A First Principles Derivation of the Galactic Radial Acceleration Relation from Measurement Quantization

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000–0001–5389–0447

Abstract

The empirical radial acceleration relation (RAR) already describes the coupling between baryonic structure and galaxy dynamics. This work asks whether its functional form and acceleration normalization can be derived from independent physics rather than inferred from rotation curves. Measurement Quantization (MQ) derives the same exponential RAR form from count-constrained gravitational realization: the acceleration scale follows from frame-resolved cosmological expansion, the square-root realization depth from the positive quadratic System Frame norm and Newtonian circular kinematics, and the exponential realization fraction from subdivision-consistent composition of unrealized gravitational availability under the Frames mapping. Every MQ component is therefore fixed upstream of galaxy data, with present-epoch

and no galaxy specific MQ fit. Across 175 SPARC galaxies (3,366 radial measurements), the point-weighted signed residual is +0.02644 km s^-1, the mean realized/observed velocity ratio is 1.00019, and pooled RMS and MAE are 22.82 and 16.20 km s^-1. For 99 highest-quality galaxies, the geometric ratio is 1.0042. A RAR/MOND-independent Milky Way test reduces Newtonian RMS from 69.03 to 8.90 km s^-1. Propagated McMillan source covariance gives generalized χ²=39.57 for 38 degrees of freedom (p=0.400). An independent 15-galaxy LITTLE THINGS test gives 0.9907 with a 95% galaxy block bootstrap interval of 0.868–1.130 and p=0.895. Because the acceleration scale inherits MQ cosmic age dependence, gravitational realization is predicted to be stronger at earlier epochs, providing a falsifiable mechanism relevant to early galaxy formation.

Keywords: Measurement Quantization, general relativity, cosmological constant, emergent spacetime, gravity, fundamental measure

1. INTRODUCTION

One of the most persistent unresolved problems in modern cosmology is the discrepancy between the gravitational behavior of galaxies and the mass inferred from directly observed baryonic matter. The central objective of this work is to determine whether that discrepancy can be derived, without a non-baryonic halo or galaxy-specific MQ fitting parameters, from a gravitational realization law fixed upstream of the galaxy data—one that recovers the Newtonian baryonic response at high acceleration but approaches a

realized gravitational response for a finite baryonic source in the weak-acceleration outer limit. The latter behavior produces an asymptotically radius-independent orbital velocity and the baryonic Tully-Fisher fourth-power scaling. Measurement Quantization (MQ) develops this result from discrete relational count structure and the Frames mapping [1]. The galactic acceleration scale is derived from independently measured non-galactic inputs and fixed before rotation curve testing. The positive quadratic spatial norm in the System Frame, together with Newtonian circular kinematics, fixes the dimensionless local realization depth, while subdivision-consistent realization of the gravitational availability remaining unrealized yields the closed-form realization fraction. The resulting galactic realization law is then applied unchanged across galaxies spanning markedly different morphologies, luminosities, gas fractions, and surface-brightness regimes.

The motivation is therefore not to introduce another empirical rotation curve interpolation. The empirical RAR functional form is already established observationally. The question is whether MQ can derive that pre-existing form, including its acceleration normalization and asymptotic behavior, from independently established physical quantities while leaving the adopted Newtonian baryonic source unchanged. In MQ, the acceleration scale follows from the frame-resolved cosmological expansion, the square-root realization depth from the positive quadratic System Frame norm and circular kinematics, and the exponential realization fraction from subdivision-consistent composition of unrealized gravitational availability under the Frames mapping. The departure from the Newtonian baryonic prediction therefore arises from bounded realization at low acceleration, with the Newtonian result recovered continuously at high acceleration.

The goals of the paper are consequently threefold. First, it derives the complete galactic realization law from these independently established MQ components rather than selecting its functional form from galaxy rotation data. Second, it tests the resulting fixed present-epoch law against independent Milky Way, SPARC, and LITTLE THINGS baryonic-source and kinematic constructions with no galaxy-specific MQ calibration. Third, because the acceleration scale inherits the cosmic age dependence of the MQ expansion construction, it derives a redshift dependent extension in which gravitational realization is stronger at earlier epochs. This provides a prospective, falsifiable connection between the same realization mechanism tested in present-day galaxies and the unexpectedly rapid emergence of dynamically organized galaxies at high redshift. The present work does not claim a complete theory of galaxy formation. It establishes the age-dependent gravitational prediction that such a theory must test. Key MQ terms and symbols used throughout are summarized in Appx. A.

The present-epoch prediction is tested against three observational constructions. The first uses the 38 Milky Way circular velocity measurements of Eilers et al. [2] with the RAR/MOND-independent Galactic baryonic source of McMillan [3]. The second uses the complete SPARC database of 175 galaxies comprising 3,366 radial velocity measurements [4]. The third is an out-of-SPARC LITTLE THINGS dwarf galaxy test constructed from the independent H I and stellar-source work of Oh et al. [5] and the published rotation curve plots of O’Brien et al. [6]. In every case, the present-epoch acceleration scale and galactic realization law are fixed before the observational target is evaluated, no galaxy specific MQ parameter is fitted, and the externally adopted baryonic source quantities are not adjusted to improve MQ agreement. For the Milky Way, application of the galactic realization law reduces the RMS residual relative to the independent Eilers measurements from 69.03 km s^-1for the Newtonian baryonic curve to 8.90 km s^-1. SPARC provides the complementary broad population test across disk galaxies spanning large ranges of morphology, luminosity, surface brightness, and gas content [4], while LITTLE THINGS tests whether the fixed MQ normalization and SPARC residual velocity behavior persist in an out-of-SPARC galaxy population.

Predictive chain and scope. Table 1 separates the external inputs, the quantities derived before the galaxy data are examined, the fixed prediction applied to each baryonic source, and the observational tests. For a direct audit of the galactic result, the shortest reading path is Sec. 3.8 for the invariant coefficient, Sec. 3.12 for the frame-resolved expansion rate, Sec. 4.2 and Appx. D for the fixed galactic realization law, and Secs. 4.3-4.10 for the Milky Way, SPARC, and LITTLE THINGS tests and controls.

Table 1. Predictive dependency and test structure for the galactic result.

This separation also defines the scope boundary for the principal claim: rotation curve agreement is not evidence that MQ succeeds at lensing, cluster merger, or cosmological perturbation scales, which remain independent physical requirements discussed in Sec. 5. For LITTLE THINGS, same-galaxy SPARC overlaps, incomplete stellar source cases, and source products unavailable for the figure reconstruction are excluded before the MQ residual is examined, leaving 15 galaxies and 288 radial comparisons. The observed circular velocities remain external to construction of the baryonic source and MQ prediction, and no fitted RAR acceleration parameter is used as a surrogate for that direct comparison.

For the statistical quantities reported below, r denotes a Pearson correlation coefficient, r_P denotes the residual velocity Pearson correlation coefficient where that subscript is used, ρ denotes the Spearman rank correlation coefficient, p denotes the associated two-sided significance probability, and χ² denotes the chi-square statistic. The externally assigned SPARC quality flag is denoted by Q, with Q = 1, Q = 2, and Q = 3 identifying the high-, medium-, and low-quality rotation curve classifications, respectively [4]. These classifications are observational metadata assigned independently of the present MQ analysis.

Table 2. Principal rotation curve validation statistics. Milky Way and SPARC results are reported separately because the two datasets employ different observational and baryonic model constructions.

The Milky Way and SPARC statistics quantify different aspects of the same fixed realization law and should not be conflated. In the Milky Way analysis, the observational rotation curve and adopted baryonic source are taken from distinct published constructions: the Eilers et al. circular velocity measurements provide the observational comparison [2], while the McMillan Galactic bulge, thin-disk, thick-disk, H I, and molecular gas components provide the Newtonian baryonic source [3].

Across the complete 5.27–24.82 kpc range, MQ reduces the Newtonian baryonic RMS discrepancy by 87.11%, with a mean absolute error (MAE) of 7.43 km s^-1. The agreement is radially nonuniform: the inner 30 measurements from 5.27–19.71 kpc have an RMS residual of 6.88 km s^-1, whereas the outer eight measurements from 20.27–24.82 kpc have an RMS residual of 14.09 km s^-1. The outer measurements also carry substantially larger observational uncertainties, giving them less weight under the adopted total error model despite their larger absolute residuals. The detailed pointwise and uncertainty analysis is developed in Sec. 4.3.

Across all 3,366 SPARC radial measurements, the point-weighted mean signed residual is +0.0264 km s^-1, while the ratio of the mean MQ realized velocity to the mean observed velocity is 1.00019, corresponding to an aggregate velocity scale difference of approximately +0.019%. These near-zero aggregate quantities coexist with pooled RMS and MAE residuals of approximately 22.82 and 16.20 km s^-1, respectively, and therefore do not imply point-by-point agreement at the same fractional level. Because radial measurements are clustered within galaxies, point-weighted aggregate quantities and galaxy-level population statistics also characterize different aspects of agreement.

