2026-07-25
Geometric origin of the square of the fine structure constant and the parameters of the Bohr atom
This work is part of a more general concept in which mathematics is viewed not only as a language for describing existing reality but also as the basis for constructing it. Reality can have many different embodiments in the form of different realities. Their physical properties and laws may differ, but the underlying mathematical relationships must remain consistent.
Within this concept, our physical reality is viewed as one possible reflection of a more general mathematical structure. Therefore, motion, mass, charge, and atomic states need not be introduced as independent primary entities. They can be different observable manifestations of a single initial wave state. In the proposed model, such an initial object is the normalized split operator, which unites the internal state of a particle and its external motion.
Introduction
The parameters of the hydrogen atom are related to the fine-structure constant by particularly simple relationships. In the first Bohr orbit, the electron's velocity is \(\alpha_{\mathrm{fs}}c\), the radius is \(\hbar/(m_ec\alpha_{\mathrm{fs}})\), and the ground-state energy is proportional to \(\alpha_{\mathrm{fs}}^2\). In standard theory, these results follow from the Coulomb interaction and the condition for quantization of angular momentum.
However, simply substituting \(\alpha_{\mathrm{fs}}\) into existing formulas does not yet explain its origin. If we assume in advance that the internal projection coefficient is equal to \(\alpha_{\mathrm{fs}}\), that the radius varies by a factor of \(1/\alpha_{\mathrm{fs}}\), and that the orbital velocity is equal to \(\alpha_{\mathrm{fs}}c\), then the parameters of the Bohr atom are indeed reproduced, but the fine-structure constant itself remains the original postulate. The geometric problem is different: first, obtain a dimensionless coefficient from the ratio of independently constructed scales and only then show that this coefficient coincides with \(\alpha_{\mathrm{fs}}\).
To do this, it is necessary to distinguish three levels. The first level is the internal electromagnetic scale of the electron \(r_e\), which we will henceforth call the zeroth orbit. The second is the reduced Compton length \(\overline\lambda_C\), related to the observed electron mass. The third is the radius of the first atomic orbit \(R_1=a_0\). The fine structure constant will initially arise as the ratio of the first two scales, and its square will result from two successive geometric transitions.
The construction sequence is as follows
\[ \tag{1} \boxed{ \left(r_e,\;\omega_{\mathrm{int}}\right) \;\longrightarrow\; \frac{r_e}{\overline\lambda_C} = \frac{\omega_C}{\omega_{\mathrm{int}}} \;\longrightarrow\; \alpha_{\mathrm{fs}} \;\longrightarrow\; R_1=a_0 \;\longrightarrow\; v_1,\;L_1,\;E_1 } \] In this chain, \(\alpha_{\mathrm{fs}}\) is not substituted before constructing the quantities \(r_e\) and \(\overline\lambda_C\). Therefore, the equality of the geometric coefficient of the fine structure constant becomes the result of comparing two scales, rather than a hidden initial assumption.
1. The complete state operator
The model is based on two mutually complementary idempotents
\[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] They define the hyperbolic unit \(\j=\ep-\em\), for which \(\j^2=1\). The complete state of the particle is written as
\[ \tag{3} \boxed{ J(a,b)=\j^a(-\j)^b = \ep e^{i\pi b}+\em e^{i\pi a} }. \] The parameters \(a\) and \(b\) play different physical roles:
\[ \tag{4} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}, \] \[ \tag{5} b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] The exponent \(a\) describes the particle's internal periodicity, while the exponent \(b\) describes its external motion in observable space. These two processes are part of the same operator but belong to different idempotent components. Their orthogonality is expressed by the equality \(\ep\em=0\), and the normalization of the total state has the form
\[ \tag{6} J\overline J=\ep+\em=1, \qquad \lVert J\rVert_i=1. \] The unit norm means the total state is preserved, but does not mean that the mass, energy, or radius are numerically equal to unity. Physical quantities arise when mapping the normalized internal state into a space with dimensional coordinates. It is in this mapping that frequency and spatial scales appear.
