Describing the Electric Constant
Using Only the Fundamental Measures

Jody A. Geiger

Informativity Institute

Chicago, IL USA 60607

jodygeiger@informativity.org

ORCID: 0000-0001-5389-0447

ABSTRACT

We derive the electric constant within Measurement Quantization (MQ) by expressing the fine structure constant, elementary charge, and vacuum permittivity through the MQ fundamental length, mass, and time measures, l_f, m_f, and t_f, together with discrete count relations realized through the Frames mapping. The construction distinguishes fundamental, Planck-like, and classical forms of the inverse fine structure constant and elementary charge, incorporates the Informativity differential, and collects the remaining count dependent geometry into the dimensionless fundamental realization parameter γ_F. The resulting elementary charge is e_c = 1.602176633(47) × 10^-19 C, corresponding to a 2.30σ concordance with the 2022 CODATA recommended value. Substitution into the electromagnetic relations reduces the electric constant to ε₀ = 4γ_F/(πl_fm_fc²), yielding ε₀ = 8.85418780(60) × 10^-12 F m^-1, in 3.93σ concordance with the 2022 CODATA value. No phenomenological parameter is fitted specifically to ε₀. Because several source and comparison quantities belong to the same algebraically related electromagnetic constant network, these agreements establish the algebraic closure and numerical consistency of the MQ realization sequence rather than an independent empirical prediction. The result provides a common fundamental measure representation of elementary charge and vacuum permittivity while making their dependency structure explicit.

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

Keywords: Electric Constant, Fundamental Measures, Planck Units, Measure, Discrete

1. INTRODUCTION

The electric constant, ε₀, is the vacuum electric permittivity appearing in the SI formulation of Coulomb's law and Maxwell's equations [1]. In the present SI, it is not an independently defined exact constant: it is related to the vacuum magnetic permeability, μ₀, and the exact speed of light, c, by ε₀=1/(μ₀c²), while the uncertainty of μ₀, and therefore of ε₀, follows from the experimentally determined fine structure constant [1,2]. Consequently, agreement with a CODATA value must be interpreted with regard to the measurement and algebraic dependencies among these constants rather than solely by digit-for-digit correspondence [2].

Measurement Quantization (MQ) [3] investigates whether ε₀ can be expressed in terms of the MQ fundamental length, mass, and time measures, l_f, m_f, and t_f. Within MQ, these fundamental measures describe discrete measure in the Internal Frame (Appx. A), while observable physical quantities arise in the System Frame through the Frames mapping. The present work applies these previously developed MQ relationships to the specific problem of the electric constant rather than re-establishing the broader MQ framework.

The derivation proceeds through the MQ representations of the fine structure constant and the elementary charge. These intermediate quantities, together with the Realization manifold and the Informativity differential (Appx. A), combine to define the dimensionless Frames mapping, γ_F. The resulting expression introduces no phenomenological fit parameter specifically for ε₀; however, its numerical evaluation depends upon the physical quantities used to determine the MQ fundamental measures and γ_F. Consequently, a quantity used to establish these inputs cannot subsequently serve as an independent experimental validation of an algebraically dependent result.

Within this construction, the electric constant is expressed using the MQ fundamental measures together with the Frames mapping. The corresponding MQ evaluation is

The result yields a 3.93σ concordance with the 2022 CODATA recommended value. The MQ derivation does not use either the elementary charge or the electric constant as source quantities in establishing the MQ fundamental measures. Instead, the construction begins from recommended values for the fine structure constant, electron mass, Bohr radius, and the defined speed of light, from which the MQ fundamental measures are obtained [1-3]. The subsequent derivation of the elementary charge and electric constant therefore represents an MQ prediction that does not use either e or ε₀ as an input, rather than a restatement of their electromagnetic defining relations. Nevertheless, because the quantities used to establish the MQ fundamental measures participate in the broader CODATA adjustment, any covariance between those source quantities and the comparison values should be evaluated when interpreting the statistical significance of the agreement [2].

The objective of this paper is therefore to develop the MQ derivation of the electric constant, identify the physical origin and dependency structure of each constituent quantity, and demonstrate that the electric constant can be represented consistently within the common fundamental measure framework used for the related electromagnetic constants. The derivation and numerical comparison are developed in the following sections, while broader interpretations of the MQ fundamental measures remain within the scope of the underlying MQ framework rather than being established by the electric constant derivation itself.

