Classical Geometric Exploration of the CMB Power Spectrum Without Λ or CDM

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine geometric domain fractions and exploratory expansion metrics in relation to the cosmic microwave background (CMB) power spectrum. The domain calculations use the dimensionless Measurement Quantization (MQ) coefficient θ_si=3.26239030504(25), calibrated from the 2022 inverse fine-structure constant. The fractions are the fundamental Ω_f=18.7913577188(29)%, presently visible Ω_vis=4.8488348909(10)%, ultimately observable Ω_obs=31.6375838776(41)%, ultimately observable but not presently visible Ω_uobs=26.7887489867(31)%, and never observable Ω_dk=68.3624161224(41)%. These are dimensionless fractions reported as percentages. They are compared with conventional dark-matter and dark-energy descriptions without identifying all of their physical properties. The quantum-epoch configuration is expressed through the Frames mapping, which relates discrete configuration to realized quantities. Seconds or years belong to the expansionary representation; the quantum epoch has no intrinsic elapsed clock duration. Its cube-root epoch relation is a geometric framework relation. The candidate CMB coordinates explore correlations rather than establish measured peak locations. The absence of the explored metric factor from the dark-domain coordinate is treated as a physical clue. Age, mass and temperature comparisons test consistency under the stated MQ realization and shared-input dependencies. No parameter is fitted to CMB peak data in this exploration.

Keywords: CMB Power Spectrum, Observational Domains, Dark Matter, Dark Energy, Visible, Observable

1. INTRODUCTION

We explore a classical geometric description of observational domains and candidate expansion metrics associated with the CMB power spectrum. Here Λ denotes the conventional cosmological constant, in m−2, and CDM abbreviates cold dark matter. Their absence refers to explicit fitted components of these geometric expressions. It does not establish that their dynamical roles in conventional cosmology have been reproduced. The domain fractions follow from the MQ partition relations; the additional coordinates explore a proposed relation to observable CMB features. The dark-domain index remains a geometric reference rather than a directly observed CMB peak. The geometric focus also motivates comparison with historical relativistic treatments [1].

Geometry motivates the MQ observational domains. The visible domain concerns what is presently visible; the observable domain includes it and what may become visible given unlimited elapsed time under the proposed expansionary description. The unobserved domain is their difference, and the dark domain is never observable relative to the specified observer. The fundamental fraction marks the pivot associated with the visible–observable relation. These domains are constrained aspects of referenceability, rather than five independent additive components. With fractions normalized to the whole, Eqs. (1)–(5) give their calculated values as percentages.

The results are compared with conventional dark-energy and dark-matter descriptions to examine their geometric correspondence. MQ describes realized length, mass and elapsed duration using the dimensionless counts n_L, n_M and n_T of the fundamental references l_f, m_f and t_f, whose realized units are m, kg and s. Here the counts refer to separation, mass and elapsed duration in the stated realization ([2], Appx. I, J, W). The speed of light c is exactly 299,792,458 m s−1. With the common 2022 calibration [3], the calculated references are l_f=1.61619991148(12)×10⁻³⁵ m, m_f=2.17643254057(17)×10⁻⁸ kg and t_f=5.39106261131(41)×10⁻⁴⁴ s. Their formal uncertainties share the inverse fine-structure input and are not independent.

The partition calculations introduce no additional parameter fitted to CMB peak data and do not invoke modified Newtonian dynamics (MOND), supersymmetry or an inflationary model [4]. Background-independence questions in Loop Quantum Gravity (LQG) [5–6] provide related theoretical context; those references do not derive the MQ count relations used here.

The MQ account uses the discrete Internal Frame for count configurations and the non-discrete System Frame for their continuous relational encoding. The Frames mapping links these descriptions (historical framework antecedents in [2], Appx. K). The resulting realization relations are used to calculate physical quantities and to express the domain partition [7]. The partition equations do not by themselves establish all properties of conventional cosmological components.

