DARK MATTER

Gravitational Realization Without a Galactic Dark-Matter Halo

The dark-matter problem begins with a persistent difference between the gravitational field generated by observed baryonic matter and the gravitational behavior inferred from astronomical observations. In spiral galaxies, orbital velocities generally remain substantially higher at large radius than a Newtonian calculation based on the observed baryonic mass alone would predict. Related missing-gravity phenomena occur in gravitational lensing, galaxy clusters, large-scale structure, and cosmological observations. The standard Lambda cold dark matter framework, LambdaCDM, attributes most of the additional gravitational response to non-baryonic dark matter.

Measurement Quantization (MQ) examines a different possibility at galactic scales. It asks whether the apparent missing-gravity phenomenon can arise from the physical realization of a baryonic gravitational source rather than from an additional galactic matter component.

MQ does not replace Newton's inverse-square source relation. Instead, the framework distinguishes the gravitational source represented in the Internal Frame from the gravitational response realized in the System Frame. The relationship between these descriptions is governed by the Frames mapping, described more generally on the MQ Frames of Reference page. At high baryonic acceleration, the realization approaches the Newtonian result. At low baryonic acceleration, the realized gravitational response departs systematically from the Newtonian baryonic prediction.

The present formulation uses a fixed acceleration scale derived upstream from the frame-resolved cosmological construction together with a local gravitational realization law. The galactic relation is therefore connected to the same MQ structure used to describe gravity, rather than being introduced as an independent galaxy-fitting prescription.

The MQ Galactic Realization Law

Let $g_{\mathrm{bar}}(R)$ denote the Newtonian gravitational acceleration generated by the enclosed baryonic source at galactocentric radius $R$, and let $a_{\mathrm{MQ}}$ denote the present-epoch MQ acceleration scale. The current galactic analysis uses $a_{\mathrm{MQ}}=1.20040492178\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$. This scale is fixed before evaluation of the galaxy rotation-curve data.

At the present epoch, the scale is related to the System Frame expansion rate $H_{\mathrm{sys}}$, the defined speed of light $c$, and the invariant MQ coefficient ${\theta }_{\mathrm{si}}$ by $a_{\mathrm{MQ}}=\frac{c{H}_{\mathrm{sys}}}{2{\theta }_{\mathrm{si}}-1}$. The coefficient ${\theta }_{\mathrm{si}}$ also enters the construction of the fundamental measures.

For an enclosed baryonic mass $M_{\mathrm{bar}}(R)$, the Newtonian source acceleration is $g_{\mathrm{bar}}(R)=G{M}_{\mathrm{bar}}(R)/R^2$, where $G$ is the gravitational constant at the applicable realized scale. The dimensionless realization coordinate is $\chi (R)=\sqrt{\frac{g_{\mathrm{bar}}(R)}{a_{\mathrm{MQ}}}}$, and the corresponding realization fraction is $P(\chi )=1-\exp (-\chi )$.

The central gravitational realization law is

For circular motion, the same mapping gives $v_{\mathrm{real}}(R)=\frac{v_{\mathrm{New}}(R)}{\sqrt{1-\exp [-\sqrt{\frac{g_{\mathrm{bar}}(R)}{a_{\mathrm{MQ}}}}]}}$, where $v_{\mathrm{New}}(R)$ is the Newtonian circular velocity generated by the baryonic source and $v_{\mathrm{real}}(R)$ is the velocity associated with the gravitational response realized in the System Frame.

The exponential realization fraction is bounded. As the baryonic acceleration becomes large relative to $a_{\mathrm{MQ}}$, $P$ approaches unity and the realized gravitational response approaches the Newtonian baryonic result. Newtonian behavior is therefore recovered in the high-acceleration limit rather than being replaced.