2. METHODS

MQ reformulates established physical relations using discrete counts of the fundamental measures together with the Frames mapping through which those counts are represented as observable quantities in the System Frame. The framework is developed from prior MQ derivations of gravitational curvature, the speed of light, escape velocity, and Heisenberg's uncertainty principle [3,4], while subsequent applications have extended the same methodology to a broad range of problems in gravitation, cosmology, and quantum physics [5–9].

The present calculation begins with the established SI relationships among the electric constant, the fine structure constant, elementary charge, the reduced Planck constant, and the speed of light [1,2]. Each constituent quantity is then replaced by its corresponding MQ representation. The reduction proceeds sequentially from the MQ fundamental measures through the Frames mapping and the associated Informativity differential to obtain the corresponding observable quantity in the System Frame. The complete mathematical development begins in Sec. 3 with the supporting expressions required for the derivation.

The MQ formulation distinguishes three successive representations of both the inverse fine structure constant and elementary charge. Their fundamental forms describe the corresponding Internal Frame quantities. Their Planck-like forms represent the mapped realization in the System Frame, while their classically realized forms additionally incorporate the Informativity differential. These successive stages are not independent physical constants but intermediate realizations within the MQ framework. The resulting expressions are subsequently combined into the dimensionless fundamental realization parameter, γ_F, which provides the final MQ reduction of the electric constant.

Nearest whole-unit realization is represented throughout using the mixed floor–ceiling operator. This notation is applied consistently to all quantization relations developed in the Results section and replaces the explicit rounding operator employed in earlier MQ publications [3,10].

The MQ framework derives physical quantities from three experimentally determined source measures – a the fine structure constant, electron mass, and Bohr radius – together with the expression and value for the speed of light. Throughout the present work, the 2022 CODATA recommended values are adopted for these source quantities to provide a common experimental baseline [15]. Although the CODATA least-squares adjustment introduces statistical covariance among many recommended constants, the MQ input and output quantities remain physically distinct. Consequently, the reported concordance values provide a meaningful assessment of agreement between the MQ framework and experimentally inferred physical constants while recognizing the covariance inherent in the CODATA adjustment.

To facilitate direct comparison with conventional formulations, MQ expressions are evaluated alongside their classical counterparts using corresponding fundamental unit representations. Classical relations are expressed in Planck units, whereas MQ relations are expressed in MQ fundamental measures derived from the right-angle triangle construction and the Frames mapping. This common normalization permits framework-to-framework comparisons that distinguish differences arising from the underlying physical models rather than from arbitrary unit conventions. Because conventional Planck-unit formulations combine constants originating from macroscopic and electromagnet determinations of physical constants without an explicit realization mapping, Planck’s treatment of units of measure differs fundamentally from that of MQ. Consequently, when electromagnetic and macroscopic constants appear within the same expression, MQ and classical frameworks exhibit markedly different concordance behavior. Throughout the MQ applications investigated to date, the MQ framework consistently provides substantially stronger agreement with experimentally inferred physical constants than the corresponding classical Planck unit formulations.

The physical basis for discrete and countable measure, together with the derivations supporting the MQ framework, is presented in the summary review publication entitled, Measurement Quantization [3]. The principal MQ derivations used directly in the present paper are summarized below for convenient reference.

Table 1. MQ derivations supporting the present work

A complete index of all MQ appendices is provided in Appx. BR of the summary paper. Numerical implementations supporting both the summary publication and the present work are available through the MQ Online Resources:

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

Supplementary Material 1 contains the complete spreadsheet implementation of every appendix presented in the summary publication. Each worksheet evaluates both the left- and right-hand sides of the corresponding equations, allowing every derivation to be reproduced numerically and independently verified.

Supplementary Material 2 extends this framework to the broader MQ research program. Individual worksheets are provided for each of the associated MQ papers, including the present work, with one worksheet devoted to each publication. These spreadsheets contain the complete numerical implementations of the reported equations together with intermediate calculations, uncertainty propagation, supporting derivations, and concordance analyses, thereby providing a transparent computational foundation for reproducing every reported result.

Several MQ derivations require numerical precision substantially exceeding the fifteen significant digits provided by conventional spreadsheet arithmetic. All reported calculations therefore employ the PrecisionCalc module of the XLPrecision Excel plug-in to maintain consistent high-precision numerical evaluation throughout the derivations. Mathematical expressions appearing in both the manuscript and the accompanying computational resources were prepared using MathType 6.9d or later to preserve consistent equation formatting.

