Resolving the Hubble Tension

68.259(19) and 73.508(20) km s⁻¹ Mpc⁻¹

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

Editorial Board Member for Scientific Reviews/Nature

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine the Measurement Quantization (MQ) interpretation of the Hubble tension through two mapping normalizations of a common inverse-age baseline. The Internal Frame supplies discrete count configurations; observable quantities arise in the System Frame through the Frames mapping. With the current MQ calibration and the measured cosmic microwave background temperature, the conditional rates are 68.259(19) and 73.508(20) km s⁻¹ Mpc⁻¹, with a baseline of 70.884(20) km s⁻¹ Mpc⁻¹. We derive the rate conversion, distinguish dimensionless coefficients from dimensional momentum and angular realizations, and state the epoch normalization needed for the age calculation. The quoted uncertainties propagate the stated inputs and exclude model discrepancy. Agreement with individual CMB and distance-ladder estimates is a consistency comparison. Establishing a resolution of the observational tension additionally requires an observable-level model and a joint analysis of the data.

Keywords: Hubble’s Constant, Hubble Tension, Expansion, Metric Expansion, Cepheids, CMB

1. INTRODUCTION

Measurement Quantization (MQ) [1] distinguishes the Internal Frame, the discrete count configuration domain, from the System Frame, its continuous relational realization. The Frames mapping connects these descriptions, and the Frames Principle assigns physical observables to the realized description. The Informativity differential is the count-dependent, non-Lorentz contraction associated with that mapping. It is distinct from the metric differential, the representation mismatch between discrete count structure and its continuous encoding. The rate comparisons in Table 1 motivate the count-dependent relations examined below; matching a calibration input does not constitute an independent experimental test.

The dimensionless circle constant π has its usual mathematical meaning throughout.

The realized fundamental measures are length l_f, mass m_f and time t_f, in m, kg and s. Internal Frame notation is representational; the physical units refer to System Frame realizations. Their dimensionless radial-separation, source-mass and elapsed-time counts are n_L,n_M,n_T, respectively. The exact vacuum light speed is c=299792458 m s^-1, with c=l_f/t_f. The distinct motion count n_Lm records the change in realized position of an observed mass, and n_Lc is the light-path count over the same interval. These roles remain distinct even where a boundary condition equates their numerical values (Appendices C–J). Conventional Planck length, mass and time are l_p,m_p,t_p (Appendix AT).

The derivations distinguish MQ relations among the fundamental measures from the conventional dimensional definitions of Planck units [2]. Recovering the same algebraic form is conditional on the coupling and realization used. Appendix AT gives that reduction, and Appendix A identifies the locations of the notation definitions.

The System Frame represents observable relational structure derived from the Internal Frame. It is not a separate external observer or a second independently measured universe. This distinction is retained when interpreting the two Hubble-rate normalizations.

MQ associates CMB and distance-ladder inference with different mapping normalizations. This is the model hypothesis tested by the rate relations below. The occurrence of a geometric factor in an observational formula alone does not derive that association.

The invariant dimensionless coefficient θ_si has momentum realization p_si in kg m s⁻¹ and angular realization Θ_si in rad. The adopted SI realization has p_si/(1 kg m s^-1)=θ_si=Θ_si/(1 rad). Equality of these numerical coefficients does not equate momentum and angle as physical quantities. The dimensional fundamental relation is l_fm_f=2p_sit_f. Upright G and ħ denote upper-count gravitational coupling and action; italic G(n_L) and ℏ(n_L) denote finite-count realizations. Conventional h is the exact SI Planck constant, and conventional ℏ=h/(2π) is used when the context explicitly concerns standard quantum mechanics.

Upright coupling and action symbols consistently identify the upper-count limit. A finite-count expression is evaluated at its stated separation, rather than replacing its count factor by the limiting value.

The proposed CMB and distance-ladder assignments are expressed as dimensionful rates only after normalizing the dimensionless mapping coefficients by a common age. Observational estimates remain dependent on their data and model assumptions.

The present universe age is A_U, in s, and its realized diameter is D_U, in m. The baseline rate H_b and its inverse-second representation H_f both represent 1/A_U. The labels H_sys and H_int identify the two MQ rate normalizations, while H₀ denotes an observationally inferred rate. The upright scalar H is dimensionless and is defined by the radius-to-light-travel-distance ratio in (63). One Julian year is 31557600 s, and one Mpc is 3.0856775814913673 × 10¹⁹ km.

For the triangles in Figs. 1 and 2, the positive dimensionless Diophantine realization residual (the geometric remainder) is Q_L=√(1+n_L²)-n_L. The letters a,b,c label geometric side lengths, with the local side label c distinct from vacuum light speed. The letters A,B,C label triangle vertices in Figs. 1–3.

Table 1 compares the conditional MQ rates with the SPT-3G base-ΛCDM inference and the Pantheon+ plus SH0ES flat dark-energy-model inference [3,4]. The latter allows a constant dark-energy equation-of-state parameter to vary. These are estimates under their respective fitted models, rather than measurements of two independently established MQ realizations. The broader observational literature provides additional comparisons [5–16].

Table 1. Conditional MQ rates and selected published inferences (km s⁻¹ Mpc⁻¹).

^a SPT-3G 2018 temperature and polarization spectra, base-ΛCDM [3].

^b Pantheon+ with SH0ES Cepheid distances and covariance, flat constant-equation-of-state model [4].

