Dark Energy – a Geometric Phenomenon

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine the geometric contribution to universal expansion within Measurement Quantization (MQ), using relations for the speed of light, escape velocity and Heisenberg’s uncertainty principle. Discrete count configurations in the Internal Frame are represented as continuous relational structure in the System Frame through the Frames mapping. Under fixed reference measures and the adopted dimensional normalization, the construction gives a diameter–age relation and a steady-rate mass allocation. The cosmic microwave background (CMB) temperature supplies an observed input to the age calculation. The quantum epoch remains non-dimensional and is assigned no physical elapsed clock time. The construction introduces no fitted cosmological variables, while its absolute age and mass values remain conditional on the stated normalization. Its domain fractions and proposed CMB correlations are geometric results, not a complete physical model of the measured spectrum.

Keywords: Dark Energy, Expansion, Metric Expansion, Space, Discrete

1. INTRODUCTION

We use the Internal Frame and System Frame ([1], Appx. K) to describe the phenomenon called dark energy. The Internal Frame contains discrete count configurations; the System Frame realizes their continuous relational structure through the Frames mapping ([1], Appx. I, J). Here ${\mathit{n}}_{\mathrm{L}}$, ${\mathit{n}}_{\mathrm{M}}$ and ${\mathit{n}}_{\mathrm{T}}$ are dimensionless length, mass and time counts. Their corresponding fundamental references ${\mathit{l}}_{\mathrm{f}}$, ${\mathit{m}}_{\mathrm{f}}$ and ${\mathit{t}}_{\mathrm{f}}$ are represented in m, kg and s in the realized framework. Those units do not assign dimensional physical properties to the configuration domain itself. The invariant coefficient ${\mathit{\theta }}_{\mathrm{si}}$ is dimensionless; its momentum realization ${\mathit{p}}_{\mathrm{si}}$ has units kg m s⁻¹. These realizations share a numerical coefficient under the adopted convention while retaining distinct dimensions. The fundamental relation is ([1], Appx. V, AZ),

Within this fixed-reference construction, expansion is described by an increasing separation count ${\mathit{n}}_{\mathrm{L}}$ of fundamental lengths ${\mathit{l}}_{\mathrm{f}}$ between galaxies. For a fixed invariant coefficient, Eq. (9) gives equal radial increments over equal realized age increments. This kinematic relation alone does not determine a cosmological scale factor, establish a complete expansion history or rule out an observational acceleration effect.

Proceeding requires that we consider the discrete count description in the Internal Frame. As noted, separation is represented by an increasing count ${\mathit{n}}_{\mathrm{L}}$ of fundamental lengths ${\mathit{l}}_{\mathrm{f}}$, correlated with the counts ${\mathit{n}}_{\mathrm{T}}$ and ${\mathit{n}}_{\mathrm{M}}$. The corresponding mass-allocation rate ${\mathit{M}}_{\mathrm{acr}}$ in kg s⁻¹ concerns available mass in the geometric construction, rather than later gravitational accretion into cosmic structures ([1], Appx. BE),

The quantum epoch denotes the pre-referenceable configuration regime; referenceability here requires an admissible, globally consistent Frames mapping to realized System Frame relations. The expansionary epoch names the realized description beyond that boundary. The quantum-epoch description corresponds to about 363,312 years when represented in the selected expansionary framework ([1], Appx. BG). We use the International System of Units (SI) for this representation. The CMB radiation mass-equivalent allocation ${\mathit{M}}_{\mathrm{CMB}}$, in kg, uses the mapped expansionary duration ${\mathit{A}}_{\mathrm{ee}}$, in seconds. The physical temperature input ${\mathit{T}}_{\mathrm{CMB}}$ is 2.72548(57) K [2]. Substituting the resulting age back into the same radiation relation is a consistency check of this input-dependent calculation. The mapping convention and geometric basis are made explicit below.

The MQ construction used here introduces no inflationary period. Once the count relations, source inputs and realization convention are fixed, it supplies the stated sequence from the configuration domain to realized expansion without fitted cosmological variables or selectable alternative event sequences. These are premises of the MQ account presented here. Inflation [3], Loop Quantum Gravity (LQG) [4, 5], String Theory [6] and other approaches adopt different theoretical premises. An assessment of MQ must therefore distinguish observational measurements from quantities inferred through a cosmological history incompatible with those premises.

