Support for a Universe without Curvature

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60605

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We examine the Measurement Quantization (MQ) interpretation of cosmological flatness using discrete counts of fundamental length, mass and time and their observable realization. Within MQ, the Internal Frame supplies integer count configurations, while the Frames mapping gives continuous relations in the System Frame. We distinguish the local fundamental relation from the cosmological count relation and compare the resulting fractions with benchmarks from the cosmological-constant and cold-dark-matter model (ΛCDM). The MQ calibration using the 2022 fine-structure input from the Committee on Data for Science and Technology (CODATA) gives the geometric coordinate 214.766864272(13). This is a conditional MQ calculation, not a measured cosmic microwave background (CMB) peak position. The age and diameter results additionally depend on an explicit International System of Units (SI) numerical-value prescription. The domain coordinates specify a proposed geometric contribution; neither the fraction comparison nor the scalar count relations independently establish the curvature of a realized cosmological metric.

Keywords: Gravitational Curvature, Space, Gravitational Mass, Curvature

1. INTRODUCTION

We examine the MQ proposal that cosmological expansion can be described through the geometry of realized count relations. Discrete reference configurations belong to the Internal Frame; continuous observable relations belong to the System Frame through the Frames mapping. The proposal builds on the fundamental-reference approach related to Planck units [1]. Absence of an intrinsic metric for a fundamental reference does not by itself establish zero curvature for an observable spacetime metric.

The realized fundamental measures are length l_f, mass m_f and time t_f, in m, kg and s, with dimensionless counts n_L, n_M and n_T (Ref. [2], Appx. I, J, V, W). The invariant dimensionless coefficient θ_si is distinct from its momentum realization p_si, in kg m s⁻¹, and angular realization Θ_si, in rad. The adopted MQ realization assigns the same numerical coefficient in these stated units; this is a realization premise, not a direct experimental identification. Equation (4) states the dimensional fundamental relation. Shwartz and Harris propose a method for producing polarization-entangled X-ray photons [3]. The later studies measure X-ray down-conversion [4] and X-ray/optical sum-frequency generation [5]. They do not report an experimental determination of θ_si or establish the MQ frame premises. Chen et al. discuss conserved quantities at null infinity within general relativity [6]; this is a different mathematical setting.

The following comparisons connect the count relations with dimensional calculations and CMB quantities. Comparisons that reuse the calibration data establish consistency of the adopted relations. An independent physical test additionally requires separately measured inputs and an observable prediction.

We also consider hypothetical changes in l_f, m_f, t_f and their counts. The local light-speed relation and the atomic relation below show which combinations would change under the stated fixed-reference assumptions. These examples test consistency of the proposed scaling; the absence of a particular change does not establish every count relation for every observer.

The dimensionless observational-domain fractions refer to the observer. The visible Ω_vis describes what is presently referenceable. The observable Ω_obs includes what can become referenceable given unlimited elapsed time. Their difference is the unobserved Ω_uobs. The dark Ω_dk lies outside that observable domain. These are MQ referenceability domains in the realized description.

Combining the cosmological count relation with these domain fractions gives a geometric description of elapsed time, expansion and accumulated mass-equivalent content. We first evaluate the dark-domain geometric multipole coordinate. Its interpretation is examined after the domain derivation. Here π is the dimensionless circle constant.

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

(1)

Equation (1) is an MQ geometric coordinate considered alongside CMB evidence on spatial flatness [7]. Its value uses the 2022 CODATA inverse fine-structure input [8] through the following MQ calibration (Ref. [2], Appx. AD). Here α_c is the dimensionless classical fine-structure coupling. In this calibration, n_L is the separation count at the fixed electromagnetic demarcation, satisfying n_Lθ_si=276, distinct from a variable light-travel count. The mixed brackets ⌊x⌉ denote rounding x to the nearest whole-unit value; the selected branch gives ⌊42θ_si⌉=137.

2 α_{c} = \frac{1}{n_{L} θ_{si}} \frac{θ_{si}}{\sqrt{{θ_{si}}^{2} + {(n_{L} θ_{si})}^{2}} - n_{L} θ_{si}} \frac{θ_{si}}{(84 θ_{si} - ⌈ 42 θ_{si} ⌋)} .

(2)

θ_{si} = 3.26239030504 (25) .