3. RESULTS

3.1 Supporting Expressions

A description of the electric constant in terms of the MQ fundamental measures is obtained by replacing the quantities in its conventional SI expression with their corresponding MQ representations. The resulting derivation uses the fundamental length, mass, and time measures l_f, m_f, and t_f, together with the count relations governing their realization through the Frames mapping ([3], Appx. W). This subsection establishes the supporting quantities required by the subsequent derivations.

The MQ quantity θ_si is defined in momentum units by

where l_f, m_f, and t_f denote the fundamental length, mass, and time measures, respectively. Equation (2) defines θ_si within the realized System Frame. The Internal Frame is dimensionless and represents the configuration domain from which observable quantities arise through the Frames mapping ([3], Appx. K, AS, BQ).

MQ also associates θ_si with the angular configuration of signal and idler photons in x-ray parametric down-conversion. Shwartz and Harris [11] developed the theoretical polarization entanglement model, while Shwartz et al. [12] experimentally measured energy-resolved coincidence rates and the associated phase-matching geometry for down-converted x-ray photon pairs. Glover et al. [13] independently demonstrated x-ray and optical sum-frequency generation, providing further experimental support for nonlinear x-ray–optical interactions. These experiments do not directly define or report the MQ quantity θ_si; rather, MQ identifies the relevant angular configuration within its own construction.

The experimental geometry provides support at approximately six-digit resolution for the angular correspondence used in the earlier MQ treatment. The present derivation does not rely on that angular result as the precision source for θ_si. Instead, θ_si is resolved from the fine structure constant, electron mass, Bohr radius, and exact speed of light, allowing the momentum value to be determined to ten significant digits ([3], Appx. AD; [1,2]). The x-ray measurements therefore provide an independent physical correspondence, while the electromagnetic constant construction supplies the higher precision value used in the calculations that follow.

The interaction length count n_L is obtained from the following construction ([3], Appx. AB),

where ℏ denotes the reduced Planck constant. The resulting value, n_L=84.60055, corresponds to the electromagnetic demarcation identified independently through the blackbody, charge coupling , and elementary charge constructions ([3], Appx. AB, AE, AN).

At the whole-count level,

so that the electromagnetic realization corresponds to n_θ=42.

The charge coupling demarcation provides the corresponding interaction length relation ([3], Appx. AE),

which independently reproduces the interaction count obtained from Eq. (3). Together, Eqs. (3), (4), and (5) establish a consistent electromagnetic realization count.

The MQ construction relates θ_si to the fine structure constant through ([3], Appx. AD),

Using the 2022 CODATA recommended value of the fine structure constant [2] gives

The value in Eq. (7) is the momentum description used throughout the remainder of this derivation. Its numerical correspondence with the angular value associated in MQ with x-ray signal-idler configurations does not constitute an independent experimental determination of the MQ relation [11–13].

The corresponding count difference produced by the Frames mapping is

This difference identifies the whole-unit offset between 42θ_si and its nearest realized count.

Application of the non-Lorentz length contraction construction ([3], Appx. D) then gives

where Q_L denotes the fractional length count contribution. The product Q_Ln_L is the Informativity differential, a dimensionless normalization arising from conserved count under the Frames mapping that quantifies the difference between Internal Frame count and its realized System Frame representation (Appx. A). These supporting quantities provide the basis for the derivation of the inverse fine structure constant presented in the following subsection.

3.2. Inverse Fine Structure Constant

Within MQ, the inverse fine structure constant is represented successively by fundamental, Planck-like, and classically realized forms. These designate successive stages of the MQ construction rather than independent physical constants ([3], Appx. AA, AS). These designate successive stages of the MQ construction rather than independent physical constants. The fundamental form is defined with respect to the Internal Frame by

where $${\mathit{\alpha }}_{\mathit{f}}^{-1}$$ denotes the fundamental inverse fine structure form. It is expressed directly in terms of the MQ quantity θ_si.