At the galaxy level, the complete SPARC population exhibits a statistically significant dependence of the MQ-to-observed normalization offset on characteristic rotational velocity. Stratification by SPARC’s independently assigned quality classification materially changes that result. Among the 99 Q=1 galaxies, the geometric MQ-to-observed velocity ratio is approximately 1.0042 and the residual dependence on rotational velocity is not statistically significant, r_P=-0.134, p=0.187. The complete sample velocity dependence is therefore not uniformly expressed across SPARC but is concentrated predominantly in the lower quality classifications.

This quality dependence does not establish that discrepancies among the lower quality galaxies are observational errors. SPARC quality is associated with observational difficulty and is also correlated in the present sample with astrophysical regime, including rotational velocity. The absence of a significant marginal velocity relation in the highest quality class therefore constrains, but does not remove, the complete catalog systematic. Section 4.8 adds a simultaneous galaxy-level partial regression test of quality, velocity, inclination, distance, luminosity, surface brightness, and gas richness. That joint test retains a significant conditional velocity term but also exposes strong velocity-luminosity collinearity, so the full-sample residual cannot presently be assigned uniquely either to the MQ realization law or to observational/source-systematic structure.

Across astrophysical systems spanning galaxies, galaxy clusters, and cosmological structure formation, gravitational observations provide evidence for a gravitating contribution beyond that attributable to directly observed baryonic matter alone [10]. Within the standard ΛCDM framework, this discrepancy is represented by a non-baryonic cold dark matter component, and cosmological fits infer a cold dark matter density substantially exceeding the baryonic matter density [11]. The particle identity of that component remains unresolved: despite extensive direct, indirect, and collider searches, no confirmed non-gravitational dark matter detection has been established [12–14]. Sections 1.2–1.4 review these observational constraints and the principal conventional responses in greater detail.

MQ investigates a different possibility by reconsidering the assumption that continuum spacetime geometry is fundamental. It interprets observable quantities as System Frame realizations of discrete Internal Frame count structure through the Frames mapping [1], building on the relational count framework established in earlier work [15]. The present paper adopts those results as its theoretical foundation. The MQ geometric construction relevant to gravitational realization is developed in Sec. 3, and its galactic consequences are tested in Sec. 4.

Within the emergent System Frame, length, mass, and time are realized quantities derived from the underlying count structure rather than primitive continuum inputs. The MQ invariant coefficient θ_si is determined from the measured fine structure constant as developed in Sec. 3.8 and earlier work ([15], Appx. AD). Throughout this paper, untagged θ_si denotes only this dimensionless invariant coefficient, while its momentum realization which has the same value is denoted p_si. In the fundamental mass relation below, p_si has dimensions of momentum, c denotes the defined speed of light, and m_f denotes the fundamental mass:

Section 3.3 makes the realization structure explicit. The untagged coefficient θ_si is dimensionless; its angular realization Θ_si carries the unit rad, while p_si carries units of kg m s^-1. These are dimensionally distinct realized representations of the same invariant MQ coefficient, with their dimensional character supplied in the System Frame. Their numerical coefficients coincide in the adopted MQ realization, as shown explicitly in Sec. 3.3. The Planck-like fine structure constant is denoted α_p; prior to application of the MQ Informativity differential – a new length contraction effect a consequence of the discrete Internal Frame - in Sec. 3.6, its inverse is

where the mixed floor-ceiling brackets denote selection of the nearest whole-unit value using the corresponding floor or ceiling result.

Using the electron mass m_e, the Bohr radius a₀, and α_p, MQ then resolves the fundamental length l_f and fundamental time t_f:

The subscript p in α_p identifies the Planck-like form of the fine structure constant. Equations (1–4) describe MQ fundamental or Planck-like quantities prior to complete finite-count realization and are therefore not direct predictions of the corresponding CODATA Planck quantities or of the classically realized fine structure constant. A concordance significance against those CODATA quantities would consequently compare different realization stages rather than like physical observables. The classical form – what is measured in the laboratory – is introduced only after derivation of the Informativity differential in Sec. 3.6, while Sec. 3.8 uses the 2022 CODATA recommended inverse fine structure constant as an input in determining θ_si. That determination is therefore not an independent CODATA concordance test. Statistical concordance becomes physically meaningful only after the complete Frames mapping and Informativity differential have been applied and the resulting classical realization is compared with an independent laboratory measurement.

Observable physics emerges through the Frames mapping from discrete count configurations in the Internal Frame to continuous relational structure in the System Frame. Geometry is therefore an emergent property of realization rather than a fundamental background. The physically meaningful counts associated with length, mass, and time are represented by n_L, n_M, and n_T, respectively, each corresponding to l_f, m_f, and t_f within the Internal Frame. Their observable physical representations arise through the Frames mapping into the System Frame. MQ thus treats the fundamental measures as units relative to which discrete count configurations are defined rather than merely as dimensional combinations of conventional constants.

The galactic application concerns two well-established properties of rotation curves: orbital velocities commonly remain approximately constant or decline only slowly over extended galactocentric radii, and observed velocities exceed those predicted from the adopted baryonic distribution in systems exhibiting a mass discrepancy [4,16]. SPARC samples these behaviors across wide ranges of luminosity, morphology, surface brightness, and gas fraction [4], while the Milky Way supplies a complementary comparison using the independent Eilers et al. circular velocity measurements [2] and McMillan Galactic baryonic model [3]. Within MQ, these phenomena are attributed to bounded realization rather than to an independently postulated non-baryonic mass distribution.

The corresponding MQ construction is developed quantitatively in Sec. 4. Section 4.1 defines the observational baryonic inputs and Sec. 4.2 derives the local predictive realization law, with the complete local derivation in Appx. D. Sections 4.3 and 4.4 provide the RAR/MOND independent Milky Way and complete SPARC tests. Secs. 4.5–4.9 characterize the residual structure and apply observational quality, stellar source, and galaxy property controls; and Sec. 4.10 provides the combined interpretation.

The realized orbital velocity v_real(R) is obtained from the adopted Newtonian baryonic source through the MQ galactic realization law. Here v_New(R) denotes the Newtonian orbital velocity predicted from the adopted baryonic mass model:

For the Milky Way, v_New(R) is reconstructed from the McMillan bulge, thin- and thick-disk, H I, and molecular gas density components by direct Newtonian force integration at the 38 Eilers radii, with the model dark halo excluded. The Eilers measurements and their uncertainties are used only for subsequent observational comparison. For SPARC, the same realization expression is evaluated without galaxy specific MQ adjustment for all 175 galaxies and 3,366 radial measurements before observational quality stratification. Equation (5) is the introductory form of the predictive relation developed in Sec. 4.2 and stated there as Eq. (137).

The gravitational coupling underlying the MQ formulation uses Q_L, the mapping-induced residual count contribution entering the realized separation under the Frames mapping; r, the radial separation between gravitating bodies; and G, the gravitational constant resolved at the upper-count limit. With p_si as defined above, the relation is

Equation (6) establishes the MQ correspondence between realized count geometry and the Newtonian gravitational field. Its derivation and realization dependence are developed in Secs. 3.2–3.6. The present section uses it only to identify the gravitational relation from which the galactic realization construction proceeds.

1.2 Observational evidence for missing gravitational mass

The empirical motivation for the present test is the established mismatch between reconstructed baryonic gravity and observed dynamics. Galaxy rotation curves provide the primary target [4,16], while gravitational lensing, merging clusters, and cosmological observations show that any general explanation must ultimately succeed outside the rotation curve regime [10–11,17]. The Milky Way comparison in Sec. 4.3 and the SPARC and LITTLE THINGS population tests developed through Secs. 4.4–4.10 are therefore the principal observational tests of this paper. The broader evidence defines independent requirements that MQ must eventually satisfy.

where v is the orbital velocity and M the enclosed gravitating mass. G and R retain their definitions established above.

For later comparison with the cosmological benchmark, MQ defines universe domains independently of the ΛCDM matter-density decomposition. In MQ notation, Ω_tot=1, Ω_obs≈0.316, and Ω_vis≈0.0488, where Ω_tot denotes the normalized total universe domain, Ω_obs the total observable domain, and Ω_vis the presently visible domain. These are MQ geometric domain quantities, not identifications with standard cosmological density parameters despite their similar numerical magnitudes. Their derivation and physical interpretation are developed in Sec. 3.9 and Appx. C. The unobserved domain is the portion of Ω_obs that is not presently referenceable,

In that all variables are constant, these geometric fractions remain constant across all epochs.

Let M_obs denote the total mass corresponding to Ω_obs, M_vis the mass corresponding to Ω_vis, and M_uobs the complementary mass corresponding to Ω_uobs. The MQ universe domain mass relations are

These relations distinguish the MQ domain construction from observational ΛCDM density accounting. They do not constitute independent observational evidence for the galactic result.

1.3 Rotation curves, lensing, CMB constraints, and cluster offsets

Four observational regimes bound the broader dark matter problem: galaxy dynamics, gravitational lensing, CMB/large-scale-structure inference, and merging cluster offsets [4,11,17]. Only galaxy dynamics is quantitatively tested by the principal analysis. The remaining regimes impose independent requirements developed prospectively in Secs. 5.3–5.9 and are not inferred from rotation curve agreement.