The different properties of the two frame descriptions provide the geometric premises for relating the fundamental measures and their counts. We investigate the resulting account of expansion under these premises. Its physical interpretation requires observational comparisons with disclosed model assumptions.

2. METHODS

We apply MQ count relations to the local comparisons involving the speed of light, escape velocity and Heisenberg’s uncertainty principle [7]. Earlier publications develop applications of the approach [8–13]; their presence in the publication record is context rather than independent validation of the present cosmological claims.

MQ supplies the discrete count relations used in this geometric construction [1].

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

The supporting resources describe the MQ relations and their source context. The calculations and comparisons below distinguish derived geometric results from physical observations and assumptions about their realization.

3. RESULTS

To understand the geometry underlying expansion, we consider two relations. First, we need the relation of the three measures assessed locally. Secondly, we need their relation between local and cosmological descriptions. This requires distinguishing discrete Internal Frame counts from their continuous System Frame realization.

The local measures satisfy the fundamental relation ([1], Appx. V). The realized cosmological diameter ${\mathit{D}}_{\mathrm{U}}$, in m, and age ${\mathit{A}}_{\mathrm{U}}$, in s, are related by the dimensionless expansion coefficient and the speed of light $\mathit{c}$, in m s⁻¹, in Eq. (6) ([1], Appx. AZ). Equations (3)–(5) retain the momentum realization introduced above.

To make the cosmological conversion explicit, let ${\mathit{n}}_{\mathrm{Tu}}$ denote the dimensionless age count and ${\mathit{n}}_{\mathrm{Lu}}$ the dimensionless diameter count. Their lower-case u identifies the universe-level counts, while ${\mathit{n}}_{\mathrm{L}}$ retains its locally specified separation role. The speed of light satisfies c=(l_f)/(t_f). The diameter and age counts follow ([1], Appx. AZ),

The diameter–age relation follows from these counts.

Equation (9) gives a geometric diameter–age relation. A statement about critical density additionally requires a mass and volume prescription; it does not follow from this scaling alone. We examine the consistency of the local fundamental measures before applying the cosmological relation. In Eq. (10), ${\mathit{c}}^{\text{'}}$ denotes the hypothetical speed in m s⁻¹ when a fundamental reference is varied. It is compared with the fixed local speed of light already defined.

For a local light-propagation comparison, the length and time counts are equal. Changing either fundamental measure while holding the other fixed changes their ratio and therefore the speed assigned by Eq. (10). This is inconsistent with the fixed local speed of light used here.

Changing only one of those local propagation counts has the same effect unless the other changes consistently. This local requirement is distinct from the cosmological diameter and age counts in Eq. (8), whose ratio includes the expansion coefficient. The argument concerns the internal consistency of the adopted relations.

A third possibility is that ${\mathit{m}}_{\mathrm{f}}$ differs while the relation describing length and time remains consistent with experiment. The hypothetical modified mass m_f^' retains units kg. It can be investigated with the MQ reduced expression for escape velocity ([1], Appx. H).

The radial separation count ${\mathit{n}}_{\mathrm{L}}$ is evaluated at the escape boundary in Eq. (11) ([1], Appx. H). Its radial interpretation refers to separation from a center of mass; a displacement interpretation uses the same count notation ([1], Appx. AW). The comparison with Heisenberg’s uncertainty principle ([1], Appx. B) must retain its own boundary assumptions.

A fourth possibility is that values have changed over time. We investigate this using the Bohr radius ${\mathit{a}}_0$, in m, and electron mass ${\mathit{m}}_{\mathrm{e}}$, in kg. The dimensionless Planck-like fine-structure form ${\mathit{\alpha }}_{\mathrm{p}}$ relates them to the fundamental measures ([1], Appx. BL, AA),

At fixed electron mass and Planck-like fine-structure form, Eq. (12) fixes the product of fundamental mass and length. A change in one fundamental measure therefore requires a compensating change in the other or in the atomic quantities entering the relation. Equation (12) alone does not establish that an isolated change in fundamental mass necessarily changes the Bohr radius. The comparison with earlier atomic spectra requires an explicit physical variation model.

Finally, p_si=(l_fm_f)/(2t_f). A change in this momentum realization requires a corresponding change in the fundamental measures. This is the dimensional relation used in the preceding comparisons; the dimensionless coefficient remains distinct.

Using the epoch distinction introduced above, the CMB radiation mass-equivalent allocation uses the mapped expansionary duration ([1], Appx. BE, BG, BH). The selected dimensional representation does not assign an intrinsic clock or dimensional physical properties to the configuration domain.