(3)

The MQ calculations of physical constants and relations use the stated reference and realization premises (Ref. [2], Appx. C, L, AA, AM, AP, AQ, AR, BA) [9, 2]. Their physical validity requires tests beyond algebraic consistency with those premises.

2. METHODS

MQ applies count-based expressions to local and cosmological quantities. The uncertainty principle [10] is part of the physical motivation for the approach; it does not by itself establish discrete spacetime or the particular MQ reference measures. Earlier MQ papers present proposed applications [9, 11–15, 2]. We use those publications as the provenance of the framework, rather than treating publication as independent confirmation of its claims.

The present calculations adopt MQ’s discreteness and countability premises for the Internal Frame counts already introduced. Their development is described in the introductory preprint [16] and the published synthesis [2]. The referenced appendices provide derivational context; the current notation and calibration used here retain the distinctions stated in Sec. 1.

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

The Online Resources collection provides background material for the MQ program. Supplemental calculations are useful for tracing an expression’s evaluation, but agreement obtained by reusing a source input is a consistency check. The comparisons below retain that distinction.

3. RESULTS

For the local light-speed application, the equal-count condition n_L=n_T gives

c = \frac{l_{f}}{t_{f}}

(3a)

, where c is the speed of light in m s⁻¹. The lower-bound escape-speed condition is n_L=2n_M (Ref. [2], Appx. H, V). Neither condition substitutes for the separate cosmological count relation used below.

Equation (4) uses the momentum realization defined in the introduction. Equations (5)–(7) then state the cosmological count construction. Here

n_{L_{U}}

(3b)

is the dimensionless count of fundamental lengths across the physical diameter D_U, in m, and

n_{T_{U}}

(3c)

is the dimensionless elapsed count of fundamental times in A_U, in s. Their ratio follows the cosmological relation; it is not obtained by multiplying independent length, mass and time factors into Eq. (4). The fundamental-count reference has n_M=1, while Eqs. (8)–(9) use a separate one-kilogram reference (Ref. [2], Appx. V; Ref. [17], Sec. 3.10).

l_{f} m_{f} = 2 p_{si} t_{f} .

(4)

n_{L_{U}} = 2 θ_{si} n_{T_{U}} .

(5)

D_{U} = n_{L_{U}} l_{f}, A_{U} = n_{T_{U}} t_{f} .

(6)

\frac{n_{L_{U}}}{2 n_{T_{U}}} = \frac{D_{U}}{2 A_{U} c} = θ_{si} .

(7)

Equations (8)–(9) use the dimensionless count corresponding to the separately chosen 1 kg reference. This is the one-kilogram endpoint illustrated here, not a universal maximum mass count. Its qualifier distinguishes it from the assignment n_M=1 and the cosmological ratio in Eq. (7). Equation (9) uses the same unrounded m_f in both factors and therefore returns exactly 1 kg.

{n_{M} |}_{1 kg} = \frac{1 kg}{m_{f}} = 4.59467491577 (35) \times 1 0^{7} .

(8)

{n_{M} |}_{1 kg} m_{f} = \frac{1 kg}{m_{f}} m_{f} = 1 kg .

(9)

\frac{D_{U}}{c} = 2 θ_{si} A_{U} .

(10)

The local relation (11) provides a consistency test. If l_f changes while t_f and the relevant counts are fixed, or conversely, their length-to-time ratio changes. These hypothetical changes concern a fixed local reference assignment.

c = \frac{n_{L} l_{f}}{n_{T} t_{f}} = \frac{2 p_{si}}{m_{f}} .

(11)

A primed quantity denotes a hypothetical changed value in these tests. Holding the other factors fixed, changing either fundamental length or fundamental time changes the speed calculated from Eq. (11). A common change that preserves their ratio is not excluded by this equation alone.

Changing one of the two local counts while holding the other fixed likewise changes that speed. The conclusion applies to the common-interval local light-speed assignment; the cosmological counts in Eqs. (5)–(7) have a different purpose.

With p_si fixed, Eq. (12) fixes m_f once c is specified. This is a consequence of the adopted momentum realization and the fundamental relation.

m_{f^{'}} = \frac{2 p_{si}}{c} .

(12)

Equation (12) therefore depends on the realization premise introduced in Sec. 1.

At the lower-bound escape-speed condition, Eq. (13) fixes the length-to-mass count ratio. This boundary relation is separate from the local light-speed equal-count condition.

n_{L} = 2 n_{M} .

