Diameter & Age of the Universe as a Function of the CMB Temperature

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We use Measurement Quantization (MQ) to calculate an age of the universe of 13.7943(38) billion Julian years and a physical diameter of 90.005(25) billion light-years from the cosmic microwave background (CMB) temperature, the 2022 CODATA fine-structure input and exact International System of Units (SI) constants. The calculation combines a proposed count-based boundary for a consistent observable description, the relation between configuration structure and its expansionary description, mass allocation and the blackbody radiation relation. The configuration domain is not assigned an intrinsic physical clock or length. Dimensionful quantities belong to the realized expansionary description. Once the measured inputs and MQ premises are specified, the calculation fits no additional epoch duration, interepoch scalar or diameter parameter. The quoted uncertainties propagate the source inputs under those premises and exclude unquantified model uncertainty. The numerical results are MQ calculations rather than direct measurements. Their comparison with cosmic-age estimates and CMB properties requires distinguishing shared-input consistency checks from independent observational tests.

Keywords: Diameter, Age, Universe, CMB Temperature

1. INTRODUCTION

The MQ physical domain considered here need not coincide with the conventional observable-universe horizon. That distinction matters when comparing the proposed cosmic scale with local observations. The Internal Frame is the discrete count configuration domain. The System Frame is its continuous relational encoding, generated through the Frames mapping ([1], Appx. K). Referenceability here means that a relational configuration supports a valid, uniquely realized observable description through that mapping. Understanding this distinction is prerequisite to the mathematical presentation. The terminology used here follows the current MQ source archive [2].

The frame definitions alone do not establish the proposed cosmological scaling. The mathematical relations and cited observations below provide the context for examining that proposal.

We begin with the fundamental measures of length ${\mathit{l}}_{\mathit{f}}$, mass ${\mathit{m}}_{\mathit{f}}$ and time ${\mathit{t}}_{\mathit{f}}$, realized in m, kg and s, and the fundamental relation ${\mathit{l}}_{\mathit{f}}{\mathit{m}}_{\mathit{f}}=2{\mathit{p}}_{\mathrm{s}\mathit{i}}{\mathit{t}}_{\mathit{f}}$ ([1], Appx. V, W). Here ${\mathit{\theta }}_{\mathrm{s}\mathit{i}}$ is the dimensionless invariant MQ coefficient and ${\mathit{p}}_{\mathrm{s}\mathit{i}}$ is its momentum realization, in kg m s⁻¹. The adopted SI realization assigns them the same numerical coefficient while retaining their different dimensions. The speed of light is $\mathit{c}={\mathit{l}}_{\mathit{f}}/{\mathit{t}}_{\mathit{f}}=299792458\mathrm{m}{\mathrm{s}}^{-1}$ [3]. For the 2022 calibration below, the reference measures are m_f=2.17643254057(17)×10^-8 kg, l_f=1.61619991148(12)×10^-35 m and t_f=5.39106261131(41)×10^-44 s. The MQ gravity construction ([1], Appx. C) relates discrete count structure to realized separations. The frame definitions above apply to these realizations ([1], Appx. K). These principles underlie the proposed calculations of physical constants ([1], Appx. BM) [4].

The presentation also considers a scaling of the fundamental relation to cosmological phenomena. In combination with an MQ account of early universe events ([1], Appx. BF-BH), we calculate the age ${\mathit{A}}_{\mathit{U}}$, in s, from the CMB temperature ${\mathit{T}}_{\mathit{CMB}}$, in K ([1], Appx. BI). The radiation constant $\mathit{a}$ has units J m⁻³ K⁻⁴. The exponential below is represented in the expansionary description using the chosen SI time basis. Its displayed one-second factor states the adopted numerical realization normalization in that unit basis and does not postulate a physical second or an elapsed clock interval in the configuration domain. The circle constant $\mathit{\pi }$ and the base $\mathit{e}$ of the natural exponential are dimensionless. Gyr denotes ${10}^9$ Julian years, with one Julian year equal to 31557600 s.