2. Two Independently Determined Electron Scales
To avoid introducing the fine-structure constant in advance, we first construct two scales, in the definitions of which \(\alpha_{\mathrm{fs}}\) is absent. The first scale is associated with the electromagnetic energy of the charge, the second with the quantum frequency of the observed mass.
The characteristic energy of electromagnetic interaction at a distance \(r\) is
\[ \tag{7} E_{\mathrm{em}}(r) = \frac{e^2}{4\pi\varepsilon_0r}. \] We define the internal scale of the electron \(r_e\) by the condition under which this energy reaches the rest energy of the electron:
\[ \tag{8} \frac{e^2}{4\pi\varepsilon_0r_e} = m_ec^2. \] Following directly
\[ \tag{9} \boxed{ r_e = \frac{e^2}{4\pi\varepsilon_0m_ec^2} }. \] This is the standard expression for the classical radius of an electron. In the present model, it does not imply that the electron is a classical hard ball of this radius. The quantity \(r_e\) is used as an internal electromagnetic scale, on which the charge energy and the rest energy are of the same order of magnitude. Since this scale precedes the construction of the atomic orbit, we will henceforth call it the zeroth orbit.
The second scale is determined by the observed Compton frequency of the electron:
\[ \tag{10} \omega_C = \frac{m_ec^2}{\hbar}. \] The corresponding reduced Compton length is
\[ \tag{11} \boxed{ \overline\lambda_C = \frac{c}{\omega_C} = \frac{\hbar}{m_ec} }. \] Thus, \(r_e\) is derived from the electromagnetic energy scale, and \(\overline\lambda_C\) is derived from the quantum frequency of the rest energy. The fine structure constant was not used in either formula (9) or formula (11).
3. Derivation of the Fine Structure Constant as a Geometric Ratio
The two constructed scales refer to the same electron, but describe different aspects of its state. The zeroth orbit \(r_e\) characterizes the inner electromagnetic level, and \(\overline\lambda_C\) characterizes the observable mass level. Therefore, the natural dimensionless conversion factor between them is the ratio
\[ \tag{12} \alpha_g = \frac{r_e}{\overline\lambda_C}. \] The subscript \(g\) emphasizes that at this stage, the geometric conversion factor for the ratio of two lengths is introduced. Substitute the independently obtained expressions (9) and (11):
\[ \tag{13} \begin{aligned} \alpha_g &= \frac{ \dfrac{e^2}{4\pi\varepsilon_0m_ec^2} }{ \dfrac{\hbar}{m_ec} } \\ &= \frac{e^2}{4\pi\varepsilon_0\hbar c}. \end{aligned} \] The right-hand side is the standard expression for the fine-structure constant. Therefore,
\[ \tag{14} \boxed{ \alpha_g = \frac{r_e}{\overline\lambda_C} = \frac{e^2}{4\pi\varepsilon_0\hbar c} = \alpha_{\mathrm{fs}} }. \] This equality is the key result of the article. The fine structure constant was not substituted into the formula for \(r_e\) to then obtain itself. First, the electromagnetic and quantum scales were determined independently, after which their ratio was found to be exactly equal to \(\alpha_{\mathrm{fs}}\).
Numerically, these scales differ significantly: \(r_e\approx2.818\cdot10^{-15}\) m, while \(\overline\lambda_C\approx3.862\cdot10^{-13}\) m. Their ratio is approximately \(7.297\cdot10^{-3}\), or \(1/137.036\). Thus, the smallness of the fine structure constant directly expresses the large difference between the internal electromagnetic and Compton scales of the electron.
It is important to precisely formulate the limit of the obtained result. Formula (14) derives the geometric and physical meaning of the fine structure constant as the ratio of two fundamental scales of the electron. It doesn't calculate the number \(1/137.035\ldots\) from abstract algebra alone, without using the physical constants \(e\), \(\varepsilon_0\), \(\hbar\), and \(c\). The numerical value remains a property of our physical realization, while the geometry explains where exactly this dimensionless coefficient arises and which levels of state it connects.