Table 1 presents the MQ mass allocations and conditional external comparisons. The visible mass ${\mathit{M}}_{\mathrm{vis}}$, in kg, uses the dimensionless visible fraction ${\mathit{\Omega }}_{\mathrm{vis}}$. The radiation constant $\mathit{a}$ has units J m⁻³ K⁻⁴, and the realized volume ${\mathit{V}}_{\mathrm{U}}$ has units m³. The CMB reference uses a separate carbon monoxide (CO) absorption analysis [14], converted at this common volume; the temperature row closes the input-dependent radiation relation. Abdullah et al. [15] determine total cluster masses and infer matter density, rather than a global visible-universe mass. We use the fast-radio-burst baryonic inventory [16] for a representative density comparison. Baryonic rest mass and the MQ visible geometric allocation have different definitions, so agreement does not establish their physical identity.

Table 1. MQ calculations and radiation comparison at a common volume.

^a The mass reference uses the CO-only temperature estimate 2.75 ± 0.11 K in Table 6 of [14], assuming the stated temperature–redshift law and excitation corrections. It is converted with the shared MQ volume. The 2.72548 ± 0.00057 K temperature input is from Fixsen [2].

^b Macquart et al. [16] infer a baryon density fraction 0.051 with a 95% interval 0.026–0.072 at their fixed reporting reference of 70 km s⁻¹ Mpc⁻¹. Using the 2022 recommended gravitational coupling from the Committee on Data of the International Science Council (CODATA) [17] and the same adopted present-day MQ sphere gives 1.52 × 10⁵³ kg, with interval 0.77–2.14 × 10⁵³ kg. The difference column gives only the central reference-minus-MQ difference. The interval is transformed at fixed normalization and volume, not reclassified as a standard uncertainty.

^c Component-fraction comparisons require their joint cosmological parameter covariance [18]. Only the directly quoted total-matter and dark-energy reference fractions are retained in Table 2.

The fast-radio-burst observations measure electron columns and host redshifts, independently of Fixsen’s CMB temperature observations. Their density inference still assumes flat matter-plus-dark-energy geometry, ionization corrections, foreground contributions and a host/population model [16]. We use the unconstrained baryon-density result, not the separate CMB-prior analysis. The inference is therefore not independent of the Lambda cold dark matter (ΛCDM) framework. For historical context, the low-redshift inventory of Fukugita et al. [19] gives a central density fraction 0.021 and conservative extrema 0.007–0.041 at the same reporting reference. Its common-volume projection is 0.62 × 10⁵³ kg, spanning 0.21–1.22 × 10⁵³ kg; those extrema are not a statistical confidence interval. Both projections describe a mean baryonic rest-mass density placed in an adopted constant-epoch sphere, not a direct weighing of the observed light cone. The common volume itself retains the paper’s Fixsen input and selected time representation.

The age inversion combines the measured CMB temperature [2] with the mass allocation and adopted spherical geometry ([1], Appx. BI). Fixsen determines temperature, not the age of the universe. The non-dimensional quantum-epoch construction ([1], Appx. BF-BG) retains no physical clocks. Let ${\mathit{A}}_{\mathrm{qe}}$ denote its temporal representation in the selected framework. The radiation constant is calculated from exact SI constants [17, 20]; Planck’s early work on irreversible radiation processes [21] provides historical context. Here $\mathit{\pi }$ is the dimensionless circle constant.

We select seconds for the dimensional representation of the configuration domain in the realized expansionary framework. The factor 1 s in Eq. (13) is an explicit normalization convention, not elapsed physical time within the quantum epoch. Its exponential uses the dimensionless Euler number $\mathit{e}$. Expressing the same reference in Julian years requires division by 31,557,600, leaving the represented extent unchanged. The dimensionless epoch ratio ${(2{\mathit{\theta }}_{\mathrm{si}})}^{1/3}$ expresses the adopted volume-to-linear geometric conversion. The orthogonal construction gives the length threshold √(3)l_f ([1], Appx. BF), but does not independently determine an absolute temporal reference. No further physical derivation of that normalization is supplied here. Equations (13)–(16) and the resulting absolute ages, sizes and mass allocations are therefore conditional on the prescribed reference. Multiplying that reference by a physical scale factor multiplies the mapped extents and CMB allocation by the same factor, and the inferred age and diameter by its cube root. A change of display units does not make this change of physical normalization.