The MQ account assigns the quantum epoch the expansionary-equivalent extent ${\mathit{A}}_{\mathit{q}\mathit{e}}$, represented here in s and corresponding to 363312.3525(25) Julian years ([1], Appx. BG, BH). This is a description of pre-referenceable configuration structure through the Frames mapping, rather than elapsed realized time in that domain. The mapped radius at the referenceability threshold is √(3)l_f. MQ interprets this closure as the onset of the referenceable physical universe and the transition to the expansionary epoch. The coefficient ${\mathit{\theta }}_{\mathrm{s}\mathit{i}}$ determines the radial rate ${\mathit{\theta }}_{\mathrm{s}\mathit{i}}\mathit{c}$, in m s⁻¹, in the realized description. This mapped cosmological radial rate is distinct from the local vacuum speed in Eq. (2). Together with the mass accretion relation, which specifies the mass-equivalent allocation per unit realized time ([1], Appx. BE), these premises enter the CMB calculation. Parenthetical digits denote one standard uncertainty in the last quoted digits, propagated from the specified inputs under the adopted MQ geometry and realization.

The present age and diameter calculation uses the fixed source inputs and the stated MQ geometry and realization premises. It fits no additional epoch parameter. It does not invoke extra dimensions, Loop Quantum Gravity (LQG) [5–6], String Theory [7], Supersymmetry or a conventional inflationary stage [8]. Comparisons with standard cosmological inferences are used as consistency tests of specified outputs. Their evidential role depends on the observations and assumptions shared with the MQ calculation.

2. METHODS

This paper uses blackbody radiation and mass-energy relations together with the MQ count geometry, speed-of-light and escape-velocity constructions and the uncertainty-principle context [9]. Examples of the MQ approach appear in [10–14]. The fixed calibration input is the dimensionless classical inverse fine-structure quantity α_c^-1=137.035999177(21) from the 2022 CODATA adjustment [15]. Its MQ inversion gives ${\mathit{\theta }}_{\mathrm{s}\mathit{i}}=3.26239030504(25)$ ([1], Appx. AD, BM). Exact SI inputs and the independent CMB temperature are specified before Eqs. (6)–(8). The reference length calculation uses the exact product identity for electron mass and Bohr radius, avoiding an assumption that those adjusted quantities are independent. The numerical results in this paper propagate the common calibration dependencies. We use 80-digit arithmetic and first-order standard-uncertainty propagation through the common input dependencies. The fine-structure input and Fixsen temperature are treated as independent. Reported uncertainties are rounded to two significant digits before aligning the central value, with ties to even. Unquantified model uncertainty is excluded rather than assigned zero.

The discrete count structure is an MQ premise used in this presentation. Earlier MQ derivations and reported measurement comparisons are presented in Measurement Quantization [1], together with the referenced appendices and a more extensive list of term and keyword definitions. They provide the background for the proposed physical interpretation; the present calculation does not independently establish discreteness from those citations. A preprint can also be found in Online Resources. The published appendices provide historical derivation context [1]. The present notation, dimensional realizations and numerical inputs are stated explicitly here [15, 2].

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

Two supplementary spreadsheets accompany the MQ presentation. Suppl. 1 documents calculations for the introductory paper and its referenced appendices. Suppl. 2 documents calculations for this paper and other MQ applications. We identify Column P of its Inputs2022 tab as the supporting calculation chain. In that direction of calculation, Fixsen’s CMB temperature fixes the MQ age, after which the mass allocation, radiation density and recovered temperature form a linked consistency check. The current values here use the fixed 2022 input ledger and retain those dependencies.

3. RESULTS

3.1. Physical Support for Scaling of the Fundamental Expression

The fundamental relation introduced above describes the fundamental measures. We ask whether the specified cosmological count relations provide a consistent scaling of that structure.

To examine the scaling hypothesis, let ${\mathit{n}}_{\mathit{L}}$ and ${\mathit{n}}_{\mathit{T}}$ denote dimensionless length and time counts in the indicated local construction. A possible change in ${\mathit{l}}_{\mathit{f}}$ or ${\mathit{t}}_{\mathit{f}}$ would give c^'=n_Ll_f^'/(n_Tt_f) or c^'=n_Ll_f/(n_Tt_f^'). A prime marks the indicated hypothetical changed value; unprimed quantities retain the reference realization. Equation (2) gives the unchanged light-count case with ${\mathit{n}}_{\mathit{L}}={\mathit{n}}_{\mathit{T}}$.

