Describing Coulomb’s Constant

Using Only the Fundamental Measures

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

Editorial Board Member for Scientific Reports/Nature

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

Online Resources available at: https://www.researchgate.net/publication/361148724_OnlineResources

Keywords: Coulomb, Coulomb’s Constant, Fundamental Measures, Planck Units, Measure, Discrete

1. INTRODUCTION

The MQ construction begins with the properties assigned to measure and the relations used to resolve them. Discrete expressions describing gravitational curvature, the speed of light, escape velocity, and Heisenberg's uncertainty principle [2] are evaluated using the dimensionless counts n_L, n_M, and n_T of the fundamental length, mass, and time measures l_f, m_f, and t_f, respectively. The parameter θ_si denotes the MQ mapping parameter relating these measures through the Frames mapping. Within MQ, the simultaneous solution of these relations identifies measure as discrete and countable in the Internal Frame.

The count values and fundamental measures are not assigned independently in the derivation. Instead, they are obtained from the joint constraints imposed by the four physical relations. The resulting fundamental measures ([3], Appx. W) refine the corresponding Planck unit constructions by incorporating the Informativity differential and the non-Lorentz length contraction contribution associated in MQ with discrete realization ([3], Appx. D).

The frame dependence of this construction is developed in ([3], Appx. K). The Internal Frame is the discrete configuration domain in which measure is represented through counts of the fundamental measures. The System Frame is the continuous physical realization produced by the Frames mapping (Appx. A). MQ attributes the difference between these descriptions to the Frames mapping, which enters its expressions for observable physical constants [4]. Coulomb's constant provides the electromagnetic example examined here.

Let c denote the speed of light in vacuum and let γ_F denote the dimensionless Frames mapping geometry factor derived later in the paper. Coulomb's constant is then obtained as

The purpose of this paper is to derive Eq. (1), including the independent construction of γ_F, and to compare the resulting value with the accepted value of Coulomb's constant.

2. METHODS

This paper applies the MQ count formalism to the established expressions for gravitational curvature, the speed of light, escape velocity, Heisenberg's uncertainty principle, the fine structure constant, elementary charge, vacuum electric permittivity, and Coulomb's constant. MQ expresses these relations in terms of counts of the fundamental measures and their realization under the Frames mapping. Previous applications of this methodology are presented in Refs. [3, 5–9].

The MQ realization adopted in this work is established from the 2022 CODATA recommended values of the fine structure constant, electron mass, and Bohr radius together with the defined value of the speed of light. These source measurements determine the MQ mapping parameter, the fundamental measures, and the Frames mapping. The same framework is then used to derive all remaining constants. None of the latter quantities participates in establishing the MQ framework used to derive them.

The derivation therefore distinguishes between source quantities used to establish the MQ realization and quantities subsequently predicted by that realization. Agreement with experimentally accepted values is assessed through concordance analysis. Although the 2022 CODATA recommended constants are obtained through a least-squares adjustment and therefore exhibit statistical covariance, the physical quantities compared remain independent observables. The reported concordance should therefore be interpreted as a measure of agreement between the MQ framework and accepted physical constants while recognizing the covariance introduced by the CODATA adjustment.

The MQ framework is also evaluated alongside the corresponding classical framework by expressing the same physical relations using their respective systems of fundamental units. Classical expressions are converted in the same manner as conventional Planck unit formulations, whereas MQ expressions are derived from the right-angle triangle construction and explicitly incorporate the Frames mapping between the Internal Frame and the System Frame. Because the MQ framework adds a prediction of non-Lorentz length contraction which greatly differs between electromagnetic and macroscopic phenomena, the resulting concordance provides a direct comparison of their ability to describe the same physical phenomena when both electromagnetic and macroscopic constants are used in the same description. In the present work the MQ framework yields an overall concordance of 8.74σ, whereas the corresponding classical framework yields no statistically significant concordance.

The MQ premises concerning discrete and countable measure are developed in the introductory summary publication Measurement Quantization [3]. Rather than reproducing those derivations here, this paper builds upon the specific MQ constructions listed in Table 1.

