The established relation and the MQ claim

The baryonic Tully–Fisher relation connects total ordinary-matter mass with a characteristic galaxy rotation speed. McGaugh, Schombert, Bothun and de Blok’s 2000 paper showed how adding gas mass to stellar mass extends the relation across gas-rich systems and described fourth-power scaling. This is historical observational work, not a discovery claimed for the 2026 MQ manuscript. [4]

The MQ contribution under examination is a proposed derivation of the local realization law and its acceleration normalization. The law has the same exponential functional shape as the previously established empirical radial acceleration relation. Flat outer curves and fourth-power scaling follow downstream. The manuscript also acknowledges the earlier deep-MOND scaling; the limiting exponent alone cannot distinguish these interpretations. [1] [2]

Define the quantities and the source domain

Let $\mathrm{𝑅}$ be positive galactocentric radius in metres; ${\mathrm{𝑔}}_{\mathrm{bar}}\left(\mathrm{𝑅}\right)$ the Newtonian baryonic acceleration; ${\mathrm{𝑔}}_{\mathrm{real}}\left(\mathrm{𝑅}\right)$ the realized inward acceleration; and ${\mathrm{𝑎}}_{\mathrm{MQ}}$ the positive MQ acceleration scale, each in m s⁻². The positive circular speed ${\mathrm{𝑣}}_{\mathrm{real}}$ has units m s⁻¹. Total baryonic mass ${\mathrm{𝑀}}_{\mathrm{bar}}$ is the finite limiting source mass in kilograms. Upright $\mathrm{G}$ is the source’s gravitational reference at the upper count limit, in m³ kg⁻¹ s⁻². These symbols refer to their stated regimes, not interchangeable finite electromagnetic realizations. [1] [2]

The Internal Frame is the discrete relational configuration domain. The System Frame is the realized encoding in which these dimensional measures are expressed through the Frames mapping. This vocabulary supplies the proposed interpretation; using it does not establish that microscopic count structure has been directly observed.

The Newtonian baryonic field must be constructed from an independently specified distribution of stars, gas and any bulge. Inside a disk, replacing that distribution with a central point mass is generally invalid. For a finite localized source, the inverse-square statement needed here is an outer limit, not a universal disk-interior formula. [1] [7]

The exact local response and its weak limit

Equation (1a) is the exact positive-acceleration realization law. The baryonic field is evaluated at the same radius as the response. The exponential argument is dimensionless and the denominator is positive. The construction’s quadratic norm, subdivision consistency and depth normalization are premises of the derivation, rather than assumptions inferred from a new mass–speed fit. [1] [8]

Define the dimensionless realization depth $\mathrm{𝜒}$ by the first part of Equation (2b). The exponential denominator divided by depth tends to one as depth tends to zero from above. The resulting normalized weak-acceleration limit is the second part. [1]

At high baryonic acceleration the denominator of Equation (1a) tends to one, recovering the Newtonian source response as a limit. At low acceleration, Equation (2b) fixes the square-root behavior. These are mathematical properties of the prescribed law. Their physical application requires the stationary galactic setting and the source reconstruction to be appropriate. [1]

From the baryonic far field to constant limiting speed

For a finite localized distribution, the enclosed baryonic mass settles to the total mass and the Newtonian far field obeys Equation (3c). The same positive, fixed-epoch acceleration scale is held constant as radius increases. The finite-source condition ensures that the baryonic acceleration then enters the weak regime. [1]

Circular motion relates the realized acceleration to speed through Equation (4d). This is the circular-motion comparison convention; gas pressure support or noncircular motion must be accounted for when inferring an observational circular speed. [1] [5]

Combining the normalized weak limit with the far-field limit gives Equation (5e). The units on both sides are m² s⁻². Thus the realized field has an inverse-radius outer behavior while the baryonic source has an inverse-square far field. [1]

Substitute Equation (4d) into Equation (5e) and square the positive limiting value to obtain Equation (6f). The right side has units m⁴ s⁻⁴, matching speed to the fourth power. The fourth power of the limiting speed is linear in total baryonic mass at fixed scale. A sixteenfold mass increase gives twice the limiting speed; there is no universal limiting speed shared by all galaxies. [1]

A numerical scale with an explicit input ledger

The current overview corrects the normalization’s frame attribution to the System Frame cosmology and gives 1.20037(33) × 10⁻¹⁰ m s⁻². Its parentheses describe conditional input propagation with the age normalization premise held fixed; they do not include model discrepancy or establish that temporal premise independently. The long manuscript and its reported rotation-curve statistics retain the frozen nominal value 1.20040492178 × 10⁻¹⁰ m s⁻². [1] [2]