The volume in Table 1 is the spherical realized volume whose radius is one half the diameter in Eq. (9) ([1], Appx. BI). Combining that volume with the radiation relation in Table 1 and the mass allocation in Eq. (15) gives the dimensionally explicit age inversion below.

The 2022 CODATA calibration uses the dimensionless classical fine-structure input α_c^-1=137.035999177(21) [17]. The electromagnetic demarcation fixes the interaction count separation at which MQ realizes electromagnetic interactions. Its relation is ${\mathit{n}}_{\mathrm{L}}{\mathit{\theta }}_{\mathrm{si}}=276$ ([1], Appx. AC). In Eqs. (17) and (20), the mixed brackets denote rounding to the nearest whole-unit value. The inversion uses the positive-real continuation of the separation count at this demarcation. The selected positive branch has $\lfloor 42{\mathit{\theta }}_{\mathrm{si}}\rceil$=137. Values for the fundamental measures and the invariant coefficient are calculated from these inputs ([1], Appx. BM).

We use the coefficient, its momentum realization, the Planck-like fine-structure form ([1], Appx. AA) and the Bohr-radius relationship ([1], Appx. BL) to resolve the fundamental length. The conventional reduced Planck constant ℏ=(h)/(2π), in J s, uses the exact SI value h=6.62607015 × 10^-34 J s [17, 20]. The adjusted-constant identity m_ea₀=(ℏα_c^-1)/(c) retains the CODATA covariance rather than treating electron mass and Bohr radius as independent inputs. Atom-interferometric determinations of fine structure include [22]; this calculation uses the 2022 recommended value [17]. Parentheses give one standard source uncertainty in the final digits, conditional on the MQ relations and prescribed normalization.

We calculate uncertainties from the independent source inputs using first-order propagation, retaining covariance between outputs that share those inputs. Repeated uses of one input are one random variable. Exact SI factors contribute no source uncertainty. The coherent 2022 calibration reduces the atomic product to the inverse fine-structure input, so independently propagating uncertainties of its derived mass and length would count the calibration more than once. Analytic implicit differentiation and a separate numerical derivative evaluation agree for the present constants and age-dependent quantities. We use these reproducible source uncertainties throughout. Unreconciled historical worksheet budgets are superseded for this paper. The adopted normalization is held fixed for this conditional calculation; its physical uncertainty and any MQ model discrepancy remain unquantified.

It follows that the diameter of the universe is,

The geometric domain partition ([1], Appx. BB) distinguishes presently visible content, content that is observable in the specified realized expansion history, its unobserved remainder, and the dark domain beyond that referenceability boundary. The visible fraction was introduced above. Here ${\mathit{\Omega }}_{\mathrm{obs}}$ denotes the observable fraction, ${\mathit{\Omega }}_{\mathrm{uobs}}$ its unobserved remainder and ${\mathit{\Omega }}_{\mathrm{dk}}$ the dark fraction; all are dimensionless. Their correspondence to conventional baryonic matter, dark matter or dark-energy densities is a physical interpretation requiring additional assumptions.

The first, second and fourth expressions below define the domain partition, and the third relates its observable and visible fractions. The dimensionless total ${\mathit{\Omega }}_{\mathrm{tot}}$ is normalized to unity. The relations concern the geometric fractions, not independently inferred physical density components.

1. Definitions

2. Calculations

We begin with the observable-to-visible relation and subtract unity after multiplying both sides by one half of the squared coefficient. Equation (31) supplies the geometric observable fraction ([1], Appx. BB). Together with the total normalization, it identifies the dark-domain expression used below.

Using the partition ${\mathit{\Omega }}_{\mathrm{dk}}+{\mathit{\Omega }}_{\mathrm{obs}}={\mathit{\Omega }}_{\mathrm{tot}}$, we replace the observable fraction ${\mathit{\Omega }}_{\mathrm{obs}}$ in the right portion of the equality to resolve the dark-domain expression.

The domain expressions depend on the dimensionless coefficient. Their numerical calibration uses the conventional fine-structure input [17]. The coefficient is distinct from the dimensional momentum realization ${\mathit{p}}_{\mathrm{si}}$ in the fundamental relation. The unity expression ([1], Appx. AX) describes a count relation at the start of the expansionary epoch. Here ${\mathit{n}}_{\mathrm{Lc}}$ is the dimensionless light-path length count; the separation count retains its earlier meaning,

The domain fractions are dimensionless relations in the geometric construction. They identify which portions are visible, observable or beyond the observable domain. The dark domain is excluded from direct observational reference by that construction. This exclusion concerns the MQ domain assignment; it does not establish that every cosmological influence outside a directly emitting component is absent from the observed CMB.