Table 1. MQ Derivations Supporting the Present Work

The complete index of MQ derivations, together with additional symbol, terminology, and concept definitions, is provided in Appx. BR of the paper. Appendix BR serves as the master navigation index for the complete MQ framework and identifies the derivations supporting all subsequent MQ applications.

The complete numerical implementation supporting the MQ framework is available through the Online Resources:

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

Supplementary Material 1 contains spreadsheet implementations of every equation presented in the summary Measurement Quantization paper, including both sides of every expression. Supplementary Material 2 contains the complete spreadsheet implementation for the present manuscript together with more than fifty additional MQ papers, with each paper organized on its own worksheet. These spreadsheets include all intermediate calculations, uncertainty propagation, concordance analysis, and numerical implementations required to reproduce the reported results.

Because several MQ derivations require numerical precision exceeding the approximately 15 significant digits provided by standard spreadsheet arithmetic, all calculations are performed using the XLPrecision PrecisionCalc Excel plug-in. Mathematical expressions are prepared using MathType 6.9d (or later) to ensure consistent equation formatting throughout the MQ publication series. The supplementary spreadsheets therefore constitute the complete computational record supporting every numerical result presented in this paper.

3. RESULTS

A solution for Coulomb's constant is obtained by successively resolving the intermediate quantities required by the MQ construction using only the fundamental measures. The derivation begins with the count relations associated with the electromagnetic demarcation and proceeds through the corresponding Frames mapping quantities.

3.1. Supporting Expressions

Within MQ, the blackbody, charge coupling, and elementary charge demarcations provide independent determinations of a common interaction length count n_L. The blackbody demarcation is developed in ([3], Appx. AB), the charge coupling demarcation in ([3], Appx. AE), and the elementary charge demarcation in ([3], Appx. AN). These determinations resolve n_L to approximately the common value of 84.60055+. For the blackbody demarcation,

Here, n_L denotes the interaction length count, l_f the fundamental length, t_f the fundamental time, ħ the reduced Planck constant, and θ_si the MQ mapping parameter.

At the whole-count level this relation yields,

The interaction length count n_L=84.60055 therefore corresponds to n_θ=42. The charge coupling demarcation ([3], Appx. AE) provides the interaction length determination adopted in this derivation:

And with these expressions we resolve the value of θ_si ([3], Appx. AD) as a function of the 2022 recommended value for the fine structure constant [10],

Evaluating this expression gives

The corresponding count then determines the metric differential,

as defined in ([3], Appx. T).

Application of the non-Lorentz length contraction construction developed in ([3], Appx. D) resolves the contraction associated with the discrete Internal Frame,

Q_L denotes the fractional length count contribution. The product 2Q_Ln_L is more commonly known as the Informativity differential.

3.2. Inverse Fine Structure Constant

The three inverse fine structure forms are described by the fundamental definition, mapped to its realized Planck-like form, and then to its classical form after accounting for the non-Lorentz length contraction associated with the discrete Internal Frame ([3], Appx. AA),

Their calculation yield a 3.45σ concordance with the 2022 CODATA recommended values. Each are dimensionless. Accordingly, θ_si is used here as the dimensionless System Frame mapping parameter.

3.3. Elementary Charge

3.3.1 Fundamental elementary charge

The MQ construction begins with the fundamental form of the elementary charge e_f. The 2022 CODATA recommended values for the fine structure constant, the mass of an electron and the Bohr radius provide the basis for the calculated value of θ_si. When comparing the MQ calculation of the fine structure constant to its measure, we use and independent determination [11] distinct from the CODATA recommendation [10]. The identification of that geometry with the mapping parameter θ_si is made within MQ through a momentum description of θ_si ([3], Appx. Q), rather than constituting an experimental determination of e_f.

For this derivation, b denotes the unmapped base quantity associated with the fundamental charge expression, while d denotes the metric differential applied in passing between the corresponding frame representations. The base is written as the fundamental charge e_f multiplied by the mapped fundamental mass function f(m_f), relative to the factor ħ/2, where ħ is the reduced Planck constant entering the MQ momentum construction. The mapping parameter satisfies θ_si=l_fm_f/2t_f, through its momentum description ([3], Appx. Q). The mapped fundamental mass function is reduced using the mass relation given in ([3], Appx. V). These substitutions provide the first principles reduction from the temporary quantities b, d, and f(m_f) to an expression involving only e_f, θ_si, and the fundamental measures.

