Classical Description of Mass in the Universe without Λ and CDM

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We describe universal mass through referenceability domains defined by what is presently referenceable, what becomes referenceable over elapsed time, and what remains non-referenceable. Using Measurement Quantization (MQ) domain relations and the 2022 inverse fine-structure input from the Committee on Data of the International Science Council (CODATA), we calculate their dimensionless fractions. With the adopted temporal mapping normalization, the total mass is $$3.05108(85)\times {10}^{54}\mathrm{kg}$$ and the visible mass is $$1.47942(41)\times {10}^{53}\mathrm{kg}$$. These absolute masses are conditional MQ calculations, and their standard uncertainties propagate the stated source inputs without quantifying model discrepancy. Comparison with cosmological-constant and cold-dark-matter (ΛCDM) density parameters concerns numerical domain fractions and does not establish identical physical contents. We also examine a conditional bound-velocity construction and discuss separate tests of the fixed MQ galactic realization law. The accompanying preprint reports a reduction of the Milky Way root mean square velocity residual from 69.03 to 8.90 km s⁻¹ using distinct baryonic-source and kinematic constructions, together with a broader galaxy population test. These results retain the source and covariance limitations stated below.

Keywords: Universal Mass, Mass, Visible Mass, Observable Mass, Referenceability Domains

1. INTRODUCTION

We describe universal mass using referenceability domains defined relative to an observer. The visible domain is presently referenceable; the unobserved domain becomes referenceable later; the observable domain contains both. The dark domain never becomes referenceable. The fundamental domain supplies a normalization pivot rather than an additional disjoint component. These domains are distinct from the physical contents and dynamics of the ΛCDM components used for numerical comparison.

The dimensionless fractions Ω_vis, Ω_obs, Ω_dk and Ω_uobs noted in Table 1 denote the visible, observable, dark and unobserved domains just described; Ω_tot denotes their total normalization. The symbol Ω_f denotes the fundamental fraction. The corresponding domain masses are M_vis, M_obs, M_dk, M_uobs, M_tot and M_f in kg. Uppercase M_f is distinct from the fundamental mass measure m_f in kg. In its comparison column, Ω_Λ, Ω_c and Ω_b denote the conventional dark-energy, cold-dark-matter and baryon density parameters. The conventional total-matter density parameter Ω_m denotes total matter. Each density parameter divides component energy density by critical energy density.

Table 1. Mass associated with referenceability domains.

The domain calculation uses classical expressions and MQ relations without adding inflation theory [1], Loop Quantum Gravity [2, 3], String Theory [4] or Supersymmetry. This does not exclude alternative cosmological physics; the numerical calibration, temporal reference and bound-count normalization remain explicit assumptions. Figure 1 is the original manuscript illustration; Ref. [5] gives related domain calculations. Its historical mass symbols beside percentages denote fractions and its kilogram labels denote masses. Table 1 gives current conditional results.

Fig 1. Historical illustration; related construction [5]. See Table 1.Figure 1 distinguishes a domain fraction from its associated mass. In MQ, the Internal Frame is the discrete count configuration domain, the System Frame is its continuous relational encoding, and the Frames mapping is the kinematic transformation between them ([6], Appx. K).

The age calculation also denotes the total mass by M_U. It retains units kg, whereas the total fraction Ω_tot is normalized to 1.

2. METHODS

Measurement Quantization uses reference measures and counts to construct the relations examined here. Classical gravitational and kinematic expressions, together with the uncertainty relation associated with Heisenberg [7], provide background for the approach. Earlier MQ applications are discussed in Refs. [5, 8–12]. Those publications document the development of the model; their existence does not by itself establish the physical validity of every application.

