Extending Precision of the Physical Constants

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We evaluate Measurement Quantization (MQ) relations for fundamental measures, gravitational and reduced Planck quantities, referenceability domains, and electromagnetic quantities using the fixed 2022 recommendation of the Committee on Data for Science and Technology (CODATA). Fundamental gamma is constructed from count relations and mapped into International System of Units (SI) charge units. The reported precision is conditional on the MQ count constraints and realization assumptions. A separate rubidium-recoil input from Morel et al. illustrates source sensitivity and is not used to assess framework accuracy. The exact-charge inversion and electromagnetic identities are consistency calculations. Source propagation excludes model uncertainty, and the domain fractions are not established cosmological density predictions.

Keywords: Physical Constants, Fundamental Measures, Planck Units, Precision, Fine Structure Constant, Gravitational Constant, Reduced Planck Constant

1. INTRODUCTION

We use MQ to relate discrete counts of length, mass and time to observable quantities. The Internal Frame is the domain of integer count configurations; the System Frame is its continuous relational encoding through the Frames mapping ([1], Appx. K). The mapping relates count configurations to physical realizations. The underlying MQ argument combines discrete gravity ([1], Appx. C), the speed of light, escape velocity and Heisenberg’s uncertainty principle [2] to motivate discrete and countable measure ([1], Appx. I, J). Here we use those relations as the premises for calculating the physical constants.

MQ uses the discrete count description and its continuous realization through the Frames mapping to organize the physical constants [1, 3]. Its count constraints are premises of this calculation. The background-independence and quantum-geometry programs discussed in [4–6] provide a different theoretical context; the present calculation does not establish an advantage over those programs.

The fundamental mass m_f, length l_f and time t_f have SI realizations in kg, m and s. The classical inverse fine-structure quantity

α_{c}^{- 1}

and the Planck-like form

α_{p}^{- 1}

are dimensionless. The Planck-like form precedes the finite-count normalization that gives the classical form. We calibrate the dimensionless invariant coefficient θ_si from

α_{c}^{- 1} = 137.035999177 (21)

, the fixed 2022 CODATA input [7]. Its momentum realization is p_si=R_p(θ_si)=l_fm_f/(2t_f), in kg m s⁻¹, where R_p denotes the MQ momentum realization operator ([1], Appx. V). The adopted SI normalization assigns this momentum the same numerical coefficient as θ_si; it does not identify a dimensionless count with momentum. The momentum context of Planck’s construction [8] is retained. Fundamental length follows from the orbital expression ([1], Appx. BL). Here m_e is electron mass in kg and a₀ is the Bohr radius in m. Their product is

m_{e} a_{0} = ℏ α_{c}^{- 1} / c

, where c=299792458 m s^-1 is the exact speed of light,

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

is the exact SI Planck constant, and ℏ=h/(2π) is the conventional reduced Planck constant in J s [7]. The Bohr identity removes electron mass as an independent calibration input. Here π is the dimensionless circle constant.

Table 1 summarizes the calculated quantities and their historical comparisons. The angular realization Θ_si, in rad, is distinct from the momentum realization. The gravitational constant G has units m³ kg⁻¹ s⁻². Upright G and ħ denote MQ realizations in the limit of unbounded radial count; their finite-count forms are italicized. Vacuum permittivity ε₀, permeability μ₀ and the Coulomb constant k_e have units F m⁻¹, H m⁻¹ and N m² C⁻². The dimensionless symbols Ω_f, Ω_dk, Ω_obs, Ω_vis and Ω_uobs denote the fundamental, dark, observable, visible and unobserved referenceability fractions, respectively, and are displayed as percentages ([1], Appx. BB). They describe the realized System Frame domains. The visible domain is presently referenceable, the unobserved domain becomes referenceable later, the observable domain contains both, and the dark domain remains non-referenceable. The fundamental fraction is a normalization pivot rather than an additional partition. Their geometric interpretation is developed in Sec. 3.1. The dimensionless cosmological comparison symbols Ω_Λ, Ω_c and Ω_b denote dark-energy, cold-dark-matter and baryonic density parameters, respectively. The MQ gravity expression ([1], Appx. C) and the fundamental gamma construction supply the gravitational and electromagnetic calculations below.

Table 1. Main MQ results from the 2022 calibration, with source roles and comparison values identified.