4. Frequency form of the same geometric transition
We associate the spatial scale \(r_e\) with the intrinsic frequency \(\omega_{\mathrm{int}}\). The primary wave process relates length and frequency by the velocity invariant:
\[ \tag{15} \boxed{ r_e\omega_{\mathrm{int}}=c }. \] For the Compton scale, formula (11) similarly implies
\[ \tag{16} \boxed{ \overline\lambda_C\omega_C=c }. \] Divide the second equality by the first. Since the product of the length and the corresponding frequency is equal to \(c\) in both cases, a decrease in frequency is accompanied by an inverse increase in spatial scale:
\[ \tag{17} \frac{\omega_C}{\omega_{\mathrm{int}}} = \frac{r_e}{\overline\lambda_C}. \] Taking into accountFrom geometric derivation (14) we obtain
\[ \tag{18} \boxed{ \frac{\omega_C}{\omega_{\mathrm{int}}} = \frac{r_e}{\overline\lambda_C} = \alpha_{\mathrm{fs}} }. \] Therefore, the same constant is simultaneously the ratio of lengths and the inverse ratio of the corresponding frequencies:
\[ \tag{19} \boxed{ \omega_C = \alpha_{\mathrm{fs}}\omega_{\mathrm{int}}, \qquad \overline\lambda_C = \frac{r_e}{\alpha_{\mathrm{fs}}} }. \] In the mass_geometric_projection geometric mass model, the decrease in frequency is interpreted as a transverse Doppler projection of the internal process between orthogonal idempotent planes. Now the coefficient of this projection does not need to be separately identified with \(\alpha_{\mathrm{fs}}\): the equality is already obtained from the ratio of the two scales.
If we denote the angle of the intrinsic mapping by \(\theta_g\) and introduce the projection parameter \(\beta_g=\sin\theta_g\), then the corresponding quantity is \(\gamma_g=1/\sqrt{1-\beta_g^2}\). Therefore, the projection coefficient can be written in geometric form.
\[ \tag{20} \widetilde\alpha = \cos\theta_g = \frac{1}{\gamma_g} = \alpha_{\mathrm{fs}}. \] The quantities \(\theta_g\) and \(\gamma_g\) characterize the internal mapping of the state. They cannot be identified with the angle and Lorentz factor of the actual external motion of the electron along the atomic orbital. The external motion is still described by a separate parameter \(b\), velocity \(\beta\), and the external factor \(\gamma\).
The rest energy is now written as a consequence of the obtained frequency ratio:
\[ \tag{21} \boxed{ m_ec^2 = \hbar\omega_C = \alpha_{\mathrm{fs}}\hbar\omega_{\mathrm{int}} }. \] Thus, the mass formula and the zeroth orbit formula are mutually consistent, but no longer form a circular derivation. First, \(r_e\) is determined by the electromagnetic condition (8), then from the ratio \(r_e/\overline\lambda_C\) we obtain \(\alpha_{\mathrm{fs}}\), and only then does the constant appear in formula (21).
5. Three Spatial Levels and the Origin of the Square
The first geometric transition has already been constructed:
\[ \tag{22} r_e \;\overset{\alpha_{\mathrm{fs}}^{-1}}{\longrightarrow}\; \overline\lambda_C, \qquad \overline\lambda_C = \frac{r_e}{\alpha_{\mathrm{fs}}}. \] It translates the internal electromagnetic scale of the electron into the quantum scale of the observable mass. When a bound state of an electron and a proton is formed, the next level appears—the spatial closure of the electron wave in the atom.
The fundamental geometric hypothesis of atomic construction is the self-similarity of two successive mappings. The transition from the internal electron level to the mass level and the transition from the mass level to the first closed atomic orbital have the same dimensionless coefficient. Therefore
\[ \tag{23} R_1 = \frac{\overline\lambda_C}{\alpha_{\mathrm{fs}}}. \] This is no longer a postulate about the numerical value of the fine structure constant: the coefficient \(\alpha_{\mathrm{fs}}\) was previously obtained in formula (14). The only new assumption here is the repetition of the same geometric relationship at the next closure level. Such a repetition means that another independent dimensionless constant is not introduced between scales.