The discrete count structure and its mapping to continuous realized quantities are MQ premises in this calculation. The introductory MQ account [6] supplies theoretical arguments and proposed comparisons. Neither the uncertainty relation alone nor the numerical results below establish universal discreteness as an independent experimental finding. The measured inputs and adopted normalizations used here are stated in Section 3.

https://www.researchgate.net/publication/361148724_OnlineResources

The published MQ account [6] identifies two calculation spreadsheets, Supplement 1 and Supplement 2, through the online resource archive linked above. Appendix citations below refer to the published article. The present numerical interpretation follows the source inputs, shared dependencies and normalizations specified in Section 3. The historical spreadsheets are not independently reproduced here, and agreement with a calibration input is not an independent test.

3. RESULTS

1. Definitions

We begin with the dimensionless domain relations in Eqs. (1)–(4). The invariant MQ coefficient θ_si is calibrated as θ_si=3.26239030504(25) from the 2022 CODATA inverse fine-structure input [13]. Its uncertainty propagates that shared calibration input through the adopted MQ realization and excludes model discrepancy. Conventional Planck units [14] and the dimensional momentum realization used below are distinguished from this coefficient ([6], Appx. Q, W, X). Parenthetical uncertainties give one standard uncertainty in the last quoted digits. The MQ entries in Tables 1 and 2 share calibration inputs and are correlated.

2. Calculations

Equation (4) fixes the ratio of observable to visible fractions. Equations (5) and (6) are algebraic rearrangements of that ratio. Equation (7) supplies the MQ geometric expression for the fundamental fraction. These steps alone do not determine the complete partition; the additional closure relation in Eq. (11) is required.

To connect that closure with the partition, Eq. (8) repeats Eq. (4). Multiplying both sides by $$\frac{{{\mathit{\theta }}_{\mathit{s}\mathit{i}}}^2}{2}$$ and subtracting 1 gives Eq. (9), which simplifies to Eq. (10).

Equation (11) is an additional MQ domain relation. Combining it with Eq. (7) fixes the observable fraction and makes the right-hand side of Eq. (10) equal to the dark fraction through Eq. (2). Substituting Ω_obs=Ω_tot−Ω_dk and the unit total normalization gives Eqs. (12) and (13). Solving yields the dark fraction in Eq. (14).

The same normalization expresses the dark fraction as the complement of the observable fraction. Equations (15) and (16) isolate the observable fraction, giving Eq. (17).

Substituting Eq. (17) into the observable-to-visible relation, Eq. (4), gives Eq. (18). Solving for the visible fraction yields Eq. (19).

The unobserved fraction then follows from the difference in Eq. (3), giving Eq. (20). Its uncertainty retains the shared dependence of the observable and visible fractions on the same coefficient.

3. Mass Scale and Cosmological Comparison

The fractions above determine relative masses. The absolute scale additionally uses the mass accretion rate M_acr in kg s⁻¹ ([6], Appx. BE). The momentum realization p_si has units kg m s⁻¹ and the same numerical coefficient as θ_si under the adopted MQ normalization in the International System of Units (SI). The fundamental length l_f and time t_f have units m and s; m_f is the mass measure introduced in Section 1. The age A_U is expressed in seconds before conversion to years ([6], Appx. BI).

Equation (22) relates the age to the cosmic microwave background (CMB) temperature T_CMB and the radiation constant a in J m⁻³ K⁻⁴. We use T_CMB=2.72548(57) K as an independent non-CODATA source input [15]. The speed of light c is exactly 299792458 m s⁻¹, and the circle constant π is dimensionless. The mapped expansionary-epoch duration A_ee and SI-equivalent quantum-epoch extent A_qe have units s. The latter encodes the pre-referenceable Internal Frame in later System Frame time units.

The reference t_ref=1 s is an adopted exact normalization of the Frames mapping. Equation (24) defines the corresponding quantum-epoch extent, and Eq. (23) maps it to the expansionary-epoch duration. Together with the rate in Eq. (21), these quantities determine the age in Eq. (22) and the total mass in Eq. (25). This calculation does not independently derive the physical temporal reference. The source-input uncertainties include the shared calibration dependence and the independent CMB-temperature uncertainty; they exclude model discrepancy. In Eq. (24), upright e is the dimensionless base of natural logarithms, not the elementary-charge symbol.