Figure 1 supplies the normalized triangle underlying the remainder construction. The hypotenuse of a triangle with integer leg counts need not be an integer; assigning physical significance to its remainder requires the MQ realization rule.

Section 3.1 develops the gravitational role of this geometric remainder. Its interpretation under successive updates is part of the MQ mapping model.

The gravitational and contraction relations offer distinct consequences of the adopted mapping. The contraction described here is not the Lorentz contraction discussed by Einstein [17]. Table 2 distinguishes the rate notation used in the comparisons.

The separate cosmological count relation and epoch normalization are stated in Section 3.5. Together with the mapping coefficients they give the two conditional Hubble rates. The baseline is their arithmetic midpoint.

Table 2 distinguishes the dimensionless scalar from the dimensional rates. Equation (1) gives the baseline implied by the age reconstructed in Section 3.5; the two mapping coefficients in Section 3.8 split this common baseline into the rates compared in Table 1.

Table 2. Dimensionless expansion scalar and dimensional Hubble-rate notation.

2. METHODS

The equations use the fundamental measures and Frames mapping of Measurement Quantization [1]. The derivation proceeds from the geometric remainder to the paired gravitational and action realizations, then to the fine-structure calibration, cosmological age reconstruction and two Hubble-rate normalizations. The uncertainty equality used in the boundary constructions is stated separately from the general quantum bound [18].

The count interpretation and its supporting measurement discussions are developed in the cited MQ literature [1,19–24]. Here each reduction retains its coupling realization and boundary assumptions. Appendix A identifies where the notation is introduced, and Appendices B–AT supply the intermediate algebra used in the main text.

The supplementary spreadsheets cited by the original MQ publication [1] document calculations within that framework. They are source calculation records, not independent experimental validations of the mapping premises. The present calculation uses the explicitly stated fine-structure and CMB-temperature inputs, exact SI constants, adopted epoch normalization and conditional uncertainty propagation. No survey likelihood is refitted here.

3. RESULTS

The results first establish the local relations required by the cosmological calculation. This order keeps the geometric count, its dimensional realization and the cosmological scaling distinct.

The MQ construction combines count relations with gravitational coupling, light-speed and uncertainty conditions [18] (Appendices C, D and I). Its interpretation uses the two frames and their mapping, with the fundamental measures as references. The boundary solutions in Appendices J and K are conditional on these relations.

The gravitational construction leads to the count-dependent coupling, while the Informativity differential supplies the associated finite-count normalization. Sections 3.1 and 3.2 relate that normalization to the gravitational and action pair. Table 3 retains the historical CODATA comparisons, and (16) identifies the common calibration used for the revised numerical results. These comparisons must be interpreted with the defining inputs and their correlations.

3.1. Gravitational Curvature

The remainder introduced with Fig. 1 is evaluated at radial separation r=n_Ll_f, in m. Rationalizing its positive root gives Q_L=1/(√(1+n_L²)+n_L), which avoids cancellation at large counts. Equation (2) is an acceleration scale. Multiplication by c/p_si gives the coupling-per-squared-distance expression in (3), and the finite-count definition of G(n_L) gives (4).

For radial counts of at least one, the remainder–count product is below one half. The count factor therefore remains explicit at finite separation. Appendix E derives its exact identities and upper-count limit.

Equation (4) is an exact identity for the MQ finite-count coupling. It does not replace a finite-count quantity with the upper-count constant. The latter is obtained only by the limit in (11)–(13).

The finite-count coupling and the upper-count constant have the same dimensions, m³ kg⁻¹ s⁻². Their distinction concerns the adopted MQ realization and separation. Classifying an experiment by apparatus alone does not establish its effective count separation.

The historical CODATA comparisons [25,26] remain comparisons with measured gravitational constants. Changes between recommended values do not establish count dependence. For a present comparison, CODATA 2022 recommends G=6.67430(15)×10^-11 m³ kg^-1 s^-2 [27]. If the upper-count value in (13) is identified with that same measured quantity, their central values differ by 1.48 times the CODATA standard uncertainty. This comparison neither determines an experimental count separation nor validates the finite-count mechanism.

Solving the finite-count coupling identity for fundamental length gives (7). The second form uses the upper-count coupling together with its explicit finite-count normalization. The values in (8)–(10) use the shared calibration stated after (15), with the detailed reduction collected in Appendix X.

Using the calibrated momentum realization and coupling gives the fundamental length, time and mass. Their uncertainties share the calibration dependence.

The momentum realization is half the product of fundamental mass and light speed. The independent-length proposals [28,29] provide context, but they are not substituted as additional calibration inputs here.

3.2. Length Contraction

The Informativity differential is the fractional departure of the finite-count normalization from unity. The normalization factor is twice the remainder–count product and tends to unity, while the differential tends to zero. Appendix E gives the exact limit.

Finite-count and upper-count relations must be evaluated with consistent realizations. Equations (12)–(15) show the gravitational and action pair. A numerical match obtained by using the measured constant in the calibration is a closure identity.

The electromagnetic demarcation used for the fine-structure calibration is specified in (21). It is a fixed interaction-count normalization, not a wavelength or a propagation distance; its noninteger calibration value does not assert an allowed fractional motion configuration. Independently inferred blackbody or charge demarcations require their own inputs and covariance information. They cannot be assigned the same value merely because their leading digits agree.