We identify this as the MQ fundamental form of elementary charge. It precedes both the non-Lorentz length contraction and the System Frame realization contribution.

3.3.2 Quantized and Classically Realized Elementary Charge

The quantization construction begins with a determination of metric differential associated with the count difference associated with the Frames mapping parameter,

This resolves the discrete count adjustment to a fundamental charge construction. The corresponding elementary charge remainder n_e is formed from e_f by the same construction as the inverse fine structure constant,

Here we take the offset due to the Frames mapping – resolved respect to θ_si in Eq. (19) – and subtract that from the initial measure (same as we do in the fine structure constant construction). The difference is a factor of two in the first term.

The Planck-like elementary charge is then obtained by

where the realization divisor is constructed from the repeating Diophantine realization sequence associated with the electromagnetic count. The whole-unit realization is specified by the integers 84 and 42, the inclusive reference count domain is given by 276+1, and the polynomial term introduces the dependence of the realization function on n_θ. The resulting mapping converts the elementary-charge residual n_e defined in the Internal Frame into the Planck-like elementary charge e_p, corresponding to its System Frame realization.

The classically realized elementary charge is resolved by application of the contraction Δ_(P-C)

The classically realized elementary charge is consequently

This completes the sequence from the MQ fundamental form e_f, through the mapped Planck-like form e_p, to the classically realized elementary charge e_c.

3.3.3. Electric Constant

The vacuum electric permittivity ε₀ is conventionally related to the vacuum magnetic permeability μ₀ and the speed of light by ε₀=1/(μ₀c²). It may also be expressed through the elementary charge, fine structure constant, Planck constant, and speed of light. Using ħ=h/(2π), the latter relation is written as

Substitution of the inverse classical form of the fine structure expression from Eq. (11) gives

and collection of common factors yields

Such that e_c follows from Eq. (24), e_c=2Q_Ln_Le_p, then

The terms describing the count geometry – Planck form of elementary charge, MQ form of the inverse fine structure constant, and the Informativity differential are then collected into the dimensionless Frames mapping factor,

This then reduces vacuum electric permittivity to

3.3.4. Calculations

Coulomb's constant is conventionally defined in SI by k_e=1/4πε₀. Substitution of Eq. (30) gives

and hence

For comparison, the 2022 CODATA recommended value is k_e = 8.9875517862(14) × 10⁹ N m² C^-2, a difference of 15.8 N m² C^-2. We can expression this in terms of the gravitational constant by introducing terms,

Such that the MQ description of the gravitational constant is

we can then substitute this into Eq. (33) to find that

This establishes the algebraic relation between Coulomb's constant and the MQ expression for gravitational curvature through the common fundamental measures and the Frames mapping factor γ_F. It does not by itself establish that electric charge is gravitational curvature; that physical interpretation requires a separate MQ derivation associated with the frame relation.

4. DISCUSSION

Coulomb's constant is conventionally related to the vacuum electric permittivity through k_e=1/(4πε₀). The factor 4π reflects the spherical solid-angle normalization of the electric flux produced by a point charge in three spatial dimensions. Within the MQ factorization k_e=γ_Fm_f/(2c²l_f), the same constant is reorganized in terms of the fundamental measures and the Frames mapping factor γ_F. This reorganization does not replace the conventional electromagnetic relation. Rather, it exposes a first principles construction based on the discrete count structure developed throughout the paper.

Unlike previous MQ implementations, the Frames mapping factor is not obtained from one of the electromagnetic constants whose values are subsequently reproduced. Instead, the derivation begins with the experimentally determined 2022 CODATA recommended values of the fine structure constant, electron mass, and Bohr radius together with the defined speed of light. These quantities determine the MQ mapping parameter, the fundamental measures, and the Frames mapping factor before the elementary charge, vacuum electric permittivity, and Coulomb's constant are derived. The resulting derivation therefore removes the circular dependency associated with determining γ_F from the same electromagnetic quantities later evaluated.