Equation (2) applies to the unchanged light-count case. At fixed fundamental measures a changed count ratio gives ${\mathit{c}}^{\text{'}}=({\mathit{n}}_{\mathit{L}}/{\mathit{n}}_{\mathit{T}})\mathit{c}$, so ${\mathit{n}}_{\mathit{L}}={\mathit{n}}_{\mathit{T}}$ preserves the light-speed relation. A change in the realized length-to-time ratio must therefore be assessed against observations. This identity alone does not exclude every coordinated change of counts and realized measures.

Cosmological light-propagation studies supply context for testing such changes. Cao et al. propose a strong-lensing and supernova method [16], while Mendonça et al. find mostly negligible speed variation under the gas-depletion assumptions they examine [17]. These are conditional observational constraints, rather than an exact comparison of identical stellar-core processes at every epoch. The orbital expression below provides a separate MQ relation ([1], Appx. BL). The gamma-ray burst spectral-lag analysis in [18] concerns possible Lorentz-invariance violation in propagation and does not directly establish the invariance of the Bohr radius or the MQ count geometry. Its inference depends on the intrinsic-emission model, and the authors report that none of their tested models adequately fits the stacked spectral-lag data.

A third possibility is a change in ${\mathit{m}}_{\mathit{f}}$ while the ratio of realized length to time remains consistent with the speed of light. Here ${\mathit{n}}_{\mathit{M}}$ is the dimensionless local mass count. For this boundary, the length count denotes radial separation from the specified center of mass. The escape-velocity expression ([1], Appx. H) gives the light-speed boundary below.

Equation (3) follows from the MQ escape-speed expression at its light-speed boundary. Here ${\mathit{n}}_{\mathit{L}}$ is the radial count of ${\mathit{l}}_{\mathit{f}}$ from the specified center of mass, distinct in role from the motion count above. The MQ equivalence discussion ([1], Appx. AW) addresses the relationship of those roles. The MICROSCOPE study [19] tests Lorentz invariance and supplies a separate experimental context; it does not itself derive the MQ boundary-count identity.

A fourth possibility concerns values that change over time. The orbital expression ([1], Appx. BL, AA) below uses the dimensionless Planck-like inverse fine-structure form α_p^-1, the electron mass ${\mathit{m}}_{\mathit{e}}$, in kg, and the Bohr radius ${\mathit{a}}_0$, in m.

Equation (4) fixes a dimensionless combination of orbital and reference measures. If ${\mathit{m}}_{\mathit{e}}$, ${\mathit{l}}_{\mathit{f}}$ and α_p^-1 are held fixed, changing ${\mathit{m}}_{\mathit{f}}$ requires a corresponding change in ${\mathit{a}}_0$. If instead m_e^'=n_Mm_f^' at fixed ${\mathit{n}}_{\mathit{M}}$, the common mass scaling cancels from their ratio. Equation (4) alone therefore does not exclude every coordinated change. Spectral-lag constraints such as [18] address propagation effects and must be distinguished from a direct orbital-geometry test.

An increasing time count is the elapsed-time property of the universe. For the light-count construction, the speed of light remains fixed when ${\mathit{n}}_{\mathit{L}}={\mathit{n}}_{\mathit{T}}$. This light-count equality does not impose the same ratio on the cosmic diameter count. The cosmic count relations are specified separately in Eqs. (24)–(28).

From ${\mathit{p}}_{\mathrm{s}\mathit{i}}={\mathit{l}}_{\mathit{f}}{\mathit{m}}_{\mathit{f}}/(2{\mathit{t}}_{\mathit{f}})$, a change in the momentum realization requires a corresponding change in at least one fundamental measure. Under the fixed realization normalization introduced above, changes in the coefficient and momentum realization are related.

The local light-count construction preserves the fundamental relation when its stated measures and counts satisfy the indicated boundary conditions. The separate cosmological count relations below supply the diameter and mass scaling.