2. METHODS

This paper uses classical expressions together with discrete count relations for gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle [8]. Earlier MQ applications appear in [7, 9–13]. The dimensionless classical fine-structure coupling α_c has the 2022 Committee on Data for Science and Technology (CODATA) inverse value 137.035999177(21) [3], which calibrates the dimensionless MQ coefficient ([2], Appx. AA). Parenthetical digits give one propagated standard source uncertainty in the final quoted digits. We retain common dependencies and covariance, using 65-digit working precision and rounding uncertainties to two significant digits, then central values to the same decimal place, with ties to even. Exact International System of Units (SI) factors contribute no source uncertainty. Missing physical-normalization or model-discrepancy uncertainties are unquantified, rather than established as zero. Published confidence intervals retain their stated interpretation and are not converted into standard uncertainties.

Discrete, countable primitive measures are premises of the MQ framework. Measurement Quantization [2] develops their proposed derivations and measurement comparisons and contains the referenced appendices and an extended terminology resource. Those comparisons must be assessed under their individual calibration and realization assumptions. Neither the uncertainty principle nor numerical agreement alone uniquely establishes a discrete ontology. The published article supplies the historical calculation context; the frame and epoch meanings used here follow the definitions above. Supporting spreadsheets are listed in Online Resources.

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

The two spreadsheet supplements accompanying [2] are listed in Online Resources. Supplement 1 contains calculations for that article; Supplement 2 also covers MQ applications. The appendix citations refer to the published article. These spreadsheets provide calculation aids within the proposed framework. Agreement among calculations that share a calibration or realization premise is not an independent experimental test of that premise.

3. RESULTS

1. Definitions

The total fraction Ω_tot is unity. Equations (6)–(7) express its partition into observable and dark fractions. Equation (8) relates the observable and visible fractions through the calibrated coefficient. Equation (9) gives the unobserved fraction by subtraction. The first two relations express normalization; Eq. (8) supplies the additional MQ geometric relation used in the following derivation. The fundamental fraction is a pivot of these relations and is not an extra component to add to their total.

The dark and unobserved domains are compared with dark energy and dark matter through their geometric roles. These comparisons do not identify every physical property of the conventional descriptions. The fundamental relation uses the momentum realization p_si, in kg m s−1, rather than treating the dimensionless coefficient as momentum. Its adopted numerical coefficient in those units is 3.26239030504(25). The relation l_fm_f=2p_sit_f links the realized references ([2], Appx. V). Expansion and mass allocation use their specified counts ([2], Appx. AZ, BE); the relation alone is not an independent cosmological mass measurement.

The MQ mass-frequency construction bounds the realized response associated with one increment of t_f ([2], Appx. O). This instantaneous bound does not determine total mass accumulated over successive increments or an accumulation dynamics. Cosmological allocation additionally uses the count relations stated below. The System Frame response and accumulated mass must therefore be distinguished [11].

Equation (10) states the fundamental relation in its momentum form. Its units agree on both sides, and Eq. (16) obtains m_f by using c=(l_f)/(t_f). This establishes the fundamental-reference calculation; cosmological allocation also requires the temporal counts and geometric relations specified below.

The extension to cosmological quantities uses the accepted expansionary realization (historical calculations in [2], Appx. BG and BH) and Frames mapping. A_qe denotes the quantum-epoch configuration extent expressed in that representation, while A_ee denotes the realized expansionary duration; both are reported using the same time unit. A_U is the current realized universe age, and D_U is its diameter. M_CMB denotes the allocated CMB mass-equivalent content and M_tot the accumulated total mass-equivalent content, both in kg. M_vis is the portion assigned to the visible domain. M_f, M_dk, M_obs and M_uobs denote the mass-equivalent allocations to the other already defined domains, also in kg. T_CMB is the CMB monopole temperature in K, and V_U is the realized spherical universe volume in m³. The blackbody radiation constant a, in J m−3 K−4, is obtained from exact SI h, the Planck constant in J s, and k, the Boltzmann constant in J K−1 [3]. Table 1 distinguishes these calculations from the specified observational projections.

The fundamental relation provides a common realization for length, mass and time. Extending it to a cosmological comparison also requires the chosen volume, temporal count and mass-allocation relation. These additional steps are made explicit below rather than inferred from a ratio of units alone.