Gravitational lensing provides an observationally distinct probe through light deflection. Let

denote the physical gravitational deflection angle and b the impact parameter. In the general relativistic weak-field point-mass limit, a photon passing a gravitating mass M is deflected according to

Strong- and weak-lensing reconstructions infer projected gravitating mass distributions that can substantially exceed those associated with observed baryonic components, particularly in galaxy clusters [17]. In merging systems such as the Bullet Cluster, lensing additionally constrains models through the observed separation between reconstructed gravitating mass and the dominant X-ray-emitting baryonic plasma [17]. Consequently, reproducing galactic rotation curves does not substitute for deriving the corresponding relativistic MQ lensing and cluster response.

The CMB provides a distinct cosmological constraint. Within standard spatially flat ΛCDM, let Ω_m denote the total matter density fraction. The baryonic matter fraction is Ω_b, and the cold dark matter fraction is Ω_c. Neglecting the small distinction associated with massive-neutrino bookkeeping for this introductory decomposition,

Planck CMB measurements constrain the physical cold dark matter density to be substantially larger than the baryonic density, making this cosmological requirement independent of the galactic rotation curve calculation [11].

1.4 Limits of particle dark matter and modified gravity

The relevant comparison is predictive structure rather than whether alternative models can fit individual galaxies. ΛCDM introduces a non-baryonic component within a cosmological framework that fits a broad range of observations [11,18], whereas MOND-like approaches modify the low-acceleration response [19–21]. MQ instead retains the adopted Newtonian baryonic source and tests the fixed galactic realization law whose acceleration scale is determined upstream of the galaxy data. The absence of a confirmed particle dark matter identification and known small-scale tensions motivate investigation of alternatives [12–14,22].

For the standard nonrelativistic MOND formulation, let a denote the centripetal acceleration, a_MOND the characteristic MOND acceleration scale, and

the interpolation function connecting the Newtonian and deep-MOND regimes. The relation is

1.5 The MQ realization of geometry and the origin of dark matter behavior

The specific proposal tested here is that the apparent excess galactic gravitational response arises from bounded realization of mapped count geometry rather than from an additional source mass. The same present-epoch galactic realization law is applied without galaxy-specific MQ calibration to the Milky Way, the complete SPARC catalog, and the out-of-SPARC LITTLE THINGS sample.

The gravitational coupling introduced in Eq. (6) relates realized count geometry to the Newtonian gravitational field through count separation and the Frames mapping. The relevant geometric construction is developed in Sec. 3, the independently specified acceleration scale and local galactic realization law in Sec. 4.2 and Appx. D, and the Milky Way, SPARC, and LITTLE THINGS tests and controls in Secs. 4.3–4.10. Relativistic lensing, cluster dynamics, cosmic-age evolution, and cosmological perturbations remain independent requirements developed in Secs. 5.3–5.9 rather than consequences of the galaxy results.

2. BACKGROUND AND COMPARATIVE FRAMEWORK

2.1 ΛCDM as a model benchmark

ΛCDM is used here as the quantitative benchmark because it jointly describes the CMB, large-scale structure, and cosmic expansion while representing the additional gravitating contribution with non-baryonic dark matter [11,18]. The comparison required for the present paper is not a global replacement test: Sec. 4 asks whether MQ reproduces galaxy dynamics with a fixed response law, while Sec. 5 identifies the independent calculations required before any broader equivalence claim is justified.

where H(t) is the Hubble expansion rate, ρ is the total mass-energy density expressed in mass-density units, k is the spatial-curvature parameter, a_FLRW(t) is the cosmological scale factor, and Λ is the cosmological constant; G and c retain their definitions established above. For the spatially flat ΛCDM benchmark, the present-day normalized density budget is decomposed into radiation, baryonic matter, cold dark matter, and dark energy according to

Here, Ω_r, Ω_b, Ω_c, and Ω_Λ denote the radiation, baryonic matter, cold dark matter, and cosmological constant density fractions, respectively. Planck 2018 base-ΛCDM constraints [11] correspond approximately to Ω_b≃0.049, Ω_c≃0.266, and Ω_Λ≃0.685, with the present radiation contribution comparatively small. Thus, most of the present cosmic energy-density budget is not contained in electromagnetically luminous baryonic matter.

where M(r) denotes the total enclosed gravitating mass, M_bar(r) the enclosed baryonic mass, and M_dm(r) the enclosed dark matter halo mass. Individual galaxy analyses then specify a halo density profile together with the baryonic mass model and observational parameters required to predict the corresponding rotation curve.

2.2 A fundamental accounting of galactic rotation curves

Galactic rotation curves expose a simple accounting problem: orbital velocities observed at large galactocentric radii generally exceed those predicted from the reconstructed baryonic mass alone. ΛCDM and MQ begin from the same discrepancy but assign it to different physical mechanisms. In ΛCDM, the additional gravitational response is attributed predominantly to a non-baryonic halo, so the gravitating source contains baryonic and dark components. MQ instead retains the Newtonian baryonic source: the adopted baryonic reconstruction specifies the gravitational availability, and the galactic realization law determines the realized gravitational response in the System Frame.

The fixed MQ construction uses the Newtonian baryonic acceleration, the independently determined MQ acceleration scale, the positive quadratic norm supplied by the realized geometry, and subdivision-consistent composition of incomplete realization. These determine the local realization depth and realization fraction used in the rotation curve prediction. Thus ΛCDM modifies the gravitating source through a halo mass distribution, whereas MQ leaves the baryonic source unchanged and tests whether a fixed galactic realization law reproduces the observed dynamics without a separately fitted non-baryonic component.

2.3 Gravitational lensing and cluster mass reconstruction

Lensing is an independent relativistic benchmark rather than part of the galactic fit. A lens reconstruction distinguishes the contribution associated with reconstructed baryonic matter from any additional gravitating contribution required by the adopted lens model. MQ must ultimately reproduce the corresponding deflection, convergence, and shear observables from the realized System Frame metric without introducing a lens-specific dark component. Section 5.3 develops the present spherical weak-field and thin-lens results and identifies the remaining extended-source covariant calculation.

For comparison among physical interpretations, let M_lens denote the total gravitating mass inferred from a lensing reconstruction, M_bar,lens the contribution associated with reconstructed baryonic matter, and M_exc,lens the additional gravitating contribution required by that reconstruction. The observational accounting is

Equation (17) is an observational accounting relation only and does not assign a physical identity to M_exc,lens. The quantities M_bar,lens and M_exc,lens are local lens-reconstruction quantities and are distinct from the MQ universe domain mass quantities developed in Appx. C.

2.4 Bullet cluster spatial offsets

Merging clusters provide a stringent nonequilibrium spatial test because the collisionless galaxy population, hydrodynamic intracluster gas, and reconstructed lensing mass distribution can separate during a merger [17,23–24]. The Bullet Cluster provides the canonical example, with the dominant X-ray-emitting plasma spatially displaced from the principal lensing mass peaks [17]. MQ must therefore evolve the baryonic system and realized metric together. The equilibrium galactic realization law alone is insufficient. Section 5.4 formulates the corresponding time-dependent problem and makes explicit that the nonlinear Bullet Cluster evolution and parameter-frozen lensing comparison remain to be performed.

Let M_bar,cl denote the reconstructed baryonic cluster mass represented here by the galaxy and gas contributions M_gal and M_gas. Then

Let r_lens,⊥ denote the projected centroid vector of a reconstructed lensing-mass peak and r_gas,⊥ the projected centroid vector of the corresponding X-ray gas component. Their plane-of-sky separation Δr_⊥ is

This separation is an observational diagnostic. Eqs. (18–19) do not by themselves provide a dynamical model of the merger.

2.5 Tensions challenging ΛCDM

Small-scale ΛCDM tensions provide comparative context. Modern baryonic galaxy formation calculations alleviate several historically identified discrepancies, while dwarf-galaxy inner structure, satellite populations, and rotation curve diversity remain active tests [25–29]. The present manuscript therefore does not treat unresolved ΛCDM phenomenology as validation of the MQ mechanism. Modern reviews similarly emphasize that distinguishing limitations of collisionless-only predictions from effects of baryonic physics or possible dark matter microphysics remains an active problem.

One long-standing example is the cusp-core problem. Collisionless cold dark matter simulations yield centrally concentrated halo profiles represented by the Navarro-Frenk-White form [25]. Let ρ_NFW(r) denote the NFW halo density at halo-centric radius r, ρ_s its characteristic density, and r_s its scale radius. The profile is

The NFW relation provides the collisionless benchmark. Whether a particular observed inner profile requires baryonic modification, altered dark matter physics, or another explanation must be assessed separately.

