must remain explicit in coverage. [2]

The claimed MQ contribution is the origin of the square-root argument, the exponential fraction and the characteristic acceleration, with the latter fixed upstream of the galaxy calculation. Recovering an existing functional form is a theoretical result to examine. It does not independently establish the microscopic interpretation or constitute a new observation of the RAR. [1] [3]

Define the baryonic source before predicting the response

Let $\mathrm{𝑅}$ denote positive galactocentric radius in metres; ${\mathrm{𝑣}}_{\mathrm{New}}\left(\mathrm{𝑅}\right)$ the positive Newtonian circular speed generated by the adopted baryonic source in metres per second; ${\mathrm{𝑣}}_{\mathrm{real}}\left(\mathrm{𝑅}\right)$ the corresponding realized circular speed; ${\mathrm{𝑔}}_{\mathrm{bar}}\left(\mathrm{𝑅}\right)$ the positive Newtonian baryonic acceleration; and ${\mathrm{𝑔}}_{\mathrm{real}}\left(\mathrm{𝑅}\right)$ the positive realized acceleration, both in metres per square second. The characteristic MQ acceleration ${\mathrm{𝑎}}_{\mathrm{MQ}}$ has the same acceleration units and is positive. Equation (1a) relates the baryonic field to its circular-speed representation. The realized field is likewise the squared realized speed divided by radius. Equation labels in this Brief are distinct from the manuscript’s labels. [1]

For a disk, the source calculation must use the spatial distributions and geometries of stars, gas and any bulge. Treating the entire disk as a central point mass at every radius is not the source reconstruction used here. The speed entering Equation (1a) is the Newtonian prediction from that reconstruction, not the observed rotation speed used later as the comparison target. [1] [4]

Stellar mass-to-light ratios, gas conversion, distance, inclination, thickness and noncircular motion can affect the comparison. A fixed MQ scale does not make these nuisance quantities disappear. If observed velocities are used to infer or tune the source being tested, that dependence must be disclosed rather than described as an independent baryonic prediction.

Why the realization depth is a square root

The current source defines $\mathrm{𝜒}\left(\mathrm{𝑅}\right)$ as the dimensionless realization depth and $\mathrm{𝑃}\left(\mathrm{𝑅}\right)$ as the dimensionless gravitational realization fraction. They belong to the realized System Frame description. The source proposes the connection to discrete Internal Frame structure through the Frames mapping. Equation (2b) specifies the depth in the positive-acceleration domain. [1] [5]

Appendix D motivates this depth using the positive quadratic norm supplied by the realized geometry. At the same radius, circular kinematics associates the baryonic acceleration with a positive speed magnitude. Applying that same map to the acceleration scale supplies a reference speed magnitude. The ratio of these norms cancels the common radius and gives the square root of the acceleration ratio. [1]

The one-half power follows within that specified norm and kinematic construction. Dimensional analysis alone would allow other dimensionless functions and does not select the square root. A reviewer must therefore examine the upstream construction of the quadratic norm and whether it applies to the stationary galactic setting. The resulting depth is not evidence that an intrinsic continuous spatial metric was directly observed in the primitive count domain.

Figure 1 identifies the assumptions that must remain visible when the derivation is summarized. In particular, choosing the additive parameter’s normalization is a separate step from establishing that a continuous exponential family exists.

How subdivision consistency gives the exponential fraction

The construction treats the unrealized fraction as one minus the realization fraction. Subdividing a fixed local depth into consecutive admissible increments must leave the final result unchanged. The source requires the unrealized residual fractions to multiply when the depths add, with continuity, positivity and a realization fraction of zero at zero depth. [1]

These conditions give an exponential family for the residual. They do not, without a normalization, fix its numerical decay rate. MQ identifies the normalized depth of Equation (2b) with the additive parameter itself, selecting a unit infinitesimal generator. This identification supplies the differential law in which the rate of increase of the realization fraction equals the remaining unrealized fraction. Integrating with the zero-depth boundary gives Equation (3c).

For positive finite depth, the fraction is between zero and one. It is a gravitational realization fraction, not a detector probability, a dark-matter mass fraction or the fraction of baryons observed. The mathematical appearance of a saturation curve does not identify those different physical quantities.

The source also gives a finite-increment argument converging to the same exponential. Its first-order increment rule is used in a vanishing-step limit; it should not be presented as an exact finite-step update for an arbitrary large increment. The limiting construction and the differential form express the same local prescription rather than two independent physical tests. [1]

The acceleration law and its circular-speed prediction

By definition in this construction, the baryonic acceleration is the realization fraction times the realized acceleration. Substituting Equations (2b) and (3c) gives Equation (4d), the fixed galactic acceleration law. Both sides describe positive inward-acceleration magnitudes in the circular-motion convention. [1]

At a common radius, each circular speed squared is that radius times its corresponding acceleration. Taking the positive speed branch therefore gives Equation (5e). The speed enhancement is the square root of the acceleration enhancement, not the acceleration enhancement itself.

