The Cosmological Constant from MQ Geometry

The cosmological constant Λ specifies a curvature contribution in Einstein’s gravitational field equation. Measurement Quantization (MQ) proposes that its value at a specified realization can be calculated from a geometric domain fraction and the realized expansion frequency. This connects a dimensionless partition of the universe to a physical curvature measured in inverse square metres. The calculation developed here uses the realized System Frame expansion rate H_sys.

The central question is why this coefficient has its observed scale. General relativity permits a cosmological term but does not, by itself, determine its numerical value. Quantum vacuum contributions introduce an additional difficulty because constant contributions to a matter Lagrangian can gravitate [6]. MQ approaches the value through discrete measure and its realized geometry [4,5]. Its proposed derivation must therefore be assessed at two levels. The geometric construction must supply a dimensionally complete coefficient, and the resulting cosmology must reproduce observations through a consistent dynamical solution.

From discrete counts to realized geometry

MQ distinguishes the Internal Frame, containing dimensionless count relations, from the System Frame, in which their relational geometry is realized as physical observables. The Frames mapping connects these descriptions subject to closure and consistency. This distinction is developed in Frames of Reference. The essential proposal is that the geometric relations underlying measure constrain physical quantities that would otherwise enter a continuum description as separate parameters.

The fundamental measures l_f, m_f and t_f are the MQ length, mass and time realizations in m, kg and s, respectively ([4], Appxs. W, BM). We first calculate the dimensionless coefficient θ_si from the 2022 CODATA classical fine-structure constant α_c [13], then evaluate the measures in the order of Appendix BM of [4]. The momentum realization

p_{s i} = l_{f} m_{f} / (2 t_{f})

has units kg m s^-1; its numerical coefficient equals θ_si in the adopted realization. The resulting measures are

m_{f} = 2.1764325406 (17) \times 10^{- 8}

kg,

l_{f} = 1.61619991148 (12) \times 10^{- 35}

m and

t_{f} = l_{f} / c = 5.39106261131 (41) \times 10^{- 44}

s. These are MQ calculations, rather than CODATA recommendations.

The local metric is reconstructed from a smooth, nondegenerate two-point realized separation [14], Sec. 3.5. We write g_μν for the dimensionless System Frame metric; μ,ν label its four spacetime coordinates, measured in length units. The reconstruction determines local geometry, while dynamics require the MQ Action, the common count-based functional, and its admissible System Frame variations [5,14]. Neither this local reconstruction nor the coefficient calculation selects a complete homogeneous cosmological solution. An age–redshift comparison therefore requires a separately specified System Frame expansion history; a complete MQ background and perturbation calculation remains a distinct test [14], Secs. 5.5, 5.8.

A local metric describes how realized separations vary from point to point. It supplies the geometric object from which connection and curvature can be constructed. A cosmological coefficient is a further ingredient in the gravitational equation. Consequently, identifying the origin of the metric and calculating Λ are connected tasks, but neither calculation alone selects the universe’s complete spacetime history. The distinction matters whenever a numerical prediction is described as a first-principles result. The premises of the count construction, the calibration inputs and the continuum identification must remain visible.

The MQ summary paper [4] compares frame-dependent forms of G and the reduced Planck constant ℏ with earlier CODATA editions [15,16,17]. Here h is Planck’s constant in J s; conventional

ℏ = h / (2 π)

is exact in the International System of Units (SI). The electromagnetic MQ expression reproduces it by the calibration identity, so this is not an independent test of discreteness. The current upper-count G result differs from the CODATA 2022 recommendation by about 1.48 standard uncertainties, indicating compatibility at that precision rather than proof of the proposed contraction mechanism.

MQ distinguishes the upper count limit from the Fine Structure demarcation ([4], Appx. AB, AE, AN). Upright fundamental measures denote the upper-count realization; electromagnetic-demarcation quantities remain italic. The dimensionless separation count at this demarcation is

n_{L} = 276 / θ_{si} = 84.6005456716 (65)

intervals of l_f. Its value depends on the source and measurement context. Here Fine Structure demarcation names the electromagnetic count demarcation; it is not an independently fixed conventional distance.

The distinction between a scalar and a dimensional realization also matters. The coefficient θ_si is dimensionless. The momentum realization p_si carries momentum units even when the two have equal numerical magnitudes under the adopted convention. Substituting one for the other without the dimensional realization would conceal the units needed by the gravitational coupling. The fundamental measures provide those units throughout the construction.