The revised SI remains important when interpreting the comparison with experiment. Following the 2019 redefinition [12], the numerical values of the speed of light c, Planck's constant h, and the elementary charge e are exact defining constants, whereas the vacuum magnetic permeability μ₀, vacuum electric permittivity ε₀, and Coulomb's constant inherit the experimental uncertainty of the fine structure constant through the CODATA adjustment. Within the MQ framework, however, neither the elementary charge, the electric constant, nor Coulomb's constant participates in constructing the derivation. Agreement between the MQ values and the 2022 CODATA recommended values therefore represents a successful prediction of these electromagnetic quantities from an independent set of source measurements rather than the reproduction of quantities already used as MQ inputs.

The practical significance of the Frames mapping factor is therefore no longer its use as a fitted electromagnetic parameter but as a quantity emerging directly from the MQ count structure. It collects the count geometry, quantization ratio, and Informativity differential into a single dimensionless quantity determined prior to the electromagnetic derivation. Coulomb's constant consequently becomes the final quantity in the derivation, allowing direct comparison with the 2022 CODATA recommended value.

The elementary charge construction also differs algebraically from the usual application of the metric differential used for the inverse fine structure constant ([3], Appx. T; Appx. AA). For the fine structure constant, the frame adjustment is represented by an additive difference of the form b-d. In the elementary charge derivation, Eqs. (14–19) instead use the quotient relation b/d=b-d at the applicable count bound. This relation permits the fundamental form of the elementary charge e_f to be reduced before the discrete adjustment is applied through the quantization ratio r_q. The quotient construction therefore changes the implementation of the metric differential while preserving the subsequent application of the Informativity differential required to obtain the classically realized elementary charge.

An additive construction analogous to that used for the inverse fine structure constant cannot presently be inferred directly from e_f=l_fc²/[4(θ_si-1)]. The MQ mapping parameter satisfies θ_si=l_fm_f/(2t_f), but the elementary charge expression has not been independently reduced to a pure whole-count expression in θ_si. Consequently, assigning an additive offset of the form b-d would introduce an unestablished count relation. The quotient construction used in Eqs. (14–19) should therefore be regarded as the canonical derivation of e_f, rather than as an interchangeable reformulation of the fine structure construction.

5. APPENDICES

APPENDIX A: DEFINITIONS OF KEYWORDS, TERMS AND SYMBOLS

For a complete set of term and keyword definitions consult Online Resources.

Keywords

Internal Frame

The Internal Frame is the discrete, count–based configuration domain in which all fundamental measures are defined relative to the fundamental units l_f, m_f and t_f. All relational structure is encoded as integer count configurations over this domain. The Internal Frame does not possess a continuous geometric structure; rather, geometry arises only through its mapping into the System Frame.

System Frame

The System Frame is the non–discrete, relational encoding of Internal Frame structure induced by the Frames mapping. It does not possess an independent fundamental measure but instead represents observable quantities as continuous relational separations derived from discrete count structure. All physically meaningful observables are defined in the System Frame.

Frames mapping

The Frames mapping is the kinematic transformation that relates discrete count configurations in the Internal Frame to continuous relational structure in the System Frame. This mapping preserves realization consistency and enforces closure conditions on relational configurations. Observable quantities arise only after application of the Frames mapping, and all geometric and interaction structure emerges from its properties.

Frames Principle

The Frames Principle asserts that all physically meaningful observables arise from the Frames mapping between the Internal Frame and the System Frame. No observable quantity is intrinsic to the Internal Frame alone. All measurement is relational and defined through the mapping. This principle replaces the assumption of a pre–existing spacetime manifold with a kinematic origin of geometry.

MQ Action

The MQ Action is the fundamental kinematic functional defined over the Internal Frame count structure. It governs the evolution and consistency of relational configurations under the Frames mapping. The action admits distinct realizations determined by count separation: fixed interaction–scale realizations, corresponding to electromagnetic behavior at the electromagnetic demarcation, and count–resolved realizations, corresponding to gravitational behavior through the mapping–induced metric structure. In both cases, observable dynamics arise from the normalization and realization of conserved count under the mapping. All emergent physical behavior, including electromagnetic interactions and gravitational curvature, therefore arises from this single action without the introduction of any independent dynamical components.