(13)

Equation (13) is applied only at that bound; it is not a relation among arbitrary cosmological counts.

For the atomic comparison, m_e is the electron mass in kg, a₀ is the Bohr radius in m and α_p is the dimensionless Planck-like MQ coupling (Ref. [2], Appx. BL, AA). Equation (14) links these quantities to the fundamental measures.

α_{p}^{- 1} = \frac{a_{0} m_{e}}{m_{f} l_{f}} .

(14)

With l_f, m_f and α_p fixed, Eq. (14) fixes the product m_e a₀. A change in electron mass would then require a reciprocal change in the Bohr radius. A change in one count alone is therefore a conditional consistency test, not an independent proof of universal scaling.

3.1 Size and Age of the Universe

To describe the size and age of the universe, we compare with age estimates [18, 19]. Equation (15) states the MQ age calculation (Ref. [2], Appx. BI) as an explicit SI numerical-value rule. The radiation constant a has units J m⁻³ K⁻⁴ and is evaluated from c, the exact SI Planck constant

h = 6.62607015 \times 10^{- 34} J s

(14a)

, and the exact Boltzmann constant

k_{B} = 1.380649 \times 10^{- 23} J K^{- 1}

(14b)

[8]. The CMB temperature is T_CMB=2.72548(57) K [20]. Dividing each input by the indicated SI unit makes the quantity inside the cube root dimensionless; multiplication by 1 s assigns the age in seconds. The numerical result is converted to Gyr using 31557600 s per Julian year. This specifies the calculation convention without asserting that its physical unit assignment has been derived independently. The dimensionless exponential base e is the base of natural logarithms. The resulting age is conditional on this prescribed seconds reference. Numerical agreement with an external age estimate would not supply the missing physical derivation of that reference. The age literature [18, 19] is contextual; the independent observational constraints are not derived in this calculation.

A_{U} = (1 s) \sqrt[3]{\frac{3 θ_{si} {(2 θ_{si})}^{\frac{1}{3}} e^{\frac{\sqrt{3} {θ_{si}}^{3}}{2}}}{4 π (\frac{a}{1 J m^{- 3} K^{- 4}}) (\frac{c}{1 m s^{- 1}}) (\frac{l_{f}}{1 m}) {(\frac{T_{CMB}}{1 K})}^{4}}} = 13.7943 (38) Gyr .

(15)

The calibration in Eq. (2) is repeated in Eq. (16) to show the input chain for these dimensional results (Ref. [2], Appx. AC, AD, BM). It uses

{α_{c}}^{- 1} = 137.035999177 (21)

(15a)

from the 2022 CODATA adjustment [8]; Eq. (17) repeats its resulting coefficient. Parentheses denote one standard uncertainty in the final displayed digits. Propagation uses the common fine-structure input and, for age-dependent quantities, the CMB temperature input. These two experimental sources are treated as independent; derived outputs retain their shared-input covariance. Model uncertainty is not quantified. In particular, a small propagated uncertainty is not an independent demonstration of the accuracy of the MQ relations.

2 α_{c} = \frac{1}{n_{L} θ_{si}} \frac{θ_{si}}{\sqrt{{θ_{si}}^{2} + {(n_{L} θ_{si})}^{2}} - n_{L} θ_{si}} \frac{θ_{si}}{(84 θ_{si} - ⌈ 42 θ_{si} ⌋)} .

(16)

θ_{si} = 3.26239030504 (25) .

(17)

The shared product of m_e and a₀ satisfies

m_{e} a_{0} = \frac{ℏ {α_{c}}^{- 1}}{c}

(17a)

, where

ℏ = \frac{h}{2 π}

(17b)

is the conventional reduced Planck constant [8]. This identity retains their shared dependence instead of treating them as independent measurements. Using this product and the momentum realization gives the following fundamental measures (Ref. [2], Appx. BL, BM).

m_{f} = \frac{2 p_{si}}{c} = 2.17643254057 (17) \times 1 0^{- 8} kg .

(18)

{α_{p}}^{- 1} = 84 θ_{si} - ⌈ 42 θ_{si} ⌋ = 137.040785623 (21) .

(19)

l_{f} = \frac{m_{e} a_{0} α_{p}}{m_{f}} = 1.61619991148 (12) \times 1 0^{- 35} m .