The geometric fraction and the density comparison

Physical curvature Λ has units m^-2. Its dimensionless normalization is

λ = Λ {l_{f}}^{2}

, where l_f is the MQ fundamental length in metres. The effective mass density ρ_Λ follows the continuum mapping in [5], Eq. (E.10), with c the speed of light in m s^-1 and G the MQ upper-count gravitational coupling in m³ kg^-1 s^-2. Multiplication by c² gives the corresponding energy density in J m^-3.

ρ_{Λ} = \frac{Λ c^{2}}{8 π G} .

The relationship in the density conversion is dimensionally complete: curvature contributes m^-2, c² contributes m² s^-2, and division by G contributes kg s² m^-3. The resulting unit is kg m^-3. No conversion from a square-root inverse-length scale is required.

The density conversion combines two quantities with proposed geometric origins in MQ: the curvature coefficient determined by the scalar fraction and expansion rate, and the coupling determined by the realized count structure. Its physical interpretation follows from their embedding in the continuum field equation, not from a new primitive density postulate.

The physical interpretation of this density relation should be kept distinct from its algebra. Curvature and effective density are alternative descriptions within a specified continuum equation. Converting between them does not identify the microscopic nature of the source. In MQ, the proposed origin of Λ is the realized geometry, while the density is its continuum representation. The question is therefore what fixes the curvature before that conversion is made.

The MQ gravitational construction [5], the corresponding relation, (52)–(53) and (E.2)–(E.7), relates the geometric domain fraction Ω_dk to the frame-resolved expansion rates. The continuum fraction Ω_Λ is the ratio of effective mass density to critical mass density ρ_c. We use the System Frame rate H_sys in km s^-1 Mpc^-1 for the realized curvature. The factor

K = 3.08567758149 \times 10^{19}

km Mpc^-1 converts this rate to the physical frequency

\frac{H_{s y s}}{K}

in s^-1. In conventional cosmology H₀ denotes the present expansion rate. The continuum density identification, applied here to the System Frame, is

Ω_{Λ} = \frac{ρ_{Λ}}{ρ_{c}} = \frac{Λ_{s y s} c^{2}}{3 {(\frac{H_{s y s}}{K})}^{2}} .

The critical mass density is

ρ_{c} = \frac{3 {(\frac{H_{s y s}}{K})}^{2}}{8 π G}

. The continuum density-fraction relation defines the continuum ratio used for comparison. the System Frame curvature relation applies the MQ identification of that ratio with the geometric fraction given below. Its numerical value uses the System Frame rate in Table 1.

The MQ scalar fraction is given by the domain-fraction relation, using θ_si=3.26239030504(25) from the 2022 CODATA inverse fine-structure input [13]. The MQ gravitational construction [5], the corresponding relation and Appx. E, identifies this geometric fraction with Ω_Λ for the continuum comparison. We retain the label Ω_dk for its MQ origin and state that identification as a model assumption. It is not a separate experimental measurement of a dark-energy fraction.

Ω_{d k} = \frac{θ_{s i}^{2} - 2}{θ_{s i}^{2} + 2} = 68.3624161224 (41) % .

The domain fraction Ω_dk describes the complementary part of the MQ domain beyond physical referenceability under the stated mapping. The conventional Ω_Λ is a ratio of effective density to critical density. Equating them is the explicit MQ identification that connects the geometric construction to the continuum calculation [5]. Their meanings are not made identical merely by a numerical match. The complementary domain construction is developed in Cosmological Domain Partitions.

Why the System Frame rate enters Lambda

The System Frame age A_U is a physical elapsed age, expressed in seconds in symbolic rate equations and in Gyr where numerical values are stated. The frame rates H_sys and H_int follow [5], Eqs. (E.4–E.5), and [18]. A numerical age in Gyr requires the Julian-year factor 3.15576×10⁷ s yr^-1 and 10⁹ yr Gyr^-1 in the denominator.

H_{s y s} = \frac{π K}{θ_{s i} A_{U}} = 68.259 (19) km s^{- 1} Mp c^{- 1} .

H_{i n t} = \frac{(2 θ_{s i} - π) K}{θ_{s i} A_{U}} = 73.508 (20) km s^{- 1} Mp c^{- 1} .

Within this interpretation, H_sys is selected because the calculated curvature belongs to realized physical observables. H_int expresses the internal relation at the same adopted diameter. Averaging the two rates produces a convenient baseline, but averaging does not create a third physical frame or an independently measured expansion rate. A midpoint should therefore not replace the realized rate in the adopted Lambda calculation. The frame interpretation and its observational comparisons are developed in the published Hubble-tension treatment [18].