Successively applying formulas (22) and (23), we obtain
\[ \tag{24} R_1 = \frac{1}{\alpha_{\mathrm{fs}}} \frac{r_e}{\alpha_{\mathrm{fs}}} = \frac{r_e}{\alpha_{\mathrm{fs}}^2}. \] Therefore,
\[ \tag{25} \boxed{ \frac{r_e}{R_1} = \alpha_{\mathrm{fs}}^2 }. \] The square of the fine structure constant appears here not as the square of the inverse Lorentz factor of some external near-light velocity. It is the product of two identical geometric ratios:
\[ \tag{26} \boxed{ \frac{r_e}{R_1} = \frac{r_e}{\overline\lambda_C} \frac{\overline\lambda_C}{R_1} = \alpha_{\mathrm{fs}} \alpha_{\mathrm{fs}} = \alpha_{\mathrm{fs}}^2 }. \] It is formula (26) that reveals the geometric origin of the square: the first factor relates to the structure of the electron itself, the second to the closure of its state in the atom.
6. Obtaining the Bohr radius
Substitute the expression for the reduced Compton length into formula (23):
\[ \tag{27} \begin{aligned} R_1 &= \frac{\overline\lambda_C}{\alpha_{\mathrm{fs}}} \\ &= \frac{\hbar}{m_ec\alpha_{\mathrm{fs}}}. \end{aligned} \] Using the equation already derived
\[ \alpha_{\mathrm{fs}} = \frac{e^2}{4\pi\varepsilon_0\hbar c}, \] we obtain
\[ \tag{28} \boxed{ R_1=a_0 = \frac{\hbar}{m_ec\alpha_{\mathrm{fs}}} = \frac{4\pi\varepsilon_0\hbar^2}{m_ee^2} }. \] The right-hand side is the standard Bohr radius. The three spatial scales form a sequence symmetric with respect to \(\overline\lambda_C\):
\[ \tag{29} \boxed{ r_e \;\overset{\alpha_{\mathrm{fs}}^{-1}}{\longrightarrow}\; \overline\lambda_C \;\overset{\alpha_{\mathrm{fs}}^{-1}}{\longrightarrow}\; a_0 }. \] On a logarithmic scale, the reduced Compton length is exactly halfway between the zeroth and first orbits:
\[ \tag{30} \boxed{ \overline\lambda_C^2 = r_ea_0 }. \] Indeed, from \(r_e=\alpha_{\mathrm{fs}}\overline\lambda_C\) and \(a_0=\overline\lambda_C/\alpha_{\mathrm{fs}}\), the product of the extreme scales is equal to the square of the mean. This is an additional expression for the self-similarity of the geometric chain.
7. Deriving the velocity in the first orbit
In the previous sequence, the equality \(v_1=\alpha_{\mathrm{fs}}c\) was taken separately. However, after obtaining the radius of the first orbit, the velocity can be derived from dynamic equilibrium without introducing it as another postulate.
For circular motion, the centripetal force must be equal to the force of Coulomb attraction:
\[ \tag{31} \frac{m_ev_1^2}{R_1} = \frac{e^2}{4\pi\varepsilon_0R_1^2}. \] Multiplying both sides by \(R_1/m_e\), we get
\[ \tag{32} v_1^2 = \frac{e^2}{4\pi\varepsilon_0m_eR_1}. \] Now let's substitute \(R_1=a_0=\hbar/(m_ec\alpha_{\mathrm{fs}})\):
\[ \tag{33} \begin{aligned} v_1^2 &= \frac{e^2}{4\pi\varepsilon_0m_e} \frac{m_ec\alpha_{\mathrm{fs}}}{\hbar} \\ &= \alpha_{\mathrm{fs}}^2c^2. \end{aligned} \] Choosing a positive velocity magnitude, we obtain
\[ \tag{34} \boxed{ \beta_1 = \frac{v_1}{c} = \alpha_{\mathrm{fs}}, \qquad v_1 = \alpha_{\mathrm{fs}}c }. \] Consequently, the orbital velocity is not specified independently of the radius and fine structure constant. It arises as a dynamical consequence of the geometrically obtained scale \(R_1\) and the Coulomb interaction.