Measure	Reference or comparison	MQ Calculation
Momentum realization p_si	Computed Θ_si=3.26239 rad; tolerance 2×10^-6 rad ^a	p_si=3.26239030504(25) kg m s^-1
Fundamental mass m_f	2022-based MQ derivation, Eq. (16) e	$2.17643254057 (17) \times 10^{- 8} kg$
Fundamental length l_f	2022-based MQ derivation, Eq. (18) ^d	$1.61619991148 (12) \times 10^{- 35} m$
Fundamental time t_f	2022-based MQ derivation, Eq. (19) ^d	$5.39106261131 (41) \times 10^{- 44} s$
Planck-like inverse fine structure $α_{p}^{- 1}$	137.040785652(11) ^d	137.040785623(21)
Upper-count gravitational constant G	6.67408(31) 10^-11 m³ kg^-1 s^-2^d	$6.6740779383 (10) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$
Upper-count reduced Planck constant ħ	-	$1.0545349844417293 (56) \times 10^{- 34} J s$
Conventional reduced Planck constant ℏ	$1.054571817 \dots \times 10^{- 34} J s$ exact ^b	$1.054571817 \dots \times 10^{- 34} J s$
Fundamental Ω_f	-	18.7913577188(29)%
Dark Ω_dk (Ω_Λ)	68.89 ± 0.56%^c	68.3624161224(41)%
Observable Ω_obs (total matter)	31.11 ± 0.56%^c	31.6375838776(41)%
Visible Ω_vis (Ω_b)	4.90% central conversion ^c	4.8488348909(10)%
Unobserved Ω_uobs (Ω_c)	26.07% central conversion ^c	26.7887489867(31)%
Electric constant ε₀	$8.8541878128 (13) \times 10^{- 12} F m^{- 1}$ ^b	$8.8541878035 (20) \times 10^{- 12} F m^{- 1}$
Magnetic constant μ₀	$1.25663706212 (19) \times 10^{- 6} H m^{- 1}$ ^b	$1.25663706345 (28) \times 10^{- 6} H m^{- 1}$
Coulomb’s constant k_e	$8.9875517923 (14) \times 10^{9} N m^{2} C^{- 2}$ ^b	$8.9875518017 (20) \times 10^{9} N m^{2} C^{- 2}$

^a Shwartz and Harris give the angle as a numerical phase-matching solution [9], not a direct measurement of the momentum realization. The 2 microradian comparison tolerance is adopted here. An MQ error estimate below 2 microradians is stated in [10], Sec. 3.3; it is not a published standard uncertainty in [9].

^b Historical 2018 CODATA comparison [11]. Current calculations use the 2022 input [7]. Parenthetical digits denote one standard uncertainty in the last displayed digits; an ellipsis denotes continuing digits of an exact value.

^c These comparisons use the final TT,TE,EE+lowE+lensing+BAO column of [12], Table 2; [13] provides the earlier Planck context. TT, TE and EE identify the temperature auto-spectrum, temperature–E-polarization cross-spectrum and E-polarization auto-spectrum of the cosmic microwave background (CMB). The lowE label identifies polarization at low multipoles, lensing identifies the CMB lensing reconstruction, and BAO denotes baryon acoustic oscillations. The cosmological symbols retain the density-parameter meanings introduced before Table 1. The total-matter comparison also includes the neutrino contribution in the base cosmology. The baryon and cold-dark-matter percentages are central conversions using the published physical-density parameters 0.02242 and 0.11933 and Hubble constant H₀=67.66 km s^-1 Mpc^-1. No transformed uncertainty is assigned without the parameter covariance. These associations with MQ domains are interpretations, not identities of definitions.

^d The gravitational comparison is the 2014 CODATA value [14]. The fundamental-measure rows marked e use the derivations in Eqs. (16), (18), (19) with the 2022 input ledger [7]. They are calculated MQ quantities, rather than direct CODATA recommendations. The Planck-like comparison marked f is transformed from the Morel et al. classical measurement [15] using Eqs. (8), (17); its full dependent calculation is given in Table 3.

The following derivation identifies the assumptions, physical realizations and shared calibration dependencies behind the current column of Table 1. Its additional digits quantify propagated source precision conditional on these premises.

2. METHODS

The calculation proceeds from the inverse fine-structure input to the invariant coefficient and its count-dependent normalization, then to fundamental measures and the derived quantities. These steps implement the MQ relations described in [1, 3, 16–19, 10]. The main calculation retains the fixed 2022 calibration [7]. We propagate its uncertainty through all dependencies, holding exact SI inputs and specified count premises fixed. Electron mass and Bohr radius are not added as independent adjusted inputs after their product has been reduced by the identity in Sec. 1. The Morel et al. calculation [15] carries that experiment’s published input and uncertainty through the same chain solely to examine sensitivity to source choice. It does not replace the CODATA calibration.

The discrete count description is a model premise in this calculation. Measurement Quantization [1] presents its derivations, reported comparisons, referenced appendices and term definitions. Neither the calibration of the relations below nor their internal consistency alone establishes the physical discreteness of measure. Earlier supporting material is identified in Online Resources.