Table 2. Comparison of Planck 2018 Density Parameters with MQ Calculations.

^aPlanck 2018 TT,TE,EE+lowE, base ΛCDM, 68% limits [16], Table 2. The total-matter row includes the source model’s matter contents and is not identified with baryons plus cold dark matter alone. Component percentages marked b are central-value conversions, not reconstructed posterior summaries. MQ masses are conditional on the adopted temporal normalization; MQ percentage and mass uncertainties are correlated. TT, TE and EE denote temperature, temperature–E-mode and E-mode polarization spectra; lowE denotes the low-multipole E-mode polarization likelihood.

^bBaryon (4.94%) and cold-dark-matter (26.56%) percentages are conversions of the published central physical-density parameters 0.02236 and 0.1202 using the same TT,TE,EE+lowE central Hubble constant of 67.27 km s⁻¹ Mpc⁻¹ [16], Table 2. They are not reconstructed posterior means. Their uncertainties have not been reconstructed from the joint posterior; no transformed uncertainty or statistical agreement with MQ is claimed.

Table 2 compares the domain fractions with historical Planck 2018 parameter estimates under the source ΛCDM model [16]. The dark-energy and total-matter entries are published posterior summaries; the component conversions have the uncertainty qualification given above. Agreement in a fraction does not establish equivalence of physical contents, perturbations or expansion dynamics. The kilogram column is calculated from Eq. (25), rather than measured by the Planck analysis.

4. Dark Energy Phenomenon

Equation (26) gives the physical diameter D_U in m from the age and the MQ geometric coefficient ([6], Appx. AZ). The factor c converts the temporal extent to length. Equation (27) defines the baseline inverse-age rate H in s⁻¹ before conversion to km s⁻¹ Mpc⁻¹ ([6], Appx. BA). Both results depend on the adopted temporal normalization. The dimensionless coefficient θ_si is not itself a dimensional expansion rate.

For the same TT,TE,EE+lowE selection used in Table 2, Planck reports H₀=67.27 ± 0.60 km s⁻¹ Mpc⁻¹ under base ΛCDM [16]. Here H₀ denotes a present-epoch rate inferred from observations, distinct from the MQ baseline H. Their comparison therefore depends on the respective models and source uncertainties. Studies of peculiar-velocity effects in local estimates [17], supernova evidence for accelerating expansion [18], and constant constraints related to the Hubble tension [19] address additional observational context. The arithmetic midpoint of different determinations is not an observational estimator that validates Eq. (27), and the equations here do not establish a resolution of the Hubble tension.

As historical context, Ref. [20] reports an earlier Carnegie Chicago Hubble Program (CCHP) determination using the tip of the red giant branch (TRGB) of H₀=69.8 ± 0.8(stat) ± 1.7(sys) km s⁻¹ Mpc⁻¹ and notes its subsequent revision. This is an independent historical comparison, not an input to the MQ calculation or evidence that its source-only uncertainty includes observational systematics. The labels stat and sys denote statistical and systematic uncertainties.

A dimensionless geometric combination follows from the dark-domain fraction in Eq. (14) ([6], Appx. BD). Under the fixed-coefficient MQ assumption, its value is constant and is given by

The CMB temperature spectrum has its first acoustic peak near multipole 220 [21]. Its peak locations depend on perturbation evolution and projection into the observed sky [22]. Equation (28) evaluates a geometric domain combination; the equations here do not establish its identification with that peak or a measurement of spatial curvature. Its parenthetical uncertainty propagates the shared 2022 calibration input and excludes uncertainty in that physical interpretation.