Table 1 distinguishes MQ calculations, shared-input checks and external comparisons. The MQ visible domain is the portion of the realized universe presently referenceable from the specified reference point. The CMB mass equivalent ${\mathit{M}}_{\mathit{CMB}}$, total mass equivalent ${\mathit{M}}_{\mathit{U}}$ and visible-domain mass ${\mathit{M}}_{\mathit{v}\mathit{i}\mathrm{s}}$ are in kg, and the geometric fraction ${\mathit{\Omega }}_{\mathit{v}\mathit{i}\mathrm{s}}$ is dimensionless. The expansionary-epoch duration ${\mathit{A}}_{\mathit{e}\mathit{e}}$ is in s, the mass-equivalent accretion rate ${\mathit{M}}_{\mathit{a}\mathit{c}\mathit{r}}$ in kg s⁻¹ and cosmic volume ${\mathit{V}}_{\mathit{U}}$ in m³. The conventional gravitational constant used for the external conversions is in m³ kg⁻¹ s⁻² and has the CODATA value specified in [15]. The external baryon-density parameter ${\mathit{\Omega }}_{\mathit{b}}$ uses the conventional critical density ${\mathit{\rho }}_{\mathit{c}}$, in kg m⁻³. The present Hubble constant ${\mathit{H}}_0$ is in km s⁻¹ Mpc⁻¹, where Mpc denotes a megaparsec. Connor et al. [20] use the dimensionless normalization ${\mathit{h}}_{70}={\mathit{H}}_0/(70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1})$. This external normalization is distinct from Planck’s constant, introduced below. Fractions with different density denominators require an explicit conversion before subtraction. The external base-ΛCDM model is the six-parameter cosmology with a cosmological constant and cold dark matter.

Table 1. MQ calculations from the fixed 2022 inputs and specified consistency comparisons.

^a Fixsen [21] supplies the temperature that fixes the MQ age. Equations (15)–(22), the radiation density and recovered temperature form a linked consistency check, not an independent CMB mass measurement.

^b The 2025 fast radio burst (FRB) result ${\mathit{\Omega }}_{\mathit{b}}{\mathit{h}}_{70}=0.051\pm 0.006$ [20] is converted to mass over ${\mathit{V}}_{\mathit{U}}$ using conventional $\mathit{G}$ [15]. Its coefficient and uncertainty are per ${\mathit{h}}_{70}$; no ${\mathit{H}}_0$ is fitted. The cluster study [22] constrains matter density, not this total mass.

^c Planck Table 8’s $0.0497\pm 0.0082$ is reionization optical depth [23]. The density check uses Table 2’s base-ΛCDM TT,TE,EE+lowE column, temperature autocorrelation (TT), temperature–E-mode cross-correlation (TE), E-mode autocorrelation (EE) and low-multipole E-mode polarization (lowE), excluding the additional CMB lensing reconstruction likelihood. Physical lensing effects on the spectra remain. MQ’s geometric fraction uses a different normalization.

For a density comparison, the conventional definition ρ_c=3H₀²/(8πG) and Planck’s baryon density expressed with the squared 100 km s⁻¹ Mpc⁻¹ Hubble normalization, $0.02236\pm 0.00015$ [23], give (4.200±0.028)×10^-28 kg m^-3 using conventional $\mathit{G}$ [15]. The MQ mass and volume above give M_vis/V_U=4.5763(26)×10^-28 kg m^-3. If these represent the same present-day baryonic density, the MQ central value is 8.96 percent higher. Applying the Planck density to ${\mathit{V}}_{\mathit{U}}$ gives (1.3577±0.0092)×10⁵³ kg. These are conditional conversions, not a direct mass measurement of the MQ physical universe. The broader FRB comparison and the Planck density test therefore have different precision and model assumptions; agreement in one does not establish agreement in the other.

3.2. Calculating Age of the Universe as a Measure of CMB Temperature

The MQ right-triangle construction places the transition to external referenceability at the dimensionless diagonal √(3), represented as √(3)l_f in the expansionary description ([1], Appx. BF). The configuration-domain extent introduced above is expressed in the same chosen unit basis. The corresponding mapped concluding expansion rate ${\mathit{v}}_{\mathit{i}}$, in m s⁻¹, is calculated below ([1], Appx. BG).

The right-triangle closure is the MQ referenceability criterion. The transition is interpreted through the mapping clarified in the Introduction and Discussion ([1], Appx. BF–BG; [2]), without assigning an intrinsic clock to the pre-referenceable domain.

The CMB mass allocation, energy density and temperature provide linked consistency checks of this construction ([1], Appx. BE, BH). In the present direction of calculation, the observed CMB temperature is an input. Combining it with the specified geometric allocation yields the cosmic age ([1], Appx. BI). Reversing the same dependent calculation does not create an independent measurement or prediction of the input temperature.