2.6 An assessment of leading models with well-established observational constraints

The comparative standard is deliberately asymmetric: ΛCDM is quantitatively established across cosmology, whereas MQ is tested here most strongly in galaxies. Comparison 1 therefore distinguishes observationally demonstrated results from compatibility within viable parameter ranges and from calculations that remain unperformed. Alternative dark matter and Modified gravity models are retained as context for the observational requirements and predictive freedom against which the fixed MQ galactic realization law should be judged.

Rotation curves, gravitational lensing, cluster dynamics, CMB anisotropies, and large-scale structure constrain the inferred gravitating contribution through different observables and physical scales. The leading frameworks do not address all of these constraints with equal quantitative maturity. At the observational accounting level, let M_grav denote the total gravitating mass inferred under the adopted gravitational reconstruction, M_bar,obs the corresponding observationally reconstructed baryonic contribution, and M_exc the additional inferred gravitating contribution. Then

Equation (21) is an observational bookkeeping relation and does not by itself assign a physical identity to M_exc. Within ΛCDM, the inferred excess is attributed predominantly to non-baryonic dark matter. Modified gravity theories instead alter the relation between the baryonic source and gravitational response. MQ retains the Newtonian baryonic source and tests whether the inferred excess arises from the realized gravitational response produced through bounded realization under the Frames mapping.

Comparison 1. Qualitative comparison of leading dark matter and modified gravity model classes against the principal observational constraints. “Compatible” denotes compatibility within viable model parameter space and should not be read as an independent quantitative validation of every realization.

ΛCDM / collisionless cold dark matter

• Rotation curves: Halo modeling provides the standard successful phenomenological description of galaxy rotation curves.

• Lensing and cluster masses: Non-baryonic halo mass supplies the additional gravitating component required by standard lensing and cluster reconstructions.

• Bullet Cluster–type offsets: Collisionless dark matter can remain spatially associated with galaxies while ram-pressure-stripped gas is displaced, consistent with the standard interpretation of merging clusters [17].

• CMB and large-scale structure: Quantitative benchmark for CMB anisotropies and large-scale structure [11,18].

• Small-scale tensions: Several classical discrepancies are alleviated by baryonic galaxy formation, while inner-halo structure and diversity remain active tests [29].

• Direct / indirect detection: No confirmed dark matter particle identification; broad classes of candidate parameter space are constrained [22].

WIMP dark matter

• Rotation curves: In the cold, effectively collisionless regime, inherits standard CDM halo phenomenology.

• Lensing and cluster masses: Compatible with the standard ΛCDM lensing interpretation when the WIMP behaves as cold collisionless matter.

• Bullet Cluster–type offsets: Compatible when self-interactions are negligible on cluster merger scales.

• CMB and large-scale structure: Thermal and non-thermal realizations are constrained by relic-density, CMB, and structure requirements; viable cold realizations reproduce CDM-like large-scale behavior.

• Small-scale tensions: Generally inherits CDM structure issues unless baryonic physics or additional particle interactions alter halo evolution.

• Direct / indirect detection: Direct, indirect, and collider searches strongly constrain broad WIMP parameter regions, with no confirmed signal [22].

QCD axion / cold axion-like dark matter

• Rotation curves: Cold axion realizations can reproduce CDM-like halo dynamics over viable parameter ranges.

• Lensing and cluster masses: Compatible with standard gravitational lensing when behaving as a cold gravitating component.

• Bullet Cluster–type offsets: Sufficiently weakly interacting cold axions behave effectively collisionlessly during cluster mergers.

• CMB and large-scale structure: Vacuum-realignment axions can constitute cold dark matter; cosmological abundance and isocurvature constraints restrict viable models [30].

• Small-scale tensions: Depends on mass, production history, and interactions; the ultralight regime has distinct wave-scale phenomenology and is listed separately below.

• Direct / indirect detection: Extensive haloscope, helioscope, laboratory, and astrophysical searches constrain axion parameter space; no confirmed dark matter axion signal has been established [30].

Sterile-neutrino / warm dark matter

• Rotation curves: Free streaming suppresses low-mass structure relative to CDM and can alter dwarf-scale halo populations [31].

• Lensing and cluster masses: Must retain sufficient clustering to reproduce galaxy- and cluster-scale lensing.

• Bullet Cluster–type offsets: Viable realizations remain effectively collisionless on merger scales.

• CMB and large-scale structure: Production mechanism and particle mass are constrained by structure formation, CMB-related cosmology, Lyman-α measurements, and radiative-decay limits [31].

• Small-scale tensions: Reduced small-scale power can lessen some low-mass-halo discrepancies but can also suppress structure beyond observationally allowed levels.

• Direct / indirect detection: No confirmed signal; X-ray decay searches and cosmological structure measurements constrain mass and mixing [31].

Self-interacting dark matter

• Rotation curves: Self-scattering can alter halo cores and reproduce a broader diversity of inner rotation curves than collisionless-only halos [32].

• Lensing and cluster masses: Viable models are constrained by cluster density profiles, strong and weak lensing, and halo-shape measurements [32].

• Bullet Cluster–type offsets: Merging clusters directly constrain the magnitude and velocity dependence of the self-interaction cross section [32].

• CMB and large-scale structure: Viable interaction models must preserve the successful near-CDM large-scale matter distribution.

• Small-scale tensions: Can lower central densities and modify halo diversity; baryonic degeneracies remain important [32].

• Direct / indirect detection: No confirmed underlying dark-sector particle signal; astrophysical observations primarily constrain the effective interaction strength.

Fuzzy / ultralight dark matter

• Rotation curves: Wave dynamics can produce solitonic or cored inner structures, depending on particle mass and composition [33].

• Lensing and cluster masses: Halo substructure and lensing constrain suppression of small-scale structure.

• Bullet Cluster–type offsets: Viable models must reproduce observed cluster-scale collision and mass-distribution behavior.

• CMB and large-scale structure: Large scales can approach CDM behavior, whereas characteristic suppression of small-scale power is tightly constrained by Lyman-α, halo-abundance, and related probes [33].

• Small-scale tensions: Potentially modifies cusp-scale structure, but parameter ranges capable of doing so are restricted by independent structure constraints [33].

• Direct / indirect detection: No confirmed signal; constraints arise from gravitational, astrophysical, black-hole, pulsar, and laboratory-sensitive effects depending on mass and coupling [33].

MOND / relativistic modified gravity

• Rotation curves: The MOND acceleration law provides strong phenomenology for many galaxy rotation curves and baryon–acceleration correlations [g,n].

• Lensing and cluster masses: Relativistic extensions can generate gravitational lensing, but galaxy clusters generally retain a residual missing-mass requirement beyond the basic MOND prescription [21].

• Bullet Cluster–type offsets: Merging-cluster observations remain a demanding constraint and generally require additional fields, unseen mass, or a sufficiently complete relativistic model [21].

• CMB and large-scale structure: Original nonrelativistic MOND has no standalone relativistic cosmology; cosmological predictions require a specified covariant completion with additional dynamical structure [21].

• Small-scale tensions: Naturally reproduces several galaxy-scale baryon–gravity regularities; its remaining difficulties occur primarily beyond the isolated-galaxy regime [21].

• Direct / indirect detection: Not intrinsically a particle dark matter model and therefore has no generic dark-particle detection target.

MQ / bounded realization

• Rotation curves: The fixed galactic realization law is applied to 175 SPARC galaxies comprising 3,366 radial points and to a prospectively screened out-of-SPARC LITTLE THINGS sample of 15 galaxies comprising 288 figure-extracted radial comparisons. LITTLE THINGS gives a galaxy-weighted geometric MQ/observed velocity ratio of 0.9907, with a 95% galaxy-block bootstrap interval of 0.868–1.130 and no significant mean-log offset (p=0.895); the characteristic-velocity normalization relation is also nonsignificant (r_P=-0.282, p=0.308), while substantial pointwise scatter remains [5–6,9].

• Lensing and cluster masses: Sec. 5.3 provides a spherical ESO 325–G004 benchmark: the frozen MQ spherical calculation predicts an Einstein radius of approximately 2.67 arcsec versus the observed 2.85 arcsec. The full resolved extended source metric/slip solution and two-dimensional deflection, convergence, and shear comparison remain untested.

• Bullet Cluster–type offsets: Sec. 5.4 formulates the time-dependent merger problem, while Sec. 5.8 provides the action-level closure used by the broader field construction. No nonlinear Bullet Cluster evolution or parameter-frozen lensing-map comparison has yet been performed; the observed offsets therefore remain an open test.

• CMB and large-scale structure: Sec. 3.9 derives MQ universe domain fractions numerically near the conventional density fractions, but these are not identified with fitted ΛCDM fluids. Section 5.8 develops the field/conservation linearization and action-level closure required to pose the perturbative problem, but no numerical Einstein–Boltzmann integration or prediction of CMB temperature/polarization spectra, CMB lensing, BAO, the matter power spectrum, or cosmological growth observables has yet been completed.

• Small-scale tensions: In the Q_SPARC=1 population, no significant residual dependence is recovered on several tested galaxy properties. This is a rotation curve control and is not itself a test of the conventional cusp-core, satellite, or halo-substructure problems.