The upper-count limit is a mathematical limit. A count rate such as the reciprocal fundamental time has units s⁻¹ and is not a dimensionless maximum count. Appendix P keeps these quantities distinct.

At the electromagnetic demarcation, the finite-count action is identified with the conventional reduced Planck constant. Equation (14) makes this calibration explicit. Equation (15) then follows by taking the upper-count limit while keeping the same momentum realization and fundamental length.

The calibration input is the 2022 inverse fine-structure constant α_c^-1=137.035999177(21) [27]. The dimensionless quantities α_c, α_p and α_f denote the classical, Planck-like and fundamental fine-structure realizations. Electron mass m_e, in kg, and Bohr radius a₀, in m, enter through the correlated identity m_ea₀=ℏ/(α_cc). They are not additional independent inputs to that identity. Equation (16) recovers the fundamental measures from the calibrated action and fine-structure normalization. Parentheses denote standard uncertainties in the final quoted digits; shared dependencies are retained when propagating them.

In the historical tables, f(ℏ,G) labels values constructed from the stated action and gravitational inputs; it does not introduce a new physical function. Table 3 retains the cited historical comparison [25,26,30]. Those values describe their original input editions. They do not provide an independent explanation for changes between CODATA adjustments, and their historical uncertainties are not transferred to the current calculations.

Table 3. Matching CODATA values for G, ℏ, and the Planck Units with MQ Calculated Values. Adapted from [1].

^a CODATA publications for 2010, 2014 and 2018 [25,26,30].

Historical values in Table 3 retain their original calibration. Finite-count evaluations and upper-count limits must not be interchanged.

The fine-structure construction uses the dimensionless Diophantine realization residual Q_nθ. For a real argument x, ⌊x⌉ denotes rounding to the nearest integer, with ties to even. This residual expresses the metric differential in the fine-structure construction. Equations (17)–(20) first form the fundamental inverse constant and its rounding residual, then combine them into the Planck-like inverse constant. The selected branch remains fixed throughout the input-uncertainty interval considered here.

The dimensionless difference Δ_P-C is the Planck-like inverse fine-structure constant minus its classical counterpart. Equations (21) and (22) evaluate the electromagnetic separation and its remainder–count product. Equations (23)–(26) then apply the finite-count normalization to obtain the classical inverse constant. Appendix AC solves the same relation for the invariant coefficient, so substitution back into (26) is calibration closure.

The fine-structure calibration is solved on the branch with nearest integer 137. Correlations among the Bohr radius, electron mass and reduced Planck constant prevent treating the alternative formulas as independent measurements of the invariant. The angular experiments [31–33] concern X-ray geometry and do not directly determine a momentum in SI units.

Table 4 preserves the historical comparison with the 2018 input. Equation (26) instead reproduces the explicitly stated 2022 calibration. These comparisons must be interpreted with their respective input editions.

The fundamental fine-structure expression encodes Internal Frame count structure. The Planck-like and classical forms describe its System Frame realization, with the finite-count normalization included in the classical form.

Table 4. Fundamental, Planck-like and Classical Solutions to the Inverse Fine Structure Constant. Adapted from [1].

^a 2018 CODATA publication [30].

3.3. Physical Significance of θ_si

The dimensionless, momentum and angular realizations defined in the Introduction remain distinct throughout this section. Their common numerical coefficient is a convention of the stated MQ realization. The following equations separate the geometric arc relation from the dimensional action relation.

Equation (27) restates the dimensional fundamental relation as the starting point of the momentum derivation.

Dividing its mass–length product by twice the fundamental time gives p_si. The product l_fm_f has units kg m, while the quotient has units kg m s⁻¹. The dimensionless count relation (28) is considered separately.

Equation (28) follows from the epoch-ratio construction in Appendix AI. The motion and light-path counts refer to the same interval, so their ratio is dimensionless. Appendix AF supplies the normalized ellipse form used in that construction and states the additional physical mapping assumption.

The X-ray analysis [31–33] motivates the angular geometry considered in Appendix B, with Bell-state context supplied by [34,35]. It does not establish an equality between physical angle and momentum. We retain separate symbols for the realized quantities.

The geometric arc and dimensional momentum relations can be compared through the common numerical coefficient. This comparison does not assert a dimensional equivalence analogous to mass–energy conversion [17].

Let L, in m, denote the arc length on the circle of radius r. Its relation to the angular realization uses the radian numerical value, as written in (29). At fundamental radius, L=l_fθ_si; this geometric length is distinct from the momentum–length product that gives action.

The count-resolved gravitational relation gives the momentum realization independently of this arc construction.

Eliminating the finite-count coupling with its normalization gives the upper-count form in (32). Substitution of the coupling relation from (13) gives (33), and the action identity from (15) gives (34). This chain obtains the momentum realization without assigning momentum units to an angle.

Equation (35) makes the SI numerical-value convention explicit. Both ratios are dimensionless.

Equations (36)–(40) combine the momentum and upper-count action identities to recover the fundamental-length expression. Equation (41) retains the conventional Planck definition with its conventional constants. The common algebraic form requires consistent choices for both factors in the coupling–action product.

Use the dimensional fundamental relation and the upper-count action identity in the following reduction.

For conventional SI constants, the Planck-length definition is instead

The product of the finite-count gravitational and action realizations equals the product of their upper-count counterparts. Thus the length product is preserved when both quantities use the same count. Mixing one finite-count quantity with one upper-count quantity does not preserve that identity.