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

The accompanying calculation record gives the source inputs, dependence, uncertainties and numerical results for this paper. Earlier MQ applications are discussed in [16–19, 10]; those applications are not additional independent tests of the common fine-structure calibration used here.

3. RESULTS

3.1. Nomenclature

The fundamental measures differ from Planck units l_p, m_p and t_p for length, mass and time ([1], Appx. W, BN). Here n_L is the positive dimensionless radial separation count in the right-triangle construction, referenced to the specified center of mass, and

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

is its dimensionless residual. Its mapped separation is n_Ll_f, in m. The finite noninteger count values in this calculation are continuum encodings of the discrete count relations, rather than integer counts of primitive configurations. The fine-structure, blackbody and charge-inversion uses retain the explicitly specified counts identified below. The Informativity differential 2Q_Ln_L is the MQ normalization for non-Lorentz length contraction associated with this count geometry ([1], Appx. D). Reference [20] supplies historical relativistic context; it is not evidence for that MQ normalization. The frame definitions in Sec. 1 apply to the count geometry ([1], Appx. K). Historical MQ comparisons for the 2010, 2014 and 2018 CODATA editions are presented in [1], Table 2.

The typography introduced with Table 1 distinguishes upper-count from finite-count realizations. Equation (1) evaluates the gravitational expression at the explicitly fixed count n_L=84.60055; Eq. (2) gives its upper-count limit ([1], Appx. F). The refined fine-structure count in Eq. (13), blackbody count in Eq. (22), and separate elementary-charge inversion in Eq. (36) retain their specified physical roles ([1], Appx. AB, AE). The charge inversion is derived in Eqs. (35)–(36).

G = {(2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}})}_{n_{L} = 84.60055} = 6.6738448317 (10) \times 1 0^{- 11} ​ m^{3} ​ k g^{- 1} ​ s^{- 2}

(1)

G = \lim_{n_{L} \to \infty} 2 Q_{L} n_{L} c^{3} \frac{t_{f}}{m_{f}} = c^{3} \frac{t_{f}}{m_{f}} = 6.6740779383 (10) \times 1 0^{- 11} ​ m^{3} ​ k g^{- 1} ​ s^{- 2}

(2)

Equations (1–2) are MQ calculations using the fixed 2022 calibration [7]. Equation (1) retains the starting count 84.60055; Eq. (2) takes the upper-count limit. Historical CODATA comparisons are separate from these updated calculations.

In cosmology, the dimensionless density parameter Ω is the ratio of energy density ρ to critical energy density ρ_c, both expressed in J m⁻³ [12]. The MQ fractions here describe referenceability domains of the realized geometry. Associating the two requires a physical interpretation beyond their similar numerical sizes. Equations (24–28) determine the MQ fractions from the calibrated coefficient; they do not supply a cosmological expansion history or independently establish the observed density parameters.

The five domain symbols introduced in Table 1 distinguish this geometric description from conventional cosmological density parameters. The visible and unobserved domains partition the observable domain; the observable and dark domains are complementary. The fundamental domain is a separate referenceability fraction, so the five entries are not summed as disjoint components. Associations with dark matter [21–23] and dark energy [24, 25] are developed in the MQ applications [16, 17].

The angular and momentum realizations introduced in Sec. 1 carry different physical meanings. In the 25-keV degenerate calculation of Shwartz and Harris [9], the signal and idler phase-matching angles include 3.26239 rad. The later Langevin-regime experiment [26] concerns coincidence rates and different detector angles; it does not give a refined uncertainty for that solution. Glover et al. [27] report an analyser rocking-curve width of 17±2 μrad and a sample width of 8.1±0.7 μrad. Those widths concern another observable and are not uncertainties on Θ_si. The Bell-state context [28, 29] and conserved-quantity discussion ([30], Sec. 4.3) remain distinct from the momentum realization, which enters the gravitational and reduced Planck expressions ([1], Appx. A, Q, S).

3.2. Calculations

We use quantization ratios ([1], Appx. AC) to compare dimensionless count products with their nearest-whole-unit values. The mixed brackets ⌊x⌉ mean rounding the dimensionless argument x to the nearest whole unit. The System Frame count n_θ=⌊n_L/2⌉ is 42, while n_L retains its radial meaning. The fine-structure remainder is 42θ_si-⌊42θ_si⌉ ([1], Appx. AA). The experimental and quantum electrodynamics (QED) determinations discussed in [31, 15, 32] provide fine-structure measurement context, rather than external derivations of this MQ remainder. Elementary-charge relations are discussed in [1], Appx. AM.

