Empirical Equation for a Fine-Structure Constant with Very High Accuracy

We proposed several empirical equations about the electromagnetic force and gravity. The main three equations were connected mathematically. However, these equations have small errors of approximately 10−3. Therefore, we attempted to improve the accuracy. Regarding the factors of 9/2 and π, we used 4.48870 and 3.13189, respectively. Then, the errors become smaller than 10−5. However, we could not show any reasons for these compensations. We noticed the following equations. 136.0113 4 4.488855 27 = × , ( ) 3 3.131777 136.0113 Rk × = . Then, we can explain the von Klitzing constant 3.131777037 4.488855463 13.5 136.0113077 Rk = × × × . It is well known that the von Klitzing constant can be measured with very high accuracy. We examined this equation for the von Klitzing constant in detail. Then, we noticed that 136.0113 should be uniquely determined. The von Klitzing constant is highly related to the fine-structure constant. After the examination of the numerical connections, we can explain the value of 137.035999081 as a finestructure constant with very high accuracy. Then, we attempt to explain this value from Wagner’s equation.

. It is well known that the von Klitzing constant can be measured with very high accuracy. We examined this equation for the von Klitzing constant in detail. Then, we noticed that 136.0113 should be uniquely determined. The von Klitzing constant is highly related to the fine-structure constant. After the examination of the numerical connections, we can explain the value of 137.035999081 as a finestructure constant with very high accuracy. Then, we attempt to explain this value from Wagner's equation.

Keywords
Fine-Structure Constant, Wagner's Equation
Equations (1), (2) and (3) were connected mathematically [3]. However, we could not establish the background theory. Then, we abandoned theoretical explanations and tried to reduce the error.
Our three equations have small errors of approximately 10 −3 and 10 −4 [3]. We attempted to reduce the error in previous reports by changing Factors 4.5, π and T c [4] [5]. Regarding the value of T c , 2.72642 K was used instead of 2.72548 K.
Regarding the factors of 9/2 and π, we used 4.48870 and 3.13189, respectively.
Then, the errors become smaller than 10 −5 .
4.48870 and 3.13189 are connected with the following equations.
However, we could not provide any reasons behind 4.48870 and 3.13189. In this report, we proposed the following two values for compensation.
The rest of the paper is organized as follows. In Section 2, we present an em-

Two Important Coefficients
Previously, we proposed the two coefficients 4.48870 and 3.13189 [4] [5]. Then, we thought that 4.48870 is the deviation from 4.5. 3.13189 is the deviation from π.
Then, we noticed the following equations.

4 4.48867 27
In Equation (14), when we believe that 4.48867 should be a deviation from 4.5, Next, we persevered for half a year and discovered the following equation.
( ) ( ) For convenience, Equation (7) is rewritten as follows: The ratio between m p and m e becomes smaller than the measured value. This  (14) and (16), which will be explained in a later section.

Relationship between
However, Equation (10) explaining α is too complex to calculate Z 0 . Therefore, an explanation for the electric constant and the magnetic constant will be published in future reports.

The Problem for the Measured Values of G and T c
The significant figures of the measured T c are only 3 or 4 in the present study.
For the value of T c , we use 2.72652 K instead of 2.72548 K. https://en.wikipedia.org/wiki/Cosmic_microwave_background.
The measured public value of G was often changed. We strongly propose that the value of G should be measured in space. On the ground, the value of G is influenced by the centrifugal force due to the rotation of the earth. We use 6.674777 × 10 −11 m 3 •kg −1 •s −2 instead of 6.6743 × 10 −11 m 3 •kg −1 •s −2 . https://en.wikipedia.org/wiki/Gravitational_constant.

Our Three Refined Empirical Equations
Equation (1) is refined as follows: Equation (41) is equal to Equation (42). Therefore, the compensation method is perfect.
Next, Equation (3) is refined as follows: for T c and G, which is slightly different from the measured value.

Discussion
The main purpose of this report is to show empirical equations. However, without any theories, this report can appear to be nothing more than numerology.
Therefore, we wrote this section. Our arguments strongly depend on numerical connections rather than logical connections. Therefore, our discussions appear to be different from the majority's opinion. This does not mean that we ignore the majority's opinions. Using simple unknown ideas, our opinions may be compatible with the majority's opinion.

Consider the Value of 136.0113077
The value of 136.0113077 is related to the mass ratio between some particle and an electron. The rest mass of a pion (π + ) is 139.57 MeV. The mass of an electron is 0.51100 MeV. Therefore, 139.57 MeV 2 136.57 2 136.011 1 0.51100 MeV = × ≈ × + When the pion consists of two quarks and an electron, the value of 136.011 is the mass ratio between a quark and an electron. This conclusion seems to disprove the concept of gluon. However, we notice that the concept of a gluon is useful for the mass ratio between a proton and an electron.

Consideration for the Value of 1836.152654
The value of 1836.152654 is related to the mass ratio between a proton and an electron.
When a proton consists of three quarks and the value of 136.011 is the mass ratio between a quark and an electron, there remains the mystery about the factor of 4.5. We believe that the factor of 4.5 derives from the degrees of freedom of the thermal energy due to the combined three quarks. Then, we must explain the value of 3.5 (=4.5 − 1).
According to Albert Einstein, E = mc 2 . Therefore, the thermal energy should be related to the weightness. A hot mass should be heavier than a cold mass. Thus, we must consider what happens concerning the degree of freedom.
When the boundary between thermal energy and the rest mass energy becomes unclear, 77% (=3.5/4.5) of the rest mass energy related to the thermal energy from the degree of freedom seems to be binding energy. This concept of binding energy may be compatible with the concept of gluons.

Consideration of the Theoretical Meaning of Equation (10)
The theory of loop quantum gravity is related to the theory of the lattice structure in space time. In the area of electrochemistry, the theory of vacancies in the lattice structure is governed by Wagner's equation. The fine-structure constant is the interference constant. When the interference constant for electrons is α, the transference number for electrons should be 1 − α. Then, the transference of quarks is α, which explains the strong interaction [6]. According to the advanced Wagner model, the diffusion time for mixed electronic and ionic currents should decrease exponentially with distance [7]. When the diffusion time for mixed electrons and quark flux should decrease exponentially with distance, the Yukawa potential can be explained.
For convenience, Equation (10) is rewritten as follows:

Consideration of the Degree of Freedom inside Electrons
In Equation (47), the number 3.131777037 is the deviation from π. The degree of freedom inside an electron seems to be 2π. We believe that 2π is related to the spin of the electron. Angrick et al. confirmed that the spin of electrons cannot be ignored thermodynamically [8]. Furthermore, Aquino et al. discovered a new method for vector analysis [9]. We hope that the degrees of freedom in electrons will be clarified experimentally in detail.

Conclusions
Previously, we proposed several empirical equations to describe the relationship between an electromagnetic force and T c [1] [2] [3]. Three equations were explained by the factors of 9/2 and π. However, these equations have small errors of approximately 10 −3 and 10 −4 [4] [5].
We tried to improve the accuracies by changing the value of 9/2 and π. Then, we persevered and discovered the following two equations. Avoiding numerology, we discuss this equation. Then, we can justify that the value of 4.488855463 is not coincidental. The explanation for the electric constant and the magnetic constant will be published in future reports.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.