The domain labels describe referenceability within the geometric construction. They do not independently determine the material composition, stress-energy or interaction history associated with an observational density estimate. A physical interpretation therefore needs a declared mapping from the geometric quantities to the relevant observables.

Equations (36)–(40) describe geometric properties in the MQ domain construction ([12], Sec. 3.5, Eqs. (82)–(90); [1], Appx. T). The metric differential describes the representation mismatch between discrete count structure and its non-discrete System Frame encoding under the Frames mapping; it is not itself a physical deformation. Here $\mathit{y}$ is the source coordinate-scale parameter and ${\mathit{\Delta }}_{\mathrm{mdif}}$ is its metric differential multiplier. For this geometric use, they are formal dimensionless multipliers of normalized coordinates; no measured spectral unit or numerical spectral calibration is assigned. A physical comparison must specify that conversion separately. Before the colons in these equations, the domain symbols label coordinate pairs; they do not assert equality between a scalar fraction and an ordered pair. The source label ${\mathit{\Omega }}_{\mathrm{v}}$ in Eq. (39) identifies the ordered pair written there, whose second component is half the unobserved fraction. It is not the visible fraction ${\mathit{\Omega }}_{\mathrm{vis}}$ or an additional independently measured domain. MQ supplies the geometric account here; the relevant physical processes and measurement response remain necessary for interpreting differences from a measured CMB peak.

Equation (36) carries no metric differential factor, whereas Eqs. (37)–(40) apply that factor to the observable-domain geometry. Within MQ, the dark domain lies beyond the observer’s observable universe and supplies no directly referenceable CMB peak requiring that modification. The unchanged application is a physical clue within the geometric account. It is not a fitted dark-component amplitude or a demonstration of quantitative agreement with measured peaks. Connecting these relations to observations requires the ordinate normalization, the applicable metric differential realization and a specified correspondence with measured spectral coordinates. Conventional cosmology describes indirect dark-energy effects on CMB anisotropies through the integrated Sachs–Wolfe effect [23]. Those model-dependent mechanisms do not constitute a measurement of the MQ dark domain.

3. Domain comparisons

Table 2 compares the MQ geometric-domain fractions with the directly quoted Planck total-matter and dark-energy fractions [18]. The conventional symbols ${\mathit{\Omega }}_{\mathrm{\Lambda }}$, ${\mathit{\Omega }}_{\mathrm{c}}$, ${\mathit{\Omega }}_{\mathrm{b}}$ and ${\mathit{\Omega }}_{\mathrm{m}}$ denote the dimensionless dark-energy, cold-dark-matter, baryonic and total-matter density parameters, respectively. They retain their conventional cosmological meaning and are distinct from the MQ geometric fractions. The total-matter reference is labeled by its published parameter rather than a restricted two-component sum. The mass column applies the MQ fractions to the total mass allocation conditional on age and normalization; the dark-domain allocation is not an independently measured particle mass or a derivation of a conventional dark-energy stress tensor. All MQ fractions share the calibrated coefficient and its covariance. The separate fundamental ratio Ω_f=(2)/(θ_si²) ([1], Appx. BB) is not another summand in the normalized partition. These are contextual comparisons, not independent validation of Eqs. (36)–(40).

Table 2. MQ geometric domains and ΛCDM-conditioned reference quantities.

^a Planck 2018 TT,TE,EE+lowE total-matter and dark-energy fractions are quoted directly [18]. TT, TE and EE denote the temperature autocorrelation, temperature–E-mode cross-correlation and E-mode polarization autocorrelation angular power spectra; lowE denotes the low-multipole E-mode polarization likelihood. Component percentages would require conversion using the joint density and expansion-parameter covariance.

MQ uses a geometric construction with the fixed premises stated in the Introduction. ΛCDM supplies a physical cosmological account with additional interaction processes [18, 24]. Equations (36–40) concern the proposed geometric contribution, so their assessment requires a fixed spectral correspondence and observational products with disclosed processing assumptions. Contextual posterior parameters cannot establish equivalence of the full models.