The MQ charge-coupling constraint fixes n_Lθ_si=276. At this specified demarcation, Eq. (3) gives the dimensionless quantization ratio r_q, and Eq. (4) states the adopted count constraint. The unit ratio follows because 276 is already a whole unit; it is not an additional measured input.

r_{q} = \frac{n_{L} θ_{si}}{⌊ n_{L} θ_{si} ⌉} = 1

(3)

n_{L} θ_{si} = 276

(4)

For the radial count introduced in Sec. 3.1, Eq. (5) is the right-triangle residual ([1], Appx. C). Multiplying it by the positive coefficient θ_si gives Eq. (6); substituting the count constraint in Eq. (4) gives Eq. (7). The count refers to the separation from the specified center of mass in this calculation.

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

(5)

Q_{L} θ_{si} = \sqrt{θ_{si}^{2} + {(θ_{si} n_{L})}^{2}} - n_{L} θ_{si}

(6)

Q_{L} = \frac{\sqrt{θ_{si}^{2} + 27 6^{2}} - 276}{θ_{si}}

(7)

The metric differential describes the mismatch between discrete count structure and its continuous realization ([1], Appx. T). In the classical fine-structure relation ([1], Appx. AA), the nearest-whole-unit term supplies the count correction and the Informativity differential supplies the finite-count normalization 2Q_Ln_L ([1], Appx. C). Rearranging Eq. (8) gives Eq. (9). Substitution of Eqs. (4), (7) gives Eq. (10), which simplifies to Eq. (11). Solving this relation with the input in Sec. 1 yields Eq. (12). The solution lies on the rounding branch ⌊42θ_si⌉=137; this branch remains unchanged across the propagated input uncertainty.

\frac{1}{α_{c}} = 2 Q_{L} n_{L} (84 θ_{si} - ⌊ 42 θ_{si} ⌉)

(8)

2 α_{c} n_{L} θ_{si} = \frac{θ_{si}}{Q_{L} (84 θ_{si} - ⌊ 42 θ_{si} ⌉)}

(9)

2 \times 276 α_{c} = \frac{θ_{si}}{\sqrt{θ_{si}^{2} + 27 6^{2}} - 276} \frac{θ_{si}}{84 θ_{si} - ⌊ 42 θ_{si} ⌉}

(10)

2 \times 276 α_{c} = \frac{θ_{si}^{2}}{(\sqrt{θ_{si}^{2} + 27 6^{2}} - 276) (84 θ_{si} - ⌊ 42 θ_{si} ⌉)}

(11)

θ_{si} = 3.26239030504 (25)

(12)

Equation (13) resolves the fine-structure length count from Eq. (4) ([1], Appx. AE). Equation (14) gives its physical separation using the fundamental length calculated in Eq. (18). The elementary-charge inversion in Eq. (36) is evaluated separately; it does not replace an explicitly specified fine-structure or blackbody count.

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

(13)

n_{L} l_{f} = 1.36731394425 (21) \times 1 0^{- 33} ​ m

(14)

Equation (15) reports the residual-count product Q_Ln_L. The Informativity differential is twice this product.

Q_{L} n_{L} = 0.4999825364172403 (27)

(15)

Table 2 summarizes the current charge construction. The quantities e_f, e_p and e_c are its fundamental, Planck-like and classical charge realizations, in C after Frames mapping. The symbol K denotes the dimensionless product Q_Ln_L at the specified count. Fundamental gamma γ_F is the dimensionless configuration-domain coefficient; γ_F,sys is its mapped realization in C². The configuration domain has no physical dimensions. The numerical normalization and charge expressions are given in Eqs. (29)–(34).

Table 2. Fundamental gamma and charge calculations at the specified counts.

Quantity	Fine-structure count	Exact-charge inversion
n_L	84.6005456716(65)	84.60159(23)
e_f	$1.60512538120 (30) \times 10^{- 19} C$	$1.60512538120 (30) \times 10^{- 19} C$
e_p	$1.60223259406 (30) \times 10^{- 19} C$	$1.60223259406 (30) \times 10^{- 19} C$
e_c	$1.60217663261 (30) \times 10^{- 19} C$	$1.602176634 \times 10^{- 19} C$ exact
γ_F,sys	$2.19846879611 (49) \times 10^{- 37} C^{2}$	$2.1984688037 (12) \times 10^{- 37} C^{2}$

The fine-structure column uses the count in Eq. (13). The charge-input column uses Eq. (36) and the exact SI elementary charge,

e = 1.602176634 \times 10^{- 19} C

[7]. Recovering this input charge is a consistency calculation. The charge-input count is not used to calculate the electromagnetic entries in Table 1.

