Modern physics boasts several highly successful theories. Classical mechanics describes the motion of bodies, electrodynamics describes electric and magnetic fields, the theory of relativity describes space, time, energy, and momentum, and quantum mechanics describes the behavior of matter at the microscopic level. However, the mathematical languages ​​of these theories differ significantly, and properties of elementary particles such as mass, charge, spin, and magnetic moment are often introduced as initial characteristics.

The series of papers in the "Wave Electricity" section represents an attempt to approach these issues from a different perspective. The starting point is a new basis constructed from two mutually complementary idempotents. Its complex expansion leads to four real directions, internal rotation, norm conservation, and a geometric separation of translational and periodic motion.

Further development of this construction allows us to establish formal connections with the Lorentz factor, the energy invariant, the time-dependent Schrödinger equation, and the particle's intrinsic frequency. In the final stage, the model is applied to the electron, linking its energy, classical radius, magnetic moment, and fine structure constant into a single geometric framework.

This paper is a popular science overview of the entire series. It does not replace the detailed mathematical conclusions presented in the original publications, but rather shows the general sequence of development of the ideas. After each significant result, references are provided to articles that discuss the relevant issue in detail.

Most physical theories were created to solve a specific class of problems. Euclidean geometry turned out to be the natural language of classical mechanics, complex numbers – of oscillatory processes and quantum mechanics, four-dimensional space-time – of special relativity. Each such language not only simplifies calculations but also determines the way we represent physical reality.

For example, the complex exponential \[ \tag{1} e^{i\omega t} = \cos\omega t+i\sin\omega t \] compactly describes rotation, harmonic oscillation, and periodic phase changes. Moreover, the complex unit \(i\) itself is not considered an additional physical direction in space. It serves as a mathematical tool to unite two mutually perpendicular components of a single process.

A similar situation can arise when describing an elementary particle. The observed translational motion may be only one projection of a more complex state, including internal periodic motion. In this case, mass, rest energy, and magnetic moment may not be independent properties, but different manifestations of a single geometric structure.

This is why the research begins not with a ready-made model of the electron, but with the search for a new mathematical language. First, the algebra is constructed, then its geometric properties are investigated, and only then is a physical interpretation introduced.

The initial construction of the new basis and its mathematical foundations are discussed in detail in the articles "A New Cartesian Basis of Two Idempotents" and "Supplement to the New Cartesian Basis".

The basis of the algebra under consideration is formed by two complementary idempotents: \[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0. \]

An idempotent is an element that does not change when squared. In this case, \(\ep\) and \(\em\) are also orthogonal in the algebraic sense: their product is zero. Together, they form unity: \[ \tag{3} 1=\ep+\em. \]

The difference of idempotents forms a hyperbolic unit: \[ \tag{4} \j=\ep-\em, \qquad \j^2=1. \] Therefore, the same algebra can be written in both the basis \[ \{1,\j\}, \] and in an idempotent basis \[ \{\ep,\em\}. \]

The key step is that the coefficients of idempotents are allowed to be complex. Then each of the two idempotent directions forms its own complex plane: \[ \tag{5} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\}. \]

Thus, the four real directions arise not as four predefined coordinates, but as a complex expansion of two idempotent components. The first plane is formed by the directions \(\ep\) and \(i\ep\), the second by the directions \(\em\) and \(i\em\).

An arbitrary state in this basis can be written as \[ \tag{6} Z = x_1\ep+x_2i\ep+x_3\em+x_4i\em, \] where \(x_1,x_2,x_3,x_4\) are real numbers.

The equivalent notation with complex coefficients is more compact: \[ \tag{7} Z=z_+\ep+z_-\em, \qquad z_+,z_-\in\mathbb C. \]

The resulting structure simultaneously contains two complex planes and a hyperbolic expansion. This allows it to describe not only position but also two qualitatively different types of motion: a change in the overall phase and a change in the internal relationship between idempotent components.

A detailed construction of the basis, the rules for multiplication, and its connection with hyperbolic numbers are given in the works "A New Cartesian Basis of Two Idempotents", "Addition to the New Cartesian Basis" and the second part of the study of the new basis.

Consider the simplest vector in which the first idempotent component remains stationary, while the second acquires a complex phase: \[ \tag{8} J(t) = \ep+\em e^{-i\omega t}. \]

Expand the exponential: \[ \tag{9} J(t) = \ep + \em\cos\omega t - i\em\sin\omega t. \]

The first component \(\ep\) is independent of time. The second component moves along the unit circle in the plane \[ \{\em,i\em\}. \] Therefore, a single object simultaneously contains a fixed direction and a rotating part.