MQ variational principle

The MQ variational principle states that physically realized configurations in the System Frame correspond to stationary points of the MQ action functional defined over admissible realizations of encoded availability derived from the Internal Frame, thereby replacing the independent postulation of dynamical laws with a single extremization condition from which all effective interactions emerge. Formally, let M:ω→ℛ denote the Frames mapping, and let ω^'⊂ω be a relational region of the Internal Frame with realized configuration Ω=M(ω^')⊂ℛ. Let ρ_R denote the corresponding coarse–grained encoded availability field. The MQ action is defined as the functional

Informativity differential

The Informativity differential is the non–Lorentz length contraction associated with the discrete count structure of the Internal Frame. It arises from the normalization of conserved count under the Frames mapping and quantifies the difference between relational separation in the Internal Frame and its realized representation in the System Frame. The magnitude of the effect is count–dependent, increasing at small separations and diminishing toward the upper–count limit. The Informativity differential is distinct from curvature and represents one of the two geometric consequences of realization in the System Frame, the other being the emergent curvature associated with count loss under successive updates.

Metric differential

The metric differential characterizes the discrete–continuous mismatch that arises when conserved Internal Frame count structure is realized as a continuous observable in the System Frame under the Frames mapping. It does not represent a physical deformation in itself but rather the necessity of expressing integer–resolved count relations as normalized, non–discrete quantities.

For a fundamental measure defined by a count n in the Internal Frame, realization in the System Frame requires application of a mapping normalization functional. This process preserves count conservation while enforcing continuity, introducing a representation–level mismatch between the discrete count structure and its continuous encoding. The metric differential refers to this mismatch as a structural consequence of the mapping.

The metric differential is therefore distinct from the Informativity differential. While the Informativity differential describes a non–Lorentz length contraction in realized separations, the metric differential captures the underlying requirement that discrete count relations be expressed in a continuous form. In this sense, it provides the count–level representation of the discrete–continuous correspondence enforced by the Frames mapping, whereas the Informativity differential describes its realized geometric manifestation in the System Frame.

Electromagnetic demarcation

The electromagnetic demarcation is the fixed interaction count distance n_L = 276/θ_si, corresponding to approximately 84.60055l_f, at which known electromagnetic interactions are realized in MQ. At this scale, the effect of the Informativity differential is still significant. Its influence increases with decreasing interaction distance and rapidly diminishes with increasing distance, becoming physically negligible beyond a few hundred counts of l_f. The electromagnetic demarcation is the fixed electromagnetic interaction scale at which that discrete measure effect is realized.

Count separation

The number of fundamental length units l_f between relational configurations in the Internal Frame. This quantity determines the scale at which mapping–induced deformation becomes significant.

Realization manifold

A realization manifold is the set of all admissible System Frame realizations compatible with a specified Internal Frame configuration under the Frames mapping ℛ(Ω_I). The realization manifold is not a primitive geometric object of the Internal Frame. Rather, it is the image of conserved Internal Frame count structure after application of the Frames mapping and any applicable realization constraints. Probabilities are assigned only after a normalized measure is defined on the admissible realization manifold.

Mapping–induced deformation

The modification of realized separation in the System Frame arising from the discrete–continuous Frames mapping between the Internal Frame and the System Frame. This deformation is fully determined by the mapping structure and does not constitute an independent physical field. It has two coupled manifestations depending on the scale of the interaction. At fixed count separation, it appears as a local, scale–invariant modification governed by the Informativity differential, corresponding to the non–Lorentz contraction of realized lengths. Over extended count separation, the same mapping–induced deformation accumulates through the mapping structure and is realized as geometric curvature in the System Frame. These two manifestations are not distinct mechanisms but represent different scale–dependent realizations of a single underlying mapping effect.

Fixed interaction scale

A fixed interaction scale is a count–defined interaction distance at which a physical interaction is realized under the Frames mapping. For electromagnetic interactions, this occurs at the electromagnetic demarcation n_L = 276/θ_si, where the interaction is evaluated at a single, fixed count separation. At this scale, both the Informativity differential (non–Lorentz length contraction) and the corresponding curvature are realized with fixed values determined by the mapping. The interaction therefore reflects a specific point in the discrete count structure rather than a limiting regime.