(20)

t_{f} = \frac{l_{f}}{c} = 5.39106261131 (41) \times 1 0^{- 44} s .

(21)

The physical diameter D_U and elapsed age A_U obey the light-time relation in Eq. (10). A distance in Gly is the distance light travels in a billion Julian years; A_U is reported in Gyr. The factor c in Eq. (22) expresses the conversion between these physical units.

D_{U} = 2 θ_{si} c A_{U} = 90.005 (25) Gly .

(22)

For the absolute mass allocations, M_acr is the MQ global realized mass-equivalent accumulation rate in kg s⁻¹ and M_tot is accumulated total mass-equivalent content in kg. These quantities describe the cosmological realization, rather than local gas accretion. Their dimensional expressions use the fundamental measures already calculated (Ref. [2], Appx. BE, BJ; Ref. [17], Sec. 3.10).

M_{t o t} = M_{a c r} A_{U} = 3.05108 (85) \times 1 0^{54} kg M_{a c r} = \frac{{θ_{si}}^{3} m_{f}}{2 t_{f}} = 7.0088794874 (27) \times 1 0^{36} kg s^{- 1} .

(23)

The duration of the earlier quantum epoch, expressed in the current expansionary-epoch representation, A_ee, in s, is evaluated with the same explicit SI numerical-value convention as Eq. (15). The exponential specifies its numerical coefficient; multiplication by 1 s supplies the reference duration. The CMB radiation mass-equivalent M_CMB, in kg, follows by multiplying this duration by M_acr (Ref. [2], Appx. BG, BH, BI). Its propagated uncertainty is conditional on that duration convention.

M_{CMB} = M_{a c r} A_{ee} = 1.501601505 (11) \times 1 0^{50} kg A_{ee} = (1 s) e^{\frac{\sqrt{3} {θ_{si}}^{3}}{2}} {(2 θ_{si})}^{1 / 3} .

(24)

For the radiation comparison, R_U is the physical radius in m and V_U is the volume in m³ of the spherical region defined by that radius. The blackbody radiation relation then gives the same M_CMB when T_CMB and the age rule (15) are used together (Ref. [2], Appx. BI).

M_{CMB} = \frac{a {T_{CMB}}^{4} V_{U}}{c^{2}} R_{U} = \frac{D_{U}}{2}, V_{U} = \frac{4}{3} π R_{U}^{3} .

(25)

Table 1 collects conditional MQ allocations and shared-input radiation checks. Its CMB mass and temperature equalities reuse T_CMB through Eq. (15). The visible mass M_vis, in kg, uses Ω_vis from Eq. (43) and M_tot from Eq. (23). The final row gives the MQ referenceability fraction; it does not equate that fraction with a measured baryonic density.

Table 1. Conditional MQ calculations and shared-input checks.

Quantity	MQ calculations and shared-input checks
Quantity	MQ expression	MQ result	Input / identity	Difference
M_CMB	M_acrA_ee	$1.501601505 (11) \times 10^{50} kg$ (25a)	$\frac{a {T_{CMB}}^{4} V_{U}}{c^{2}}$ (25b)	0 (identity) ^a
M_vis	Ω_visM_tot	$1.47942 (41) \times 10^{53} kg$ (25c)	— ^b	—
T_CMB	${(\frac{M_{CMB} c^{2}}{a V_{U}})}^{\frac{1}{4}}$ (25d)	2.72548(57) K	2.72548(57) K ^a	0 (identity)
Ω_vis	$\frac{2}{θ_{si} ({θ_{si}}^{2} + 2)}$ (25e)	4.8488348909(10)%	— ^c	—

^a T_CMB is the input measured by Fixsen [20]. The two radiation calculations use the common volume and temperature, so their zero differences are identities.

^b An absolute visible-mass comparison requires a physical volume and density assignment. The cluster-abundance analysis [21] constrains matter density, rather than measuring this universe-wide visible mass.

^c MQ referenceability fractions and the ΛCDM baryonic density parameter have distinct definitions [22]. No direct baryonic measurement is assigned to this row.

The two zero differences in Table 1 are algebraic identities under the adopted age prescription. They do not provide independent evidence for that prescription. The domain partition is considered with the supported model-based benchmarks in Table 2.

We now discuss geometry and its relation to cosmological measure.