The equal-count speed-of-light relation links l_f and t_f. Equations (11), (42) then calculate the same dimensionless geometric index, with π the circle constant and the factor 100 expressing the dark fraction in percentage units. The index is exploratory and does not independently measure spatial curvature or identify a directly detected dark-domain CMB peak ([2], Appx. BD).

Equation (12) retains the fixed historical age illustration, with c explicit so that D_U remains a length and A_U a time. Gyr denotes a billion Julian years and Gly a billion light-years; a Julian year is exactly 31,557,600 s, and a light-year is c times that interval. Equation (13) gives the same general dimensional relation with D_U in m, A_U in s and c in m s−1. The current CMB-derived age is obtained below; the historical 13.794 Gyr example is not an independent age measurement.

To connect the epoch allocation to age, use the mass relation in Eq. (35), the blackbody relation in Table 1 and the spherical volume with radius D_U/2. Substituting Eq. (13) makes that radius θ_siA_Uc. The factors θ_si³ cancel, giving the positive age solution in Eq. (14). The cube-root argument has units s³. Thus the reconstruction uses the realized expansionary duration and fundamental mass/time ratio; it introduces no clock intrinsic to the configuration domain (historical temperature–age construction in [2], Appx. BI).

The current conditional age is 13.7943(38) Gyr and the diameter from Eq. (13) is 90.005(25) Gly. They use the prescribed MQ realization and measured temperature, rather than supplying independent timing and distance measurements. The fine-structure and Fixsen temperature source inputs are treated as uncorrelated across their separate experiments; derived outputs retain their shared covariance. Substitution back into the same radiation relation recovers the input temperature. Equation (15) separately adopts the accumulated-mass allocation law, rather than deriving it from the instantaneous response bound. The normalization convention is held fixed in this source-uncertainty calculation.

Equation (15) gives 3.05108(85)×10⁵⁴ kg. Multiplication by each dimensionless domain fraction gives the current mass allocations in Table 2. The historical figure is retained to show the earlier illustration; it is not a second calibration or an independent total-mass measurement.

Table 1. MQ allocations and specified observational comparisons.^a

^a The reference masses are observational inferences projected into the same adopted MQ volume, not independent measurements of that volume. Differences mean reference minus MQ. The fast radio burst (FRB) confidence interval and the conditional standard source uncertainty on the carbon monoxide (CO) projection retain different statistical meanings.
^b The CO absorption estimate of the present CMB temperature is 2.75 ± 0.11 K [14, Table 6], obtained using the standard temperature–redshift law and excitation corrections. Applying the radiation relation at the common volume gives the quoted CMB mass reference. Its difference uncertainty retains the shared-volume dependence and the separate CO and Fixsen temperature inputs.
^c The unconstrained FRB baryon census [15] gives a central density fraction 0.051 and 95% interval 0.026–0.072 at a fixed expansion-rate reporting reference of 70 km s−1 Mpc−1, where Mpc denotes a megaparsec. Using the conventional 2022 Newtonian gravitational constant, 6.67430(15)×10⁻¹¹ m³ kg−1 s−2, and the same MQ sphere gives the mass reference shown. The interval is transformed at fixed normalization and volume; it is not a standard uncertainty or a total interval over the MQ premises. Baryonic rest mass and the MQ visible allocation have different physical definitions.
^d Fixsen supplies the temperature input 2.72548 ± 0.00057 K [16]. The recovered value and zero difference record shared-input closure. The earlier 13.794 Gyr example in Eq. (12) is not used as an independent temperature test. Abdullah et al. [17] infer matter density from cluster abundances, not the global visible mass formerly quoted here. A direct Planck baryon-fraction comparison is not reported without the density-to-fraction conversion and its joint covariance.

At the fixed reporting normalization and common volume, the FRB 95% interval projects to 0.77–2.14×10⁵³ kg. Its central mass exceeds the MQ visible allocation by 0.04×10⁵³ kg. The FRB observations supply electron-column and redshift information separate from Fixsen’s temperature observations, while their inference retains ionization, foreground, host-population and cosmological modelling [15]. The shared MQ volume prevents interpreting this as an independent weighing of the physical universe.

Equation (16) gives the same fundamental mass introduced above, using the momentum realization and exact c. The following calculations then derive the five domain fractions from their normalization and the visible–observable relation.