In the split operator, this real external velocity corresponds to the exponent
\[ \tag{35} \boxed{ b_1 = \frac{\arcsin\alpha_{\mathrm{fs}}}{\pi} }. \] The external Lorentz factor of an electron in the first orbit is
\[ \tag{36} \boxed{ \gamma_1 = \frac{1}{\sqrt{1-\alpha_{\mathrm{fs}}^2}} \approx 1.0000266 }. \] Orbital motion is nonrelativistic to a first approximation. The coefficient \(1/\alpha_{\mathrm{fs}}\approx137\) is not its Lorentz factor, and the quantity \(1/\alpha_{\mathrm{fs}}^2\) especially cannot be interpreted as the Lorentz factor of the total atomic motion.
8. Angular Momentum of the First Orbit
Now both the radius and velocity are obtained independently of the Bohr condition. Let's check the angular momentum:
\[ \tag{37} L_1=m_ev_1R_1. \] Substituting formulas (28) and (34), we find
\[ \tag{38} \begin{aligned} L_1 &= m_e \left(\alpha_{\mathrm{fs}}c\right) \frac{\hbar}{m_ec\alpha_{\mathrm{fs}}} \\ &= \hbar. \end{aligned} \] Thus, the Bohr condition is satisfied for the first orbit:
\[ \tag{39} \boxed{ m_ev_1a_0=\hbar }. \] In this sequence, the condition \(L_1=\hbar\) is not used to calculate the radius or velocity. Instead, it arises as a test of the consistency of the electromagnetic scale, geometric closure, and dynamic equilibrium.
9. Ground State Energy
The kinetic energy of an electron in the first orbit is
\[ \tag{40} T_1 = \frac{m_ev_1^2}{2} = \frac12m_ec^2\alpha_{\mathrm{fs}}^2. \] The Coulomb potential energy at radius \(a_0\) is given by
\[ \tag{41} U_1 = -\frac{e^2}{4\pi\varepsilon_0a_0}. \] From the dynamic equilibrium condition (31) it follows
\[ \frac{e^2}{4\pi\varepsilon_0a_0} = m_ev_1^2. \] Therefore
\[ \tag{42} U_1 = -m_ev_1^2 = -m_ec^2\alpha_{\mathrm{fs}}^2 = -2T_1. \] The total energy of the bound state is
\[ \tag{43} \boxed{ E_1 = T_1+U_1 = -\frac12m_ec^2\alpha_{\mathrm{fs}}^2 }. \] The minus sign indicates that the state is bound. To release an electron, it must be supplied with energy \(|E_1|\). The square of the fine structure constant determines the relative depth of the bound state compared to the electron's rest energy.
10. Orbital Frequency
The angular frequency of motion along the first orbit is determined by the ratio of the velocity to the radius:
\[ \tag{44} \omega_1 = \frac{v_1}{a_0}. \] After substituting formulas (28) and (34), we obtain
\[ \tag{45} \boxed{ \omega_1 = \alpha_{\mathrm{fs}}^2 \frac{m_ec^2}{\hbar} = \alpha_{\mathrm{fs}}^2\omega_C }. \] Since \(\omega_C=\alpha_{\mathrm{fs}}\omega_{\mathrm{int}}\), the orbital frequency is related to the original intrinsic frequency by another relationship:
\[ \tag{46} \boxed{ \omega_1 = \alpha_{\mathrm{fs}}^3\omega_{\mathrm{int}} }. \] When moving from the internal state of the electron to the first atomic orbit, the spatial scale increases, and the observed frequency of motion decreases. These changes occur in a coordinated manner and are determined by powers of the same coefficient.
11. Frequency and Length Invariants
For the zeroth orbit and the Compton scale, the same mutually inverse pair of transformations applies:
\[ \tag{47} \overline\lambda_C = \frac{r_e}{\alpha_{\mathrm{fs}}}, \qquad \omega_C = \alpha_{\mathrm{fs}}\omega_{\mathrm{int}}. \] Therefore, the product of the corresponding spatial and frequency quantities is conserved:
\[ \tag{48} \boxed{ r_e\omega_{\mathrm{int}} = \overline\lambda_C\omega_C = c }. \] Formula (48) shows two sides of the same transformation. If the observed frequency decreases by a factor of \(\alpha_{\mathrm{fs}}\), the corresponding spatial scale increases by a factor of \(1/\alpha_{\mathrm{fs}}\). The product remains equal to \(c\).