3.2 Domain Definitions

The observational fractions introduced in Sec. 1 are evaluated below (Ref. [2], Appx. BB). They refer to the observer’s realized referenceability. The total Ω_tot is dimensionless and is assigned the value 1; the dark and observable fractions partition this total.

Equations (26)–(29) state closure, the total assignment and the visible-to-observable mapping (Ref. [2], Appx. BB, V). The mapping in Eq. (28) is an MQ domain relation, not a consequence of closure alone.

Ω_{tot} = Ω_{obs} + Ω_{dk} .

(26)

Ω_{dk} + Ω_{obs} = 1 .

(27)

Ω_{obs} = 2 θ_{si} Ω_{vis} .

(28)

Ω_{uobs} = Ω_{obs} - Ω_{vis} .

(29)

3.3 Observational Domains

The fundamental domain Ω_f is a dimensionless geometric invariant, rather than an additional additive matter or energy component. Substituting the mapping in Eq. (30) into the ratio in Eq. (31) gives Eq. (32). The sum of all five listed fractions is therefore not assigned the value 1.

2 θ_{si} Ω_{vis} = Ω_{obs} .

(30)

\frac{2}{θ_{si}^{2}} = 4 \frac{Ω_{vis}}{θ_{si} Ω_{obs}} = Ω_{f} .

(31)

Ω_{f} = \frac{2}{{θ_{si}}^{2}} = 18.7913577188 (29) % .

(32)

We now use the same visible-to-observable mapping in the dark-domain construction. Equations (34)–(35) multiply both sides of Eq. (33) by the same factor and subtract 1.

2 θ_{si} Ω_{vis} = Ω_{obs} .

(33)

2 θ_{si} Ω_{vis} \frac{θ_{si}^{2}}{2} - 1 = Ω_{obs} \frac{θ_{si}^{2}}{2} - 1 .

(34)

θ_{si}^{3} Ω_{vis} - 1 = Ω_{obs} \frac{θ_{si}^{2}}{2} - 1 .

(35)

The MQ dark-domain relation is

Ω_{dk} = {θ_{si}}^{3} Ω_{vis} - 1

(35a)

(Ref. [2], Appx. BB, BC). Using this relation on the left of Eq. (35), and the closure relation on its right, gives Eq. (36). Thus closure and the visible-to-observable ratio alone do not determine the dark fraction; this additional MQ relation is required.

Ω_{dk} = (Ω_{tot} - Ω_{dk}) \frac{θ_{si}^{2}}{2} - 1 = (1 - Ω_{dk}) \frac{θ_{si}^{2}}{2} - 1 .

(36)

Ω_{dk} = (1 - Ω_{dk}) \frac{θ_{si}^{2}}{2} - 1 .

(37)

Ω_{dk} = \frac{{θ_{si}}^{2} - 2}{{θ_{si}}^{2} + 2} = 68.3624161224 (41) % .

(38)

Solving Eq. (37) gives Eq. (38). Using the complementary observable fraction then gives Eqs. (39)–(41). Equation (42) recovers the visible fraction from the earlier mapping, yielding Eq. (43); subtracting it from the observable gives Eq. (44).

1 - Ω_{obs} = Ω_{obs} \frac{θ_{si}^{2}}{2} - 1 .

(39)

2 = Ω_{obs} \frac{θ_{si}^{2}}{2} + Ω_{obs} .

(40)

Ω_{obs} = \frac{4}{{θ_{si}}^{2} + 2} = 31.6375838776 (41) % .

(41)

Ω_{obs} = 2 θ_{si} Ω_{vis} = \frac{4}{θ_{si}^{2} + 2} .

(42)

Ω_{vis} = \frac{2}{θ_{si} ({θ_{si}}^{2} + 2)} = 4.8488348909 (10) % .

(43)

Ω_{uobs} = Ω_{obs} - Ω_{vis} = 26.7887489867 (31) % .

(44)

Applying the dark-domain coordinate prescription to Eq. (38) reproduces Eq. (1), as shown in Eq. (45) (Ref. [2], Appx. BD).

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

(45)

For fixed θ_si, Eq. (22) makes the physical diameter proportional to elapsed age. Under the adopted MQ cosmological relation, its time derivative is constant. This geometric statement does not make θ_si a speed or establish the curvature tensor of an observable metric. The accumulated mass-equivalent content follows the rate relation used in Table 1 (Ref. [17], Sec. 3.10).