5. Dark Matter Phenomenon

The galactic construction uses the Frames mapping between discrete Internal Frame counts and continuous System Frame quantities ([6], Appx. K, T). We denote by M_e the mass in kg associated with the calculated circular velocity v_e in m s⁻¹. The adopted count bound supplies a velocity scale. The relations below describe the response to a supplied model velocity profile; they do not independently reconstruct that profile.

For the bound construction, n_M is the fundamental-mass count and n_L is the radial fundamental-length count. We adopt $$\frac{{\mathit{n}}_{\mathit{M}}}{{\mathit{n}}_{\mathit{L}}}=\frac{{\mathit{m}}_{\mathit{f}}}{1\mathrm{kg}}$$ ([6], Appx. AF). The denominator is a fixed 1 kg reference; converting it with the numerator preserves the dimensionless ratio. The bound velocity v_b in Eq. (29) follows from this normalization. The supplied model velocity v_m and the calculated velocity use the same units. This assignment is an adopted model premise, rather than an independently measured count ratio.

At orbital radius R in m, Eq. (30) converts the bound velocity to the bound mass M_b in kg through the circular-velocity relation ([6], Appx. AG). We use the upright upper-count MQ gravitational realization $$\mathrm{G}=6.6740779383(10)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$$. It is derived from the same calibration as the bound velocity. Their covariance is therefore retained when combining them; an independent conventional gravitational-constant uncertainty is not added.

The dimensionless velocity differential Δ_f(vm) in Eq. (31) is the modeled-minus-bound velocity divided by the bound velocity. It expresses departure from the normalization scale and supplies the ratio used in Eq. (32).

Equation (32) expresses the mass associated with the calculated response through the bound mass and the dimensionless modeled-to-bound velocity ratio ([6], Appx. AK). It uses the same radius and gravitational realization as Eq. (30). Here the masses are parameters inferred through the adopted circular-speed relation. Identifying them with actual enclosed Galactic mass also requires a justified mass geometry; that identification is not established for the disk here [23].

Nonnegative mass and squared speed require $${\mathit{v}}_{\mathrm{m}}\ge \frac{{\mathit{v}}_{\mathrm{b}}}{2}$$. Substituting Eq. (32) into the circular-velocity relation and using Eq. (30) eliminates radius and gravitational coupling, giving Eq. (33) ([6], Appx. AK).

Equation (33) maps an input velocity profile to a calculated response. It does not make those velocities equal for arbitrary input. Imposing v_e=v_m gives (v_m−v_b)²=0, so equality occurs only at the bound velocity. A discrepancy between the curves cannot by itself be assigned to measurement error.

Fig. 2. Original manuscript curves, retained without recalculation. Model-source context is given in [24, 23, 25].The original illustration was described as using a McGaugh Milky Way profile over the first 84,000 lightyears ([6], Appx. AK). Ref. [24] discusses earlier modified Newtonian dynamics (MOND) mass models. The linked file in Data Availability instead identifies the 2016 model [23] with a 2018 update [25]. It tabulates component and total model velocities, rather than a raw catalogue of stellar measurements. Its total profile includes a modeled halo contribution, so reuse of that profile cannot independently establish the absence of dark matter.

Figure 2 retains the original manuscript curves as a historical illustration. The modeled, calculated, bound and classical curves have not been recomputed here. Figure 2 provides historical context for the conditional construction above. The linked profile contains inner-radius total velocities below half the adopted bound velocity, outside the nonnegative branch of Eq. (33). Stronger evidence is supplied by the separate galactic realization law developed in Ref. [26]. That work derives the acceleration normalization from upstream MQ inputs and applies the same law without galaxy-specific MQ calibration. Its high-acceleration limit recovers the Newtonian baryonic response; for a finite baryonic source, its weak-acceleration outer limit yields a radius-independent orbital velocity and the baryonic Tully–Fisher fourth-power scaling. These are consequences of the preprint’s full law, not independent results established by Eq. (33).