3.4. Observational Domains

The cosmological construction next uses dimensionless domain fractions. Their partition and closure relations are stated before the age–diameter relation, so the role of each additional MQ relation is explicit.

In conventional cosmological notation, Ω denotes a density parameter and ρ_c denotes critical mass density, in kg m⁻³. The MQ domain fractions below describe the stated geometric partition. In Tables 5 and 20, Ω_Λ,Ω_c,Ω_b and Ω_m denote conventional dark-energy, cold-dark-matter, baryonic and total-matter density parameters. Total matter includes the neutrino contribution in the cited Planck model; it is not exactly the sum of cold dark matter and baryons.

The dimensionless domains are the presently visible Ω_vis, observable Ω_obs, unobserved Ω_uobs, dark Ω_dk and total Ω_tot fractions. Observable means referenceable given unlimited elapsed time; unobserved is the observable remainder beyond the presently visible domain. The dark domain is its permanently non-referenceable complement. The fundamental fraction Ω_f is the normalization pivot in (46), rather than an additional disjoint part of the total partition. Equations (42)–(45) give the partition identities, and (46) supplies the additional closure.

The partition identities (42)–(45) do not by themselves determine the domain fractions. Equation (46) supplies the additional MQ closure. Combining it with the partition gives (47)–(51), as expanded in Appendix AM.

The resulting fractions are

Table 5 compares the MQ geometric partition with selected Planck density parameters [5]. Similar fractions do not establish equality of the underlying physical quantities. Baryon and cold-dark-matter percentages are central-value conversions of the published physical densities using the same fitted Hubble parameter; no separate uncertainties are assigned without the joint posterior.

Table 5. MQ geometric fractions and Planck TT,TE,EE+lowE density parameters [5]. Comparison structure adapted from [1]. Mass allocations use the current conditional age.

a Planck 2018 TT,TE,EE+lowE base-ΛCDM, 68% limits [5]. Converted baryon and cold-dark-matter entries give central values only.

The domain fractions give a geometric partition, not a complete energy-density or acoustic-spectrum model [24]. The following cosmological count relation is an additional MQ relation. The contraction identities in Appendices AF and AG are discussed with their assumptions [17].

3.5. Scaling the Fundamental Relation

The local fundamental relation is restated in (52). The additional cosmological relation is (53), with length count n_LU=D_U/l_f and time count n_TU=A_U/t_f. Substitution of these definitions and c=l_f/t_f gives (54)–(56). Independent multiplication of the local relation by arbitrary counts would not establish (53).

Diameter is a length and age is a time. The factor of light speed therefore remains in (56). Expressing diameter in light years and age in years absorbs its numerical conversion, but does not remove it from the physical equation. The resulting D_U is an MQ realized diameter; identifying it with a cosmological particle-horizon diameter would additionally require a propagation and expansion-history calculation.

Table 6 distinguishes conditional allocations from independently inferred quantities [1]. Here M_tot is total mass-equivalent content, M_vis its visible-domain allocation and M_CMB the CMB mass equivalent, all in kg. The fundamental mass-allocation rate M_acr, in kg s⁻¹, is derived in Appendix AN. It describes geometric mass availability rather than subsequent gravitational accretion. The quantum epoch is the pre-referenceable count regime, not ordinary elapsed System Frame time. Its normalization A_qe is a configuration extent represented in seconds under the adopted realization; A_ee is the realized expansionary-epoch duration, in s. Their relation is given in (64). The CMB monopole temperature is T_CMB, in K; radiation energy density is ρ, in J m⁻³; and the radiation constant is a, in J m⁻³ K⁻⁴. These definitions also apply to Tables 7 and 21.

Table 6. Conditional mass allocations, input closure and cosmological benchmarks. Historical comparison structure adapted from [1].

^a The CMB temperature is the measured input [36]; mass and reconstructed temperature are derived from the stated construction.

^b Reference [37] constrains cosmological parameters from cluster abundances; it does not independently measure the global visible mass listed here.

^c Planck TT,TE,EE+lowE baryon central-value conversion [5]. SPT-3G rate inference is cited separately in Table 1 [3].

Table 6 distinguishes derived allocations from observational inputs. Where the CMB temperature determines the age, reconstructing that temperature is a shared-input closure test. It is not an independent prediction of the temperature.

The local propagation relation is constrained by the measured constancy of vacuum light speed. The tests discussed in [38,39] concern that propagation property. They do not establish the separate cosmological count relation.

Constancy of the mapping coefficient is an assumption of this realization. CMB temperatures at different epochs must be interpreted through a temperature–redshift law and its observational model. The light-propagation test in [38] does not establish that law or an invariant cosmological mapping coefficient.

Atomic spectra constrain combinations of constants, with interpretation dependent on the transition sensitivities and physical model [40]. Equation (57) identifies the combination involving the electromagnetic finite-count action and the momentum realization. Absence of variation in the combination does not independently establish constancy of each constituent.

Equations (58)–(62) restate the cosmological count relation in dimensional and count form. The factor of light speed converts the age to a length before a dimensionless diameter ratio is formed.

The count ratio in (61) and (62) is cosmological. A light-path count equal to a time count describes propagation at light speed over a common interval and supplies a separate condition.

Equation (63) identifies the upright expansion scalar introduced in the Introduction with the invariant coefficient. Its denominator contains age times light speed, so the scalar is dimensionless.