3.4 Observational Comparisons

Table 2 compares the MQ fractions with selected density fractions inferred in the flat base-ΛCDM model and gives the conditional MQ mass-equivalent allocations (Ref. [2], Appx. BJ). For the external quantities, Ω_Λ is the dark-energy fraction, Ω_c the cold-dark-matter fraction, Ω_b the baryon fraction and Ω_m the total matter fraction, including the neutrino contribution in the cited Planck model [22]. The fraction labels describe a proposed numerical comparison between different physical definitions. They do not establish that MQ domains are the corresponding material components.

Table 2. Selected Planck density fractions and conditional MQ allocations.

Phenomenon / MQ Term	Domain
Phenomenon / MQ Term	Planck % ^a	(MQ) %	(MQ) kg
Dark energy Ω_Λ / Dark Ω_dk	68.34±0.84%	68.3624161224(41)	$2.08579 (58) \times 10^{54}$ (45a)
Dark Matter Ω_c / Unobserved Ω_uobs	—	26.7887489867(31)	$8.1735 (23) \times 10^{53}$ (45b)
Ω_m / Observable Ω_obs	31.66±0.84%	31.6375838776(41)	$9.6529 (27) \times 10^{53}$ (45c)
Baryonic Ω_b / Visible Ω_vis	—	4.8488348909(10)	$1.47942 (41) \times 10^{53}$ (45d)
- / Fundamental Ω_f	-	18.7913577188(29)	$5.7334 (16) \times 10^{53}$ (45e)

^a Planck 2018 TT,TE,EE+lowE base-ΛCDM results [22], with flatness assumed. TT, TE and EE denote temperature and E-mode polarization auto- and cross-spectra; lowE denotes the low-multipole E-mode polarization likelihood. Dashes indicate that no observational benchmark is asserted here. MQ masses inherit the age prescription; overlapping and fundamental allocations are not added to the total.

The dark and observable fractions lie within the two retained Planck intervals, which are complementary constraints rather than independent tests. Because those intervals were inferred assuming a flat base-ΛCDM model, their agreement cannot independently establish flatness. The MQ mass allocations share the accumulated content in Eq. (23), including its conditional age assignment. The fundamental allocation is an invariant reference, not an additional component.

We next define geometric coordinates associated with the MQ domain fractions. The metric differential describes the representation mismatch when Internal Frame count structure is encoded continuously in the System Frame under the Frames mapping. Its dimensionless factor for the present coordinate construction is (Ref. [2], Appx. T, Eq. (T.13))

Δ_{m d i f} = {(\frac{π}{θ_{si}})}^{2 / 3} .

(46)

Equation (46) supplies the metric differential used in the following coordinate definitions (Ref. [2], Appx. T, Eq. (T.13)). In each ordered pair, x is the horizontal geometric multipole coordinate and y is the vertical MQ domain fraction. The domain named before the colon identifies the pair; the domain symbol retains its scalar fraction meaning. Domain subscripts distinguish the pairs. The factor 100 converts each domain fraction to its percentage value when calculating x; y remains a fraction. The differential multiplies the horizontal coordinate for the observable, unobserved, fundamental and visible domains. The dark-domain definition omits it. These equations define the geometric contribution considered here.

Ω_{dk} : (x_{dk}, y_{dk}) : = (100 π y_{dk}, \frac{{θ_{si}}^{2} - 2}{{θ_{si}}^{2} + 2}) .

(47)

Ω_{obs} : (x_{obs}, y_{obs}) : = (100 \frac{π^{5}}{18} y_{obs} Δ_{m d i f}, \frac{4}{{θ_{si}}^{2} + 2}) .

(48)

Ω_{uobs} : (x_{uobs}, y_{uobs}) : = (100 π^{3} y_{uobs} Δ_{m d i f}, \frac{2 (2 θ_{si} - 1)}{{θ_{si}}^{3} + 2 θ_{si}}) .

(49)

Ω_{f} : (x_{f}, y_{f}) : = (100 \times 9 π^{2} y_{f} Δ_{m d i f}, \frac{2}{{θ_{si}}^{2}}) .

(50)

Ω_{vis} : (x_{vis}, y_{vis}) : = (100 π^{5} y_{vis} Δ_{m d i f}, \frac{2}{θ_{si} ({θ_{si}}^{2} + 2)}) .

