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A First Principles Description of Coulomb's Constant

Coulomb's constant is one of the fundamental constants of electromagnetism. In conventional physics it is introduced through the vacuum electric permittivity,

making it appear as a quantity derived from another physical constant. Measurement Quantization (MQ) approaches the problem from a different perspective. Rather than beginning with electromagnetic constants, MQ begins with the fundamental measures of length, mass, and time and the discrete count structure from which observable quantities emerge. The objective is to determine whether Coulomb's constant can be constructed from first principles rather than defined in terms of previously measured electromagnetic constants.

The derivation proceeds through six logical stages. First, the electromagnetic interaction count is established. Next, the fine structure constant is recovered from the MQ mapping parameter. This provides the foundation for deriving elementary charge, followed by the vacuum electric permittivity, and finally Coulomb's constant.

Determining the Electromagnetic Interaction Count

The first step is identifying the discrete count associated with electromagnetic interactions. Within MQ, this count measures how the fundamental units are organized before they are realized in the observable world.

The count alone is insufficient because realized measurements are not exact whole-number counts. MQ therefore introduces the metric differential, a normalization arising from the transition between the Internal Frame, where measure is represented as discrete counts, and the System Frame, where those counts appear as continuous physical quantities.

The metric differential is a recurring quantity throughout MQ and provides the mapping when converting discrete count structure into measurable physical quantities. Its form differs for each physical application.

Recovering the Fine Structure Constant

The fine structure constant provides one of the most stringent tests of any electromagnetic theory. Rather than treating it as an isolated empirical quantity, MQ derives three successive forms corresponding to different stages of realization.

The fundamental form identifies the description of the phenomenon with respect to the discrent Internal Frame.

Applying the Metric differential gives the Planck-like form, while applying the non-Lorentz length contraction associated with the discrete Internal Frame (aka. the Informativity differential), produces the classically realized value,

This progression illustrates a central feature of MQ: observed constants emerge through successive transformations from discrete count relations rather than being inserted independently. The implementation yields a 3.45σ concordance with the 2022 CODATA recommended value.

Constructing Elementary Charge

Once the fine structure constant has been established, MQ proceeds to the elementary charge.

The derivation begins with the fundamental form, denotes the MQ fundamental length measure and c is the defined speed of light.

The corresponding electromagnetic count is what determines the quantization ratio. The elementary charge count remainder then allows construction of the Planck-like elementary charge, before application of the Informativity differential, which produces its classically realized form,

The result yields a 2.30σ concordance with the 2022 CODATA recommended value. This sequence demonstrates an important distinction between the MQ treatment of the fine structure constant and the elementary charge. The fine structure constant employs an additive Metric differential, whereas elementary charge uses a quotient relation before the Informativity differential is applied. Although the algebra differs, both derivations follow the same physical principle: observable quantities arise from discrete count structure realized through the Frames mapping.

Deriving the Vacuum Electric Permittivity

Elementary charge is then substituted into the conventional electromagnetic relationship to recover the vacuum electric permittivity,

The result yields a 3.93σ concordance with the 2022 CODATA recommended value. Unlike the conventional presentation, the MQ derivation reaches this expression only after resolving the fundamental measures and elementary charge. Vacuum permittivity is therefore not a starting point but a consequence of the preceding construction.

Applying The Frames Mapping

A central feature of the derivation is the dimensionless Frames mapping geometry factor,

Rather than representing an independently fitted parameter, γ_F collects the geometric effects of the MQ count structure into a single quantity. It combines the mapping parameter, the Informativity differential, the electromagnetic count remainder, and the quantization ratio. Once these quantities have been determined, the geometry follows directly from the MQ framework.

Deriving Coulomb's Constant

The final result follows immediately.

The result yields a 1.93σ concordance with the 2022 CODATA recommended value. This expression depends only upon the MQ fundamental measures and the Frames mapping. The conventional dependence on vacuum permittivity has disappeared, replaced by a first principles description rooted in the discrete structure of measure itself.

In the present implementation, the MQ fundamental measures are determined using the 2022 CODATA recommended value of the fine structure constant together with the electron mass, the Bohr radius, and the defined value of the speed of light. From these quantities, MQ successively derives the mapping parameter, the fundamental measures, the fine structure constant, elementary charge, the vacuum electric permittivity, and finally Coulomb's constant. Neither elementary charge, the electric constant, nor Coulomb's constant is used as an input to the derivation.

The final calculated value agrees with the 2022 CODATA recommended value of Coulomb's constant within the reported uncertainty. This agreement demonstrates that the MQ framework reproduces the accepted electromagnetic constant through a derivation based on the discrete organization of the fundamental measures rather than by treating Coulomb's constant as a quantity defined only through the electric constant. As a result, the derivation provides an alternative first principles interpretation of Coulomb's constant while preserving consistency with the established SI description of electromagnetism.