For the first orbit, the product of the radius and the orbital frequency is equal to the actual orbital velocity:
\[ \tag{49} \boxed{ a_0\omega_1 = v_1 = \alpha_{\mathrm{fs}}c }. \] 12. The Unified Role of the Squared Fine Structure Constant
After the geometric derivation of \(\alpha_{\mathrm{fs}}\) and the construction of the first orbit, the quantity \(\alpha_{\mathrm{fs}}^2\) acquires several interrelated meanings.
It defines the ratio of the zeroth and first spatial scales:
\[ \tag{50} \frac{r_e}{a_0} = \alpha_{\mathrm{fs}}^2. \] It is the square of the dimensionless orbital velocity:
\[ \tag{51} \frac{v_1^2}{c^2} = \alpha_{\mathrm{fs}}^2. \] It defines the ratio of the orbital and Compton frequencies:
\[ \tag{52} \frac{\omega_1}{\omega_C} = \alpha_{\mathrm{fs}}^2. \] Finally, it defines the energy scale of the ground state:
\[ \tag{53} \frac{2|E_1|}{m_ec^2} = \alpha_{\mathrm{fs}}^2. \] All four relations can be combined:
\[ \tag{54} \boxed{ \frac{r_e}{a_0} = \frac{v_1^2}{c^2} = \frac{\omega_1}{\omega_C} = \frac{2|E_1|}{m_ec^2} = \alpha_{\mathrm{fs}}^2 }. \] The same number links the spatial, kinematic, frequency, and energy aspects of the first orbit. This coincidence no longer appears as a set of independent formulas: all relations go back to two successive transitions with the coefficient \(\alpha_{\mathrm{fs}}\).
13. The Complete Geometric Chain
The main results can be presented in a single sequence:
\[ \tag{55} \boxed{ \begin{aligned} r_e &= \frac{e^2}{4\pi\varepsilon_0m_ec^2}, \\ \overline\lambda_C &= \frac{\hbar}{m_ec}, \\ \alpha_{\mathrm{fs}} &= \frac{r_e}{\overline\lambda_C} = \frac{e^2}{4\pi\varepsilon_0\hbar c}, \\ \omega_C &= \alpha_{\mathrm{fs}}\omega_{\mathrm{int}}, \\ a_0 &= \frac{\overline\lambda_C}{\alpha_{\mathrm{fs}}} = \frac{r_e}{\alpha_{\mathrm{fs}}^2}, \\ v_1 &= \alpha_{\mathrm{fs}}c, \qquad b_1 = \frac{\arcsin\alpha_{\mathrm{fs}}}{\pi}, \\ L_1 &= m_ev_1a_0 = \hbar, \\ E_1 &= -\frac12m_ec^2\alpha_{\mathrm{fs}}^2. \end{aligned} } \] The order of the rows in formula (55) is crucial. First, two scales are constructed without using \(\alpha_{\mathrm{fs}}\). Then, the constant is obtained as their ratio. After this, the same coefficient is used for the second geometric transition, and the velocity, angular momentum, and energy are derived as consequences of the constructed radius and dynamic equilibrium.
14. What is assumed and what is derived
To correctly evaluate the model, it is necessary to separate the initial physical and geometric assumptions from the obtained results.
Accepted:
1. The primary state of the electron is described by the normalized operator \(J(a,b)=\j^a(-\j)^b\), where \(a\) is responsible for the internal periodicity, and \(b\) is responsible for the external motion.
2. The internal electromagnetic scale \(r_e\) is determined by the condition that the characteristic charge energy \(e^2/(4\pi\varepsilon_0r_e)\) is equal to the rest energy \(m_ec^2\).
3. The spatial scale and the corresponding angular frequency are related by the wave invariant \(r\omega=c\).
4. When transitioning from the Compton scale to the first closed atomic orbit, the same dimensionless coefficient that relates the zeroth orbit to the Compton scale is repeated. This is the hypothesis of self-similarity of two successive geometric mappings.