The arithmetic midpoint of the two frame rates equals the baseline metric rate [5]. It is distinct from the upright dimensionless radial coefficient used in Appendix B. This midpoint is a comparison quantity, whereas the physical curvature calculation uses the System Frame frequency. Explicitly,

\frac{H_{s y s} + H_{i n t}}{2} = \frac{K}{A_{U}} = 70.884 (20) km s^{- 1} Mp c^{- 1} .

\frac{H_{s y s}}{K} = \frac{π}{θ_{s i} A_{U}} = 2.21212 (62) \times 1 0^{- 18} s^{- 1} .

Λ_{s y s} = \frac{3}{c^{2}} (\frac{θ_{s i}^{2} - 2}{θ_{s i}^{2} + 2}) {(\frac{H_{s y s}}{K})}^{2} = 1.11665 (62) \times 1 0^{- 52} m^{- 2} .

The difference is amplified because curvature depends on the square of the rate. Table 1 retains the preprint’s rate comparisons. Its Planck row is a hybrid calculation that inserts a published expansion rate into the MQ fraction; it must not be read as Planck’s own inferred density. The other rows share inputs and therefore are not independent confirmations of the geometric construction.

Table 1. Expansion rates, normalized curvature and effective continuum mass density. Uncertainties are one standard uncertainty. The last column uses the density conversion, with the same MQ upper-count gravitational coupling in every row.

Frame or comparison	H (km s⁻¹ Mpc⁻¹)	H/K (10⁻¹⁸ s⁻¹)	λ (10⁻¹²²)	ρ_Λ (10⁻²⁷ kg m⁻³)
Planck rate in MQ formula	67.66 ± 0.42	2.193(14)	2.866(36)	5.879(73)
System Frame	68.259(19)	2.21212(62)	2.9168(16)	5.9831(33)
Arithmetic midpoint	70.884(20)	2.29718(64)	3.1454(18)	6.4521(36)
Internal Frame	73.508(20)	2.38224(66)	3.3827(19)	6.9387(39)

^a Planck 2018 TT,TE,EE+lowE+lensing+BAO (temperature and E-mode polarization spectra, low-multipole polarization, lensing and BAO) gives 67.66±0.42 km s^-1 Mpc^-1 ([1], Table 2). This row applies that input to the MQ domain fraction and current fundamental measures; its calculated density is not a published Planck vacuum-density result.

^b System Frame rate at fixed diameter, with dimensionless radial coefficient π ([5], Eq. (E.4)).

^c Arithmetic midpoint of the two frame rates, equal to the baseline metric rate

\frac{K}{A_{U}}

([4], Eq. (BA.1)). It has no separate observational identification in this calculation.

^d Internal Frame rate at fixed diameter, with dimensionless radial coefficient 2θ_si-π ([5], Eq. (E.5)).

The System Frame physical curvature is

Λ_{s y s} = 1.11665 (62) \times 10^{- 52} m^{- 2}

. Its the density conversion density is

ρ_{Λ} = 5.9831 (33) \times 10^{- 27}

kg m^-3. Substitution of the midpoint and Internal Frame rates into the same geometric formula gives 1.20417(68)×10^-52 m^-2 and 1.29499(70)×10^-52 m^-2, respectively. Their corresponding densities are 6.4521(36)×10^-27 kg m^-3 and 6.9387(39)×10^-27 kg m^-3. The adopted physical result uses H_sys, because the System Frame contains the realized observables in the stated MQ interpretation. The association of that rate with CMB inference is a model assignment, rather than an observational demonstration that the two MQ frames are separately measured.

The expanded expression and its physical scale

Substitution removes the intermediate observational rate from the final expression and exposes the complete dependence on the geometric coefficient and the System Frame age. The speed of light remains explicit so that the units of curvature can be checked directly. No replacement coefficient is needed to hide the combination of symbols.

Λ_{s y s} = \frac{3}{c^{2}} (\frac{θ_{s i}^{2} - 2}{θ_{s i}^{2} + 2}) {(\frac{π}{θ_{s i} A_{U}})}^{2} = 1.11665 (62) \times 1 0^{- 52} m^{- 2} .

The first parenthesis is the geometric domain fraction. The second is the realized expansion frequency written in terms of the age and the same geometric coefficient. The expression shows precisely how geometry enters this proposed value of Λ. It does not identify all spacetime curvature with Λ, nor does it determine an evolving metric simply by providing this coefficient.

The measured CMB temperature T_CMB=2.72548(57) K [19] controls most uncertainty in the age-dependent results; A_U=13.7943(38) Gyr. Shared dependencies are propagated consistently. The electron mass m_e and Bohr radius a₀ satisfy the calibration identity

m_{e} a_{0} = ℏ / (α_{c} c)

and are not independent inputs here. The exact SI constants h, c, elementary charge e and Boltzmann constant k_B contribute no measurement uncertainty.