The preprint’s Milky Way test compares the fixed prediction with 38 circular-velocity measurements from Eilers et al. [27], using the separately published McMillan baryonic density model [28] with its dark halo force excluded. Ref. [26] reports that the root mean square (RMS) residual decreases from 69.03 km s⁻¹ for the Newtonian baryonic curve to 8.90 km s⁻¹ for MQ. With the specified McMillan source covariance propagated, the generalized chi-square is 39.57 for 38 degrees of freedom, with a probability value of 0.400. This compatibility is conditional on that covariance model. The diagonal error treatment rejects the complete sample, and neither the full observational covariance nor the complete source posterior is propagated jointly.

The complementary Spitzer Photometry and Accurate Rotation Curves (SPARC) sample [29] tests the fixed law across 175 galaxies and 3,366 radial measurements. Ref. [26] reports a pooled RMS residual of 22.82 km s⁻¹ and a geometric calculated-to-observed velocity ratio of 1.0042 for the 99 highest-quality galaxies. The larger population supplies a stronger test of a common normalization than a single historical curve. It nevertheless retains significant source sensitivity and a velocity-dependent residual in the complete sample. The empirical radial acceleration relation predates the MQ derivation; these results test the proposed normalization and physical account, rather than establish a new discovery of that empirical relation or complete precision validation.

The supplied model velocity is not defined here by a universal mass-to-radius formula. A circular-speed relation and an escape-speed relation have different factors and physical meanings. Equations (30) and (33) use the circular relation; replacing the input model profile by an escape-speed expression would change the construction.

4. DISCUSSION

The domain relations determine relative fractions once the calibration and closure assumptions are specified. The absolute masses additionally depend on the temporal reference and the CMB input, while the galactic response depends on the adopted bound-count normalization and a supplied velocity profile. These dependencies separate the calculation from tests of cosmological dynamics, galaxy structure and the physical interpretation of the domains. Inflation and the flatness, horizon and monopole questions concern broader cosmological models; the equations reviewed here do not establish their resolution.

The resulting mass scale is a conditional MQ calculation based on the stated 2022 CODATA inputs and independent CMB temperature. Its total mass is a model-defined assignment, not an independently measured global mass or a demonstrated relativistic mass observable. Numerical domain comparisons do not establish equivalent physical contents or cosmological dynamics. The peak/curvature mapping remains unresolved. The separate galactic tests in Ref. [26] strengthen the observational case for the fixed realization law while leaving the source, covariance and disk-mass interpretation limitations explicit.

5. APPENDICES

APPENDIX A: TERMS

Background definitions are given in Measurement Quantization [6]. The entries below specify the quantity roles used in this paper.

The coefficient and momentum realization are distinguished in Section 3. The angular realization is Θ_si in rad. The cited X-ray studies [30–32] concern polarization and wave mixing; they are not used as inputs to the 2022 fine-structure calibration. Chen et al. [33] discuss conserved quantities in general relativity; that reference supplies no calibration input here.

For the speed of light introduced in Section 3, the MQ count expression is $$\mathit{c}=\frac{{\mathit{n}}_{\mathit{L}}{\mathit{l}}_{\mathit{f}}}{{\mathit{n}}_{\mathit{T}}{\mathit{t}}_{\mathit{f}}}$$. Here n_T denotes the fundamental-time count; the length count has the local displacement meaning. The unit-count case n_L=n_T=1 gives the same exact SI speed. Ref. [34] is historical CODATA context; the current input ledger uses the 2022 adjustment [13].

The upper-count gravitational realization is $$\mathrm{G}=6.6740779383(10)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$$. Upright G denotes the upper count limit; italic G denotes a finite-count realization. At the fixed interaction scale, the coherent 2022 electromagnetic demarcation parameter n_L=84.6005456716(65), the latter is $$\mathit{G}=6.6738448317(10)\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$$. This noninteger calibration parameter is distinct from an asserted integer motion count; the historical blackbody count is not used here. The uncertainties propagate the shared calibration input.

