A First Principles Description of the Electric Constant

The electric constant, vacuum permeability, Coulomb's constant, elementary charge, and the fine structure constant are among the most recognizable quantities in electromagnetism. Although these constants appear in different equations throughout classical and quantum physics, they are closely related through established electromagnetic relationships. Modern SI definitions express several of these quantities in terms of one another, while others are determined through precision measurement and the international CODATA least-squares adjustment.

Measurement Quantization (MQ) approaches the electromagnetic constants from a different perspective. Rather than beginning with the electric constant, elementary charge, or fine structure constant themselves, MQ asks whether these observable quantities can emerge from a smaller collection of more fundamental relationships.

The answer begins with a careful distinction between count and measure.

Within MQ, the most fundamental mathematical object is not physical space, nor spacetime, nor even length. It is the Internal Frame — a dimensionless configuration domain that contains only discrete relational structure. This domain possesses neither distance, duration, direction, nor continuous geometry. Instead, it consists entirely of admissible count relationships governed by the constraints of the MQ Action and the Frames mapping.

Because the Internal Frame contains no physical geometry, the quantities $l_f$, $m_f$, and $t_f$ should not be interpreted as conventional lengths, masses, or times existing within that domain. Rather, they are the primitive units of count from which observable measures are ultimately realized. They establish the elementary count structure upon which the realization process operates but do not themselves imply an underlying geometric manifold.

Observable physics appears only after these discrete count relationships are realized through the Frames mapping. This mapping transforms Internal Frame count structure into the continuous quantities measured within the System Frame, where familiar concepts such as distance, mass, time, momentum, and electromagnetic fields acquire their conventional physical meaning.

This distinction also clarifies the relationship between the Internal Frame and the configuration domain discussed in the MQ theory of universe initiation. The abstract configuration domain, denoted by $\omega$, represents the complete space of admissible relational configurations before any universe is realized. In this state, no physical geometry, metric structure, or observable measures exist. A persistent mapping defect introduces a realizable relational region whose conserved count structure can be mapped into a System Frame. The Internal Frame therefore represents the discrete relational structure associated with a realized universe rather than the unreduced configuration domain itself. Although both descriptions are dimensionless, the former supports a realizable Frames mapping while the latter represents the broader space of possible configurations.

Discussion

The electromagnetic constants can be described in part through the fine structure constant, elementary charge, and Planck's constant. Other familiar electromagnetic quantities — the electric constant, magnetic constant, and Coulomb's constant — are connected through established electromagnetic relations. Within MQ, the objective is to replace these interdependent constant-to-constant descriptions with expressions resolved from the fundamental measures and the realization geometry connecting the Internal Frame and System Frame.

Most of the descriptions used in the derivation are counts of the fundamental measures or expressions composed from the fundamental measures. The electromagnetic constants are not introduced as component terms in the final first-principles definition. Two geometric effects enter the realization sequence.

The first, the metric differential, describes the count-level offset introduced when a discrete quantity is represented through the Frames mapping. For the elementary-charge construction, the corresponding count remainder is

The second effect is the Informativity differential. It describes the non-Lorentz length contraction associated with discrete measure. The effect depends on count separation and decreases in magnitude with increasing distance. In MQ it follows from the same Pythagorean count geometry used throughout the realization construction:

Understanding these effects is important to resolving the fundamental, Planck-like, and classical forms of the inverse fine structure constant. The forms are successive stages of one realization sequence. The fundamental form is established from the discrete count construction, the Planck-like form incorporates the metric differential, and the classical form additionally incorporates the Informativity differential.

Inverse Fine Structure Constant

The inverse fine structure constant provides the starting point for the electromagnetic realization sequence. Within Measurement Quantization, this quantity is represented by three successive forms that describe a single physical quantity viewed at different stages of realization rather than three independent physical constants.

These three expressions illustrate one of the central ideas of MQ: observable physics emerges through a defined realization sequence. Rather than introducing separate definitions for fundamental and classical descriptions, MQ interprets them as successive representations connected through the Frames mapping. This realization sequence becomes the foundation for the electromagnetic quantities that follow.

Reduced Planck Constant

The reduced Planck constant occupies a central role in quantum mechanics, relating energy, momentum, and angular measure. Within MQ, the reduced Planck constant also participates in the realization sequence that transforms Internal Frame quantities into their observable System Frame representations.

MQ distinguishes measures evaluated at different count separations because the Informativity differential is distance dependent. At finite count separations, including the electromagnetic demarcation, the non-Lorentz contraction associated with discrete measure remains nonzero. As count separation increases, that correction decreases. At the upper count bound, the contraction approaches zero. MQ therefore uses the typography of the symbol to identify the count scale of the measure rather than a different physical constant.

For the electromagnetic derivation, the reduced Planck constant is evaluated at the electromagnetic demarcation:

This expression identifies how the same measure is represented at a finite count scale where the mapping-associated contraction has not yet diminished to its upper-bound limit.

Fundamental Elementary Charge

Having established the inverse fine structure constant and reduced Planck constant within the MQ framework, the next objective is to derive the elementary charge. Unlike its conventional SI definition, where the elementary charge is an exact defining constant, MQ derives it as the realized outcome of the underlying discrete-measure construction.

The derivation begins with the fundamental elementary charge. Several intermediate quantities preserve the discrete count realization while maintaining the mapping between the Internal Frame and System Frame. The sequence begins with

The mapping-parameter count remainder is

The corresponding quantization ratio is

The elementary-charge remainder and Planck-like form are then

Application of the Informativity differential gives the Planck-to-classical difference and the classically realized elementary charge:

The final result is the classically realized elementary charge, obtained after application of both the Frames mapping and the Informativity differential. The intermediate quantities are realization parameters rather than additional physical constants.

Electric Constant

With the elementary charge now expressed within the MQ framework, the electric constant can likewise be reduced to the fundamental measures. This derivation replaces the conventional electromagnetic quantities one by one with their corresponding MQ representations until the remaining expression depends on the realization sequence established above.

An important feature of this reduction is that every substitution preserves the physical interpretation of the preceding quantity. The derivation does not introduce a phenomenological fitting parameter specific to the electric constant. Instead, the result follows from the MQ realization procedure already established for the inverse fine structure constant and elementary charge.

The Fundamental Realization Parameter

The remaining realization terms appearing in the electric constant derivation are collected into a single dimensionless quantity, ${\gamma }_F$, the fundamental realization parameter.

The parameter ${\gamma }_F$ encapsulates the realization geometry associated with the Frames mapping. It combines the Planck-like inverse fine structure form, the Informativity differential, the elementary-charge remainder, and the quantization ratio into a single canonical electromagnetic realization parameter. This separates the realization geometry from the underlying electromagnetic phenomenon.

Final MQ Expression for the Electric Constant

Substituting the fundamental realization parameter into the previous reduction produces the final MQ expression for the electric constant:

Within the MQ framework, this equation represents the culmination of the realization sequence. Beginning with the fundamental measures, proceeding through the Frames mapping, and incorporating the Informativity differential, the derivation produces the observable electric constant without using either elementary charge or the electric constant itself as source quantities.

Comparison with the 2022 CODATA recommended value demonstrates close numerical agreement. That comparison should be interpreted carefully. Several source quantities participate in the broader CODATA least-squares adjustment, so covariance among adjusted constants must be acknowledged when assessing statistical significance. The agreement therefore supports the MQ derivation while experimentally independent measurements of the source quantities would provide the strongest future validation.