We use the CMB temperature from Fixsen [21], who combined the measurements available in that analysis. The Boltzmann constant $\mathit{k}$, in J K⁻¹, and Planck constant $\mathit{h}$, in J s, are exact SI inputs in the 2022 CODATA adjustment [15]. Their values are k=1.380649×10^-23 J K^-1 and h=6.62607015×10^-34 J s. The radiation constant is obtained from these exact inputs and $\mathit{c}$ [15]. Its application to the CMB uses the blackbody radiation description discussed in [24]. Its decimal ellipsis denotes continuing digits of an exact derived value.

The radius ${\mathit{R}}_{\mathit{U}}$, in m, enters the volume relation in Eq. (10). The radiation energy density ${\mathit{E}}_{\mathit{\rho }:\mathit{c}\mathrm{m}\mathit{b}}$, in J m⁻³, and radiation energy ${\mathit{E}}_{\mathit{CMB}}$, in J, then give the mass equivalent already introduced for Table 1. Equations (9)–(14) combine the realized geometry with the blackbody relation and mass-energy conversion. All quantities use the same mapped description and unit system.

The epoch representatives introduced above are related by the dimensionless scalar $(2{\mathit{\theta }}_{\mathrm{s}\mathit{i}}{)}^{1/3}$ ([1], Appx. BH, Eq. (BH.3), and Appx. BK, Eqs. (BK.1)–(BK.3)). Their units belong to the realized description. Converting both representatives from seconds to years preserves the scalar; it does not merely relabel the same numerical exponential. For the mass accretion rate introduced before Table 1, the current MQ relation is M_acr=θ_si³p_si/l_f ([1], Appx. BE). Thus Eq. (18) allocates CMB mass from the expansionary-epoch duration. Its calculated value is M_CMB=1.501601505(11)×10⁵⁰ kg. Equation (18) adopts the MQ allocation of this epoch-associated mass to the CMB. Additional early-expansion accretion and the photon-production mechanism discussed in [14] are not independently determined here. Their omission from this calculation is a model assumption, and their possible contribution is excluded from the quoted propagated input uncertainty. The cubic coefficient dependence in Eq. (15) follows the epoch form in [1], Eqs. (BG.7) and (BH.2).

The geometric closure supporting this interpretation uses three independent unit generators represented by orthogonal unit directions in the realized geometry. Applying the right-triangle construction successively gives the minimum three-dimensional diagonal below ([1], Appx. BF). Multiplication by the already defined ${\mathit{l}}_{\mathit{f}}$ gives its mapped length. The count identity is geometric; identifying its threshold with physical referenceability is the MQ premise.

Equation (19) substitutes the epoch expressions into the mass allocation. Equating that result to the radiation mass in Eq. (14) gives Eq. (20). Multiplying both sides by the positive factors in the denominators gives Eq. (21). The positive cube root then yields Eq. (22), with the epoch quantities interpreted in the chosen expansionary unit representation.

The fixed 2022 inverse fine-structure calibration specified in Methods, together with exact SI constants and the independent CMB temperature [21, 15], gives the age in Eq. (22). The uncertainty retains their shared dependencies and excludes an unquantified MQ model component. For this calculation the radius is 4.2576(12) ×10²⁶ m, the volume is 3.2328(27) ×10⁸⁰ m³ and the present radiation energy density is 4.1747(35) ×10⁻¹⁴ J m⁻³. These values satisfy Eqs. (9)–(14) with the same inputs. The cross-disciplinary age estimates discussed in [25] remain comparison results rather than calibration inputs.

An external age comparison is provided by the base-ΛCDM Planck TT,TE,EE+lowE result $13.800\pm 0.024$ Gyr [23], using the same choice to omit the additional lensing reconstruction likelihood. A complementary stellar-population analysis by Valcin et al. [26] infers an age of 13.5_-0.14^+0.16 Gyr statistically, with a separate systematic uncertainty of $0.5$ Gyr at the reported 68 percent level. The MQ age is compatible with these quoted ranges under their respective assumptions. The stellar inference depends on stellar modelling and the delay between the origin of the universe and cluster formation; it is not wholly independent of cosmological assumptions. Age compatibility does not establish agreement with the separate baryon-density comparison.