The fundamental measures used in Section 3 are distinct from the conventional Planck units denoted l_p, m_p and t_p for length, mass and time.

Conventional Planck units provide comparison scales; the present calculations use the MQ realizations and calibration specified above.

The mass and radial-length counts in Eq. (29) have the meanings and adopted ratio specified in Section 3. The time count used above counts fundamental time units. These count labels must be interpreted with their stated realization.

In a displacement context, n_L counts fundamental lengths representing a change in an observable’s position relative to the observer. This local usage differs from the radial count used in the galactic construction.

The radial usage is the separation from a center of mass, as specified before Eq. (29). A displacement count and a radial count are not interchangeable merely because they share a label.

n_Lc is the dimensionless count of l_f representing light displacement relative to the observer.

n_θ is the dimensionless System Frame count of θ_si resolved from the Internal Frame demarcation after the metric differential, the mismatch between discrete count structure and its continuous encoding. The nearest-whole-unit operation uses the mixed brackets $$\lfloor \frac{84.60055}{2}\rceil =42$$.

The mixed left-floor and right-ceiling brackets denote rounding to the nearest whole-unit value; they do not denote ordinary floor or ordinary ceiling.

Q_L is the dimensionless residual length count in the Frames mapping; in the right-triangle construction, n_L is the Internal Frame separation count and n_L+Q_L gives the realized hypotenuse count.

2Q_Ln_L is the dimensionless normalization factor associated with the Informativity differential, which describes the non-Lorentz contraction of realized separations under the Frames mapping. The departure of this factor from unity characterizes that contraction.

h is the exact SI Planck constant in J s. The conventional reduced Planck constant is $$\mathit{\hslash }=\frac{\mathit{h}}{2\mathrm{\pi }}$$ in J s. The distinct upper-count MQ realization is $$\mathrm{ħ}=1.0545349844417293(56)\times {10}^{-34}\mathrm{J}\mathrm{s}$$. Upright ħ denotes the upper count limit; italic ℏ denotes the finite-count realization in MQ contexts. The conventional SI quantity is distinguished here by its stated calibration role. The quoted uncertainty is conditional on the current MQ calibration.

m_e is the electron rest mass in kg [13].

a₀ is the Bohr radius, an atomic length scale in m [13].

$${{\mathit{\alpha }}_{\mathit{f}}}^{-1}$$, $${{\mathit{\alpha }}_{\mathit{p}}}^{-1}$$ and $${{\mathit{\alpha }}_{\mathit{c}}}^{-1}$$ are the dimensionless fundamental, Planck-like and classical forms of the inverse fine-structure constant. $${{\mathit{\alpha }}_{\mathit{f}}}^{-1}=42{\mathit{\theta }}_{\mathit{s}\mathit{i}}$$ is defined with respect to the Internal Frame; the remaining two use the System Frame. The difference between $${{\mathit{\alpha }}_{\mathit{c}}}^{-1}$$ and $${{\mathit{\alpha }}_{\mathit{p}}}^{-1}$$ accounts for the length contraction associated with discrete measure.

STATEMENTS AND DECLARATIONS

The author has no relevant financial or non-financial interests to disclose.

The author has no competing interests to declare that are relevant to the content of this article.

DATA AVAILABILITY

The source inputs and conditional calculations are specified in Section 3. The original Figure 2 analysis is retained as historical material, and its exact data version and calculation procedure have not been reconstructed. The galactic statistics discussed from Ref. [26] are reported preprint results, not a reanalysis performed in this manuscript. Their source constructions, covariance treatment and observational inputs are documented in that preprint and Refs. [27–29]. The model file is available at https://astroweb.case.edu/ssm/models/MW2018model.dat. Its header identifies the component velocities, units and model history described in Refs. [23, 25]. It is model output, not an independent set of star-velocity measurements.

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