The age reconstruction uses T_CMB=2.72548(57) K [36]. The radiation constant is obtained from exact SI h, c and the Boltzmann constant k_B, in J K⁻¹, as shown in Appendix AR [27]. The quantum-epoch reference value is A_qe,0=1 s, and R_qe(A_qe) denotes the mapped radius representing the quantum epoch, in m. The condition R_qe(A_qe,0)=0 fixes the integration constant in Appendix AP. The one-second factor specifies the numerical realization basis; it does not assign a clock interval intrinsic to the Internal Frame. With these assumptions, (64) supplies the epoch normalization used in (65). The comparison literature [41–44] provides context.

Equation (65) equates the radiation-law CMB mass equivalent to the epoch allocation in Appendices AN–AR. The diameter follows from (56), and (67) restates the same momentum–mass identity. These results retain their shared-input covariance. The adopted reference A_qe,0=1 s sets the absolute scale of this reconstruction; the geometric boundary does not independently derive that temporal normalization. The quoted uncertainties propagate the fine-structure and temperature inputs with the normalization held fixed. Uncertainty in the mapping assumptions and model discrepancy has not been quantified.

The baseline rate and its inverse-second representation describe the same inverse-age normalization. A megaparsec is converted to kilometers only when displaying the astronomical rate unit.

3.6. Why do Different Studies Get Different Results?

The fine-structure example illustrates a mapping normalization, but reuse of its calibration input does not independently validate the mapping. Its application to Hubble inference requires the explicit rate equations that follow.

Substitution into (70) closes the calibration used in Section 3.2.

The baseline inverse-age relation is the midpoint of the two MQ rate realizations. It is a model normalization, rather than a consequence of spatial flatness alone.

The dimensionless mapping offset is Δ_H=θ_si-π. The mathematical constant π is exact. This offset becomes a rate difference only after multiplication by the baseline rate divided by the invariant coefficient.

The two rate normalizations are model assignments for different inference procedures. They do not imply that the observational procedures measure intrinsic observables inside the Internal Frame. Observable quantities remain System Frame realizations under the Frames Principle.

Equation (72) gives the power-law flux-density conversion used as an illustration of inference conventions. Radio spectral modeling supplies the context [45]. A geometric factor in this conversion does not select the MQ normalization for a distance-ladder analysis.

For (72), L_ν is rest-frame spectral luminosity in W Hz⁻¹, S_obs observed flux density in W m⁻² Hz⁻¹, D_L luminosity distance in m, z redshift and α the power-law spectral index. This locally standard unsubscripted index is distinct from the fine-structure constants.

Equation (72) uses flux density proportional to frequency raised to the spectral index. Cepheid inference instead requires calibrated photometry, extinction and stellar-population treatment [4]. Its MQ assignment is the separate hypothesis stated in (81)–(89).

CMB inference compares the observed angular spectrum with a model, including its acoustic scale. The conventional spectrum normalization below does not itself determine a Hubble rate [3,5].

The nonnegative integer multipole index is l. The angular temperature-power coefficient C_l and its rescaled ordinate D_l have the same temperature-squared units, commonly μK². Equation (73) changes the conventional ordinate normalization and does not specify an MQ rate assignment.

The MQ coordinates x_j and y_j in (74)–(78) are dimensionless. The index j takes the domain labels dk, obs, uobs, f and vis; each x_j is an abscissa and each y_j is the corresponding domain-fraction ordinate. The dimensionless metric differential Δ_mdif is the coordinate-normalization factor defined by (79). These geometric coordinates follow the domain construction [24] and are distinct from the temperature-power ordinate in (73).

The coordinate-normalization factor appearing in (75)–(78) is

The observable, unobserved, fundamental and visible abscissae include the metric differential as a multiplicative factor. The dark abscissa omits it. The factor 100 expresses the percentage normalization while the underlying domain fractions remain numbers between zero and one.

The MQ domain coordinates provide a geometric comparison with the conventional density parameters in Section 3.4. Predicting the CMB spectrum additionally requires transfer functions and an observable likelihood connecting these coordinates to temperature and polarization correlations [3,5,46].

The denominator in the conventional spectrum ordinate is a normalization choice. It cannot establish the direction of a Hubble-rate shift; (83) and (88) specify the proposed MQ assignment. The two rate assignments can be summarized through the explicit relations in Section 3.8.

The calculation predicts a pair of rates conditional on the MQ mapping and epoch normalization. Input uncertainties are propagated, but model discrepancy and observational likelihood uncertainties are separate. The calculation alone does not reproduce the full analysis of any survey.

3.7. Nomenclature

The upright scalar expansion magnitude is dimensionless. The rates H_b,H_sys,H_int have inverse-time dimensions, conventionally reported in km s⁻¹ Mpc⁻¹. The label H_int denotes the complementary mapping normalization assigned to distance-ladder inference; it does not denote a directly observable rate intrinsic to the Internal Frame. Both observational inferences belong to the System Frame. An inferred observational rate retains the symbol H₀.

The geometric scalar and inverse-age rates must be kept separate from the measured recession history and from peculiar velocities.

3.8. Tension by Source

The dimensionless offset from (71) is restated in (80). Equations (81) and (82) multiply the common baseline by complementary dimensionless coefficients. Equations (83)–(85) then expose the resulting exact rate relation.