5. The stable circular state satisfies the balance of the Coulomb and centripetal forces.
Derived:
1. The coefficient of the first geometric transition is equal to \(r_e/\overline\lambda_C\) and coincides with the fine-structure constant \(\alpha_{\mathrm{fs}}\).
2. The frequency projection of the internal state has the same coefficient: \(\omega_C/\omega_{\mathrm{int}}=\alpha_{\mathrm{fs}}\).
3. Two successive transitions yield the ratio \(r_e/a_0=\alpha_{\mathrm{fs}}^2\).
4. The radius of the first orbit coincides with the Bohr radius, and dynamic equilibrium leads to the velocity \(v_1=\alpha_{\mathrm{fs}}c\).
5. The angular momentum of the first orbit is \(\hbar\), and the energy of the bound state is \(-m_ec^2\alpha_{\mathrm{fs}}^2/2\).
This distinction is important. The model derives the fine-structure constant as the ratio of independently introduced electromagnetic and quantum scales, but does not claim to calculate its numerical value from the idempotent algebra alone. The next, deeper task remains to obtain the quantities \(e\), \(\hbar\), and the electromagnetic scale itself directly from the internal dynamics of the operator \(J\).
15. Connection with the Further Construction of the Atom
This article establishes the general framework of the atomic problem: the zeroth orbit \(r_e\), the Compton intermediate scale \(\overline\lambda_C\), the first orbit \(a_0\), its velocity, angular momentum, and energy. The next step consists not in re-obtaining these quantities, but in investigating the internal structure of the first orbit.
In the first part of constructing the first orbit, the transition from the internal motions of the electron and proton to a closed joint state is considered. In the next section, we examine the splitting of this state into closely spaced branches and the energy gradient that arises between them.
The ratios \(r_e/a_0=\alpha_{\mathrm{fs}}^2\) and \(v_1/c=\alpha_{\mathrm{fs}}\) define the overall scale of the first orbit, while its two-branch structure is the next level of geometric construction. The presence of two idempotent components does not in itself automatically imply the existence of two different physical radii: specific branches appear only after the atomic closure conditions are specified.
Conclusions
This work reconstructs a fundamentally important stage—the derivation of the fine-structure constant before its use in the Bohr atom formulas. First, the internal electromagnetic scale of the electron and the reduced Compton length are determined independently:
\[ \tag{56} r_e = \frac{e^2}{4\pi\varepsilon_0m_ec^2}, \qquad \overline\lambda_C = \frac{\hbar}{m_ec}. \] Their dimensionless ratio is
\[ \tag{57} \boxed{ \frac{r_e}{\overline\lambda_C} = \frac{e^2}{4\pi\varepsilon_0\hbar c} = \alpha_{\mathrm{fs}} }. \] Due to the length-frequency invariant, the same number is the coefficient of the internal frequency projection:
\[ \tag{58} \boxed{ \frac{\omega_C}{\omega_{\mathrm{int}}} = \frac{r_e}{\overline\lambda_C} = \alpha_{\mathrm{fs}} }. \] Repeating the same ratio in the next geometric transition yields the square of the fine-structure constant:
\[ \tag{59} \boxed{ r_e \;\longrightarrow\; \overline\lambda_C \;\longrightarrow\; a_0, \qquad \frac{r_e}{a_0} = \alpha_{\mathrm{fs}}^2 }. \] The actual orbital velocity is then obtained from dynamic equilibrium, rather than being taken separately. As a result, the main parameters of the Bohr atom are reconstructed:
\[ \tag{60} \boxed{ \left\{ \begin{aligned} v_1 &= \alpha_{\mathrm{fs}}c, \\ a_0 &= \frac{\hbar}{m_ec\alpha_{\mathrm{fs}}}, \\ L_1 &= \hbar, \\ E_1 &= -\frac12m_ec^2\alpha_{\mathrm{fs}}^2. \end{aligned} \right. } \] Thus, the fine-structure constant is no longer introduced at the beginning of the construction as an unknown internal projection coefficient. It arises as the ratio of the electromagnetic and quantum scales of a single electron. Its square, in turn, appears as the composition of two successive geometric transitions and simultaneously determines the spatial, velocity, frequency, and energy scales of the first orbit.