A First Principles Description of Coulomb's Constant

Coulomb's constant is one of the fundamental constants of electromagnetism. In conventional SI physics it is introduced through the vacuum electric permittivity, $k_e=\frac{1}{4\pi {\epsilon }_0}$, which makes it appear as a quantity derived from another electromagnetic constant. Measurement Quantization (MQ) approaches the problem from a different direction. It begins with the fundamental measures of length, mass, and time and the discrete count structure from which observable quantities are constructed, then asks whether Coulomb's constant can be obtained without using the electric constant or Coulomb's constant itself as an input.

The construction proceeds through a linked electromagnetic sequence. The discrete interaction count establishes the realization structure; the fine-structure constant fixes the electromagnetic mapping; elementary charge is then constructed; the vacuum electric permittivity follows; and the same count geometry finally yields Coulomb's constant. The purpose of the sequence is not to replace the SI relation above, but to expose a common first-principles structure beneath the electromagnetic constants.

Determining the Electromagnetic Interaction Count

The first step is the discrete count associated with electromagnetic realization. Let $n_{\theta }$ denote the electromagnetic count difference, ${\theta }_{si}$ the MQ mapping parameter, and $\lfloor x\rceil$ the floor of $x$. The count relation is

Realized measurements are not represented only by whole-number counts. MQ therefore introduces the metric differential, a normalization associated with the transition from the discrete Internal Frame description to the observable System Frame. Let $n_L$ be the length count and $Q_L$ the corresponding length-count differential. Their relation is

Recovering the Fine-Structure Constant

The fine-structure constant provides a stringent electromagnetic consistency condition. MQ distinguishes fundamental, Planck-like, and classically realized inverse forms. Let ${\alpha }_f^{-1}$, ${\alpha }_p^{-1}$, and ${\alpha }_c^{-1}$ denote these three forms, respectively. The fundamental form is ${\alpha }_f^{-1}=42{\theta }_{si}$, while application of the count offset gives ${\alpha }_p^{-1}=84{\theta }_{si}-\lfloor 42{\theta }_{si}\rceil$.

The classically realized form then incorporates the length-count differential:

This progression illustrates a central feature of MQ. The observed constant is reached through successive transformations of a discrete count relation rather than introduced as an independent empirical parameter. The comparison with the 2022 CODATA recommended value is therefore best interpreted in light of the input dependencies used to establish the fundamental measures.

Constructing Elementary Charge

Once the electromagnetic count structure has been established, MQ proceeds to elementary charge. Let $l_f$ denote the fundamental length, $t_f$ the fundamental time, $m_f$ the fundamental mass, and $c$ the defined speed of light. The fundamental charge form is

Let $n_e$ denote the electromagnetic realization count, $e_p$ the Planck-like elementary charge, and $e_c$ the classically realized elementary charge. The discrete realization uses $n_e=84e_f-\lfloor 42e_f\rceil$, after which the Planck-like charge is obtained from the charge-count realization polynomial.

The Informativity differential then supplies the contraction between the Planck-like and classical forms. If ${\Delta }_{p-c}$ denotes that difference, then ${\Delta }_{p-c}=e_p(1-2Q_L{n}_L)$ and $e_c=e_p-{\Delta }_{p-c}$. Consequently,

Deriving the Vacuum Electric Permittivity

Elementary charge can then be inserted into the electromagnetic relation for the electric constant. Let ${\epsilon }_0$ denote the vacuum electric permittivity and ${\gamma }_F$ the dimensionless Frames mapping geometry factor. After the count-dependent terms are collected into ${\gamma }_F$, the result reduces to

In this construction the electric constant is not the starting point for Coulomb's constant. It is an intermediate consequence of the fundamental measures and the electromagnetic realization geometry. This reverses the conventional explanatory order while preserving the established SI relationship among the constants.

Applying the Frames Mapping

The count geometry is collected into the dimensionless factor ${\gamma }_F$. It combines the electromagnetic count offset, the length-count differential, the elementary-charge remainder, and the realization polynomial. In this way the Frames mapping separates frame-dependent realization geometry from the fundamental measures that enter the physical expression.

Deriving Coulomb's Constant

Let $k_e$ denote Coulomb's constant. Substitution of the MQ electric-constant expression into the conventional relation $k_e=1/\left(4\pi {\epsilon }_0\right)$ eliminates the electric constant from the final first-principles form:

The expression depends on the MQ fundamental measures and the Frames mapping geometry factor rather than on vacuum permittivity as an independent input. In the present implementation, the source quantities used to establish the fundamental measures ultimately trace to the 2022 CODATA adjustment and defined SI constants. The numerical agreement with the CODATA value should therefore be read as a concordance and structural-consistency result, not automatically as an independent experimental prediction.

This dependency distinction is important because the elementary charge and speed of light are exact defining constants in the present SI, while the electric and magnetic constants and Coulomb's constant are linked to the experimentally determined fine-structure constant. A calculation that uses one member of this algebraically connected network to fix the realization geometry and then reproduces another member demonstrates consistency of the MQ construction with the electromagnetic constant network. A statistically independent test requires a source measurement whose determination does not already fix the comparison quantity through the same SI/CODATA dependencies.