• Direct / indirect detection: MQ is not a particle dark matter model and therefore predicts no dark matter particle detection.

3. MQ FOUNDATIONS

3.1 Right–triangle construction and count geometry

Figure 1. Count of l_f describing realized separation in the System Frame, with discrete count evaluated in the Internal Frame. Modified from [18].

The count structure underlying the Frames Principle is made explicit through the right–triangle realization construction shown in figure 1. The triangle is not a geometric structure within the Internal Frame. It is the realized representation used to relate discrete count structure to measurable separation under the Frames mapping. In this representation, side a specifies the unit reference count and is fixed to unity, establishing the normalization used for count comparison. Side b corresponds to the variable Internal Frame count n_L. The labels side a, side b, and side c identify roles within this construction and do not introduce additional physical frames.

Side c, the hypotenuse, represents the realized separation l in the System Frame. The mapping-consistent count combination entering this separation combines the Internal Frame count n_L with the mapping-induced residual count contribution Q_L, giving n_L+Q_L. Multiplication by the fundamental length l_f then gives the realized physical separation l=(n_L+Q_L)l_f where l denotes the realized separation in the System Frame, l_f the fundamental length, n_L the Internal Frame count, and Q_L the mapping-induced residual count contribution entering the realized separation. The right–triangle construction determines Q_L through

.

Thus, Q_L is not an independent primitive count but is fixed uniquely by n_L within the realization construction. As n_L increases, Q_L decreases while remaining positive, providing the finite residual contribution required for mapping-consistent realization of the count relation. The same construction satisfies the right–triangle identity 1+n_L²=(n_L+Q_L)².

The orthogonality of side a and side b belongs to this realized right–triangle representation and encodes the algebraic independence of the unit reference contribution and the variable count contribution used in constructing the hypotenuse. It does not assign an intrinsic Euclidean geometry to the Internal Frame. The hypotenuse instead provides the mapping-consistent realized separation through the count combination n_L+Q_L.

The role of this representation is operational. It defines the count relation from which subsequent MQ separation constructions are developed. A separation measured in the System Frame is therefore a mapping-normalized realization of conserved Internal Frame count rather than a direct geometric projection of that count. Consequently, the realized separation contains the residual contribution Q_L required by the Frames mapping. The gravitational significance of this residual is developed in Sec. 3.2, while its relation to the emergent continuum metric is developed independently in Sec. 3.5.

3.2 Gravitational curvature using the right-angle triangle construct

The right-angle triangle construct provides the count geometry used to describe gravitational realization. Side b carries the Internal Frame count n_L, while side c represents the mapping-consistent realized separation in the System Frame. The mapping-induced residual count contribution Q_L is uniquely determined by n_L and preserves the exact right-triangle relation under the Frames mapping.

MQ identifies this mapping-induced deformation as the geometric origin from which gravitational curvature is subsequently realized. At this stage, “curvature” denotes the MQ count-resolved scalar measure associated with gravitational coupling. It is not the tensorial curvature of the emergent continuum metric developed in Sec. 3.5. In the curvature-resolved regime, the scalar measure is constructed from the dimensionless residual ratio

and the fundamental acceleration scale

:

Equation (22) preserves the count structure explicitly. The factor

gives the fractional residual contribution associated with count separation, while

supplies the dimensional acceleration scale. No continuum radial coordinate is required at this stage.

To express the relation as a gravitational field per unit source mass, MQ applies the normalization factor

, where p_si is the momentum realization of the dimensionless invariant coefficient θ_si. Because

has dimensions of inverse mass, multiplication of Eq. (22) converts the acceleration measure to the dimensions of

. Using

and defining the radial count coordinate associated with side b as r=n_Ll_f, the expression transforms as

The coordinate r=n_Ll_f is the radial count coordinate associated with side b. It is distinct from the complete mapping-consistent realized separation l=(n_L+Q_L)l_f associated with side c.

The resulting MQ scalar measure may then be compared dimensionally with the Newtonian inverse-square gravitational field per unit source mass,

where

has the same dimensions as the normalized MQ expression. Equation (24) states only an exact dimensional equivalence. It is neither a physical equality nor an approximate dynamical law. The physical relation is stated separately in Eq. (25).

Throughout this paper, upright constants denote quantities evaluated at upper-count-limit realizations, while italicized constants denote quantities evaluated at the electromagnetic demarcation (Appx. B) or, more generally, at the applicable finite-count realization scale. MQ predicts that electromagnetic determinations are most sensitive to the Informativity differential, whereas that contribution becomes negligible at sufficiently large count separation. When both sides are evaluated within the same realization regime, the corresponding MQ gravitational relation is

Equation (25) is the canonical MQ count-resolved gravitational coupling relation used throughout the remainder of the paper. It establishes the scalar physical relation between realized count geometry and the Newtonian gravitational field in the curvature-resolved regime. The tensorial metric interpretation is developed separately in Sec. 3.5.

3.3 Geometric correlation of electromagnetic and gravitational phenomena

The dimensional distinction must be fixed before the common MQ coefficient is used in a physical relation. Untagged θ_si denotes the invariant dimensionless coefficient. Its angular realization is Θ_si and its momentum realization is p_si. The realization operators R_L and R_p denote the angular and momentum representations of θ_si, respectively; the realized representation, not θ_si itself, supplies the physical dimensions. The reduced Planck constant is denoted by ℏ. The definitions are

Equations (26–30) are representation definitions, not equalities among unlike dimensions. The angular and momentum realizations inherit the same invariant numerical coefficient through their respective realization operators. In the adopted SI realization, their displayed numerical coefficients therefore coincide with θ_si, while C_pθ supplies the dimensional conversion between angle and momentum. A plane angle and a momentum are not thereby identified as the same physical quantity.

The gravitational coupling relation may be reduced at the upper-count limit using

, as shown in Fig. 2. With r=n_Ll_f, the relation becomes

Equation (31) is written dimensionally with p_si, not θ_si. It identifies the momentum realization as the quantity relating the upper-count gravitational normalization to the fundamental length scale. The invariant coefficient carried by that realization remains θ_si. The finite-count gravitational relation remains Eq. (25).

Using the MQ fundamental length relation, Eq. (31) gives

Equation (32) has the required dimensions of momentum and does not assign momentum dimensions to θ_si. Note that the reduced Planck and gravitational constants are not italicized. All symbols are taken at the upper-count limit.

Figure 2. Arc-length construction for a circle of radius l_f. The source schematic’s local label θ=S denotes the illustrated plane angle and its numerical radian measure. It is distinct from the MQ invariant coefficient θ_si. Modified from [18].

(S18)

A distinct angular realization follows from the conventional plane-angle construction illustrated in Fig. 2. The angular realization Θ_si, defined by Eq. (27), carries the unit radians. The corresponding arc length L_si on a circle of radius l_f is

Figure 2 makes the dimensional distinction explicit: θ_si is the invariant dimensionless coefficient, Θ_si its angular realization, and p_si its momentum realization. Their common numerical coefficient expresses the shared invariant carried through realization, not an equality of physical dimensions.

These realizations provide the geometric correlation relevant to electromagnetic and gravitational phenomena. MQ does not identify electromagnetism and gravitation as the same physical field, nor Θ_si and p_si as the same dimensional quantity. The momentum realization enters the gravitational coupling developed above, while the dimensionless and angular realizations enter the fixed-count and geometric constructions developed in Secs. 3.4 and 3.7. Their common coefficient therefore provides a structural correlation within the Frames mapping while the physical representations remain distinct.

Equating the two forms of p_si in Eq. (32) gives

The gravitational construction developed in Sec. 3.2 provides the MQ relation from which the Planck-scale relations and fundamental measures are subsequently derived. As established in Ref. [15], these results follow algebraically from the MQ gravitational coupling together with the MQ fundamental relation, rather than being introduced as independent Planck-scale assumptions. Equation (35) has the conventional Planck length form, constructed from the reduced Planck constant, gravitational constant, and defined speed of light [7]. This algebraic coincidence identifies the standard Planck-scale form recovered by the MQ reduction.

3.4 Electromagnetic and gravitational interaction demarcations
The gravitational curvature relation developed in Sec. 3.2 establishes that gravitational realization depends on the mapping-induced deformation associated with count separation. Within MQ, electromagnetic behavior is realized at a fixed interaction scale defined by the electromagnetic demarcation, whereas gravitational behavior occupies the curvature-resolved regime and is evaluated across varying count separations. These are scale-dependent realizations of the same underlying mapping-induced deformation. The electromagnetic demarcation is not a fixed electromagnetic wavelength, energy, propagation distance, or QED renormalization scale. It is the fixed interaction count distance at which electromagnetic realization is evaluated under the Frames mapping.

The electromagnetic demarcation is not a galaxy- or interaction-specific fitting parameter. It is derived from the invariant coefficient θ_si, introduced in Sec. 3.3, together with the MQ count geometry. In the source construction used here, θ_si and the MQ fundamental measures ultimately depend on the measured fine structure constant α, electron mass m_e, Bohr radius a₀, and defined speed of light c. Quantities derived from this source set are therefore not subsequently treated as independent validation when they share that upstream dependence.