Application of the Frames mapping gives the Planck-like form,

where $${\mathit{\alpha }}_{\mathit{p}}^{-1}$$ denotes the mapped inverse fine structure form and the mixed floor–ceiling brackets specify nearest whole-unit realization. The Planck-like form is evaluated with respect to the System Frame ([1], Appx. AS). The classically realized form is obtained by incorporating the Informativity differential established from the supporting count construction,

where $${\mathit{\alpha }}_{\mathit{c}}^{-1}$$ denotes the classically realized inverse fine structure form and 2Q_Ln_L is the dimensionless Informativity differential ([3], Appx. D, AA). The classical form differs from the Planck-like form through the normalization associated with the non-Lorentz length contraction of discrete measure. Together, these three expressions establish the MQ sequence used in the elementary charge derivation that follows.

3.3. Elementary Charge

3.3.1 Fundamental Elementary Charge

The MQ elementary charge construction begins with its fundamental form, e_f. This quantity is an MQ intermediate and is distinct from the SI elementary charge e, whose numerical value is fixed exactly in the present SI [1].

The reduction begins with two auxiliary quantities, b and d, constrained at the count bound by

where b is the auxiliary base quantity, d is the corresponding offset quantity, and b/d denotes their mapped ratio. The mapped ratio is then expressed in terms of the fundamental elementary charge form and the fundamental mass function within the mapped relation,

where f(m_f) denotes the function of the MQ fundamental mass m_f, ħ is the reduced Planck constant, and l_f is the MQ fundamental length. Substitution of the MQ momentum and mass relations reduces the expression to the form shown in Eq. (14) ([3], Appx. Q, V, BN).

Equating the ratio and difference representations gives

which, after collecting terms, becomes

Solving for the MQ fundamental elementary charge yields

thereby eliminating the auxiliary quantities b, d, and f(m_f). Equation (17) expresses the MQ fundamental elementary charge solely in terms of θ_si and the MQ fundamental measures l_f, m_f, and t_f ([3], Appx. AM). Finally, substitution of the momentum relation from Eq. (2) gives

This expression defines the MQ fundamental elementary charge that serves as the starting point for the realization procedure developed in the following subsection.

3.3.2 Quantized and Classical Elementary Charge

The realization of the fundamental elementary charge begins with the metric differential associated with the nearest whole-unit count of θ_si,

where n_θ denotes the System Frame count difference between 42θ_si and its nearest whole-unit realization. This quantity is identical to the count differential introduced in Eq. (8) and provides the count offset required for the elementary charge realization.

The elementary charge remainder n_e is then obtained from e_f using the same nearest whole-unit construction employed for the inverse fine structure constant,

where n_e denotes the residual count remaining after removal of the nearest whole-unit contribution from 84e_f.

The Planck-like elementary charge is then obtained by

where the realization divisor describes the repeating Diophantine realization structure associated with the electromagnetic interaction count. The whole-unit realization is defined by the integers 84 and 42, while the reference inclusive count domain is specified by 276+1. The polynomial correction introduces the dependence of the realization function on n_θ. Collectively, these terms transform the elementary charge remainder n_e from its continuous Internal Frame representation into the Planck-like elementary charge e_p, corresponding to its realization in the System Frame.

The contraction between the Planck-like and classical charge forms is

where Δ_(P−C) denotes the difference between the Planck-like form and the classical elementary charge. The factor 2Q_Ln_L represents the Informativity differential evaluated at the electromagnetic demarcation ([3], Appx. D, AE). The classical elementary charge is therefore

Substituting Eq. (23) gives

and hence

The calculated value corresponds to a 2.30σ concordance with the 2022 CODATA recommended value [2]. These expressions complete the realization sequence from the MQ fundamental form e_f, through the Planck-like form e_p, to the classical elementary charge e_c.

3.3.3 Electric Constant

The electric constant ε₀ (vacuum electric permittivity) is conventionally related to the vacuum magnetic permeability μ₀ and the speed of light c through ε₀=1/μ₀c² [1,2]. It may also be expressed in terms of the elementary charge, the fine structure constant, Planck's constant, and the speed of light. Using ℏ=h/(2π), this relation becomes

Substituting the classical inverse fine structure form from Eq. (12) gives

Collecting common factors yields

Substituting the classical elementary charge from Eq. (24), e_c=2Q_Ln_Le_p, gives

The count-dependent terms – the Planck-like inverse fine structure, the Planck-like elementary charge, the quantization ratio, and the Informativity differential – are then collected into the dimensionless fundamental realization parameter,

where γ_F is the fundamental realization parameter combining the Planck-like inverse fine structure form, the Informativity differential, the charge count remainder, and the quantization ratio into the canonical MQ electromagnetic realization parameter.

