Geometric Origin of Cosmological Curvature in Measurement Quantization

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Keywords: Cosmological Curvature, Measurement Quantization, Geometric Coupling

1. INTRODUCTION

Cosmological measurements constrain distances, redshifts, anisotropies and the expansion history. Their interpretation in a specified gravitational and cosmological model leads to a curvature coefficient or an effective dark-energy density [1,2,3]. These inferences do not, by themselves, identify a microscopic vacuum substance. We investigate whether the curvature coefficient can instead be fixed by the discrete-to-continuum geometry of Measurement Quantization (MQ) [4,5].

In this interpretation Λ is a geometric coefficient, not a postulated independent quantum vacuum energy. General relativity already places Λ in its geometric field equation; the proposed MQ contribution is a count-based origin for its value and for the gravitational coupling [5]. The classical vacuum-energy correspondence can be retained as an effective continuum parametrization. This choice of ontology does not by itself show that all matter-sector quantum vacuum contributions decouple from gravity; the present calculation addresses the geometric coefficient, rather than claiming a demonstrated quantum cancellation [6].

The relationship in Eq. (1) 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.

Equation (1) 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 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.

We first specify the geometric scalar fraction and frame-resolved expansion rates, calculate the resulting physical curvature, and then apply Eq. (1) for a conventional density comparison. We next discuss the count geometry of the coupling and the conservation condition for an evolving curvature coefficient. All density values in this paper follow the same continuum mapping.

2. METHODS

This paper uses classical expressions in a way that accounts for new properties of measure resolved using discrete expressions describing gravitational curvature, the speed of light, escape velocity and Heisenberg’s uncertainty principle. We identify this approach with the name Measurement Quantization (MQ). Notably, several examples in the publication record exist [4,7,8,9,10,11,12] whereby MQ is applied to problems in modern theory.

The MQ framework assigns discrete counts to the Internal Frame and realizes their relational geometry in the System Frame through the Frames mapping ([5], Secs. 1.3.1–1.3.3, 2.1–2.2). The Internal Frame is the discrete count domain; the System Frame is its non-discrete relational encoding, where physical observables are defined. The Frames mapping connects them subject to closure and consistency. We use these as model premises. The calculation evaluates a geometric coefficient at a specified realization; it does not infer the universe’s complete expansion history from that coefficient alone. Independent empirical support must distinguish predictions from relations calibrated using shared source constants.

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

3. RESULTS

For those not familiar with MQ, we will briefly go over nomenclature and terms commonly used in this paper.

1. Nomenclature

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.

2. Cosmological Curvature from the Geometric Construction

The MQ scalar fraction is given by Eq. (4), using θ_si=3.26239030504(25) from the 2022 CODATA inverse fine-structure input [13]. The MQ gravitational construction [5], Eq. (50) 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.

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.

Table 1 retains the published Planck-plus-BAO expansion-rate input and the three MQ frame calculations. Curvature follows Eq. (3), and the density column now follows Eq. (1) with the MQ upper-count coupling. The Planck row inserts a published rate into the MQ construction and is therefore a hybrid benchmark, not Planck’s independently inferred density. Appendix B retains the distinct age-offset construction. None of the shared MQ calculations is counted as an independent empirical confirmation.

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

^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 Baseline metric expansion rate, with dimensionless radial coefficient θ_si ([4], Eq. (BA.1)).

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

Writing the frame-specific frequencies explicitly as the respective observational rates divided by K gives

4. DISCUSSION

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

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.

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.

Equation (11) 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 spectral test of that identification requires a stated observational comparison and a calculation of how the jointly varied quantities affect atomic observables. Equation (11) 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. Equations (4) and (12) 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.

Within the assumed fixed-coefficient model, Ω_dk is constant. The quantity in Eq. (13) is a dimensionless geometric combination; its constancy follows from the same assumption.

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:

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.

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], Eq. (37), gives L_m→L_m+C and hence T_μν→T_μν+Cg_μν. In Eq. (9) 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.

5. APPENDICES

APPENDIX A: TERMS

The following nomenclature uses the count and measure descriptions in [4] and the current realized-geometry definitions in [5], Appx. A. The Informativity differential follows [14], Sec. 3.6.

• 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

  $$\mathit{c}={\mathit{n}}_{\mathrm{L}}{\mathrm{l}}_{\mathrm{f}}/({\mathit{n}}_{\mathrm{T}}{\mathrm{t}}_{\mathrm{f}})$$

  (19a)

  ([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

  $$\mathrm{G}={\mathit{c}}^2{\mathrm{l}}_{\mathrm{f}}/{\mathrm{m}}_{\mathrm{f}}=6.67407793828(10)\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\text{.}$$

  (19b)

  At the Fine Structure demarcation the italicized electromagnetic realization is

  $${\mathit{G}}_{\mathrm{EM}}=2{\mathit{Q}}_{\mathrm{L}}{\mathit{n}}_{\mathrm{L}}\mathrm{G}=6.67384483165(10)\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}\text{.}$$

  (19c)

  These MQ results differ from the CODATA 2022 recommendation

  $$\mathrm{G}=6.67430(15)\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$$

  (19d)

  [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 Section 3.1 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

  $$\lfloor {\mathit{n}}_{\mathrm{L}}/2\rceil =42$$

  (19e)

  .

• 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

  $$\mathit{\hslash }=\mathit{h}/(2\mathit{\pi })$$

  (19f)

  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

  $${\mathrm{ħ}}_{\mathrm{up}}=2{\mathit{p}}_{\mathrm{si}}{\mathrm{l}}_{\mathrm{f}}=1.054534984442\times {10}^{-34}\text{J s,}$$

  (19g)

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

  $${\mathit{p}}_{\mathrm{si}}{\mathrm{l}}_{\mathrm{f}}/({\mathit{Q}}_{\mathrm{L}}{\mathit{n}}_{\mathrm{L}})$$

  (19h)

  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

  $${{\mathit{\alpha }}_{\mathrm{f}}}^{-1}$$

  (19i)

  ,

  $${{\mathit{\alpha }}_{\mathrm{p}}}^{-1}$$

  (19j)

  and

  $${{\mathit{\alpha }}_{\mathrm{c}}}^{-1}$$

  (19k)

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

  $${{\mathit{\alpha }}_{\mathrm{f}}}^{-1}=42{\mathit{\theta }}_{\mathrm{si}}$$

  (19l)

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

  $${{\mathit{\alpha }}_{\mathrm{c}}}^{-1}$$

  (19m)

  and

  $${{\mathit{\alpha }}_{\mathrm{p}}}^{-1}$$

  (19n)

  accounts for the discrete-measure contraction.

APPENDIX B: HUBBLE TENSION CALCULATIONS

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 Eqs. (29–30) 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.

Combine the first two expressions and reduce.

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.

For the distinct age-offset rates in Eqs. (29)–(30), the denominator uses 2θ_siA_U+ΔA, with ΔA set to the corresponding apparent-age offset. The dimensionless numerator coefficients follow Eqs. (21) and (24). 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 Eqs. (21)–(26) remains dimensionless.

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 datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request. Calculations can also be accessed at https://www.researchgate.net/publication/361148724_OnlineResources.

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