3.3. Correlating Age and Diameter of the Universe

Equation (23) restates the fundamental relation. For the cosmic application, ${\mathit{n}}_{{\mathit{T}}_{\mathit{U}}}={\mathit{A}}_{\mathit{U}}/{\mathit{t}}_{\mathit{f}}$, ${\mathit{n}}_{{\mathit{L}}_{\mathit{U}}}={\mathit{D}}_{\mathit{U}}/{\mathit{l}}_{\mathit{f}}$ and ${\mathit{n}}_{{\mathit{M}}_{\mathit{U}}}={\mathit{M}}_{\mathit{U}}/{\mathit{m}}_{\mathit{f}}$ denote dimensionless realized time, diameter and total mass-equivalent count parameters. Here ${\mathit{D}}_{\mathit{U}}$ is the physical diameter in m; the total mass equivalent was introduced before Table 1. These continuous encodings are distinct from primitive integer configurations in the Internal Frame. The current cosmic count relations [2] specify Eqs. (24)–(25); [1], Appx. AZ, BE, provide the historical cosmological context. Using ${\mathit{l}}_{\mathit{f}}/{\mathit{t}}_{\mathit{f}}=\mathit{c}$ in Eq. (24) yields Eq. (26).

Dividing Eq. (25) by Eq. (24), with positive elapsed age, gives Eq. (27). Multiplying the mass count by ${\mathit{m}}_{\mathit{f}}$ gives Eq. (28). The latter yields a total mass equivalent of M_U=3.05108(85)×10⁵⁴ kg for the calculated age. A reciprocal mass has units kg⁻¹ and is not the dimensionless cosmic mass count. The visible-domain relation ${\mathit{M}}_{\mathit{v}\mathit{i}\mathrm{s}}={\mathit{\Omega }}_{\mathit{v}\mathit{i}\mathrm{s}}{\mathit{M}}_{\mathit{U}}$ then supplies the corresponding Table 1 entry ([1], Appx. BJ).

The diameter relation is a System Frame realization of the cosmic count relation. The light-travel quantity ${\mathit{D}}_{\mathit{U}}/\mathit{c}$ is a time, whereas ${\mathit{D}}_{\mathit{U}}$ is a length. Using the age calculated from the CMB temperature gives the diameter below. Gly denotes a billion light-years, with one light-year equal to $\mathit{c}$ times one Julian year.

Equations (26), (29) relate diameter to age under the stated scalar and mapping. Both outputs therefore share the same source uncertainties. Equation (30) independently restates the reference mass calibration from momentum and the exact speed of light; it is not an independent observational determination from the cosmic diameter. The two calculations connect reference measures and cosmological outputs through common MQ premises rather than supplying a new fitted mass parameter.

4. DISCUSSION

The interpretation of Eqs. (15)–(22) raises a specific question. Does writing a quantum-epoch description in seconds or years attribute an intrinsic physical clock to a configuration domain that has no realized temporal or geometric qualities, and does the calculation consequently introduce a freely chosen epoch duration? The distinction between configuration structure and its realized description is essential to answering this question.

Within MQ, the answer follows from the Frames mapping introduced above. The configuration domain supplies count structure; dimensionful observables belong to the System Frame realization. The quantity ${\mathit{A}}_{\mathit{q}\mathit{e}}$ is therefore the expansionary-unit representative of that configuration-domain extent. Seconds or years describe the representative. They do not describe a clock operating before referenceability. The interepoch scalar specifies the relation between descriptions and remains unchanged by a consistent change of units. The unit factor displayed in Eqs. (1), (15), (19) states the fixed realization normalization adopted in this calculation. A consistent conversion of units preserves that representative. The unit-count closure does not by itself independently verify this absolute normalization. This qualification concerns the realization rule, not a physical clock in the configuration domain or an additional fitted epoch parameter.

The scalar is constrained by relations already used in the paper. The coefficient ${\mathit{\theta }}_{\mathrm{s}\mathit{i}}$ is the invariant coefficient whose momentum realization ${\mathit{p}}_{\mathrm{s}\mathit{i}}$ enters the fundamental relation in Eq. (23). Thus the factor $2{\mathit{\theta }}_{\mathrm{s}\mathit{i}}$ in the geometric description is linked to the same coefficient as the length, mass and time relation, while the dimensional fundamental relation retains ${\mathit{p}}_{\mathrm{s}\mathit{i}}$. The dimensionless scalar and its dimensional realization must remain distinct. This connection prevents the epoch factor from being introduced as an unrelated adjustment to reproduce the age result ([1], Appx. V, W, BH).