Substituting Eq. (30) into Eq. (29), together with the MQ fundamental elementary charge relation from Eq. (18), reduces the electric constant to

The resulting value is in 3.93σ concordance with the 2022 CODATA recommended value [2]. This completes the MQ reduction of the electric constant from the MQ fundamental measures through the fundamental realization parameter.

4. DISCUSSION

Immediately noticeable, the fundamental realization parameter γ_F describes the realization geometry between the Internal and the System Frame. It collects the realization terms introduced by the Frames mapping while remaining external to the underlying electromagnetic phenomenon, thereby separating the frame dependent geometry from the fundamental measures.

Care must be taken when comparing MQ results with CODATA recommended values. The elementary charge e is an exact defining constant within the present SI [1]. The electric constant ε₀ is related to the magnetic constant μ₀ through ε₀=1/μ₀c², while Coulomb's constant is given by k_e=1/4πε₀ [1,2].

Consequently, many electromagnetic constants are algebraically correlated through the same defining relationships [2]. When quantities participating in the CODATA least-squares adjustment are used to establish MQ source measures, any remaining covariance with subsequently predicted quantities should be considered when assessing statistical significance [2]. Such comparisons therefore test internal algebraic consistency rather than independent predictive capability.

The reader may naturally ask why the derivation adopts γ_F rather than the earlier gravitational construction presented in the paper entitled, Measurement Quantization [3]. The principal advantage is that the realization parameter encapsulates the realization geometry associated with the Frames mapping without requiring an explicit solution for the electromagnetic demarcation count n_L. It combines the Planck-like inverse fine structure form, the charge count remainder, the quantization ratio, and the Informativity differential into a single dimensionless quantity. Throughout the MQ electromagnetic derivations, this parameter exhibits stable numerical behavior while preserving a direct connection to the underlying MQ realization sequence.

The elementary charge construction is physically more involved than the corresponding inverse fine structure construction because the fundamental elementary charge expression contains an offset in θ_si, whereas the MQ fundamental inverse fine structure form is simply $${\mathit{\alpha }}_{\mathit{f}}^{-1}=42{\mathit{\theta }}_{\mathit{s}\mathit{i}}$$.

To preserve discrete count realization at the count bound, MQ introduces the auxiliary quantities b and d, together with their associated quantization ratio b/d. This construction allows the offset to be incorporated into the quantization ratio before application of the quantization ratio r_q. The procedure avoids direct realization of a non-integer count associated with θ_si while preserving the discrete realization sequence required for the elementary charge derivation. The resulting Planck-like elementary charge form is then transformed into the classical elementary charge through application of the Informativity differential.

Under this framework, agreement with the 2022 CODATA recommended values provides evidence supporting the MQ derivation while recognizing that any covariance introduced through the CODATA least-squares adjustment of the source quantities should be considered when interpreting the statistical significance of the reported concordance [2]. Future comparisons with experimentally independent determinations of the underlying source quantities will provide a more stringent assessment of the predictive capability of the MQ framework.

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

$$\mathit{S}[{\mathit{\rho }}_{\mathcal{ℛ}}]=\underset{\mathcal{ℛ}}{\int }\mathcal{L}({\mathit{\rho }}_{\mathcal{ℛ}},\partial {\mathit{\rho }}_{\mathcal{ℛ}},\dots )\mathit{d}\mathit{\mu },$$ where L is constructed from count–based relations induced by the mapping M, and dμ is the induced measure on ℛ. The MQ variational principle asserts that admissible realizations satisfy δS=0 under variations consistent with the constraints of the Frames mapping and the conservation relations defined on ω^'.

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 [11-13]. 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 [14] 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 [15] 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 [2] 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] Bureau International des Poids et Mesures, The International System of Units (SI Brochure), 9th edn. (BIPM, Sèvres, 2019), updated version, doi:10.59161/AUEZ1291.

[2] P.J. Mohr, D.B. Newell, B.N. Taylor, E. Tiesinga, CODATA recommended values of the fundamental physical constants: 2022, Rev. Mod. Phys. 97, 025002 (2025). https://doi.org/10.1103/RevModPhys.97.025002

[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] 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.

[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] 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.

[11] 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.

[12] 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.

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

[14] 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.

[15] 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.