Multiply the mapping coefficients by the same baseline normalization. This operation preserves dimensions and gives the two rate assignments.

Substituting the mapping offset reduces the System Frame expression.

The complementary rate follows by adding twice the offset contribution to the System Frame rate. Equation (86) gives the rate ratio and midpoint identity, making the shared baseline explicit.

The baseline rate can be expressed through either age or diameter. The light-speed factor is required in the latter form.

The two rate realizations follow by multiplying the baseline by their respective dimensionless coefficients.

Equations (88) and (89) supply the two conditional rate values. Their comparison with published SPT-3G and Pantheon+ data [3,4] is made in Section 3.9, keeping the inference model and shared calibration explicit.

3.9. Multi-frame Comparisons and Relation to Observational Data

Friedmann evolution [47] requires an expansion history, density evolution and curvature prescription. A generic cosmological model does not identify H₀ with inverse age; the relation depends on that history. Here H_b=1/A_U is the MQ normalization. Equations (90)–(92) invert its rate assignments at the stated epoch. A test against CMB, baryon acoustic oscillation and distance data must also preserve the linked acoustic and distance constraints [46].

For a conditional comparison, assign an observed CMB rate to the System Frame model rate and invert (88). This supplies a model-implied age, rather than a new independent measurement of age.

We would then use the calculated value for A_U to resolve the corresponding values for H_int and H_b.

Assigning the published SPT-3G value H_sys=68.3±1.5 km s^-1 Mpc^-1 [3] to (90) gives a conditional age near 13.79 Gyr, with a first-order propagated uncertainty of 0.30 Gyr. Equation (91) gives a complementary rate near 73.6±1.6 km s^-1 Mpc^-1. This is a conversion of the same inference, not an independent prediction with a second observational uncertainty. Pantheon+ [4] and the temperature input [36] provide separate comparisons with their respective assumptions.

Table 1 uses 68.3±1.5 from SPT-3G base-ΛCDM [3] and 73.5±1.1 km s^-1 Mpc^-1 from Pantheon+ plus SH0ES with a flat, constant-equation-of-state dark-energy model [4]. The MQ values lie within the quoted individual uncertainties. This selected agreement does not establish a universal conversion between their likelihoods. Planck TT,TE,EE+lowE base-ΛCDM instead gives H₀=67.27±0.60 km s^-1 Mpc^-1 [5], emphasizing that the full dataset comparison remains necessary. No joint significance is inferred by counting the two correlated MQ rates as independent successes.

The tip of the red giant branch provides another distance-ladder calibration [48,49]. The preliminary 2024 report [50] was superseded by a published 2025 analysis [51]. Its HST-plus-JWST TRGB result is H₀=70.39±1.22 (stat)±1.33 (sys)±0.70 (SN) km s^-1 Mpc^-1, with NGC 4258 supplying the geometric zero point. Its proximity to the MQ midpoint is a numerical comparison, not evidence that source distance selects that normalization.

3.10. Alternative Approaches to a Measure of Metric Expansion

For a trial signal path of length d, let H_d denote the recession speed assigned by a uniform-expansion ansatz, Δ_d its accumulated distance increment over one light-travel time and Δ_t the corresponding delay. Here H_d has units m s⁻¹, d and Δ_d have units m, and Δ_t has units s. These are local path-estimate labels; H_d is a speed, not another Hubble rate. Equations (93)–(95) give this conditional estimate. A bound Solar System link additionally requires a spacetime solution, endpoint dynamics and a tracking observable; cosmological expansion cannot simply be inserted as a local recession law [52].

The clock performance in [53] cannot be compared directly with a delay in seconds. A test needs a time-transfer or frequency-transfer observable, its averaging interval, calibrated propagation effects and a noise budget. Equations (93)–(95) alone establish neither detectability nor a predicted anomaly.

4. DISCUSSION

Recovering familiar classical identities does not establish that all MQ mapping assumptions are already tested. Comparison with modified-gravity models [54] requires specified dynamics and distinct observable predictions. The count constructions in Appendices J and K retain their stated premises and do not independently prove discreteness. A covariant action and its observational consequences are not derived in this paper.

An action-based formulation, in the tradition of analytical mechanics [55], could provide such a dynamical test. Rewriting classical notation alone would not supply the missing physical derivation.

The Bohr-radius expression constrains a combination of the momentum realization, fundamental time, coupling and electron mass. It does not isolate the cosmological coefficient without assumptions about the remaining quantities.

A spectral test would therefore constrain this combination under its specified physical assumptions [40].

The SPT-3G comparison [3] and other rate comparisons in Section 3.9 are conditional consistency checks. The two MQ rates share the reconstructed age and calibration, so their errors are correlated. An observational resolution requires a joint likelihood, including survey calibration and the acoustic, distance and growth predictions of the proposed dynamics [46].

The quantum-epoch calculation retains the reference-time and mapping assumptions of Appendix AP. The geometric three-axis boundary does not establish an absolute physical time. Table 7 preserves historical benchmarks, while Appendix AQ identifies the temperature closure [1,24]. These calculations do not replace an inflationary or recombination history [56].

Table 7. Conditional MQ quantities and historical cosmological benchmarks. Adapted from [1].

^a A consolodated study of CMB temperature measurements prior to 2009 [36].

^b WMAP first temperature-spectrum peak multipole [57]. This published quantity differs from the MQ geometric coordinate. A matched observational inference for the MQ epoch normalization is not supplied.