This is fundamentally different from a conventional complex exponential. In the expression \(e^{-i\omega t}\), the entire vector in the complex plane rotates. In expression (8), the rotation applies only to one idempotent component, while the other remains unchanged.

If we multiply \(J(t)\) by some velocity \(u\), we obtain the velocity vector: \[ \tag{10} V(t) = u\ep+u\em e^{-i\omega t}. \]

Integrating over time yields the trajectory: \[ \tag{11} R(t) = ut\ep + \frac{iu}{\omega}\em e^{-i\omega t} + R_0. \]

The first part of the trajectory increases linearly with time and can be interpreted as translational motion. The second part describes a circle of radius \[ \tag{12} r=\frac{u}{\omega}. \]

When mapping such a trajectory into familiar three-dimensional space, the image of a helical motion emerges: the center of the system moves translationally, and the internal component rotates around it. Importantly, this separation is not introduced by an additional postulate. It is already contained in the algebraic form of the vector.

The geometry of the motion vector and the separation of its translational and rotational components are studied in detail in the articles "The New Cartesian Basis. Part 2" and "The New Cartesian Basis. Part 3".

A circle differs from an arbitrary curve in that the distance from its center to the moving point remains constant. Therefore, rotation can be considered not as an independent law of motion, but as a geometric consequence of the conservation of vector length.

For an ordinary real vector \(X(t)\), norm constancy means: \[ \tag{13} X(t)\cdot X(t)=\operatorname{const}. \]

Differentiating this equality with respect to time, we obtain: \[ \tag{14} \frac{d}{dt} \left( X\cdot X \right) = 2X\cdot\dot X = 0. \] Therefore, \[ \tag{15} X\cdot\dot X=0. \]

This means that the derivative of a vector is orthogonal to the vector itself. Geometrically, the derivative is directed tangent to the trajectory, while the original vector is directed radially. This orthogonality is characteristic of circular motion.

A similar result arises for the vector \(J(t)\). Its rotational component has a constant magnitude: \[ \tag{16} \left| e^{-i\omega t} \right| = 1. \] Therefore, changing the phase does not change its length, but only rotates its direction in the plane \((\em,i\em)\).

The derivative of the rotational component is equal to \[ \tag{17} \frac{d}{dt} e^{-i\omega t} = -i\omega e^{-i\omega t}. \] The factor \(-i\) rotates the complex vector by a right angle, and the factor \(\omega\)determines the speed of this rotation.

Therefore, in the construction under consideration, the chain of reasoning is as follows:

norm conservation  →  orthogonality of the vector and its derivative  →  rotation.

This allows us to consider rotation as a natural kinematic state of the system with a conserved norm, rather than as a motion that must be introduced separately.

A detailed derivation of orthogonality and an interpretation of rotation as a consequence of norm conservation are given in the paper "Rotation as a Consequence of Norm Conservation in an Idempotent Basis". The connection of this result with mass and the time-dependent Schrödinger equation is discussed in the paper "The Schrödinger Equation and the Origin of Mass in a New Idempotent Basis".

One of the central results of the cycle is the formal connection between the derivative of the internal rotation and the time part of the Schrödinger equation.

The original vector has the form \[ \tag{18} J(t) = \ep+\em e^{-i\omega t}. \] Its derivative is \[ \tag{19} \dot J(t) = -i\omega\em e^{-i\omega t}. \]

Multiply expression (19) by \(i\hbar\): \[ \tag{20} i\hbar\dot J(t) = \hbar\omega\em e^{-i\omega t}. \]

The quantum energy appears on the right-hand side. \[ \tag{21} E_\omega=\hbar\omega. \] Therefore, the expression can be rewritten as follows: \[ \tag{22} i\hbar\dot J(t) = E_\omega\em e^{-i\omega t}. \]

If we isolate the rotational component \[ \Psi(t)=\em e^{-i\omega t}, \] then we obtain for it \[ \tag{23} i\hbar \frac{\partial\Psi}{\partial t} = E_\omega\Psi. \]

This coincides with the time form of the Schrödinger equation for the steady state: \[ \tag{24} i\hbar \frac{\partial\Psi}{\partial t} = \widehat H\Psi, \qquad \widehat H\Psi=E\Psi. \]

Thus, in this model, the time-domain operator of quantum mechanics receives a simple geometric interpretation. Differentiation determines the rate of change of the internal phase, the factor \(i\) reverses the derivative back to the original rotating component, and \(\hbar\) converts the angular frequency into energy.