Curvature–resolved regime

A curvature–resolved regime refers to the dependence of realized geometric structure on count separation under the Frames mapping. In this description, curvature is a direct consequence of the mapping evaluated at a given count distance. Different separations correspond to different realized curvature values in the System Frame, providing a geometric description of interaction that varies with count scale. This behavior characterizes gravitational phenomena, where measurements probe a range of separations rather than a fixed interaction scale.

Terms

α— is the fine structure constant, a dimensionless fundamental physical constant that characterizes the strength of the electromagnetic interaction between electrically charged particles. The recommended numerical value is published by CODATA based on the least-squares adjustment of the best available experimental and theoretical data.

m_e — is the mass of an electron.

a₀ — is the ground state orbital of an atom, also known as the Bohr radius.

θ_si — can be measured as the polarization angle of quantum entangled X–rays at the degenerate frequency of a maximal Bell state. As an angle θ_si=3.26239 rad ± 2 μrad [15-17]. As a momentum θ_si = 3.262390305(89) kg m s^–1. And with respect to the System Frame, θ_si has no units.

l_f,m_f,t_f — are the fundamental measures, more precise expressions for Planck’s units–length, mass, and time–that consider the effects of length contraction associated with discrete measure.

2Q_Ln_L — also known as the Informativity differential describes the length contraction associated with discrete measure. It is a dimensionless normalization modifier arising from the discrete count structure of the Internal Frame, representing the deviation between integer–resolved count measures and their realized System Frame encoding under the Frames mapping M:ω→ℛ. It quantifies how conserved Internal Frame count N_I(ω',A) is redistributed and normalized when expressed as a System Frame observable.

G — is the gravitational constant, G=6.67407794(17) × 10^–11 m³ kg^–1 s^–2 [13] such that its value considers the effects of length contraction associated with discrete measure at the upper count limit. Italicized G identifies a measure (e.g., G=6.67384483(17) × 10^–11 m³ kg^–1 s^–2 [14] at the blackbody demarcation ([3], Appx. AB)).

c is the speed of light which may also be written as c=n_Ll_f/n_Tt_f =299,792,458 m/s [10] such that n_L=n_T=1 is physically significant.

l_p, m_p and t_p — are Planck’s units for length, mass, and time.

n_L, n_M and n_T — are physically significant discrete counts of l_f, m_f and t_f respectively.

n_L — describes the count of l_f representative of a change in position of an observable measured with respect to the observer’s frame of reference.

n_θ — is the System Frame count of θ_si resolved as a function of the Internal Frame demarcation adjusted for the metric differential (i.e., ⌊84.60055/2⌉ = 42).

h — is Planck’s constant.

ℏ — is the reduced Planck constant, 1.054571817 10^–34 m² kg s^–1. When accounting for the Informativity differential at the upper count bound, this term is not italicized (i.e., ħ=1.054534984(26) 10^–34 m² kg s^–1).

α_f^–1, α_p^–1, and α_c^–1 — are the fundamental, Planck–like, and classical forms of the inverse fine structure constant. α_f^–1=42θ_si is defined with respect to the Internal Frame; the remaining two against the System Frame. α_c^–1 differs from α_p^–1, accounting for the length contraction associated with discrete measure.

γ_F — is the fundamental gamma, a first principles realization parameter that replaces the conventional gamma transform within the MQ framework. Unlike γ, which is resolved from experimentally measured electromagnetic constants, γ_F is derived entirely from fundamental quantities and serves as the canonical electromagnetic realization parameter from which the electromagnetic constants are independently determined.

ɛ₀ — is the electric constant (vacuum permittivity), which characterizes the ability of free space to permit electric field formation. Conventionally, ɛ₀ = 1/u₀c² = 8.854 187 8188(14) × 10-12 F m^-1.

u₀ — is the magnetic constant (vacuum permeability), which characterizes the magnetic properties of free space. Conventionally, u₀ = 2αh/e²c = 1.256 637 06127(20) × 10^-6 N A^-2.

k_e — is Coulomb's constant, the proportionality constant governing the electrostatic force between two point-charges. Conventionally, k_e = 1/4π ɛ₀ = 8.987 551 7923(14) x 109 N m² C^-2.

e — is the elementary charge, the magnitude of the electric charge carried by a proton (or equivalently the magnitude of the negative charge carried by an electron). In the SI, e = 1.602 176 634 x 10^-19 C exactly.