With the SI momentum normalization in Sec. 1, the fundamental relation l_fm_f=2p_sit_f ([1], Appx. V) and l_f/t_f=c give the fundamental mass in Eq. (16).

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

(16)

Equation (17) evaluates the Planck-like inverse quantity introduced in Sec. 1 ([1], Appx. AA). It is the factor inside the parentheses in Eq. (8), before the non-Lorentz normalization supplied by the Informativity differential. Applying that normalization gives the classical inverse fine-structure quantity measured in the laboratory. Solving Eq. (8) from the 2022 published input and then evaluating Eq. (17) is therefore a consistency calculation. Section 3.3 retains a separate experimental input as a source-sensitivity comparison.

α_{p}^{- 1} = 84 θ_{si} - ⌊ 42 θ_{si} ⌉ = 137.040785623 (21)

(17)

The orbital expression ([1], Appx. BL) gives Eq. (18). The 2022 publication reports

m_{e} = 9.1093837139 (28) \times 10^{- 31} kg

and

a_{0} = 5.29177210544 (82) \times 10^{- 11} m

[7]. These are correlated adjusted quantities. The product identity in Sec. 1 is used before uncertainty propagation, avoiding an unsupported independence assumption and preserving the shared inverse fine-structure dependence.

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

(18)

The electron-mass dependence cancels in m_ea₀ under the identity introduced in Sec. 1. Equation (18) consequently uses the 2022 inverse fine-structure input and exact h and c, together with the MQ momentum realization and Planck-like relation. Equation (19) converts the calculated length to fundamental time. Neither result is a direct CODATA recommendation for an MQ fundamental measure.

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

(19)

The upper-count gravitational constant follows from the fundamental measures ([1], Appx. F).

G = c^{2} \frac{l_{f}}{m_{f}} = 6.6740779383 (10) \times 1 0^{- 11} ​ m^{3} ​ k g^{- 1} ​ s^{- 2}

(20)

The same measures determine the upper-count reduced Planck realization in Eq. (21) ([1], Appx. L). This value is distinct from the conventional exact input used in Eq. (22).

ħ = 2 p_{si} l_{f} = 1.0545349844417293 (56) \times 1 0^{- 34} ​ J ​ s

(21)

The blackbody expression uses the conventional exact reduced Planck constant ([1], Appx. AB). Substitution of the shared calibration into Eq. (22) gives the same count as Eq. (13). The expressions are evaluated separately, but their numerical equality follows from their shared dependencies and is not an independent calibration.

n_{L} = p_{si} l_{f} \sqrt{\frac{1}{ℏ (ℏ - 2 p_{si} l_{f})}} = 84.6005456716 (65)

(22)

Equation (23) returns the conventional reduced Planck constant exactly under these shared dependencies. The ellipsis indicates continuing digits, not a standard uncertainty.

ℏ = \frac{p_{si} l_{f}}{Q_{L} n_{L}} = 1.054571817 \times 1 0^{- 34} ​ J ​ s

(23)

Equations (24)–(28) calculate the five referenceability-domain fractions from θ_si ([1], Appx. BB). Equation (27) selects the visible fraction within the observable domain, and Eq. (28) gives its unobserved remainder. Their percentages in Table 1 use the same source uncertainty as the preceding constants.

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

(24)

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

(25)

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

(26)

Ω_{vis} = \frac{Ω_{obs}}{2 θ_{si}} = 4.8488348909 (10) %

(27)

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

(28)

We now apply the fundamental gamma construction introduced with Table 2. Equation (29) expresses the dimensionless coefficient through the Planck-like charge and the residual-count product; Eqs. (30)–(33) specify its numerical SI realization. Division by the displayed SI units in Eq. (30) extracts dimensionless numerical coefficients, rather than assigning physical dimensions to the configuration domain. The dimensionless residual

Q_{n_{θ}}

in Eq. (31) expresses the metric differential and differs from the integer count n_θ. The quantity n_e is a charge-valued intermediate in C. Equation (32) defines the adopted cubic charge construction, and substitution into Eq. (29) gives the fundamental gamma relation of the unpublished Unification manuscript ([33], Appx. H, Eq. (H.6)), with the residual notation made explicit.