One-half of the electromagnetic interaction count is

giving approximately 42.3. Its nearest whole-unit count is 42. This value provides the discrete reference associated with one-half of the electromagnetic demarcation and reappears in subsequent electromagnetic constructions, including the fine structure constant realization developed in Sec. 3.7.

The electromagnetic demarcation may first be obtained from the blackbody construction developed in Appx. B. Beginning with the gravitational curvature relation and substituting the count-separation relation

, the construction gives

Equation (37) defines the blackbody realization of the electromagnetic demarcation. Its complete derivation is given in Appx. B, Eqs. (B.1–B.9).

A second construction follows from the charge-coupling relation,

which expresses the electromagnetic demarcation directly as a function of the invariant coefficient θ_si. Its derivation is given in Appx. B, Eq. (B.10).

A third, historically earlier construction follows from the elementary charge solution developed using the gamma approach. Here, e_c denotes the classically realized MQ elementary charge, γ is the conventional electromagnetic transformation parameter used in the historical gamma approach, and r_q41 denotes the historical approximate discrete-offset quantity used in that construction. In this approach, γ was deliberately resolved from a measured electromagnetic constant different from the quantity being calculated, avoiding direct use of the prediction target as its own input. The construction nevertheless depended on an additional experimentally measured electromagnetic source quantity rather than solely on MQ fundamental quantities. The corresponding demarcation relation is

The γ in Eq. (39) is distinct from the later fundamental gamma γ_F. The latter is a first principles realization parameter introduced after the elementary charge realization eliminated the earlier discrete-offset and auxiliary-γ construction. Unlike the conventional γ, which is resolved from experimentally measured electromagnetic constants, γ_F is derived entirely from MQ fundamental quantities and serves as the canonical electromagnetic realization parameter in later MQ derivations.

The blackbody, charge-coupling, and historical elementary charge constructions provide different derivational routes to the same electromagnetic interaction count. Their numerical agreement is an internal closure test of the shared MQ count geometry, not three statistically independent laboratory determinations. The historical route is additionally distinguished by its auxiliary measured electromagnetic input, whereas the later γ_F construction removes that dependence. Table 3 summarizes the three demarcation values.

Table 3. Electromagnetic demarcation obtained from three MQ constructions

ᵃ Here e denotes the defined SI elementary charge [7]. The associated interaction count is solved as a function of the fine structure constant and this defined value. This historical gamma-based construction is retained as a cross-consistency route and is distinct from the later MQ γ_F realization.

The physical significance of the electromagnetic demarcation is not that electromagnetic processes occur at a single conventional physical distance. Electromagnetic realization under the Frames mapping is evaluated at the fixed interaction scale

, whereas gravitational phenomena occupy the curvature-resolved regime, in which realized geometric structure varies with count separation. Both arise from the same mapping-induced deformation but represent different scale-dependent realizations of that mapping effect. MQ therefore does not identify electromagnetic and gravitational phenomena as the same physical field or replace their conventional scale-dependent dynamics.

3.5 Emergent metric from coarse–graining

The right-triangle count construction developed in Sec. 3.1 establishes the mapping-consistent count combination n_L+Q_L, where n_L is the Internal Frame count and Q_L is the mapping-induced residual count contribution introduced by the Frames mapping. Multiplication by the fundamental length l_f gives the corresponding realized separation l in the System Frame,

The quadratic separation is therefore

. This finite System Frame realization supplies the realized separation functional from which the local continuum metric is reconstructed. The metric is not introduced independently of the mapped count structure.

For the local continuum description, let x^μ denote realized System Frame coordinates and dx^μ their differential displacements. The differential interval dl has the standard quadratic form

where

denotes the emergent System Frame metric tensor evaluated at x.

To reconstruct the metric covariantly from separation data, let l(x,x') denote the realized two-point separation between neighboring System Frame points x and x'. In a normal neighborhood where l²(x,x') is smooth and admits a nondegenerate continuum limit, the mixed-endpoint coincidence relation gives [34]

The minus sign follows from differentiation with respect to opposite endpoints of the squared separation. Equivalently, with Synge’s world function

, the standard coincidence identity is [σ_;μν']=-g_μν [34]. Within MQ, the distinction lies in the origin of l²(x,x'): it is the realized separation functional supplied by mapped count structure rather than a separation assumed from a primitive metric field.

The continuum coordinates arise from discrete count increments through x^μ=n^μl_f, where n^μ is the realized count coordinate corresponding to x^μ. Since l_f is constant,

Applying Eq. (43) to both coordinate arguments of Eq. (42) cancels the explicit

in the realized quadratic separation. Let n=(n^μ) and n'=(n'^μ) denote the neighboring count-coordinate arguments corresponding to x and x'. The scalar count quantities evaluated between them are written n_L(n,n') and Q_L(n,n'). The count-resolved metric reconstruction is then

Equation (44) reconstructs the local System Frame metric from the second-order response of the realized separation functional. It does not by itself select a global metric, signature, or curvature field. It specifies the local continuum metric when coarse-graining of the realized count structure admits a smooth, nondegenerate continuum representation.

The defining count relation follows directly from the right-triangle construction,

where the scalar count separation satisfies the exact mapping-consistent identity established in Sec. 3.1. Equations (40–45) therefore connect the finite right-triangle count relation to its local continuum metric representation. The metric is not a primitive structure of the Internal Frame, but a System Frame continuum object reconstructed from the realized separation functional through the Frames mapping.

Equations (40–45) establish metric emergence only. They are a kinematic reconstruction and do not, by themselves, supply a gravitational field equation, matter source law, geodesic dynamics, or photon propagation. In the broader MQ construction summarized in Ref. [1], the reconstructed metric enters the MQ Action, whose stationary configurations are governed by the MQ variational principle. Metric variation defines the stress-energy response, while the Levi-Civita connection, curvature, diffeomorphism compatible variation, and contracted Bianchi identity provide the corresponding geometric dynamics and covariant conservation. The metric therefore acquires dynamics through this separately derived variational construction, not merely because it can be reconstructed from the realized separation functional.

The relativistic completion in Ref. [1] extends the realized spatial geometry, under its stated causal order and regularity conditions, to a time-oriented Lorentzian System Frame geometry supporting locally Lorentz covariant matter dynamics and matter geometry feedback. These downstream results do not make Eqs. (40–45) dynamical. They establish that the metric reconstructed here is the geometric input to the separate MQ dynamical completion.

For the present dark matter analysis, the galactic realization law developed in Sec. 4.2 is the nonrelativistic weak-field result used for orbital dynamics. Rotation curve agreement alone does not derive the complete relativistic metric solution for an extended galaxy. The independent lensing problem in Sec. 5.3 instead requires solving the System Frame field dynamics for the specified baryonic source and propagating null geodesics through that solution. Metric emergence, metric dynamics, and a system-specific relativistic solution are therefore separate requirements.

3.6 Non-Lorentz length contraction
Using the right-triangle count construction introduced in Fig. 1, MQ predicts a non-Lorentz length contraction arising from discrete Internal Frame count structure and its realization in the System Frame. Unlike Lorentz contraction, whose magnitude depends on relative velocity between inertial observers, the MQ effect is the Informativity differential: a count-dependent geometric manifestation of normalization under the Frames mapping. It is therefore not a relativistic velocity effect, but a manifestation of the mapping-induced deformation quantified by Q_L.

Realized separation is obtained from the mapping-consistent combination n_L+Q_L rather than by direct projection of the Internal Frame count n_L. Beginning with the count identity established in Eq. (45),

the right-triangle relation is

The exact relation in Eq. (49) has the upper-count behavior

Since Q_L→0 as n_L→∞, the quadratic contribution vanishes asymptotically, giving

Thus, the normalization factor 2Q_Ln_L approaches unity, while the Informativity differential, 1-2Q_Ln_L, approaches zero. The classical large-count limit is recovered, whereas finite-count realization retains a nonzero mapping-induced contraction.

This distinguishes the realization regimes established in Sec. 3.4. Electromagnetic interactions are evaluated at the fixed interaction scale defined by the electromagnetic demarcation, where the Informativity differential remains finite. Gravitational realization occupies the curvature-resolved regime, in which increasing count separation drives the same mapping-induced deformation toward its upper-count limit.

Two applications are the reduced Planck constant and gravitational coupling. At the electromagnetic demarcation, the finite-count reduced Planck constant ℏ contains the non-Lorentz normalization,

while the finite-count gravitational coupling G follows from the same count geometry,

Equation (53) shows that finite-count G is constructed from the momentum realization p_si, the mapping-induced residual Q_L, the count separation n_L, and the fundamental measures. Its numerical value therefore depends on the count regime at which the expression is evaluated.

At the electromagnetic demarcation, the finite-count evaluation is

This evaluates to

, where italic G denotes the gravitational coupling evaluated at this finite interaction count.