Table 1 and Figure 1 use that explicitly frozen nominal scale and an upper-count gravitational reference of 6.6740779383 × 10⁻¹¹ m³ kg⁻¹ s⁻² from the current MQ calibration. The latter is a derived framework reference, not a new laboratory measurement. The masses are selected hypothetical inputs in kilograms. Numerical precision in this calculation is not observational accuracy. [2]

At fixed mass, changing only the acceleration from the frozen nominal value to the current shared-input central value reduces the limiting speed by about 0.000749%. At fixed limiting speed it increases the inferred mass by about 0.002997%. Those algebraic shifts do not imply that catalog residuals have been rerun. The common normalization uncertainty is correlated across the entire sample, rather than independent noise for each galaxy. [2]

Why a finite-radius measurement can miss the limit

To isolate the approach to the limit, consider an ideal exterior field that is exactly inverse-square over the radii under study. Its transition length is the square root of gravitational reference times total mass divided by the acceleration scale. In this idealization, depth is that transition length divided by radius. Substituting into Equation (1a), using circular motion and dividing by Equation (6f)’s right side yields Equation (7g). This is an exact algebraic consequence for that ideal exterior, not a replacement for the general disk source. [1]

The right side exceeds one at positive finite depth and tends to one as radius increases. Thus this idealized realized speed approaches its asymptote from above, while the Newtonian baryons-only circular speed continues to fall. Figure 2 and Table 2 quantify the distinction without attaching it to an identified galaxy. An actual disk can have a different finite-radius approach because its source is extended and nonspherical.

At ten transition lengths the speed is still 2.510% above its asymptotic value, producing a 10.425% fourth-power excess. At one hundred transition lengths the corresponding differences are 0.250% and 1.004%. A small velocity-definition or finite-radius mismatch can therefore matter substantially when mass is compared with a fourth power. These percentages are deterministic model calculations, not observed population scatter.

What a dedicated observational comparison requires

Lelli and colleagues’ 2019 study of 153 SPARC galaxies explicitly compared velocity definitions. Its tightest relation used the mean speed along the flat part of the rotation curve. It reported a best-fit slope of 3.85 ± 0.09, with systematic uncertainties allowing slopes from 3.5 to 4.0. Such a fitted finite-sample slope is a different object from MQ’s exact limiting exponent. Neither a universal measured slope of exactly four nor an MQ-specific empirical signature follows from this comparison. [5]

Mass estimation also changes the test. The 2016 small-scatter study of 118 disk galaxies found that the adopted stellar mass-to-light ratio affected the slope, normalization and scatter. For a new MQ comparison, stellar and gas masses need their source assumptions, conversion factors and uncertainties documented. A baryonic mass inferred by imposing the relation being tested cannot serve as an independent mass input for that same test. [6] [7]

Table 3 separates the necessary inputs to a dedicated comparison. Distance affects photometric mass estimates and physical radii; inclination affects deprojected speed. Their uncertainty can move both axes coherently. Selection criteria, pressure-support corrections and treatment of galaxies without a well-measured flat portion should be set before interpreting residuals. A likelihood must keep common calibration and within-galaxy covariance distinct from intrinsic population scatter. [5] [7]

The MQ preprint reports resolved-curve tests in the Milky Way, SPARC and LITTLE THINGS. This Brief does not present those statistics as a newly completed baryonic Tully–Fisher fit. The current overview retains material population residuals and source sensitivity. Radial acceleration, resolved rotation and mass–speed comparisons can use the same galaxies and baryonic reconstructions, so their evidentiary dependence must be reported. [1] [2]

Chronology, scope and reporting context

September 8, 2026 is the completed-source milestone recorded by this retrospective archive. It is not a verified first discovery, public announcement or journal publication date. The public manuscript includes later revisions, which this October 30, 2026 Brief uses for accurate definitions. The 2000 empirical paper and the later observational studies retain their separate authorship and publication histories. [1] [4] [5] [6]

For interviews, request the upstream normalization ledger and a dedicated mass–speed residual analysis with its velocity definition fixed in advance. Ask how the source’s finite-mass and circular-motion assumptions apply to the selected galaxies. The derived limit is a concrete cross-check on the local law; agreement with this one galactic scaling does not establish a complete covariant theory or agreement with lensing, cluster and cosmological observations. [1] [2] [9]