^c Corresponds to about 411 photons/cm³ [58].

The reconstructed temperature in Table 7 is a closure result because that temperature supplied the age input. Its radiation density is the corresponding blackbody-law conversion, rather than an independent datum. The geometric coordinate x_dk has not been derived as a CMB peak multipole, and A_qe has not been identified with a fitted recombination age. Thus the juxtaposed numbers do not constitute like-for-like statistical tests. Equations (97)–(99) rewrite the same conditional epoch ratio.

The count-ratio relation has the normalized ellipse form discussed in Appendix AF. Substituting a dimensional momentum expression requires an explicit momentum-unit normalization, as retained in (97).

The motion-to-light-path count ratio refers to a common time interval. A count and a count rate are not interchangeable. The resulting dimensionless relation is

The speed v_u, in m s⁻¹, is the motion speed assigned to the epoch count ratio over a common interval. Substituting n_Lm/n_Lc=v_u/c into (99) gives (100). The resulting power has the algebraic form of the cube of a Lorentz factor; this algebra alone does not supply a spacetime metric.

The count reductions in Appendices C–J are conditional on minimum uncertainty and the specified coupling. In particular the upper-count gravitational boundary and the finite-count boundary give different minimum mass counts.

Under the stated assumptions, the paper establishes an internally checkable reduction and conditional numerical reconstruction. The absolute epoch normalization, the assignment of observed inference procedures to the two rates and an observable-level dynamical model remain to be independently tested. The numerical comparisons reported here do not by themselves resolve the observational Hubble tension.

5. APPENDICES

APPENDIX A: TERMS

Notation is introduced at first use and reused throughout the derivation. The Introduction establishes the fundamental measures, count roles, realizations, Hubble-rate notation and triangle labels. Section 3.1 develops the remainder relation, and Sections 3.2–3.5 introduce the calibration, domains and cosmological quantities. Appendix B supplies the angular and polarization notation. Later appendices state their local coordinate or realization roles before the associated equations.

APPENDIX B: ANGULAR DESCRIPTION OF θ_si

For Tables 8, 13 and 14, θ_p,θ_s,θ_i are pump, signal and idler angles, in rad. The branch label θ_max identifies the larger signal–idler angular solution. The polarization states H_s,V_s,H_i,V_i denote horizontal and vertical polarization of the signal or idler photon. The two terms in Table 8 are normalized by 1/√(2).

In the angular construction [31–33], θ is the common signal–idler angle, n_sx,n_sy,n_ix,n_iy,n_px,n_py are the normalized Cartesian signal, idler and pump components, and G⃗=(G_x,G_y) is the normalized reciprocal-lattice vector. This vector is distinct from gravitational coupling. The inverse trigonometric equations must be evaluated on the same physical branch and require their usual real domains. The component identities alone do not fix a numerical angle.

The vector construction concerns angles. The following momentum identity is separate; its numerical realization is related to the angular coefficient only through (35).

The following normalized count choice belongs to the upper-coupling boundary construction of Appendix J, not to the vector identity.

Table 8 presents the Shwartz and Harris theoretical proposal for polarization-entangled X-ray generation [31]. X-ray down-conversion and optical wave-mixing studies [32,33] provide experimental context; they do not constitute a measurement of the MQ invariant or validate the tabulated angular precision.

Table 8. Shwartz and Harris model of radian measures of the wave vectors of the pump, signal and idler for maximally entangled Bell state at the degenerate frequency. Reprinted from [1].

APPENDIX C: HEISENBERG’S UNCERTAINTY PRINCIPLE

The standard deviations σ_X, in m, and σ_P, in kg m s⁻¹, concern position and momentum distributions of a quantum state, with observables X and P. Equation (C.3) is the canonical quantum lower bound [18,59]; (C.4) imposes a minimum-uncertainty state as an additional assumption. The positive square roots in (C.5) represent the corresponding standard-deviation scales. Substitution of these scales and conventional Planck identities (C.1)–(C.2) gives the conditional count relation (C.7); it does not prove a universal discreteness rule.

APPENDIX D: DISCRETE GRAVITY

Figure 2 illustrates the radial separation count. Equations (D.1)–(D.3) retain the finite-count coupling and momentum realization.

APPENDIX E: LENGTH CONTRACTION (INFORMATIVITY DIFFERENTIAL)

Appendix E uses the geometric side-length labels introduced with Fig. 1. Its dimensionless side counts are n_La=1,n_Lb=n_L and n_Lc=n_L+Q_L. Here n_Lc is the hypotenuse count of this local triangle, rather than a new assignment of the light-path count in the motion relations. Equation (E.3) substitutes these counts into (E.2); solving its positive branch gives (E.4)–(E.6).

Squaring the positive-root relation and canceling the radial-count square gives (E.8) and (E.9). Equations (E.10) and (E.11) are equivalent rearrangements that expose the bounded remainder–count product.

As the radial count tends to infinity, the positive remainder tends to zero. Equation (E.10) therefore gives the exact limit

APPENDIX F: UPPER-COUNT SOLUTION FOR GRAVITATIONAL COUPLING

The source mass is M=n_Mm_f and v_e denotes the escape speed. The upper-count coupling is used explicitly in Appendices F and H; Appendix I instead keeps its finite-count form.