γ_{F} = α_{p}^{- 1} {((Q_{L} n_{L})^{2} \frac{e_{p}}{1 ​ C})}^{2}

(29)

\frac{e_{f}}{1 ​ C} = \frac{\frac{l_{f}}{1 ​ m} {(\frac{c}{1 ​ m ​ s^{- 1}})}^{2}}{4 (θ_{si} - 1)}

(30)

Q_{n_{θ}} = 42 θ_{si} - ⌊ 42 θ_{si} ⌉, ​ n_{e} = (84 \frac{e_{f}}{1 ​ C} - ⌊ 42 \frac{e_{f}}{1 ​ C} ⌉) (1 ​ C)

(31)

e_{p} = \frac{n_{e}}{84 + \frac{42}{277 - 3 Q_{n_{θ}} - 7 Q_{n_{θ}}^{2} - 14 Q_{n_{θ}}^{3}}}

(32)

γ_{F, sys} = (1 ​ C)^{2} γ_{F} = α_{p}^{- 1} {((Q_{L} n_{L})^{2} e_{p})}^{2} = 2.19846879611 (49) \times 1 0^{- 37} ​ C^{2}

(33)

The numerical value in Eq. (33) uses the fine-structure count in Eq. (13), which also satisfies the blackbody expression in Eq. (22). Equation (34) applies the Informativity differential to the Planck-like charge. Its square-root form follows directly from Eq. (33); the positive root is selected because charge magnitudes and the residual-count product are positive. For the separate elementary-charge inversion, the exact input e specified below Table 2 replaces the predicted charge, giving Eqs. (35)–(36). This produces the default charge demarcation; a phenomenon-specific count takes precedence when explicitly specified.

e_{c} = 2 Q_{L} n_{L} e_{p} = \frac{2}{Q_{L} n_{L}} \sqrt{\frac{γ_{F, sys}}{α_{p}^{- 1}}} = 1.60217663261 (30) \times 1 0^{- 19} ​ C

(34)

K = Q_{L} n_{L} = \frac{e}{2 e_{p}}, ​ Q_{L} = \frac{1}{\sqrt{1 + n_{L}^{2}} + n_{L}}

(35)

n_{L} = \frac{K}{\sqrt{1 - 2 K}} = \frac{e}{2 e_{p} \sqrt{1 - e / e_{p}}} = 84.60159 (23)

(36)

For the exact-charge inversion, K is evaluated at the charge-input count in Eq. (36); the finite positive solution requires 0<K<1/2. Squaring

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

gives

n_{L}^{2} (1 - 2 K) = K^{2}

, hence Eq. (36). The electromagnetic calculations in Eqs. (37)–(39) evaluate Q_L and n_L at the fine-structure count in Eq. (13), with its mapped fundamental gamma. With Eq. (23),

α_{c}^{- 1} = 2 Q_{L} n_{L} α_{p}^{- 1}

and Eq. (34), the conventional relation

ε_{0} = α_{c}^{- 1} e_{c}^{2} / (4 π ℏ c)

reduces to Eq. (37). Equations (38)–(39) then follow from μ₀=1/(ε₀c²) and k_e=1/(4πε₀). Here these symbols label the MQ charge-based calculations; recommended SI comparison values retain their identified source roles. Their uncertainties retain the common source dependencies.

ε_{0} = \frac{4 γ_{F, sys}}{π l_{f} m_{f} c^{2}} = 8.8541878035 (20) \times 1 0^{- 12} ​ F ​ m^{- 1}

(37)

μ_{0} = \frac{π l_{f} m_{f}}{4 γ_{F, sys}} = 1.25663706345 (28) \times 1 0^{- 6} ​ H ​ m^{- 1}

(38)

k_{e} = \frac{l_{f} m_{f} c^{2}}{16 γ_{F, sys}} = 8.9875518017 (20) \times 1 0^{9} ​ N ​ m^{2} ​ C^{- 2}

(39)

3.3. Calibration and source sensitivity

We use θ_si resolved from the 2022 CODATA recommendation [7] for the authoritative calculation. Future recalibration should use a specified later CODATA edition consistently throughout the dependent calculation. The rubidium-recoil measurement of Morel et al. [15],

α_{c}^{- 1} = 137.035999206 (11)

, remains the selected experimental comparison. Its value is independent of the MQ construction and contributes to the CODATA adjustment, so the two inputs are not statistically independent. Its column in Table 3 illustrates source sensitivity; the paper does not require the same concordance from an individual experimental input or use that comparison alone to rank framework accuracy.

For each input, Eqs. (4), (8), (17) determine the count and Planck-like quantity, equivalently

α_{p}^{- 1} = α_{c}^{- 1} / (2 Q_{L} n_{L})

. The Morel calculation uses the matched product

m_{e} a_{0} = ℏ α_{c}^{- 1} / c

throughout the same dependency chain. Its results are the sensitivity column in Table 3. CODATA imposes observational relations and includes covariance among correlated inputs, but adjusts G separately because it is independent of the other constants in that adjustment [7], Sec. XV.A. This separation does not imply that all electromagnetic and other constants are uncorrelated, or that the adjustment enforces additional MQ cross-constant relations. Substituting one experiment does not establish concordance of an entire set of recommended constants under an additional theoretical relation. Recovery of a calibration input remains a consistency calculation.