The realized three-dimensional construction supplies a second constraint. Cubic scaling converts the specified relative volume factor into the linear scalar $(2{\mathit{\theta }}_{\mathrm{s}\mathit{i}}{)}^{1/3}$ used in Eq. (17). Three dimensionality determines the cube-root exponent once the MQ relative factor and its mapping direction are specified; it does not independently determine that relative factor. Equation (16) supplies the unit-count diagonal closure. The mapped threshold √(3)l_f, together with the quantum-epoch radius relation and its exponential inversion, connects that construction to Eq. (15) ([1], Appx. BF, BG). MQ interprets this boundary as the initiation of the referenceable physical universe. The radius is an expansionary representation of the threshold rather than a physical length possessed by the pre-referenceable domain.

These relations form the proposed geometric closure of the calculation. The fundamental relation links the invariant coefficient to its realized measures; the unit-count right-triangle construction specifies the proposed referenceability boundary; the interepoch mapping relates the corresponding descriptions; and the mass allocation and radiation relations connect them to the measured CMB temperature. Equations (18)–(22) then give the age, and Eq. (29) gives the diameter. Once the measured inputs and these MQ premises are fixed, the present calculation introduces no independently fitted epoch duration, interepoch scalar or diameter parameter. The choice of a time unit is separate from those geometric premises. It neither adds a physical property to the configuration domain nor independently proves the premises.

The empirical role of this construction is to expose a constrained set of consistency tests. Fixsen’s CMB temperature [21] fixes the age in the present calculation. The resulting mass allocation, density and recovered temperature test the consistency of that MQ interpretation, as documented by the supporting calculation chain. The age comparisons [25–26] and external baryon constraints [23, 20] provide further checkpoints whose information depends on their observations, domain definitions and model assumptions. Table 1 therefore separates the shared-input closure from the external comparisons. The conditional Planck density comparison described above must be accounted for in any concordance assessment; the broader FRB result does not make all checkpoints independent or equally precise.

The same care applies to distance comparisons. The MQ observable domain includes the visible domain and portions that become referenceable later. These MQ domain definitions must be distinguished from the conventional observable-universe horizon. The physical diameter defined in Sec. 3.3, the extent of a visible or observable domain, and the sound horizon are different quantities. A sound-horizon comparison must use the distance traversed by acoustic disturbances in the specified early-universe history, rather than identifying that distance with the total diameter. CMB acoustic-scale constraints are discussed in [23]. Such comparisons can extend the suite of MQ consistency tests when the corresponding observable, domain, source assumptions and shared inputs are specified. The sound horizon is a prospective additional test in this paper; no numerical sound-horizon prediction or direct measurement of the total physical-universe diameter is established here.

The significance of the proposed closure is its restriction of the allowed outcomes. The same fixed construction must remain compatible with each applicable comparison, without readjusting an epoch factor for each observable. Concordance across distinct observations would support that interpretation, while a reproducible discrepancy could challenge it. The absence of an additional fitted parameter in this calculation makes that restriction explicit. It does not by itself establish agreement with every available dataset or superiority to a cosmological model evaluated against a different set of observations. For example, the Planck analysis tests a specified six-parameter base cosmology against temperature, polarization and lensing data [23]. A comparative assessment must account for the observables explained, input dependencies, uncertainty and predictive performance, as well as the number of adjusted parameters.

From the MQ perspective, the density discrepancy motivates the hypothesis that the base-ΛCDM interpretation omits a relation captured by the MQ construction. This does not require an arithmetic error in the Planck analysis, whose reported parameters are conditional on its cosmological assumptions [23]. The present comparison does not identify which assumption would account for the difference. The exact unit-count geometry in Eq. (16) supports the proposed construction; its application to physical referenceability, the epoch relation and the mass allocation also requires the MQ premises stated above. In this calculation the measured CMB temperature fixes the age, so recovery of that temperature establishes consistency of the linked calculation rather than independent evidence locating the density discrepancy in ΛCDM. The geometric derivation introduces no additional fitted epoch parameter. Establishing an incompleteness of the conventional interpretation would additionally require a quantitative MQ account of the same observational constraints, including this density comparison.