4. DISCUSSION

The domain fractions describe the MQ geometric partition and its referenceability conditions. Equation (9) supplies a diameter–age scaling, but it does not by itself establish physical homogeneity or isotropy at the Planck scale, or a complete dynamical explanation of cosmic acceleration. These physical claims require additional evidence and an explicit connection between the geometric quantities and the measured observables.

We clarify the frame interpretation of space. The Internal Frame describes discrete counts, while the System Frame provides their non-discrete relational realization. Such that there is no change in the reference measure ${\mathit{l}}_{\mathrm{f}}$, the expanding separation corresponds to an increasing length count.

With this, expanding space is described by the realized diameter-age relation in Eq. (9) and the correlated cosmological counts in Eqs. (7–8) ([1], Appx. AZ, BE). The dimensional fundamental relation retains the momentum realization. This construction distinguishes cosmological expansion from the local reference measures used to express it.

For Eqs. (36–40), a measured spectrum contains physical interactions beyond the proposed geometric contribution. A quantitative test needs a fixed correspondence to spectral coordinates, the applicable metric differential realization and the relevant measurement covariance. Detector records, maps and spectra also need calibration, instrument-response and foreground treatment. The published high-multipole Planck temperature display fixes foreground and nuisance parameters to a base-ΛCDM best fit [18, 25]. Such processing must be disclosed before using that product to assess MQ geometry. Identifying and quantifying the physical processes contributing to any residual remains future work. We do not assign a fitted residual correction or claim a complete physical peak prediction.

The retained calculation gives an age of 13.7943(38) Gyr under the prescribed normalization. Stellar chronology provides a separate observational check. Valcin et al. [26] infer 13.39 ± 0.10 Gyr statistical and ±0.23 Gyr systematic for their oldest globular-cluster population using stellar models, Hubble photometry and Gaia distances. The MQ central age exceeds that population’s central age by 0.40 Gyr. Turning stellar ages into a universe age also requires a formation interval; their published estimate is 13.57 Gyr with statistical bounds −0.14/+0.16 Gyr and ±0.23 Gyr systematic. This evidence is compatible with the conditional MQ value but does not independently determine its normalization. Stellar-model systematics and the formation-interval assumptions are distinct from the Fixsen temperature uncertainty.

For context, Planck’s TT,TE,EE+lowE+lensing analysis gives 13.797 ± 0.023 Gyr within base ΛCDM [18], 0.0027 Gyr above the MQ central result. This is a model-conditioned inference with shared CMB assumptions, not an independent timing observation or a replacement for the Fixsen temperature input. Neither this agreement nor the stellar comparison turns the adopted MQ normalization into a derived physical scale. The present result is a conditional age estimate, not a fully normalized first-principles determination of temporal passage.

5. APPENDICES

APPENDIX A: TERMS

Foundational symbol and term definitions are given in Measurement Quantization [1], Appx. BP–BQ.

The X-ray phase-matching and wave-mixing studies [27–29] concern different theoretical solutions and experimental observables; they do not establish a standard experimental uncertainty on the MQ invariant coefficient. The 2 μrad tolerance used in the earlier comparison is author-adopted, not a measured uncertainty. Conserved quantities at null infinity [30] concern asymptotic general-relativistic geometry and supply no such angular error bar.

The speed of light $\mathit{c}$ is exactly 299,792,458 m s⁻¹ in SI [17]. The local propagation relation uses equal length and time counts; this requirement was distinguished from the cosmological diameter–age count ratio above.

Here upper-count identifies the limiting large-separation-count realization, distinguished from the fixed finite electromagnetic demarcation. The upper-count gravitational realization is G=(c³t_f)/(m_f)=6.6740779383(10) × 10^-11 m³ kg⁻¹ s⁻². Its finite value at the electromagnetic demarcation is G=6.6738448317(10) × 10^-11 m³ kg⁻¹ s⁻². Both are MQ calculations from the 2022 CODATA calibration, rather than the CODATA recommended Newtonian constant 6.67430(15) × 10⁻¹¹ m³ kg⁻¹ s⁻² [17]. The upper-count value uses upright G, while the finite demarcation value uses italic G. Their source uncertainties retain the common calibration covariance.

The fundamental length, mass and time were introduced in Sec. 1. Their current MQ calculated values are given in Eqs. (19), (21) and (22). Their propagated source uncertainties are conditional on the MQ relations and should not be interpreted as independent measurement uncertainties of conventional Planck units.