The normalized curvatures follow by multiplying the physical expressions by

{l_{f}}^{2}

, as in the MQ gravitational construction [5], Eqs. (E.8)–(E.9). Restoring this length factor is essential when expressing a physical frequency and physical age: the normalized coefficients are dimensionless.

Λ_{s y s} l_{f}^{2} = 3 Ω_{d k} {(\frac{H_{s y s} t_{f}}{K})}^{2} = 2.9168 (16) \times 1 0^{- 122} .

Λ_{i n t} l_{f}^{2} = 3 Ω_{d k} {(\frac{H_{i n t} t_{f}}{K})}^{2} = 3.3827 (19) \times 1 0^{- 122} .

For either rate substitution, physical curvature has dimensions m^-2, while multiplication by

{l_{f}}^{2}

gives a dimensionless normalization. Applying the density conversion then yields mass density. The same source chain closes the units of curvature, normalized curvature and effective density. Only the System Frame substitution is adopted as the physical curvature in this discussion.

Λ_{s y s} l_{f}^{2} = 3 (\frac{θ_{s i}^{2} - 2}{θ_{s i}^{2} + 2}) {(\frac{π t_{f}}{θ_{s i} A_{U}})}^{2} = 2.9168 (16) \times 1 0^{- 122} .

The geometric origin of the gravitational coupling

The coupling has a geometric origin within MQ [5], Sec. 2.7. The dimensionless Diophantine realization residual Q_L enters the realized hypotenuse count n_L+Q_L. The dimensionless normalization factor 2Q_Ln_L modifies the finite-count gravitational coupling and tends to unity at the upper count limit. The Informativity differential is its finite-count departure 1-2Q_Ln_L, which tends to zero in that limit ([14], Sec. 3.6). This count-dependent, non-Lorentz contraction is distinct from curvature. In the upper count limit

G = c^{3} t_{f} / m_{f} = c^{2} l_{f} / m_{f}

. The effective density therefore combines curvature with a coupling derived from count geometry.

The underlying right-triangle relation takes one reference interval and a separation of n_L intervals. The realized hypotenuse exceeds that separation by Q_L. Thus the residual is fixed by the geometry rather than added as a freely fitted gravitational parameter [4,14].

Q_{L} + n_{L} = \sqrt{1 + n_{L}^{2}}

Rationalizing the difference gives a form that exposes the upper-count behavior without subtracting two nearly equal numbers. The product approaches one half as the separation count grows, so twice that product approaches unity. The finite-count departure therefore vanishes at the upper-count limit, while the dimensional coupling remains finite [4,14].

Q_{L} n_{L} = \frac{n_{L}}{\sqrt{1 + n_{L}^{2}} + n_{L}}, lim_{n_{L} \to \infty} Q_{L} n_{L} = \frac{1}{2}

Reduction to fundamental measures does not remove physical dimensions: G retains m³ kg^-1 s^-2. Substitution into the density conversion gives

ρ_{Λ} = Λ m_{f} / (8 π l_{f})

, with kg m^-3. This direct reduction is the dimensional bridge from the geometric MQ construction to its continuum density representation. It follows algebraically from the density conversion, the upper-count coupling and

c = l_{f} / t_{f}

The Diophantine realization residual describes a finite-count difference in the realized right-triangle relation. Its normalization factor and the Informativity differential distinguish finite-count measure from its upper-count limit. They are not interchangeable names for spacetime curvature. Keeping that distinction prevents a finite-count correction to the coupling from being mistaken for a separate cosmological source. Further treatment is available in Informativity Differential.

The field coupling k relates stress-energy to curvature and has units s² kg⁻¹ m⁻¹. It is distinct from the unit-conversion factor K and Boltzmann’s constant k_B. Its reduction to the fundamental measures keeps the dimensional momentum realization explicit.

The gravitational coupling traces to the MQ gravitational construction [5], Eq. (58). To keep its dimensions explicit, we use the momentum realization p_si in place of the scalar θ_si in the final reduction:

k = \frac{8 π G}{c^{4}} = \frac{8 π t_{f}^{2}}{l_{f} m_{f}} = \frac{4 π t_{f}}{p_{s i}} .

This chain makes the dimensional content of the geometric coupling explicit. It uses the upper-count value of G consistently with Table 1. A finite-count realization retains its residual-count factor and must use that same factor in both the field coupling and the effective density. The reduction changes the origin assigned to the coupling within MQ, while preserving its units and its role in the continuum equation.