5. APPENDICES

APPENDIX A: TERMS

Historical symbol and term definitions appear in [1], Appx. BP–BQ. The current definitions and realization distinctions used here follow [2].

The coefficient and momentum realization are introduced in the Introduction. The angular realization ${\mathit{\Theta }}_{\mathrm{s}\mathit{i}}$, in rad, is distinct from both ([1], Appx. A, Q, S). The X-ray studies [27–29] concern nonlinear optical processes, while Chen et al. [30] discuss conserved quantities at null infinity in general relativity. These are separate physical contexts; they are not used here as measurements of the MQ coefficient or tests of the interepoch mapping.

The speed of light and its exact SI value are introduced in the Introduction [3].

For the gravitational coupling, upright $\mathrm{G}$ denotes the upper-count realization and italic $\mathit{G}$ a finite-count realization, both in m³ kg⁻¹ s⁻². The current calibration gives G=6.6740779383(10)×10^-11 m³ kg^-1 s^-2. At the Fine Structure demarcation, the radial argument is $276/{\mathit{\theta }}_{\mathrm{s}\mathit{i}}$ and the finite-count value is G=6.6738448317(10)×10^-11 m³ kg^-1 s^-2. These are MQ calculations from the 2022 calibration [15], rather than CODATA recommendations for the gravitational constant ([1], Appx. M, AD, BM).

The fundamental measures and their SI units are introduced in the Introduction.

l_p, m_p and t_p are Planck’s units for length, mass, and time.

The local length, mass and time counts are introduced in Sec. 3.1. The separate cosmic count parameters are introduced in Sec. 3.3.

The motion-count use of ${\mathit{n}}_{\mathit{L}}$ is introduced in Sec. 3.1.

The radial-count use of ${\mathit{n}}_{\mathit{L}}$ at the escape boundary is specified before Eq. (3).

The dimensionless light-motion count ${\mathit{n}}_{\mathit{L}\mathit{c}}$ describes a change in position represented in units of ${\mathit{l}}_{\mathit{f}}$ with respect to the observer’s frame of reference.

The dimensionless System Frame count ${\mathit{n}}_{\mathit{\theta }}$ is resolved at the specified demarcation through the metric differential, which describes the discrete-to-continuous representation mismatch under the mapping. The mixed brackets $\lfloor \mathit{x}\rceil$ denote rounding the dimensionless argument $\mathit{x}$ to its nearest whole-unit value. The retained count example is $\lfloor 84.60055/2\rceil =42$.

The mixed nearest-whole-unit bracket convention is defined immediately above.

The residual used in the next entry is a dimensionless quantity of the radial count construction ([1], Appx. C).

The Informativity differential describes the MQ non-Lorentz contraction of realized separations. Its dimensionless radial form is $2{\mathit{Q}}_{\mathit{L}}{\mathit{n}}_{\mathit{L}}$, with residual Q_L=√(1+n_L²)-n_L for positive ${\mathit{n}}_{\mathit{L}}$.

The conventional reduced Planck constant is ℏ=h/(2π)=1.054571817646…×10^-34 J s and is exact under the SI definition of $\mathit{h}$ [15]. Upright $\mathrm{ħ}$ denotes the distinct MQ upper-count realization, ħ=2p_sil_f=1.0545349844417293(56)×10^-34 J s ([1], Appx. L). The ellipsis denotes continuing digits of the exact conventional value. The calculated upper-count value carries the propagated calibration uncertainty.

Electron mass is introduced before Eq. (4).

The Bohr radius is introduced before Eq. (4).

The additional fundamental inverse fine-structure form α_f^-1 is dimensionless and satisfies α_f^-1=42θ_si in the Internal Frame count description. The Planck-like and classical forms retain the meanings introduced before Eq. (4) and in Methods. Their System Frame realizations differ through the finite-count normalization ([1], Appx. AA, AD).

The symbol $\mathit{\gamma }$ denotes the historical electromagnetic transformation parameter used in the supplementary gamma approach ([1], Appx. AS). It is resolved from an electromagnetic input in that historical construction and is distinct from the later fundamental gamma parameter. No gamma parameter enters the cosmological equations in this paper.

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

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