The symbols ${\mathit{l}}_{\mathrm{p}}$, ${\mathit{m}}_{\mathrm{p}}$ and ${\mathit{t}}_{\mathrm{p}}$ denote the conventional Planck length, mass and time, in m, kg and s. They use the conventional SI reduced Planck constant and the CODATA recommended Newtonian constant, rather than the MQ upper-count realizations.

The dimensionless counts ${\mathit{n}}_{\mathrm{L}}$, ${\mathit{n}}_{\mathrm{M}}$ and ${\mathit{n}}_{\mathrm{T}}$ and their fundamental references were introduced in Sec. 1. Their local or cosmological interpretation follows the stated measurement context.

The separation count ${\mathit{n}}_{\mathrm{L}}$ uses the interpretation stated for the local comparison in Sec. 3. A radial distance and an observer-reported displacement must retain their respective measurement contexts.

The radial use of the length count refers to separation from a center of mass, as distinguished in the discussion of Eq. (11).

The light-path length count retains its definition before Eq. (35). Its displacement is expressed using the same fundamental length reference.

The dimensionless System Frame count ${\mathit{n}}_{\mathrm{\theta }}$ is resolved from the Internal Frame demarcation adjusted for the metric differential. For the stated example, ⌊(84.60055)/(2)⌉=42. The decimal argument is an illustrative count conversion, not a new 2022 CODATA estimate or an independent datum.

The nearest-whole-unit operation uses the mixed brackets defined before Eq. (17).

The residual ${\mathit{Q}}_{\mathrm{L}}$ is a dimensionless count contribution. In the discrete right-triangle construction it is the realized hypotenuse count in excess of the separation count ${\mathit{n}}_{\mathrm{L}}$; multiplying a count by ${\mathit{l}}_{\mathrm{f}}$ gives a realized length.

The Informativity differential describes the non-Lorentz contraction of realized spatial separation arising from discrete count structure through the Frames mapping. It is distinct from the metric differential introduced above.

The conventional SI reduced Planck constant is ℏ=(h)/(2π)=1.054571817646156… × 10^-34 J s and is exact [17]. Accounting for the Informativity differential gives the MQ upper-count realization ħ=2p_sil_f=1.0545349844417293(56) × 10^-34 J s. The latter uses upright ħ and has the propagated source uncertainty of the shared calibration. It is a calculated action realization, not a revised SI defining constant.

The electron mass was introduced in Sec. 3 and is used in the adjusted-constant atomic product.

The Bohr radius ${\mathit{a}}_0$ is the conventional atomic length scale introduced above, not a universal ground-state orbital radius for every atom.

The inverse fine-structure forms introduced above remain distinct. Their realized forms and the finite-count contraction factor are described in [1], Appx. AA.

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 manuscript has data included as electronic supplementary material.

REFERENCES

[1] J.A. Geiger, Measurement quantization, International Journal of Geometric Methods in Modern Physics 20(4), 2350069 (2023), doi:10.1142/S021988782350069X.

[2] D.J. Fixsen, The Temperature of the Cosmic Microwave Background, (2009), arXiv: 0911.1955, doi:10.1088/0004-637X/707/2/916.

[3]Alan H. Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins (1997), Basic Books. pp. 233–234. ISBN 978-0201328400.

[4] L. Smolin, The Case for Background Independence, In D. Rickles, S. French, J. T. Saatsi, The Structural Foundations of Quantum Gravity, p. 196 (Clarendon Press, 2006), doi:10.1093/acprof:oso/9780199269693.001.0001.

[5]Abhay Ashtekar and Jerzy Lewandowski, Background Independent Quantum Gravity: a Status Report, Class. Quant. Grav. 21, 15 (2004), doi:10.1088/0264-9381/21/15/R01.

[6]Becker, Katrin; Becker, Melanie; Schwarz, John (2007). String theory and M-theory: A modern introduction, Cambridge University Press, ISBN 978-0-521-86069-7.

[7] Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 (3–4): 172–198 (1927), Bibcode:1927ZPhy.43.172H, doi:10.1007/BF01397280, S2CID 122763326.

[8] J.A. Geiger: Measurement Quantization Unites Classical and Quantum Physics, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 2, 262–311 (Apr 2018), doi:10.4236/jhepgc.2018.42019.

[9] J.A. Geiger: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 4 (Oct 2018), doi:10.4236/jhepgc.2018.44038.

[10] J.A. Geiger: Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture, Journal of High Energy Physics Gravitation and Cosmology. Vol. 5, No. 2 (Apr 2019), doi:10.4236/jhepgc.2019.52028.