Table 3. MQ results for the authoritative 2022 CODATA calibration and Morel source-sensitivity comparison.

Quantity	2022 CODATA calibration	Morel 2020 sensitivity
p_si	3.26239030504(25) kg m s^-1	3.26239030538(13) kg m s^-1
θ_si	3.26239030504(25)	3.26239030538(13)
n_L fine structure	84.6005456716(65)	84.6005456627(34)
n_L blackbody	84.6005456716(65)	84.6005456627(34)
$α_{p}^{- 1}$	137.040785623(21)	137.040785652(11)
m_f	$2.17643254057 (17) \times 10^{- 8} kg$	$2.176432540797 (87) \times 10^{- 8} kg$
l_f	$1.61619991148 (12) \times 10^{- 35} m$	$1.616199911307 (65) \times 10^{- 35} m$
t_f	$5.39106261131 (41) \times 10^{- 44} s$	$5.39106261074 (22) \times 10^{- 44} s$
n_Ll_f	$1.36731394425 (21) \times 10^{- 33} m$	$1.36731394396 (11) \times 10^{- 33} m$
Q_Ln_L	0.4999825364172403(27)	0.4999825364172366(14)
G upper count	$6.6740779383 (10) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$	$6.67407793687 (54) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$
G at 84.60055	$6.6738448317 (10) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$	$6.67384483026 (54) \times 10^{- 11} m^{3} {kg}^{- 1} s^{- 2}$
ħ upper count	$1.0545349844417293 (56) \times 10^{- 34} J s$	$1.0545349844417215 (30) \times 10^{- 34} J s$
ℏ conventional	$1.054571817 \dots \times 10^{- 34} J s$	$1.054571817 \dots \times 10^{- 34} J s$
Ω_f	18.7913577188(29) %	18.7913577148(15) %
Ω_dk	68.3624161224(41) %	68.3624161281(21) %
Ω_obs	31.6375838776(41) %	31.6375838719(21) %
Ω_vis	4.8488348909(10) %	4.84883488952(52) %
Ω_uobs	26.7887489867(31) %	26.7887489824(16) %
e_f	$1.60512538120 (30) \times 10^{- 19} C$	$1.60512538078 (16) \times 10^{- 19} C$
e_p	$1.60223259406 (30) \times 10^{- 19} C$	$1.60223259364 (16) \times 10^{- 19} C$
e_c predicted	$1.60217663261 (30) \times 10^{- 19} C$	$1.60217663220 (16) \times 10^{- 19} C$
γ_F,sys	$2.19846879611 (49) \times 10^{- 37} C^{2}$	$2.19846879544 (26) \times 10^{- 37} C^{2}$
n_L charge inversion	84.60159(23)	84.60191(12)
γ_F,sys charge inversion	$2.1984688037 (12) \times 10^{- 37} C^{2}$	$2.19846880533 (61) \times 10^{- 37} C^{2}$
ε₀	$8.8541878035 (20) \times 10^{- 12} F m^{- 1}$	$8.8541878008 (10) \times 10^{- 12} F m^{- 1}$
μ₀	$1.25663706345 (28) \times 10^{- 6} H m^{- 1}$	$1.25663706383 (15) \times 10^{- 6} H m^{- 1}$
k_e	$8.9875518017 (20) \times 10^{9} N m^{2} C^{- 2}$	$8.9875518045 (10) \times 10^{9} N m^{2} C^{- 2}$

The charge-inversion rows use the exact SI charge; the predicted charge and electromagnetic rows use the explicitly specified fine-structure count. The conventional reduced Planck value is an exact closure in both columns. Parenthetical digits describe propagated source uncertainty, conditional on the MQ premises. Covariance within each calculation is retained; cross-column statistical independence is not assumed.

4. DISCUSSION

The SI outputs depend on the inverse fine-structure input, the count constraints and the specified momentum and charge realizations. Their uncertainties describe source propagation conditional on these premises, rather than experimental accuracy or model uncertainty. In particular, assigning the same numerical coefficient to the momentum realization in the adopted SI units is a normalization assumption. It is not established by dimensional analysis or by the Morel measurement.

The SI speed of light and Planck constant are exact [7]. Their finite decimal specifications do not impose a digit limit on the calculation. Uncertainty comes from the inverse fine-structure input and propagates jointly through its dependent quantities.