The exact asymptotic fourth-power velocity relation is

Thus, asymptotically flat rotation curves and fourth-power baryonic Tully-Fisher scaling follow from the same realization law. This distinction between gravitational source and gravitational realization is central to the MQ interpretation. The baryonic source continues to obey the Newtonian inverse-square relation. What changes is the response realized through the Frames mapping as the source enters the low-acceleration regime.

The Milky Way

The principal Milky Way comparison combines an independently specified baryonic source model with an independently measured Galactic rotation curve. The observational target is the Eilers et al. circular-velocity curve, derived from more than 23,000 luminous red-giant stars over galactocentric distances of approximately 5 to 25 kpc. The MQ calculation compares those measurements with a baryonic source reconstructed from the McMillan Galactic mass model after excluding its dark-halo contribution.

This separation is important. The Newtonian calculation and the MQ calculation begin with the same baryonic source. The difference between them is the gravitational realization law, not the addition of a fitted MQ halo.

The statistical interpretation requires an important qualification. Uncertainty in a reconstructed Galactic baryonic source is correlated across radius. Treating every rotation-curve point as statistically independent would therefore overstate the information contained in the comparison. When the specified McMillan baryonic-normalization covariance is propagated, the comparison gives ${\chi }_{\mathrm{generalized}}^2=39.57$ for 38 degrees of freedom, with $p=0.400$.

The Complete SPARC Test

A broader population test is provided by the SPARC database. SPARC contains 175 nearby disk galaxies spanning wide ranges of morphology, luminosity, surface brightness, gas fraction, and rotation velocity. The published catalog combines Spitzer 3.6-micron photometry with high-quality rotation curves and provides baryonic mass models suitable for testing gravitational relations across a heterogeneous galaxy population.

The MQ analysis applies the same fixed realization law to all 175 SPARC galaxies and all 3,366 radial velocity measurements before subdivision by observational quality or galaxy properties. No galaxy-specific MQ acceleration scale or galaxy-specific MQ dark-matter distribution is introduced.

These statistics measure different aspects of the comparison. The ratio of mean velocities and the small pooled signed residual show that the complete catalog has very little aggregate velocity-scale displacement. The MAE and RMS show that individual radial measurements have substantially larger scatter. Near-unity aggregate normalization must therefore not be interpreted as point-by-point agreement at the same fractional level.

The 3,366 measurements are nested within 175 galaxies and share observational and source-reconstruction quantities. They are not 3,366 statistically independent systems. The pooled values are consequently descriptive catalog statistics rather than a significance calculation based on 3,366 independent trials.

Rotation-curve shape provides an additional diagnostic. The median per-galaxy Pearson correlation between the MQ realized and observed rotation curves is $r_{P,\mathrm{median}}=0.96418$. The high median correlation indicates that the fixed realization law reproduces much of the radial structure of the observed curves across the catalog, while the nonzero MAE and RMS preserve the substantial pointwise and galaxy-to-galaxy residual structure.

The remaining residuals have been examined against observational quality, inclination, distance, luminosity, velocity, and stellar-mass assumptions. These tests show that baryonic source reconstruction remains a material systematic limitation. Changing the adopted stellar-mass construction can alter residual trends without changing the MQ realization law. Such sensitivity must be separated from evidence for additional MQ freedom because the realization law and $a_{\mathrm{MQ}}$ remain fixed.

An Independent Dwarf-Galaxy Population

The LITTLE THINGS survey provides an additional population outside the principal SPARC sample. The published program contains high-resolution rotation curves and mass models for nearby dwarf galaxies, making it useful for testing the same frozen MQ relation in a substantially different galactic environment.

The out-of-SPARC MQ test uses 15 LITTLE THINGS galaxies and 288 radial comparisons. Its galaxy-weighted geometric MQ-to-observed velocity ratio is 0.9907, with a 95% galaxy-block bootstrap interval from 0.868 to 1.130. The corresponding test for zero mean logarithmic offset gives $p=0.895$.