[11] J.A. Geiger: Physical Significance of Measure, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 1 (Jan 2020), doi:10.4236/jhepgc.2020.61008.

[12] J.A. Geiger: Measurement Quantization Describes History of Universe – Quantum Inflation, Transition to Expansion, CMB Power Spectrum, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 2 (Apr 2020), doi:10.4236/jhepgc.2020.62015.

[13] J.A. Geiger: Measurement Quantization Describes the Physical Constants, International Journal of Theoretical and Mathematical Physics, 11(1), pp. 29-59 (Jan 2021),

[14] V. V. Klimenko, A. V. Ivanchik, P. Petitjean, P. Noterdaeme and R. Srianand, Estimation of the CMB temperature from atomic C I and molecular CO lines in the interstellar medium of early galaxies, Astronomy Letters 46, 715–725 (2020), doi:10.1134/S1063773720110031, arXiv:2106.00119.

[15]Mohamed Abdullah, Anatoly Klypin, Gillian Wilson, Cosmological Constraint on Ωm and σ8 from Cluster Abundances using the GalWCat19 Optical-Spectroscopic SDSS Catalog (2020), arXiv:2002.11907, doi:10.3847/1538-4357/aba619.

[16] J.-P. Macquart et al., A census of baryons in the Universe from localized fast radio bursts, Nature 581, 391–395 (2020), doi:10.1038/s41586-020-2300-2, arXiv:2005.13161.

[17] P.J. Mohr, D.B. Newell, B.N. Taylor and E. Tiesinga, CODATA Recommended Values of the Fundamental Physical Constants: 2022 (2024), arXiv:2409.03787, https://arxiv.org/abs/2409.03787.

[18] Planck Collaboration, Planck 2018 results. VI. Cosmological parameters, Astronomy & Astrophysics 641, A6 (2020), arXiv:1807.06209, doi:10.1051/0004-6361/201833910.

[19] M. Fukugita, C. J. Hogan and P. J. E. Peebles, The Cosmic Baryon Budget, Astrophysical Journal 503, 518–530 (1998), doi:10.1086/306025, arXiv:astro-ph/9712020.

[20] BIPM, Information for users about the redefinition of the SI (updated 20 May 2019), https://www.bipm.org/en/committees/cc/ccm/redefinition-kg.

[21] M. Planck, Über irreversible Strahlungsvorgänge, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu, Berlin, 5, p. 480 (1899).

[22] Morel, L., Yao, Z., Cladé, P. et al. Determination of the fine-structure constant with an accuracy of 81 parts per trillion. Nature 588, 61–65 (2020), doi:10.1038/s41586-020-2964-7.

[23] Planck Collaboration, Planck 2015 results. XXI. The integrated Sachs–Wolfe effect, Astronomy & Astrophysics 594, A21 (2016), doi:10.1051/0004-6361/201525831, arXiv:1502.01595.

[24]Planck Collaboration, Planck 2013 Results. XVI. Cosmological Parameters, Astronomy & Astrophysics, 571, A16 (2014), doi:10.1051/0004-6361/201321591.

[25] Planck Collaboration, Planck 2018 results. V. CMB power spectra and likelihoods, Astronomy & Astrophysics 641, A5 (2020), doi:10.1051/0004-6361/201936386.

[26] D. Valcin, R. Jimenez, U. Seljak and L. Verde, The age of the universe with globular clusters. Part III. Gaia distances and hierarchical modeling, Journal of Cosmology and Astroparticle Physics 2025 (10), 030 (2025), doi:10.1088/1475-7516/2025/10/030, arXiv:2503.19481.

[27] S. Shwartz and S. E. Harris, Polarization Entangled Photons at X-Ray Energies, Phys. Rev. Lett. 106, 080501 (2011), arXiv: 1012.3499, doi:10.1103/PhysRevLett.106.080501.

[28] S. Shwartz, R. N. Coffee, J. M. Feldkamp, et. al., X-ray Parametric Down-Conversion in the Langevin Regime, Physical Review Letters, 109, 013602 (July 6, 2012), doi:10.1103/PhysRevLett.109.013602.

[29] T. E. Glover et al., X-ray and optical wave mixing, Nature, 488, 7413, 603–608 (2012), doi:10.1038/nature11340.

[30] Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau, Conserved quantities in general relativity -- the view from null infinity (Apr. 8, 2022), doi:10.48550/arXiv.2204.04010.