Comparison with community calculations

Prat et al. [2] report a combined-data mass density

(6.03 \pm 0.13) \times 10^{- 27}

kg m^-3. Their low-redshift analysis gives

({6.33}_{- 0.29}^{+ 0.37}) \times 10^{- 27}

kg m^-3 with a tip of the red giant branch (TRGB) calibration and

({7.03}_{- 0.31}^{+ 0.27}) \times 10^{- 27}

kg m^-3 with a Cepheid calibration. These are posterior inferences under an assumed flat cosmological-constant plus cold-dark-matter (ΛCDM) model; differing calibrations alter the likelihood and are not direct measurements of different vacuum substances.

The interpretation must be tested through the observables from which cosmological parameters are inferred. The Dark Energy Spectroscopic Instrument second data release (DESI DR2) measures distance ratios using baryon acoustic oscillations (BAO) and constrains dark-energy models in combination with other datasets [3]. Such model comparisons probe expansion dynamics; they do not directly identify the microscopic origin of the effective source. A numerical curvature value at one epoch is consequently a necessary benchmark, rather than a complete test of the MQ cosmology.

Table 2 compares published flat ΛCDM inferences with the MQ result. A CMB-derived Hubble constant is an inference of today’s expansion rate under the fitted cosmological model, not the expansion rate at recombination. In the community calculation, the present density fraction and Hubble constant must be taken from the same joint fit. The conversion is

Λ = \frac{3}{c^{2}} Ω_{Λ} {(\frac{H_{0}}{K})}^{2} = \frac{8 π G ρ_{Λ}}{c^{2}} .

Table 2. Community cosmological-constant calculations reconstructed from published parameters. Hubble rates are in km s^-1 Mpc^-1; the last column is in 10^-52 m^-2. These values are conditional on flat ΛCDM. The two Planck rows give central values only.

Dataset and reference	H₀	Ω_Λ or density input	Λ (10^-52 m^-2)
Planck 2018 CMB + lensing [1]	67.36	Ω_Λ = 0.6847	1.0891
Planck 2018 + BAO [1]	67.66	Ω_Λ = 0.6889	1.1056
DESI DR2 + CMB [3]	68.17 ± 0.28	Ω_Λ = 0.6973	1.136 ± 0.015
Prat et al., combined data [2]	Joint density inference	ρ_Λ = (6.03 ± 0.13) × 10^-27 kg m^-3	1.125 ± 0.024

The Planck values use Table 2 of [1]. For DESI DR2 + CMB, the corresponding relation of [3] gives the matter fraction Ω_m=0.3027±0.0036 and a correlation of −0.975 between it and the fitted Hubble constant. The listed curvature uncertainty is a first-order propagation of that covariance, with the present radiation fraction neglected below the displayed precision; it is not a separately published Λ posterior. The Prat row converts the combined-data mass density in Table 2 of [2] using the CODATA 2022 conventional gravitational constant 6.67430×10^-11 m³ kg^-1 s^-2 [13]. The uncertainty in that constant is negligible at the displayed density precision.

The System Frame result lies within the range of these community central values. This agreement is a benchmark at the stated realization and does not replace a joint MQ fit to the expansion and perturbation observables. Changing the inferred Hubble constant alone while holding the density fraction fixed does not reproduce a new community fit. For pure de Sitter expansion the fraction is unity; that special case does not describe the present matter-containing universe assumed in Table 2.

Table 2 compares conditional inferences within flat ΛCDM, the cosmological-constant plus cold-dark-matter model. The comparison must use the expansion rate and density fraction from the same fit, including their covariance when an uncertainty is calculated. Replacing the fitted rate with a value from another analysis while retaining the original fraction would create a new hybrid estimate. It would not reproduce either analysis. The present MQ value falls among these benchmark values; establishing explanatory success requires matching the observables from which they were inferred.

From a coefficient to a cosmological solution

The physical curvature coefficient enters Einstein’s field equation [20] as

R_{μ ν} - \frac{1}{2} R g_{μ ν} + Λ_{s y s} g_{μ ν} = k T_{μ ν} .

Here R_μν is the Ricci tensor and R its scalar contraction, both with curvature dimensions m^-2 in length coordinates. The tensor T_μν is the realized energy stress tensor, with components in J m^-3 in a normalized local frame. We use the original Einstein-equation symbol

k = 8 π G / c^{4}

for the gravitational coupling, with units s² kg^-1 m^-1; it is distinct from both conversion factor K and Boltzmann constant k_B. The metric is the System Frame metric defined above. The physical field equation uses physical Λ;

λ = Λ {l_{f}}^{2}

is dimensionless.