The domain derivation depends on the calibrated coefficient and the partition relations, independently of the numerical age illustration.

2. Calculations

Start with the visible–observable relation in Eq. (8). Rearrangement gives Eqs. (17)–(18), and substitution into the fundamental-fraction expression gives Eq. (19), reproducing Eq. (1).

Equations (20)–(22) restate the visible–observable relation and introduce the complementary dark fraction. Substituting the normalization in Eq. (7) gives Eqs. (23)–(24); solving yields Eq. (25), the same result as Eq. (2).

The dark domain lies outside the observer’s ultimate observable domain in this construction. Equation (23) substitutes Ω_tot=1 from Eq. (7); Eq. (24) collects terms. The interpretation is relative to the stated observer and expansionary realization.

Subtracting the dark fraction from unity gives Eq. (26); rearrangement gives Eq. (27), and solving produces Eq. (28).

Insert the observable fraction from Eq. (28) into Eq. (8). Equation (29) rearranges this relation and Eq. (30) gives the visible fraction.

Finally, Eq. (31) subtracts the visible fraction from the observable fraction, reproducing the unobserved fraction in Eq. (5).

Fig. 1 adapts the layout of [11], Fig. 3, with historical allocations from [2], Appx. BJ. Its percentage legend gives masses divided by M_tot; the bars give kg. Table 2 gives current results.

3. Observational Comparisons

Table 2 compares MQ referenceability fractions with Planck 2018 base-ΛCDM TT,TE,EE+lowE results [18, Table 2], excluding the additional CMB lensing reconstruction likelihood. TT, TE and EE denote temperature auto-, temperature–E-mode polarization cross-, and E-mode polarization auto-spectra; lowE denotes the large-angular-scale E-mode polarization likelihood. That exclusion does not remove lensing effects from the temperature and polarization spectra themselves. The source directly reports Ω_Λ and Ω_m, where Ω_m is total matter and includes the source model’s neutrino contribution; it is not merely Ω_c+Ω_b. The cold-dark-matter and baryonic density parameters Ω_c and Ω_b are not obtained here by copying optical depth or attaching a common percentage error. Such conversions require the joint parameter covariance. MQ fractions use the common calibration [3], and their mass allocations additionally use the temperature-derived age and Eq. (15). Fig. 1 retains its historical total 3.0510×10⁵⁴ kg and allocations ([2], Appx. BJ); Table 2 gives current results. Similar numerical fractions do not establish a common physical definition or greater experimental precision.

Table 2. Planck without the additional CMB lensing likelihood and MQ domain calculations.

^a The quoted 68.34 ± 0.84% and 31.66 ± 0.84% are the directly reported dark-energy and total-matter fractions in Table 2 of [18], with the source’s 68% parameter intervals. These two comparisons are related, not independent confirmations. Table 8’s no-CMB-lensing value 0.0497 ± 0.0082 is optical depth in an effective field theory (EFT) model, not a baryonic density fraction. It is therefore excluded from the fraction comparisons. A dash marks an unreported component comparison.

Recent research identifies physical considerations relevant to interpreting this comparison. Connor et al. [19] locate much of the cosmic baryon inventory in diffuse ionized gas outside galaxies. Material that is difficult to detect through emitted light can therefore remain measurable through radio-wave dispersion. Such observational incompleteness must be distinguished from the MQ unobserved domain, which concerns referenceability relative to the observer. The study informs the physical census; it does not derive a new MQ domain fraction. Sharma et al. [20] further show that baryon redistribution by astrophysical feedback affects the small-scale matter power spectrum. This supplies a physical contribution beyond the geometric partition considered here, while retaining dependence on gas and cosmological modelling. Neither study identifies that contribution with the MQ metric differential, the mismatch between discrete count structure and its continuous realization.

Equations (32)–(42) explore dimensionless expansion metrics and possible CMB correlations. The coordinate pairs in Eqs. (36)–(40) are manuscript-local exploratory definitions. Their first component supplies a geometric coordinate and their second a domain fraction; y is a dimensionless coordinate parameter, with no numerical value selected here. The auxiliary Ω_v in Eq. (39) has half the unobserved fraction as its second component and does not add a sixth domain. These expressions address a proposed geometric contribution. Quantitative comparison with a measured spectrum additionally requires a fixed correspondence to multipoles and power normalization, the applicable metric differential realization, physical interaction effects and measurement covariance.