The normalization and what “fixed” means

The current source uses a present-epoch central value of ${\mathrm{𝑎}}_{\mathrm{MQ}}$ equal to 1.20040492178 × 10⁻¹⁰ metres per square second. Section 4.2 and Appendix D connect it to the defined speed of light, the Internal Frame local expansion-rate realization and the dimensionless invariant MQ coefficient. In the current Appendix D expression, the product of speed and local rate is divided by twice that coefficient minus one. The expansion rate must be converted to inverse seconds before this supplies an SI acceleration. [1]

The frame label matters. The current public derivation uses the Internal Frame local-rate realization for this normalization; a different expansion-rate realization cannot be substituted while keeping the same claimed numerical normalization. The source value is an MQ model output under its upstream cosmological construction, not a CODATA-recommended constant or a direct independent measurement of the galactic scale.

“Fixed” here means specified upstream of the rotation-curve comparison and applied unchanged across galaxies at the selected epoch. It does not mean independent of every measured cosmological input, immune to a physical normalization uncertainty or constant across all cosmic epochs. A reproducible test should freeze the complete upstream dependency chain and record which source version, age and realization choices produced the adopted scale. [1] [6]

This Brief uses dimensionless examples to expose the response law without attaching unsupported precision or uncertainty to that upstream scale. The examples are conditional calculations, not new fits or measured galaxies.

A worked acceleration and velocity example

Table 1 evaluates the realization fraction, acceleration enhancement and positive speed enhancement for four baryonic accelerations relative to the fixed scale. Each row follows the same equations. All displayed entries are rounded to six decimal places; the calculations retain additional precision. No row supplies an experimental confidence interval.

When the baryonic acceleration equals the scale, the depth is one. The realization fraction is about 0.632121, the acceleration ratio 1.581977 and the speed ratio 1.257767. The same-radius comparison is essential. These are deterministic evaluations of the prescribed function, not evidence that a galaxy with these exact properties has been observed.

At one ten-thousandth of the scale, the acceleration enhancement is about 100.500833. Nevertheless, the realized acceleration is only about 0.010050 times the scale. A growing ratio at low acceleration does not mean a divergent absolute gravitational field. Figure 2 plots this distinction over the dimensionless example domain.

High and low acceleration limits

At high baryonic acceleration relative to the scale, the depth becomes large, the unrealized fraction tends to zero and the realization fraction tends to one. The realized acceleration and circular speed approach their Newtonian baryonic values. At low baryonic acceleration, the realization fraction divided by depth tends to one. An exact normalized statement of the low-acceleration result is Equation (6f). [1]

For a finite localized baryonic source, the distant exterior baryonic acceleration falls with inverse radius squared. The low-acceleration MQ response then falls with inverse radius. Circular motion therefore tends to a radius-independent speed. Its fourth power tends to the upper-count gravitational constant times total baryonic mass times the fixed acceleration scale. These conclusions require the exterior, finite-source and circular-motion assumptions; they are not exact flat-curve claims at every radius. [1]

The baryonic Tully–Fisher scaling and the RAR limit are linked consequences of the same law. They should not be multiplied together as statistically independent confirmations. Nor can one infer a universal relativistic potential, light-deflection law or cosmological perturbation solution from this local circular prescription alone. [6]

What a rigorous galaxy comparison must preserve

An observational test needs the baryonic reconstruction, its uncertainties and the comparison kinematics specified independently enough to avoid reconstructing the answer from the velocity curve. Freeze the MQ scale, response law and source choices before examining residuals. Preserve the selection criteria and the individual galaxies with difficult source geometry, rather than reporting only an aggregate normalization. [1] [4]

Points in the same galaxy share distance, inclination and source assumptions. Treating every radial point as an independent galaxy-level test can understate uncertainty. A mean signed residual near zero can coexist with large scatter or coherent radial errors. Report scatter, source covariance and between-galaxy structure as well as any population normalization statistic. This Brief’s worked rows and curve are not a reanalysis of SPARC data.

Table 2 is a guide to the source claims, not a ranking of astronomy datasets. Stronger evidence comes from a comparison that could distinguish alternatives under controlled inputs. Further tests may involve different source populations or epoch dependence, but their physical assumptions and data overlap must be assessed separately. [6]

Chronology and useful interview questions

The archive uses September 8, 2026 as the completed-source milestone for this retrospective entry. It is not a journal publication or verified original discovery date. The public preprint includes later revisions, which this October 26, 2026 Brief uses where needed for accurate definitions. The 2016 RAR authors and their published phenomenology retain their separate historical attribution. [1] [2]

For a technical interview, request the construction establishing the positive quadratic norm, the reason local depth is additive, the residual composition law and the identification fixing its generator. Request the complete cosmological normalization with source uncertainties, then executable predictions from a frozen baryonic reconstruction. Ask which statistic was specified before inspecting galaxy residuals, which tests share data and which broader gravitational calculations remain to be done.