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.

REFERENCES

[1] C. de Coulomb, Premier mémoire sur l’électricité et le magnétisme, Histoire de l’Académie Royale des Sciences (1785), doi:10.5479/sil.304245.39088000647479.

[2] Heisenberg, W., Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Zeitschrift für Physik, 43 (3–4): 172–198 (1927), Bibcode:1927ZPhy.43.172H, doi:10.1007/BF01397280, S2CID 122763326.

[3] J.A. Geiger, Measurement Quantization, Intl. J. Geom. Methods Mod. Phys., Vol. 20, No. 04, 2350069 (Jan. 2023), CC BY 4.0, doi:10.1142/S021988782350069X.

[4] J.A. Geiger: Measurement Quantization Describes the Physical Constants, International Journal of Theoretical and Mathematical Physics, 11(1), pp. 29-59 (Jan 2021) , CC BY 4.0, doi:10.4236/jhepgc.2020.62015.

[5] J.A. Geiger: Measurement Quantization Unites Classical and Quantum Physics, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 2, 262–311 (Apr 2018), doi:10.4236/jhepgc.2018.42019.

[6] J.A. Geiger: Quantum Model of Gravity Unifies Relativistic Effects, Describes Inflation/Expansion Transition, Matches CMB Data, Journal of High Energy Physics Gravitation and Cosmology. Vol. 4, No. 4 (Oct 2018), doi:10.4236/jhepgc.2018.44038.

[7] J.A. Geiger: Measurement Quantization Describes Galactic Rotational Velocities, Obviates Dark Matter Conjecture, Journal of High Energy Physics Gravitation and Cosmology. Vol. 5, No. 2 (Apr 2019), doi:10.4236/jhepgc.2019.52028.

[8] J.A. Geiger: Physical Significance of Measure, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 1 (Jan 2020), doi:10.4236/jhepgc.2020.61008.

[9] J.A. Geiger: Measurement Quantization Describes History of Universe – Quantum Inflation, Transition to Expansion, CMB Power Spectrum, Journal of High Energy Physics Gravitation and Cosmology. Vol. 6, No. 2 (Apr 2020), doi:10.4236/jhepgc.2020.62015.

[10] CODATA recommended values of the fundamental physical constants: 2022, (June 2025), Reviews of Modern Physics, https://arxiv.org/abs/2409.03787.

[11] Dinko Milakovic, Chung-Chi Lee, Robert Carswell, et. al., A new era of fine structure constant measurements at high redshift (March 2021), arXiv: 2008.10619.

[12] BIPM statement: Information for users about the proposed revision of the SI (May 5, 2018), https://www.bipm.org/utils/common/pdf/SI-statement.pdf.

[13] P. Mohr, B. Taylor, D. Newell, CODATA Recommended Values of the Fundamental Physical Constants: 2014, p. 3, (2015), arXiv: 1507.07956v1, doi:10.1063/1.4954402.

[14] P. Mohr, B. Taylor, and D. Newell, CODATA Recommended Values of the Fundamental Physical Constants: 2010, p. 73 (2012), arXiv:1203.5425v1, doi:10.48550/arXiv.1203.5425.

[15] S. Shwartz and S. E. Harris, Polarization Entangled Photons at X-Ray Energies, Phys. Rev. Lett. 106, 080501 (2011), arXiv: 1012.3499, doi:10.1103/PhysRevLett.106.080501.

[16] S. Shwartz, R. N. Coffee, J. M. Feldkamp, et. al., X-ray Parametric Down-Conversion in the Langevin Regime, Physical Review Letters, 109, 013602 (July 6, 2012), doi:10.1103/PhysRevLett.109.013602., doi:10.1103/PhysRevLett.109.013602.

[17] T. E. Glover et al., X-ray and optical wave mixing, Nature, 488, 7413, 603–608 (2012), doi:10.1038/nature11340.