At the upper-count limit, Q_L→0, n_L→∞, and 2Q_Ln_L→1, giving the upright upper-count gravitational constant,

The upper-count value is

. Equations (54) and (55) are finite-count and upper-count evaluations of the same MQ gravitational coupling. Relative to the 2022 CODATA recommendation

[7], they are distinct MQ realization values rather than separate constants recognized by conventional gravitational metrology.

The historical CODATA sequence provides a retrospective numerical comparison. The 2010 recommendation differs from the finite-count value of Eq. (54) by 0.006σ, the 2014 recommendation differs from the upper-count value of Eq. (55) by 0.007σ [35–36], and Ref. [15] obtains a Planck-coupled evaluation close to the 2018 value retained in the 2022 adjustment [7,37], with a reported correspondence of approximately 0.08σ. Because CODATA adjustments combine heterogeneous measurements and were not constructed according to MQ realization regimes, Appx. E classifies these historical matches as one retrospective concordance family rather than three statistically independent confirmations. Measurements of G also remain unusually dispersed relative to other fundamental constants [38]. The CODATA sequence therefore does not establish that its changes were caused by the Informativity differential.

A temporally later external test became available in 2026 from the NIST redetermination using the BIPM torsion balance [39]. The measurements themselves were acquired in 2024, substantially after publication of the MQ Informativity differential and its realization-dependent gravitational predictions. Schlamminger et al. obtained [value], approximately [value] below the 2022 CODATA recommendation and [value] from the finite-count prediction of Eq. (54), using the combined quoted uncertainties. The upright notation identifies the macroscopic gravitational determination, whereas comparison with the finite-count Gof Eq. (54) tests the previously published MQ realization prediction.

The significance of this experiment extends beyond numerical agreement in the combined value. The BIPM torsion balance determines the same gravitational interaction by two physically distinct closure methods. In free-deflection operation, all relevant components are electrically grounded and the gravitational torque is inferred from the mechanical response of the torsion balance. In electrostatic-servo operation, the pendulum electrode is grounded while driven electrodes supply an electrostatic counter-torque that balances the gravitational torque. These are precisely the contrasting experimental conditions relevant to the MQ distinction between curvature-resolved macroscopic gravitational realization and gravitational determination closed through a fixed electromagnetic interaction scale.

The realization-dependent distinction predates the NIST/BIPM measurements and was not introduced in response to their outcome. MQ predicts that a determination whose torque closure depends explicitly on the electromagnetic interaction should exhibit the finite-count realization more strongly than a free mechanical deflection determination, shifting the inferred gravitational coupling toward the finite-count value. Schlamminger et al. subsequently found this ordering independently: for both copper and sapphire source masses, the electrostatic-servo determination was lower than the corresponding free-deflection determination. Their paper explicitly identifies this repeated ordering across the two source-mass materials.

The experimental group subjected the discrepancy to extensive investigation. The analysis considered source- and test-mass geometry, pressure-dependent effects, autocollimator nonlinearity, capacitance and voltage calibration, mass integration, and other method- and configuration-dependent systematic contributions [39]. The distinction between the two operating modes was itself a longstanding motivation of the BIPM program: the apparatus was designed to determine G by both free deflection and electrostatic compensation in part to investigate the persistent dispersion among precision measurements of G. The 2026 analysis did not identify a conventional experimental mechanism that converts the observed mode ordering into the MQ prediction.

The NIST/BIPM experiment therefore provides a direct experimental test of a prediction published by MQ before these measurements were performed. The experiment found the predicted direction of the realization-dependent shift and, within the stated comparison, the predicted gravitational-coupling scale.

As this effect is directly a consequence of the MQ proposed Frames mapping and those properties recognized by the System and Internal Frames, the result is a definitive finding in support of the MQ approach. Notably, the result carries substantially greater evidentiary weight than the retrospective correspondence with historical CODATA recommendations because the MQ prediction preceded the measurements used to test it. The NIST/BIPM authors did not analyze their result within the MQ framework or cite the earlier MQ prediction. Consequently, the experiment should not be described as an endorsement of MQ by those authors. Rather, their independently obtained measurements provide direct experimental evidence consistent with—and quantitatively testing—the prior MQ realization prediction.

With this realization structure established, the conventional Planck unit expressions follow from the MQ count-geometric construction ([15], Appx. BN),

These equations are terminal conventional forms, not starting definitions of l_f, m_f, or t_f. Ref. [15] first resolves the gravitational relation and fundamental measures from the right-triangle count construction, invariant coefficient θ_si, and fundamental count relations; those relations are then combined in Appx. BN of Ref. [15] to recover Eqs. (56–58). The G and ħ appearing in these Planck-coupled expressions can then by generalized from their finite-realization typography thus giving us Planck’s expressions without assuming the dimensional expressions at the outset.

3.7 An Example of the Frames mapping via the fine structure constant
The preceding sections established the count geometry, electromagnetic demarcation, and Informativity differential required by the Frames mapping. This subsection follows those elements through successive realization stages to the experimentally accessible inverse fine structure constant. Because α is also used upstream to determine θ_si in Sec. 3.8, the calculation is an internal closure test of the MQ realization structure rather than an independent determination of α.

As established in Sec. 3.4, electromagnetic realization occurs at the fixed interaction scale n_L. One-half of the demarcation is approximately 42.3, whose nearest whole-unit realization is the System Frame count n_θ=42. MQ associates this count with the fundamental inverse fine structure form through 42θ_si. The realized count n_θ=42 and the quantity 42θ_si are distinct: the former is the selected whole-unit count, whereas the latter is the corresponding inverse fine structure quantity before subsequent realization stages.

The invariant coefficient θ_si was introduced in Sec. 3.3 and is treated here as established. Section 3.8 determines its numerical value by solving the inverse MQ realization relation using the measured fine structure constant. Here the direction is reversed: the established θ_si is propagated through the whole-count relation, Frames mapping, metric differential, and finite-count Informativity differential to recover the classically realized inverse fine structure constant. Comparison with the same CODATA fine structure input used in Sec. 3.8 therefore tests algebraic closure rather than providing an independent prediction.

Equation (59) defines the fundamental Internal Frame form,

Application of the Frames mapping produces the continuous System Frame representation. The corresponding fine structure metric differential is

where the mixed floor-ceiling brackets denote nearest whole-unit realization. The metric differential is the representation-level mismatch required when the discrete count relation is expressed continuously in the System Frame. It is distinct from the physical contraction described by the Informativity differential.

Adding

to the fundamental form gives the Planck-like inverse fine structure form,

At this stage the construction includes the discrete-to-continuous metric differential but not the finite-count normalization established in Sec. 3.6.

At the electromagnetic demarcation, the charge-coupling relation gives

while the residual count product from the right-triangle construction in Sec. 3.1 is

Together, Eqs. (63) and (64) determine 2Q_Ln_L at the fixed interaction scale.

Let Δ_(int-sys) denote the finite-count correction carrying the Planck-like inverse fine structure form to its classically realized value under the Informativity differential. Equation (65) defines

Subtracting this correction gives

which reduces to

Substitution of Eq. (62) gives the classically realized form,

The resulting

is the MQ quantity directly comparable with a laboratory determination of α^-1.

Table 4. Fundamental, Planck-like and classically realized forms of the inverse fine structure constant

Morel et al. measured the inverse fine structure constant by rubidium atom-recoil matter-wave interferometry, obtaining α^-1=137.035999206(11) with a relative accuracy of 81 parts per trillion [40]. The fundamental and Planck-like entries are intermediate MQ realization stages rather than corresponding Morel observables. The classical MQ value differs from the Morel result by approximately 0.19σ when the quoted standard uncertainties are combined. This agreement is a closure benchmark, not an independent experimental confirmation, because the Morel result contributes to the fine structure constant measurement family incorporated into the CODATA adjustment used upstream.

The like-for-like comparison with the 2022 CODATA recommendation used in Sec. 3.8, α^-1=137.035999177(21) [7], gives a difference of approximately 0.63σ from the classically realized MQ value 137.035999200(30). Because that CODATA value is used to determine θ_si, this is likewise an internal concordance result rather than an independent prediction test.

Table 4 therefore records successive MQ realization stages, not three separate experimental predictions. Equations (59–68) provide a worked example in which the discrete count relation, metric differential, and Informativity differential combine under the Frames mapping to recover the experimentally accessible inverse fine structure form.

3.8 Calculating the invariant coefficient as a function of the fine structure constant

Section 3.7 demonstrated the forward MQ realization sequence by propagating an established θ_si through the Frames mapping to the classically realized inverse fine structure constant

. The present subsection performs the inverse operation: the 2022 CODATA recommended inverse fine structure constant, α^-1=137.035999177(21) [7], is identified with the corresponding classically realized MQ quantity

and used to determine θ_si. Equation (76) is therefore a parameter determination from the measured fine structure constant, not an independent prediction of it.

The inversion begins with the right-triangle count construction established in Sec. 3.1 and illustrated in Fig. 1. The mapping-induced residual count contribution is

Multiplication by θ_si gives

and division by θ_si gives

The inverse fine structure relation is the classically realized MQ form established in Sec. 3.7. It incorporates the representation-level metric differential associated with the Frames mapping and the finite-count normalization associated with the Informativity differential,

where the mixed floor-ceiling brackets denote nearest whole-unit realization.