For the observable-domain exploration we use the factor in Eq. (32), denoted Δ_mdif. It is dimensionless and specifies the direction of the exploratory coordinate rescaling defined here. The reciprocal factor corresponds to reversing that rescaling. The factor is not asserted to be a Lorentz time-dilation factor between specified inertial clocks.

The explored factor links π with θ_si through the exploratory coordinate prescription in Eq. (32). Equation (33) gives the dimensionless unity constraint ([2], Appx. AX, Eq. (AX.6)). Here n_Lc is the dimensionless light-displacement count in the same realization as n_L. The constraint determines their squared ratio once θ_si is fixed; it does not independently determine both counts. An equal-count assumption from a different relation is not transferred to this construction.

The historical epoch-scaling relations appear in [2], Appx. BG and BK. Here the quantum-epoch configuration is represented through its scalar relation to the expansionary epoch. Seconds or years belong to the realized representation and do not specify an intrinsic configuration-domain clock. Equation (34) states the accepted mapping used in Eq. (14) and the CMB mass calculation.

The factor 2θ_si is linked to the fundamental references; its cube root expresses the adopted volume-to-linear conversion and its mapping direction. Three dimensionality fixes that exponent once the relative factor is specified, rather than independently deriving the factor. The unit-count construction reaches diagonal count √(3) and mapped radial threshold r, in m, satisfying r=√(3)l_f ([2], Appx. BF, BG). Identifying this threshold with physical referenceability is an MQ premise. The quantum-epoch representative is evaluated using a fixed SI realization normalization, giving 363312.3525(25) years and mapped duration 678894.2574(47) years. Converting both representatives preserves their scalar relation. This fixed convention neither assigns an intrinsic clock to the configuration domain nor supplies an independently verified absolute physical scale. Its uncertainty is excluded from the conditional source budget, not established as zero; no freely fitted epoch parameter or additional reference symbol is introduced.

Equation (35) uses the mapped duration A_ee, while Eq. (15) uses the present age A_U. In each relation, division by t_f gives the relevant dimensionless temporal count, and multiplication by m_f and the allocation factor yields kg ([2], Appx. BE and BH). The current CMB allocation is 1.501601505(11)×10⁵⁰ kg. It extends the expansionary representation through the Frames mapping and does not assign intrinsic mass or time to the configuration domain.

The dark domain lies outside the specified observer’s observable domain. Equation (36) is defined without Δ_mdif, while the other exploratory coordinates retain the factor. This asymmetry is the physical clue examined here; exclusion from referenceability does not algebraically require that particular coordinate choice. A directly received CMB feature belongs to the observable description, so the dark-domain index is not assigned a directly observed peak. Non-referenceability does not imply destruction of the primitive count substrate, and it does not by itself calculate or exclude every indirect physical influence. The proposed clue therefore remains distinct from a proof of the partition or a dynamical spectrum prediction.

For an angular power spectrum, l is the dimensionless multipole order and C_l is its power coefficient in the chosen normalization. For temperature anisotropies expressed in K or μK, C_l has units K² or μK²; for fractional temperature anisotropies it is dimensionless. Equation (41) retains those units and gives the customary power ordinate. It is not itself a temperature or a calibration of y. No numerical spectrum is selected for the present geometric exploration.

Equation (42) evaluates the same dark-domain geometric index as Eq. (11), using the recalculated domain fraction. The fixed factor 100 converts the fraction to the percentage convention used in this index ([2], Appx. BD). Its numerical proximity to a CMB multipole scale motivates further comparison; the index is not identified with a directly detected dark-domain feature.