At the selected upper-coupling boundary, impose light-speed escape, radial count one, mass count one half and equal unit light-path and time counts. Substitution into (F.2), followed by (F.1), gives (F.3)–(F.4).

APPENDIX G: FINITE-COUNT SOLUTION FOR GRAVITATIONAL COUPLING

Equation (G.1) retains the finite-count coupling from (D.3). Substituting the dimensional fundamental relation yields its count-factor form. Taking the limit from Appendix E gives the upper-count coupling in (G.2).

APPENDIX H: ESCAPE VELOCITY WITH UPPER-COUNT G

Substituting the upper-count coupling from (G.2) into the escape-speed relation gives (H.1)–(H.3). The strict subluminal condition gives the mass-to-radius inequality.

APPENDIX I: ESCAPE VELOCITY WITH FINITE-COUNT G

The finite-count counterpart uses (G.1). Equations (I.2) and (I.3) retain its remainder factor, so their bound must not be replaced by the upper-count form at finite separation.

APPENDIX J: SOLVING THE MINIMUM MEASURE COUNTS

For the upper-coupling boundary only, combine the minimum-uncertainty equality with light-speed motion and radial count twice the mass count. These are assumptions of the construction, not consequences of the uncertainty inequality alone.

Within this upper-coupling construction, mass count one half fixes radial count one. These values are normalized reference counts, not a proof that every reference count must be a positive integer. Unit time count is a further elementary normalization. Equation (J.9) records the distinct finite-coupling result.

The finite-coupling result (J.9) uses the finite-count action and coupling together. It differs from the upper-coupling value one half. These alternative normalizations must not be combined.

APPENDIX K: DISCRETENESS OF MEASURE

The examples illustrate the assumed discreteness of realized length. The fractional differences in (K.1)–(K.3) contrast with the integer multiple in (K.4); the arithmetic itself does not independently establish discreteness.

Not measurable

Not measurable

Measurable

APPENDIX L: MEASUREMENT FRAMEWORKS

Figure 3 illustrates the relational construction used by the Frames mapping. The geometric segment AC joins the observer vertex A to the target vertex C; B is the third triangle vertex. The fundamental measures define the reference for the Internal Frame configuration, while the realized geometry belongs to the System Frame.

The reference in Fig. 3 specifies the fundamental measures used by the Internal Frame count configuration.

Observable geometry is the System Frame realization under the Frames mapping. The reference convention is not a third independent observable domain.

APPENDIX M: PRECISE DESCRIPTION OF ℏ

Substitution of the upper-count length identity from (40) gives

Equations (M.2)–(M.4) retain the finite-count gravitational normalization and dimensional momentum relation. Substitution of the finite-count action definition gives (M.5), which can be solved for (M.6). Its upper-count limit yields (M.7). Table 9 retains the historical action comparison.

Table 9. Historical finite-count and upper-count action values. Reprinted from [1]. These retain the original source calibration.

APPENDIX N: PRECISE DESCRIPTION OF G

Solving the finite-count identity (D.3) gives (N.1)–(N.3). Appendix E gives the limit (N.4); Table 10 retains the historical gravitational comparison.

Table 10. Historical finite-count and upper-count gravitational comparison. Reprinted from [1].

^{a The Bohr radius, as defined in Section 3.2.}

^b Exceeds the physical significance of the source measures.

APPENDIX O: SOLUTIONS TO G USING THE CODATA

Table 11 collects calculations of G from conventional Planck-unit expressions using the 2010–2018 CODATA inputs.

Table 11. Calculations of G using Planck’s Unit Expressions. Reprinted from [1].

^a Planck’s Unit expressions are resolved using those values provided in CODATA publications for 2010, 2014 and 2018 [25,26,30].

Table 12. Matching CODATA values for G, ℏ, and the Planck Units with MQ Calculated Values. Reprinted from [1].

^a CODATA publications for 2010, 2014 and 2018 [25,26,30].

Table 12 retains the demarcations and inputs of its source edition.

The historical arithmetic mean is G_avg, in m³ kg⁻¹ s⁻². Equation (O.1) retains it as a numerical comparison of the two printed inputs. Averaging those values does not establish a count-dependent mechanism or supply their covariance.

APPENDIX P: BOUNDS TO MEASURE

A rate of count accumulation has inverse-time units. The light-path count rate and the subluminal mass-to-radius bound are therefore written separately.

Equation (P.3) gives a count rate. The following upper-coupling mass-to-radius bound is dimensionless.

The ratio twice mass count over radial count equals squared escape speed over squared light speed only under the upper-coupling substitution.

APPENDIX Q: ELECTROMAGNETIC DESCRIPTION OF θ_si

The finite-count and upper-count identities reduce to the same momentum realization when coupling and action are paired consistently. Equation (Q.1) collects this reduction. Table 13 retains the historical angular comparison using the notation introduced in Appendix B.

Table 13. MQ predicted radian measures of the wave vectors of the pump, signal, and idler for the maximally entangled Bell state of X-rays at the degenerate frequency. ^a Reprinted from [1].

a Historical angular comparison from [1], motivated by the theoretical proposal [31]. References [32,33] establish related nonlinear X-ray processes, not an independent MQ angle measurement.

APPENDIX R: MOMENTUM DESCRIPTION OF θ_si

Equation (R.1) collects the dimensional momentum identity from (32)–(34). Table 14 repeats the historical angular comparison, which concerns the angular realization rather than momentum in SI units.