No independently measured interaction count enters the inversion. The electromagnetic demarcation established in Sec. 3.4 fixes n_Lθ_si=276, equivalently

. Thus, the measured

is the sole experimental input to the numerical determination, while n_L, Q_L, and their relation are supplied by the MQ construction.

For the following rearrangement, α_c denotes the classically realized fine structure constant and

its inverse. Rearranging Eq. (72) gives

Substitution of Eq. (71) into Eq. (74), with n_Lθ_si=276, gives

The nearest-whole-unit operation makes Eq. (75) piecewise across its admissible count realizations. The applicable branch is fixed by the electromagnetic demarcation and nearest-whole-unit realization established in Secs. 3.4 and 3.7 rather than selected by fitting the resulting value. Solving Eq. (75) on that branch with the 2022 CODATA value α^-1=137.035999177(21) [7] gives

The quoted uncertainty propagates from the CODATA fine structure input through the nonlinear inversion. No independent determination of n_L is required because the charge-coupling relation fixes n_Lθ_si=276.

Equation (76) establishes the numerical invariant coefficient used in subsequent MQ calculations. It is complementary to the forward construction in Sec. 3.7: Sec. 3.8 uses the measured fine structure constant to determine θ_si, whereas Sec. 3.7 propagates that coefficient through the realization sequence to test internal algebraic closure. Reconstructing the fine structure constant from the same upstream CODATA source is therefore an internal closure test rather than an independent prediction.

3.9 Referenceability-domain fractions of the expanding frame

The expanding frame domain relations provide a cross-scale consequence of the same upstream MQ invariant coefficient used elsewhere. They are mapping-defined referenceability fractions, not independently conserved ΛCDM fluids. Their numerical proximity to Planck density fractions is therefore a correspondence benchmark. A dynamical cosmological test additionally requires the background and perturbation evolution developed in Sec. 5.8.

Here Ω_f is the fundamental domain quantity, Ω_dk the dark domain, Ω_obs the observable domain, Ω_vis the visible domain, and Ω_uobs the unobserved domain. The realized referenceability partition satisfies Ω_obs=Ω_vis+Ω_uobs and Ω_dk+Ω_obs=1, as established in Appx. C. The fundamental domain Ω_f is not an additional additive component of this partition; it constrains the referenceability geometry.

Once θ_si has been determined in Sec. 3.8, Eqs. (77–81) follow algebraically without fitting matter-density, energy-density, or astrophysical population data. Their uncertainties therefore propagate the source uncertainty in θ_si through these closed-form relations.

The MQ fractions may be compared numerically with fractional density parameters inferred under ΛCDM, but the quantities have different physical definitions. In the standard cosmological analysis, baryonic, cold dark matter, total matter, and dark energy fractions are inferred jointly within a dynamical background and perturbation model from the CMB spectra and other cosmological data [11]. Equations (77–81) instead describe referenceability geometry. Table 5 therefore reports numerical correspondence rather than dynamical equivalence.

Table 5. Numerical comparison of MQ referenceability domain fractions with Planck 2018 ΛCDM density fractions density fractions and associated MQ mass equivalent allocations

The stronger test is dynamical: the MQ field equations must generate a homogeneous cosmological background and perturbations that reproduce CMB, BAO, CMB-lensing, and structure-growth observables without inserting the corresponding ΛCDM density fractions as MQ inputs. Section 5.8 establishes the field/conservation linearization and regular source formulation needed for that calculation but does not yet provide the numerical Einstein–Boltzmann evolution. The Table 5 fractions therefore must not be assigned directly to independently conserved System Frame stress-energy components.

Conversion of the percentages to the absolute mass-equivalent allocations shown in Table 5 additionally requires the elapsed universe age. The realized expanding frame construction uses the MQ mass accretion rate derived in Sec. 3.10,

where p_si is the momentum realization introduced in Sec. 3.3. Equation (82) is stated here because it supplies the Table 5 mass column; its derivation follows in Sec. 3.10. For a specified universe age, integration of this realized rate gives the accumulated total mass-equivalent content M_tot, to which the dimensionless fractions are applied. Thus θ_si fixes the domain fractions, the expanding frame construction fixes M_acr, elapsed age fixes M_tot, and the fractions fix the associated mass-equivalent allocations.

The fundamental domain Ω_f has no direct ΛCDM counterpart and is not an additional matter or energy component.

3.10 Count allocation before realization and mass accretion after realization

This subsection records the allocation/accretion consequence of the expanding frame count relations. Before System Frame realization, the relations describe count allocation; after realization, the same algebraic structure acquires a temporal mass-equivalent interpretation. These relations are retained for cosmological provenance and are not required to evaluate galaxy rotation curves. The galactic dependency proceeds instead through the frame-resolved expansion rate developed in Sec. 3.12.

The dimensionally explicit fundamental relation uses the momentum realization p_si,

Equation (83) is a relation among the fundamental measures and their momentum realization. The universe-scale length, mass, and time counts are not obtained by multiplying independent count factors into both sides of Eq. (83). Instead, the expanding frame construction fixes separate count relations for spatial extent and accumulated mass-equivalent content.

For the universe-age count

, the realized diameter count is

while the total mass-equivalent count is

where

is the universe-scale count of l_f,

is the count of m_f corresponding to the accumulated total mass-equivalent content, and

is the elapsed count of t_f. Their ratio is therefore

and the dimensionless diameter-age relation may be written

with

,

, and

. Equation (87) is dimensionless throughout and supplies the count identity used later in Sec. 3.12.

The elapsed universe-age count is

where n_Tᵤ denotes the elapsed count of fundamental time units t_f. Restoring the fundamental length-to-time conversion gives the physical radius,

and the corresponding fundamental domain mass allocation is

where M_f denotes the allocation associated with the fundamental domain and is distinct from the fundamental mass measure m_f. Prior to System Frame realization, Eq. (90) specifies count-based allocation rather than physical mass production through elapsed time.

The domain relations required for the global allocation are

Equation (97) establishes the normalized relation between the fundamental and observable domains. The corresponding mass allocations are related to the normalized fractions by

rather than by substituting dimensional masses directly for the dimensionless Ω quantities.

Using the universe-scale mass count from Eq. (85), the accumulated total mass-equivalent content is

Equation (99) shows that after System Frame realization, accumulated mass-equivalent content is proportional to elapsed fundamental time count. Before realization, the same relation specifies count allocation. Equations (90), (92), and (99) are mutually consistent because

Let

denote the realized elapsed duration. Dividing the accumulated mass equivalent by T gives

The final form in Eq. (100) is equivalently a rate of

fundamental mass units m_f per fundamental time unit t_f. Using the dimensionally explicit momentum relation

yields

Equation (101) defines the MQ mass accretion rate M_acr for the realized expansionary epoch. It is distinct from the fundamental mass m_f, the pre-realization fundamental domain allocation M_f, and accumulated total mass M_tot. It is not a conventional astrophysical accretion rate such as gas infall or halo growth. It is the global MQ rate associated with the expanding frame count relation.

For the adopted universe age used in this work, the count relations give approximately

with

The final numerical line is included only to make the count interpretation explicit. The defining relations are Eqs. (84) and (85).

For the dark matter application, Sec. 3.9 supplies the global referenceability fractions, while Eqs. (98-101) provide the corresponding allocation/accretion accounting. Local gravitational evolution may subsequently redistribute realized mass-equivalent content into galaxies, clusters, and other structures, but such structure formation is distinct from the global MQ mass accretion law. The observationally resolved expansion rate representations relevant to the galactic construction are developed separately in Sec. 3.12.

3.11 CMB temperature from the MQ count-geometric construction

The CMB-temperature calculation is retained as a cross-scale consistency check rather than as part of the galaxy prediction. It propagates the expanding frame construction to a present radiation density and blackbody temperature without using the observed CMB monopole temperature as an input. The calculation does, however, inherit the adopted Planck cosmic age and is therefore not statistically independent of all CMB information. No CMB-spectrum validation is claimed.

The derivation begins with the minimum three-dimensional count configuration at which a globally referenceable realization first becomes admissible. Let n_a and n_b denote orthogonal unit-count components and n_bn the resulting two-dimensional diagonal count entering the third-dimensional extension. These labels specify components of this local count construction and introduce no additional MQ count primitives.

Using this minimum count with the previously established count relations gives the kinematically resolved expansion quantity v' associated with this referenceability construction. The universe-scale counts

,

, and

denote counts of the fundamental length, mass, and time measures l_f, m_f, and t_f, respectively. The time count

is the universe age count established in Sec. 3.10, while

denotes the boundary-resolved length count entering the minimum referenceability construction. The quantity v' acquires a dimensionful velocity interpretation only upon realization.

Dimensional consistency requires the momentum realization p_si introduced in Sec. 3.3. Thus,

and for the boundary-resolved length count,

Let