This smaller sample is consistent with zero population-level normalization offset under the stated analysis, but its uncertainty is much broader than that of the complete SPARC comparison. It also does not significantly reproduce the conditional velocity ordering found in portions of SPARC. Figure digitization and reconstructed-source uncertainties further limit its precision evidentiary weight.

What MQ Changes and What It Does Not

At galactic scales, MQ proposes a specific alternative interpretation of the gravitational phenomenon conventionally attributed to a dark-matter halo. In the standard galactic interpretation, the gravitational response exceeding the baryonic Newtonian prediction is attributed primarily to an additional non-baryonic matter distribution. In MQ, the underlying source used in the calculation remains baryonic, while the gravitational response realized in the System Frame differs from the classical Internal Frame source description when the baryonic acceleration becomes small relative to $a_{\mathrm{MQ}}$.

An observer who interprets $g_{\mathrm{real}}$ exclusively through the Newtonian source equation would therefore infer a gravitating mass larger than the baryonic source mass. MQ interprets this inferred galactic excess as a consequence of gravitational realization through the Frames mapping rather than immediately identifying it with an additional galactic matter component.

This construction also explains why the MQ relation shares low-acceleration phenomenology with empirical radial-acceleration relations. The important distinction is provenance. The present-epoch value of $a_{\mathrm{MQ}}$ is fixed upstream by the MQ cosmological construction, and the exponential realization function is part of the mapping. Neither is introduced as a separate galaxy-by-galaxy fitting parameter.

Dark Matter Is More Than Rotation Curves

Successful reproduction of galactic rotation curves would not, by itself, resolve the full dark-matter problem. A physical alternative to particle dark matter must confront gravitational lensing, galaxy clusters, structure formation, and cosmic microwave background phenomenology in addition to orbital dynamics. These probes test different aspects of the gravitational and cosmological framework and cannot be replaced by agreement with galaxy rotation curves.

Gravitational lensing is especially important because photon deflection provides a gravitational probe independent of stellar circular motion. In MQ, a successful relativistic extension must connect the same gravitational realization responsible for galactic dynamics to the spacetime response governing photon propagation. Introducing an unrelated lensing-specific correction would weaken the predictive content of the framework.

If the gravitational realization law that accounts for galactic orbital dynamics cannot also generate the gravitational field required by well-characterized lensing systems, the present MQ description is incomplete. The same requirement extends beyond individual galaxies. Cluster dynamics and lensing, the growth of cosmic structure, and detailed CMB power-spectrum phenomenology remain independent observational tests. Agreement with rotation curves cannot substitute for those measurements.

Present Evidentiary Status

The current result is more specific and more constrained than the original dark-matter argument. MQ supplies a fixed galactic gravitational realization law with a present-epoch acceleration scale of 1.20040492178 × 10⁻¹⁰ m s⁻². The law recovers the Newtonian baryonic response in the high-acceleration limit and yields exact asymptotic relations for flat outer rotation curves and fourth-power baryonic Tully-Fisher scaling.

Applied without galaxy-specific MQ adjustment, the same construction reduces the Milky Way baryonic RMS residual from 69.0279 to 8.8965 km s⁻¹, gives a covariance-qualified Milky Way consistency result of ${\chi }^2=39.57$ for 38 degrees of freedom with $p=0.400$, produces a mean realized-to-observed velocity ratio of 1.0001888 across all 175 SPARC galaxies and 3,366 radial measurements, and remains consistent with zero population-level normalization offset in the separate 15-galaxy LITTLE THINGS test.

Those results establish a quantitative galactic test of MQ. They do not establish that non-baryonic dark matter has been eliminated from cosmology. That broader conclusion would require the same fixed construction to account successfully for lensing, clusters, cosmological structure formation, and the cosmic microwave background without introducing phenomenon-specific fitting freedom.

The central empirical question is therefore broader than whether MQ can reproduce galaxy rotation curves. It is whether one conserved measurement structure, realized through the same Frames mapping and the same fixed constants, can reproduce the independent gravitational phenomena presently attributed to dark matter.