With fixed model inputs, the curvature coefficient calculated from the expansion frequency can evolve with age. A time-dependent physical cosmological coefficient also constrains stress-energy conservation through the field equation. Spatial flatness does not imply a vanishing Ricci tensor, so the separate curvature and source evolution cannot be inferred merely by fixing spatial curvature.

For a fixed fundamental time t_f, the normalized System Frame coefficient could remain constant during a change of H_sys only if the geometric domain fraction compensated that change. The following expressions hold θ_si fixed. They evaluate the stated realization and do not establish that the same inverse-age prescription yields a time-independent cosmological constant throughout one evolving spacetime.

The dimensionless coefficient θ_si=3.26239030504(25) is calibrated from the fine-structure input [13]. Its momentum realization is

p_{si} = l_{f} m_{f} / (2 t_{f})

([4], Appx. S), with units kg m s^-1. The equality of numerical magnitudes refers to the adopted MQ realization and does not equate a dimensionless scalar with a dimensional momentum. For a fixed radial coefficient, the physical diameter D_U and age A_U obey the following light-time relation ([4], Appx. AZ).

\frac{D_{U}}{c} = 2 θ_{s i} A_{U} .

Here the flat, constant-radial-coefficient construction is a model assumption. The observable motion of galaxies, the metric expansion described by a scale factor, and a whole-universe radius must be distinguished before comparing expansion histories.

We distinguish stellar motion from the baseline metric construction. To establish a dynamical cosmology, the latter must specify a spacetime metric, its curvature and a compatible stress-energy source; the geometric coefficient alone supplies no field solution.

The Bohr-radius relation is the MQ Bohr-radius relation ([4], Appx. BL). Its connection to cosmic expansion uses the model identification of the atomic and expansion coefficients. This identification is an additional physical claim to be tested, rather than a consequence of the atomic equation alone.

a_{0} = \frac{1}{α_{c} m_{e}} \frac{p_{s i} t_{f}}{Q_{L} n_{L}} .

A spectral test of that identification requires a stated observational comparison and a calculation of how the jointly varied quantities affect atomic observables. The diameter–age relation alone does not separate those variations or determine the spatial curvature of the universe.

The MQ dark-domain fraction Ω_dk describes the complementary, permanently non-referenceable part of the realized domain: it cannot become observable within the Frames mapping ([4], Appx. BB; [14], Sec. 3.9). The conventional Ω_Λ describes a ratio of energy densities. The two occurrences of the domain relation repeat the same domain relation at their respective uses; similar numerical magnitudes do not alone establish that the geometric fraction is a general-relativity source density fraction.

Ω_{d k} = \frac{θ_{s i}^{2} - 2}{θ_{s i}^{2} + 2} = 68.3624161224 (41) % .

Within the assumed fixed-coefficient model, Ω_dk is constant. The quantity in the dimensionless geometric combination is a dimensionless geometric combination; its constancy follows from the same assumption.

100 π \frac{θ_{s i}^{2} - 2}{θ_{s i}^{2} + 2} = 214.766864272 (13) .

The atomic relation introduces a₀, the Bohr radius, m_e, the electron mass, and α_c, the classical fine-structure constant; their roles are shared with the calibration discussed earlier. Connecting that relation to cosmic expansion requires the additional identification stated in the model. A common coefficient is a proposed physical connection whose observational consequences must be worked out, rather than a substitute for that test.

The variational source in [5] is defined on the realized metric. At a fixed realization the computed curvature can be compared with a continuum coefficient. If the age dependence in the expanded curvature relation instead describes evolution within one spacetime, the contracted Bianchi identity requires a conserved total source. Define the effective geometric tensor as the term

- Λ g_{μ ν} / k

assigned to the source side of the physical field equation; the matter tensor plus this term must be covariantly conserved. A varying Λ then requires an exchange contribution and cannot coexist with separately conserved matter at fixed k. That evolving interpretation requires an additional dynamical solution; it is not inferred from the coefficient calculation.

l_{f}^{2} (R_{μ ν} - \frac{1}{2} R g_{μ ν} + Λ_{s y s} g_{μ ν}) = l_{f}^{2} (\frac{8 π G}{c^{4}}) T_{μ ν} .

Within MQ the effective density represents geometry: the expansion construction supplies Λ, and count normalization supplies G. To identify what a constant does, let L_m be the realized continuum matter Lagrangian density in J m^-3 and let C be a metric-independent constant with the same units. The local source expression in [5], the corresponding relation, gives L_m→L_m+C and hence T_μν→T_μν+Cg_μν. In the physical field equation this contribution can be assigned to the geometric coefficient. The simultaneous representation changes T_μν→T_μν+Cg_μν and Λ→Λ+kC leave that field equation unchanged. We interpret the calculated curvature as the total effective coefficient in the stated continuum convention, without separately adding the same constant contribution again. This assignment does not establish physical decoupling of an arbitrary matter constant or determine its magnitude.