The fundamental gamma construction uses the count and charge premises in Eqs. (29)–(34), without measured permittivity, permeability or Coulomb constants as inputs. Its authoritative calculations retain the 2022 CODATA inverse fine-structure calibration [7]. Morel et al. [15] provides the separate source-sensitivity comparison in Sec. 3.3 and Table 3. Differences under that substitution alone establish neither improved nor reduced framework accuracy. Transforming either input into the Planck-like quantity remains conditional on MQ.

The charge prediction must be distinguished from recovery of an exact input. The 2022 calculation in Table 3 gives

e_{c} = 1.60217663261 (30) \times 10^{- 19} C

, whereas the exact SI elementary charge is

e = 1.602176634 \times 10^{- 19} C

[7]. The difference is 4.6 propagated source uncertainties in magnitude. The Morel sensitivity calculation gives a corresponding ratio of 11.5. These ratios omit unquantified model uncertainty and are conditional numerical diagnostics, rather than complete statistical significances or rankings of framework accuracy. The 2022 construction has a residual against the exact charge within these stated assumptions. The separate adjustment of G does not explain a charge residual against an exact SI definition. Substituting the exact charge into Eqs. (35)–(36) recovers it by construction; the electromagnetic outputs retain their predicted-charge dependence.

The upper-count gravitational calculation can also be compared with the 2022 recommendation,

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

[7]. Either calibration places the upper-count result about 1.5 published standard uncertainties below that recommendation. This comparison does not justify replacing its experimental uncertainty by the much smaller conditional MQ uncertainty. The finite-count result in Eq. (1) has its specified count premise and is not separately validated by the upper-count comparison.

Eliminating the common mapped fundamental gamma from Eqs. (37)–(39) gives Eqs. (40)–(41). These identities check internal consistency, rather than independently testing the count construction or its physical realization. Their mapping context is given in [1], Appx. T.

4 π ε_{0} k_{e} = 1

(40)

4 π k_{e} = μ_{0} c^{2}

(41)

Fundamental gamma belongs to the configuration-domain count construction. Its SI realization is specified in Eq. (33), and the Informativity differential supplies the charge normalization in Eq. (34). Physical dimensions are assigned through the Frames mapping, rather than to the configuration domain ([1], Appx. K, T).

The coefficient and its momentum and angular realizations retain the distinct meanings introduced earlier ([1], Appx. A, Q, S). The conserved-quantity discussion ([30], Sec. 4.3) is a separate physical context.

5. APPENDICES

APPENDIX A: TERMS

Foundational term definitions are given in Measurement Quantization [1]. The entries below identify the meanings used in this paper and refer back to their introductions.

The coefficient and its angular and momentum realizations are introduced in Secs. 1 and 3.1. The X-ray references [9, 27, 26] and conserved-quantity context ([30], Sec. 4.3) retain their separate roles.

The speed of light and its exact source status are introduced in Sec. 1.

The upper-count gravitational value is given in Eqs. (2), (20); the finite-count value at the starting count is given in Eq. (1). These realizations use upright and italic G, respectively.

The fundamental measures and their units are introduced in Sec. 1.

The Planck units are introduced in Sec. 3.1.

The dimensionless mass and time counts n_M and n_T count m_f and t_f, respectively, in the Internal Frame. The radial length count is introduced in Sec. 3.1.

For this paper, the length count has the radial separation meaning given before Eq. (5). It is not an interchangeable motion count.

The radial use of n_L refers to the specified center-of-mass separation introduced before Eq. (5).

The dimensionless light-path count n_Lc counts fundamental lengths along the change in position of light relative to the observer’s frame. It is distinct from the radial separation used here.

The count n_θ and its nearest-whole-unit operation are defined before Eq. (3).

The mixed bracket convention is introduced before Eq. (3).

The residual count is introduced in Sec. 3.1.

The Informativity differential and its normalization factor are introduced in Sec. 3.1.

The conventional reduced Planck constant and its exact source status are introduced in Sec. 1. Equations (21), (23) give its distinct MQ upper-count and finite-count realizations.

Electron mass is introduced in Sec. 1; its cancellation in the orbital product is explained before Eq. (19).

The Bohr radius is introduced in Sec. 1.

The dimensionless fundamental inverse fine-structure quantity

α_{f}^{- 1} = 42 θ_{si}

belongs to the Internal Frame count description. The Planck-like and classical forms introduced in Sec. 1 are System Frame realizations, with the classical form incorporating the finite-count normalization.

Fundamental gamma and its mapped realization are introduced before Table 2. Their count and SI forms are specified in Eqs. (29), (33), respectively.

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 source inputs, dependencies, propagated uncertainties and numerical results are supplied in the accompanying calculation and validation record.

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