Planck 2018 [18] remains the comparator. Atacama Cosmology Telescope (ACT) Data Release 6 (DR6) temperature and polarization spectra [21] provide a complementary observational test and are consistent with the conventional cosmological model within that analysis. They do not establish the coordinate-to-multipole mapping required here. The coefficient constructing Eq. (42) is calibrated from the fine-structure input, so its index is not an independent measurement of θ_si. The X-ray studies [22–24] provide polarization and wave-mixing context ([2], Appx. A), rather than the uncertainty propagated from the 2022 calibration. The conserved-quantity construction of Po-Ning Chen and collaborators [25] supplies a separate relativistic context rather than an independent calibration of these coordinates.

A further consideration is the distinction between acoustic geometry and oscillation dynamics. Montefalcone, Wallisch and Freese [26] isolate the CMB acoustic phase shift associated with freely propagating neutrinos and their gravitational influence. This provides an example of a physical contribution to peak structure that is not captured by a geometric coordinate alone. Absence of a direct luminous signal does not, by itself, exclude an indirect influence on observable CMB structure. This conventional result does not identify neutrinos or conventional dark matter with the MQ dark domain. Within the MQ interpretation, the missing metric factor in Eq. (36) remains a clue about the geometric description associated with the observer; the present construction supplies no calculation of these additional physical effects.

The research considered here clarifies possible sources of differences between a geometric account and a physical survey inference. Diffuse baryons, feedback and acoustic phase shifts are distinct effects and cannot be inserted as interchangeable corrections to a domain percentage. No reviewed study derives the MQ coordinate mapping or establishes an exact correspondence between its referenceability fractions and Planck density parameters. We therefore retain the historical Planck reference and use these studies to delimit the comparison. No extra correction factor, fitted peak coordinate or new MQ equation is introduced.

4. DISCUSSION

The epoch construction, the referenceability criterion and the mass-allocation law supply the proposed geometric closure ([2], Appx. BF–BH). Equations (34)–(35), (14)–(15) connect that construction to the measured CMB temperature and conditional age, diameter and mass. Once the realization convention and MQ premises are fixed, no epoch duration is adjusted separately for each observable. Their shared dependencies make the recovered temperature a consistency check. External temperature, baryon and stellar-age observations provide additional comparisons under their respective physical definitions and models. Scalar closure alone does not establish a complete dynamical CMB spectrum or a horizon-problem solution.

For an external age comparison, Planck TT,TE,EE+lowE gives 13.800 ± 0.024 Gyr [18, Table 2], omitting the additional lensing reconstruction likelihood. Its central value is 0.0057 Gyr above the conditional MQ result. Valcin et al. [27] infer 13.39 ± 0.10 Gyr statistically and ±0.23 Gyr systematically for the oldest globular-cluster population. Their universe-age inference is 13.57 Gyr with statistical bounds −0.14/+0.16 Gyr and a separate ±0.23 Gyr systematic uncertainty, including a formation-delay model. The MQ result lies within these quoted comparison ranges. We retain the statistical and systematic budgets separately, without constructing an unsupported combined significance or using either age to recalibrate the Fixsen-based result [16].

Independent comparisons require the same physical quantity, observation domain and uncertainty definition on both sides. A measurement protocol is needed to test a realized MQ quantity; an algebraic representation alone does not supply that protocol. The accepted numerical realization provides a conditional calculation, while its empirical validation remains a separate task.

The partition relations in Eqs. (1)–(31) establish internal algebraic consequences of the stated MQ definitions. Equations (32)–(42) explore geometric coordinates. Their dark-domain asymmetry motivates further investigation, but a coordinate-to-multipole relation and covariance-aware comparison to a specified spectrum are needed before a peak correlation is quantitatively established.

No parameter in these exploratory coordinates is fitted to CMB peak data. A comparison of physical interpretations would additionally require their applicable dynamics and likelihoods. The present consistency results do not establish uniqueness or superiority over a model fitted to the full spectrum.

5. APPENDICES

APPENDIX A: TERMS

Historical symbol and term definitions appear in Measurement Quantization [2], Appx. BP and BQ; the quantities needed here are defined at first use.

Θ_si denotes the angular realization in rad for the comparison discussed above. In Table I of Shwartz and Harris [22], 3.26239 rad is a calculated angle of each signal and idler wave vector relative to the diamond atomic planes in one degenerate configuration. It is not their angular separation or a measurement of the MQ coefficient. The calculation assumes phase matching with a reciprocal lattice vector. Its numerical proximity motivates the comparison; the author-adopted 2 μrad tolerance is not a published measurement uncertainty. References [23–24] concern other X-ray processes, and [25] concerns relativistic conserved quantities. None independently calibrates the coefficient used here.