A decreasing Hubble rate is compatible with an increasing size. For example, a constant radial speed gives a size proportional to elapsed time and a rate per unit distance proportional to the inverse age. This distinction is needed when interpreting the baseline expansion used here.

For equal time intervals, the successive radii of a constant-speed model increase by equal amounts. This describes expansion with zero radial acceleration. It does not, by itself, derive the observed acceleration of a cosmological scale factor.

A comparison with observed accelerated expansion requires a defined scale factor and its time derivatives, together with the model’s relation to observational distances. The inverse-age Hubble dependence alone supplies no such acceleration test.

The age dependence is especially informative. With the geometric coefficient fixed, the expanded expression scales with inverse age squared. If that prescription describes evolution within a single spacetime, its coefficient evolves; it cannot simultaneously be a time-independent cosmological constant without an additional compensating mechanism. A changing coefficient must also satisfy the field equation’s conservation requirement. A viable extension therefore needs a compatible source and metric evolution. Calling Λ geometric changes its proposed origin, but it does not remove these dynamical conditions.

The distinct apparent-age construction

Here the upright symbol H denotes the dimensionless scalar radial coefficient equal to θ_si; upright CMB and Cepheid subscripts identify its corresponding dimensionless realizations. Italicized H with those subscripts in the physical age-offset relations denotes the physical age-offset comparison rates in km s^-1 Mpc^-1. These rates differ from the fixed-diameter rates H_sys and H_int in Table 1. The dimensionless offset Δ_H=θ_si-π follows [18,5]. Recent TRGB determinations [22,23] provide observational comparisons but do not fix a unique MQ radial offset.

The apparent-age comparison in the preprint uses a different denominator from the fixed-diameter frame rates. Its intermediate equalities are retained here to make the distinction explicit. The radial coefficient H is upright and dimensionless; the physical rates introduced at the end of the construction are italicized and have expansion-rate units.

Δ_H=θ_si-π.

H_CMB=H-Δ_H=π.

H_cep=H+Δ_H.

Combine the first two expressions and reduce.

H_cep-Δ_H=H_CMB+Δ_H.

H_cep=H_CMB+2Δ_H=π+2(θ_si-π)=2θ_si-π.

The physical diameter D_U gives light-time

D_{U} / c = 2 θ_{s i} A_{U} = 90.005 (25)

Gyr. Using the dimensionless CMB coefficient π or Cepheid coefficient 2θ_si-π yields the respective apparent ages A_CMB and A_cep, in Gyr:

A_{C M B} = \frac{D_{U}}{2 c H_{C M B}} = \frac{90.00503708}{2 π} = 14.3247 (40) Gyr .

A_{c e p} = \frac{D_{U}}{2 c H_{c e p}} = \frac{90.00503708}{2 (2 θ_{s i} - π)} = 13.3018 (37) Gyr .

The apparent-age offsets are ΔA_CMB=A_CMB-A_U and ΔA_cep=A_cep-A_U, in Gyr. They compare the respective apparent ages at fixed physical diameter.

ΔA_CMB=A_CMB-A_U=0.53041(15)Gyr.

ΔA_cep=A_cep-A_U=-0.49253(14)Gyr.

For the distinct age-offset rates in the physical age-offset relations, the denominator uses 2θ_siA_U+ΔA, with ΔA set to the corresponding apparent-age offset. The dimensionless numerator coefficients follow the CMB radial relation and the Cepheid radial relation. Numerical ages here are in Gyr, so the factor 10⁹ and 3.15576×10⁷ s yr^-1 convert the denominator to seconds. Italicized H denotes these physical comparison rates; upright H in the dimensionless radial and age relations remains dimensionless.

H_{C M B} = \frac{2 π K}{(2 θ_{s i} A_{U} + Δ A_{C M B}) s / y \times 1 0^{9}} = 67.859 (19) km s^{- 1} Mp c^{- 1} .

H_{c e p} = \frac{2 (2 θ_{s i} - π) K}{(2 θ_{s i} A_{U} + Δ A_{c e p}) s / y \times 1 0^{9}} = 73.913 (21) km s^{- 1} Mp c^{- 1} .