(51)

The coordinates in Eqs. (47)–(51) are manuscript-local definitions of the proposed geometric contribution, rather than independently measured peak locations. Their vertical fractions do not give the temperature-power heights in μK² plotted in the measured CMB spectrum [22]. A quantitative spectrum test additionally requires the physical contributions and a prescription relating these coordinates to measured multipoles and powers. The fundamental point uses the fraction in Eq. (32); no domain is identified one-for-one with an acoustic peak.

4. DISCUSSION

The scalar relations establish the stated count and domain geometry conditional on MQ’s premises. They do not specify a complete cosmological metric, a full CMB spectrum or an independent test of spacetime discreteness. The absence of an intrinsic metric in the Internal Frame is a premise about reference configurations; observable curvature belongs to their System Frame realization. Galactic-rotation calculations are presented in a separate MQ application (Ref. [2], Appx. AK) [13], and are not validated by the fraction comparison in this paper.

CMB evidence for spatial flatness is model- and data-dependent. The early BOOMERanG result [7] was interpreted through cosmological models; it did not measure the MQ dark-domain coordinate. In Planck’s extension allowing curvature, the CMB spectra alone favor a closed geometry, whereas adding lensing reconstruction and baryon acoustic oscillation data brings the constraints into consistency with flatness [22]. These are constraints on observable cosmological geometry, not proof that curvature is exactly zero. Age and curvature dependence are also discussed in [18].

5. APPENDICES

APPENDIX A: TERMS

The entries below supplement the definitions introduced in the text. Broader MQ terminology is developed in Measurement Quantization [16].

The three MQ realizations follow the introduction. The conserved-quantity construction in [6] is formulated at null infinity within general relativity and supplies no experimental calibration of the MQ coefficient.

The exact speed of light used in the local and cosmological relations is c=299792458 m s^-1 [23, 8].

G denotes the upper-count MQ gravitational realization, in m³ kg⁻¹ s⁻². Its value is

6.6740779383 (10) \times 10^{- 11} m³ kg⁻¹ s⁻².

(51a)

At the electromagnetic demarcation, the italic G gives

6.6738448317 (10) \times 10^{- 11} m³ kg⁻¹ s⁻².

(51b)

Both use the common 2022 calibration [8] and are conditional MQ outputs, not direct CODATA recommendations.

The fundamental measures and their physical units follow the introduction and Sec. 3. Their numerical values are evaluated in Eqs. (18), (20) and (21).

l_p, m_p and t_p are the conventional Planck reference length, mass and time, in m, kg and s, respectively.

The fundamental-measure counts follow the introduction. Local light-speed counts, cosmological counts and the one-kilogram reference are distinguished in Sec. 3.

The dimensionless count n_Lc describes the light displacement in fundamental lengths over the observer’s stated common time interval.

The count n_θ is the System Frame count of the invariant coefficient, resolved from the Internal Frame demarcation with the metric differential (Ref. [2], Appx. T).

The nearest-whole-unit convention is introduced before Eq. (2). Matched floor brackets and matched ceiling brackets retain their usual distinct mathematical meanings.

For positive separation count n_L,

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

(51c)

is the dimensionless right-triangle residual. Its use does not subdivide a fundamental reference.

The normalized factor 2Q_Ln_L enters the Informativity differential, describing the non-Lorentz contraction of realized separation. The metric differential used in Eq. (46) is distinct.

The conventional ℏ introduced with the calibration is h divided by twice π and is exact in SI. The distinct upright upper-count MQ realization ħ is 2p_sil_f, giving

1.0545349844417293 (56) \times 10^{- 34} J s

(51d)

from the same 2022 inputs [8]. This calculated MQ quantity does not change the exact SI defining value of h.

The electron mass is introduced before Eq. (14); the calibration uses its shared product with the Bohr radius.

The Bohr radius is introduced before Eq. (14) and enters the shared-product identity preceding Eq. (18).

The dimensionless fundamental inverse coupling

{α_{f}}^{- 1}

(51e)

=42θ_si belongs to the configuration representation. The Planck-like and classical couplings were introduced before Eqs. (14) and (2), respectively. Their distinction accounts for the finite-count Informativity differential in the calibration (Ref. [2], Appx. AA).

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 manuscript has data included as electronic supplementary material.

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