The earlier speed-of-light definition uses the exact SI value, also present in the retained 2018 compilation [28]. Equal fundamental counts give c=(l_f)/(t_f); the current numerical inputs in this paper use the 2022 edition [3].

G denotes the upper-count gravitational coupling, while italic G denotes its finite-count realization. The current upper-count calculation is G=6.6740779383(10)×10⁻¹¹ m³ kg−1 s−2. At the Fine Structure demarcation, with count n_L=(276)/(θ_si), the finite value is G=6.6738448317(10)×10⁻¹¹ m³ kg−1 s−2. These share the current MQ input chain and are calculations, not CODATA recommendations or independent measurements. Here Q_L is the dimensionless right-triangle residual for that count. The count-dependent normalization is 2Q_Ln_L.

The fundamental-reference values and their units were introduced before Eq. (10). They remain distinct from conventional Planck quantities and share the coefficient calibration.

l_p, m_p and t_p denote the conventional Planck length, mass and time, in m, kg and s. They are distinct from the fundamental MQ references already introduced.

The counts n_L, n_M and n_T retain the dimensionless meanings introduced above; a count’s stated physical realization and reference measure determine the corresponding dimensional quantity.

n_Lm denotes the dimensionless count of l_f for a change in realized position of an observed mass relative to the reference point.

The radial count n_L already introduced can describe separation from a center of mass when that reference point is specified. It is distinguished from the motion count n_Lm.

n_Lc retains the light-displacement count defined beside Eq. (33), using the same fundamental length reference as n_L.

n_θ denotes the dimensionless System Frame count selected from the electromagnetic count argument. At the adopted demarcation, the nearest-whole-unit evaluation of (n_L)/(2) gives 42. The underlying count is 84.6005456716(65). The same branch holds over a stated ±10-standard-uncertainty sensitivity range of the source input. This range is a branch-stability check, not a hard probability bound or a Gaussian assumption.

For a generic dimensionless argument n, the same operation is written ⌊n⌉. No half-unit tie is reached by the present count argument.

The residual defined in the gravitational-demarcation note has the stable evaluation Q_L=(1)/(√(1+n_L²)+n_L), avoiding subtraction of nearly equal large quantities.

The finite-count realization factor is 2Q_Ln_L and approaches unity at the upper count limit. Its dimensionless departure 1−2Q_Ln_L tends to zero and describes the non-Lorentz correction associated with the Informativity differential. The factor and its departure must not be interchanged; neither is a Lorentz transformation or curvature.

For the conventional reduced Planck constant, ℏ=(h)/(2π), with exact h=6.62607015×10⁻³⁴ J s [3]. Upright ħ denotes the MQ upper-count action realization. Its calculated value is ħ=1.0545349844417293(56)×10⁻³⁴ J s, obtained from 2p_sil_f under the same calibration. The conventional finite-count value and the upright upper-count realization must not be interchanged.

m_e denotes electron mass, in kg. It does not supply an additional independent input to the calibration used in this paper.

a₀ denotes the Bohr radius, in m. Electron mass and Bohr radius are dependent conventional quantities when used to evaluate the fundamental-length relation; their exact product identity eliminates that shared dependence in the adopted input chain.

α_f and α_p are the dimensionless fundamental and Planck-like fine-structure couplings. The classical coupling α_c was introduced with the current input. The fundamental inverse is α_f⁻¹=42θ_si; the Planck-like inverse precedes the count-residual correction to the classical form. Their distinct configuration and realization roles are retained rather than assigning the same numerical value to all three.

γ denotes the conventional electromagnetic transformation parameter in the historical gamma construction, with units m⁶ C−1 s−4. It is resolved from an electromagnetic source measurement and is distinct from the later fundamental gamma parameter. It is not an independent input to the CMB calculation integrated here.

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 calculated results and governing relations are included in the manuscript. The cited supplementary resources describe the supporting MQ calculations.