These apparent-age rates are comparison outputs of their own construction. They do not replace H_sys in the Lambda result above. Retaining both derivations makes it possible to distinguish a change of frame relation from a change in the adopted age prescription. Observational calibration differences, including the tip of the red giant branch comparisons [22,23], must be assessed through their actual likelihoods and distance measurements.

Additional notation and calibration relations

The following source relations complete the notation used by the preprint and preserve its supplementary count and calibration definitions. Numerical results retain the same input realization used above.

The dimensionless coefficient θ_si=3.26239030504(25) comes from the fine-structure calibration [13]; its momentum realization uses the adopted dimensional convention. Shwartz and Harris [21], Table I, calculate phase-matched signal and idler angles of 3.26239 rad in a Bell-state construction. Their paper does not report a measurement of the MQ coefficient with a ±2 μrad uncertainty. This numerical comparison must therefore be distinguished from an independent experimental calibration.

The speed c may also be expressed as

c = n_{L} l_{f} / (n_{T} t_{f})

([4], Sec. 1), where n_L and n_T are dimensionless length and time counts. At the fundamental bound n_L=n_T=1.

At the upper count limit

G = c^{2} l_{f} / m_{f} = 6.6740779383 (10) \times 10^{- 11}

m³ kg^-1 s^-2. At the Fine Structure demarcation the italicized electromagnetic realization is

G_{EM} = 2 Q_{L} n_{L} G = 6.6738448317 (10) \times 10^{- 11}

m³ kg^-1 s^-2. These MQ results differ from the CODATA 2022 recommendation

G = 6.67430 (15) \times 10^{- 11}

m³ kg^-1 s^-2 [13].

The fundamental measures l_f, m_f and t_f are the MQ length, mass and time realizations. Their values and the order of calculation are given in the fundamental-measure discussion above using Appendix BM of [4].

The conventional Planck measures l_p, m_p and t_p denote length, mass and time, in m, kg and s, respectively.

The dimensionless counts n_L, n_M and n_T count intervals of l_f, m_f and t_f, respectively.

The separation count n_L specifies intervals of l_f relative to the reference appropriate to the observable, such as the observer or the center of mass. The subscript L is used consistently for this count.

The dimensionless light count n_Lc describes a light displacement in units of l_f relative to the observer’s frame.

The Internal Frame nearest-whole-unit count at the Fine Structure demarcation is

⌊ n_{L} / 2 ⌉ = 42

The mixed brackets ⌊x⌉ denote nearest-whole-unit rounding of the dimensionless count argument x. This declared MQ operation differs from floor or ceiling alone.

The residual Q_L is the fractional count contribution in the right-triangle realization. It is dimensionless and resolves the difference between n_L and the hypotenuse count.

The dimensionless normalization factor is 2Q_Ln_L; Q_Ln_L is half that factor. The Informativity differential is the departure 1-2Q_Ln_L, which vanishes at the upper count limit ([14], Sec. 3.6).

Conventional

ℏ = h / (2 π)

is exactly 1.054571817646…×10^-34 J s in SI. The upright MQ upper-count realization, labelled ħ_up here to distinguish it from the conventional value, is

ħ_{up} = 2 p_{si} l_{f} = 1.054534984442 \times 10^{- 34}

J s, with propagated standard uncertainty 5.6×10^-49 J s. The electromagnetic realization

p_{si} l_{f} / (Q_{L} n_{L})

reproduces conventional ℏ through the calibration identity; it is not an independent measurement.

The electron mass is m_e, in kg.

The ground-state Bohr radius is a₀, in m.

The dimensionless inverse constants

{α_{f}}^{- 1}

{α_{p}}^{- 1}

and

{α_{c}}^{- 1}

denote the fundamental, Planck-like and classical fine-structure forms. The relation

{α_{f}}^{- 1} = 42 θ_{si}

uses the Internal Frame description; the remaining forms use the System Frame description. The difference between

{α_{c}}^{- 1}

and

{α_{p}}^{- 1}

accounts for the discrete-measure contraction.

What the geometric interpretation predicts

The proposed contribution is a constrained numerical relationship between a geometric partition, a realized expansion frequency and a curvature coefficient. The domain fraction is fixed once the calibrated geometric coefficient is fixed. The selected frame and physical age then determine the curvature at the stated realization, while the count-based coupling determines its continuum density representation. This provides a calculable benchmark rather than an independently adjustable density inserted at the last step.

A stronger cosmological claim requires the same construction to predict a consistent expansion history and perturbation observables. The distinction between a realized coefficient and an evolving solution is therefore central to the interpretation. It preserves the content of the geometric calculation while making clear what a further empirical test must establish. Related discussions of General Relativity and Dark Energy develop the wider context.

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