REFERENCES

[1] A. Einstein, The Collected Papers of Albert Einstein, Vol. 6, The Berlin Years: Writings, 1914–1917, English translation of selected texts, Princeton University Press, ISBN 978-0-691-01734-1. Volume information: https://www.einstein.caltech.edu/what-we-do/published-volumes/volume-6.

[2] J. A. Geiger, Measurement Quantization, International Journal of Geometric Methods in Modern Physics 20(4), 2350069 (2023), published 25 January 2023, CC BY 4.0, doi:10.1142/S021988782350069X.

[3] Peter J. Mohr, David B. Newell, Barry N. Taylor and Eite Tiesinga, CODATA recommended values of the fundamental physical constants: 2022, Reviews of Modern Physics 97, 025002 (2025), doi:10.1103/RevModPhys.97.025002.

[4] A. H. Guth, The Inflationary Universe: The Quest for a New Theory of Cosmic Origins, Basic Books (1998), ISBN 978-0-201-32840-0.

[5] L. Smolin, The Case for Background Independence, in D. Rickles, S. French and J. T. Saatsi (eds.), The Structural Foundations of Quantum Gravity, Clarendon Press, Oxford (2006), pp. 196–239, doi:10.1093/acprof:oso/9780199269693.003.0007.

[6] Hermann Nicolai; Kasper Peeters; Marija Zamaklar, "Loop quantum gravity: an outside view", Classical and Quantum Gravity. 22 (19): R193 (2005), doi:10.1088/0264-9381/22/19/R01.

[7] J. A. Geiger, Measurement Quantization Describes the Physical Constants, International Journal of Theoretical and Mathematical Physics 11(1), 29–59 (2021), doi:10.5923/j.ijtmp.20211101.03. Publisher journal archive: http://www.sapub.org/Journal/articles.aspx?journalid=1070.

[8] 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.

[9] 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.

[10] 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.

[11] 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.

[12] 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.

[13] 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.

[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] 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.

[16] D. J. Fixsen, The Temperature of the Cosmic Microwave Background, The Astrophysical Journal 707, 916–920 (2009), arXiv:0911.1955, doi:10.1088/0004-637X/707/2/916.

[17] M. H. Abdullah, A. Klypin and G. Wilson, Cosmological Constraints on Ωm and σ8 from Cluster Abundances Using the GalWCat19 Optical-spectroscopic SDSS Catalog, The Astrophysical Journal 901, 90 (2020), arXiv:2002.11907, doi:10.3847/1538-4357/aba619.

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

[19] Liam Connor et al., A gas-rich cosmic web revealed by the partitioning of the missing baryons, Nature Astronomy 9, 1226–1239 (2025), doi:10.1038/s41550-025-02566-y.

[20] Kritti Sharma et al., Signatures of suppressed matter clustering revealed by fast radio bursts, Nature Astronomy (2026), published 8 September 2026, doi:10.1038/s41550-026-02957-9.

[21] Thibaut Louis et al., The Atacama Cosmology Telescope: DR6 power spectra, likelihoods and ΛCDM parameters, Journal of Cosmology and Astroparticle Physics 2025 (11), 062, doi:10.1088/1475-7516/2025/11/062; arXiv:2503.14452.

[22] 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.

[23] 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.

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

[25] P.-N. Chen, M.-T. Wang, Y.-K. Wang and S.-T. Yau, Conserved quantities in general relativity – the view from null infinity, in The Physics and Mathematics of Elliott Lieb, EMS Press (2022), pp. 211–224, doi:10.4171/90-1/9; author version arXiv:2204.04010.

[26] Gabriele Montefalcone, Benjamin Wallisch and Katherine Freese, Free-Streaming Neutrinos and Their Phase Shift in Current and Future CMB Power Spectra, Journal of Cosmology and Astroparticle Physics 2025 (08), 051, doi:10.1088/1475-7516/2025/08/051.

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

[28] E. Tiesinga, P. J. Mohr, D. B. Newell and B. N. Taylor, CODATA recommended values of the fundamental physical constants: 2018, Reviews of Modern Physics 93, 025010 (2021), doi:10.1103/RevModPhys.93.025010.