It is important, however, to distinguish between formal coincidence and a complete derivation of quantum mechanics. Expression (23) directly yields the time-domain equation for a single internal mode. The full space-time Schrödinger equation additionally requires the definition of a spatial energy operator, boundary conditions, potential, and measurement rules.

Therefore, the new basis does not replace all of quantum mechanics, but it does offer a geometric origin for one of its most important elements—the time-phase evolution of a stationary state.

A complete derivation, the choice of the sign of the complex phase, and a discussion of orthogonality are given in the article "The Schrödinger Equation and the Origin of Mass in a New Idempotent Basis". The mathematical basis for internal rotation is discussed in the article "Rotation as a Consequence of Norm Conservation".

In quantum mechanics, the angular frequency \(\omega\) corresponds to the energy \[ \tag{25} E=\hbar\omega. \] In the theory of relativity, the rest energy of a particle is defined by the expression \[ \tag{26} E_0=mc^2. \]

If the internal rotational mode is an eigenstate of the particle and its energy corresponds to the rest energy, we can identify \[ \tag{27} \hbar\omega_0=mc^2. \]

Hence, the mass is expressed in terms of the eigenfrequency: \[ \tag{28} m = \frac{\hbar\omega_0}{c^2}. \]

Considering \[ \omega_0=2\pi\nu_0 \] we obtain: \[ \tag{29} m = \frac{h\nu_0}{c^2}. \]

In this interpretation, mass ceases to be solely an external coefficient of inertia. It becomes an energy measure of the internal periodic process. The higher the natural frequency, the greater the energy of the internal mode and the corresponding mass.

The relationship can be written directly in terms of the vector derivative. Since the absolute value of the derivative of the rotational component is proportional to \(\omega_0\), we have: \[ \tag{30} m = \frac{\hbar}{c^2} \left\| \frac{dJ}{dt} \right\|. \]

This entry requires clarification of the norm and the isolation of a physically significant rotational component, butThis conveys the basic idea well: mass is determined by the intensity of an internal change of state.

This interpretation does not mean that the existence of mass is already deduced from algebra alone. The equality \(\hbar\omega_0=mc^2\) is a physical identification. Algebra yields an internal frequency and its corresponding energy, and identifying this energy with the rest energy connects the model with mass.

In this sense, the result consists of two parts. Mathematically, from \(J(t)\) arises the frequency and magnitude \(\hbar\omega\). Physically, it is assumed that for a proper internal mode, this energy is the rest energy of the particle.

The relationship between mass and the derivative of the internal state and the derivation of the time-dependent Schrödinger equation are discussed in detail in the article "The Schrödinger Equation and the Origin of Mass in a New Idempotent Basis".

After constructing the general mathematical apparatus, the model is applied to the electron. The basic idea is to separate two motions: the translational motion of the system's center and the internal motion of the rotational component.

The internal state vector of the electron is written in the form \[ \tag{31} J_e(t) = \ep+\em e^{-i\omega_e t}. \]

If we multiply this vector by the speed of light, we get: \[ \tag{32} V_e(t) = c\ep+c\em e^{-i\omega_e t}. \] The first component is associated with the translational direction, and the second with the internal periodic motion.

Integrating the rotational component yields the geometric radius: \[ \tag{33} r_e^{(J)} = \frac{c}{\omega_e} = \frac{c}{2\pi\nu_e}. \]

If we choose an internal frequency \[ \tag{34} \nu_e \approx 1.69\cdot10^{22}\;\text{Hz}, \] then the radius (33) turns out to be close to the classical radius of an electron: \[ \tag{35} r_e = \frac{1}{4\pi\varepsilon_0} \frac{e^2}{m_ec^2} \approx 2.818\cdot10^{-15}\;\text{m}. \]

This numerical coincidence is used as the basis for the hypothesis: \[ \tag{36} r_e^{(J)}\approx r_e. \] In this case, the classical radius receives a geometric interpretation as the scale of the internal rotational component.

The model also distinguishes between the center of translational motion and the center of rotation of the charge. This distinction is reminiscent of some interpretations of Dirac's zitterbewegung (energy center), where the motion of the charge center may differ from the motion of the averaged center of energy.

However, there is a fundamental difference between these approaches. In Dirac's theory, zitterbewegung (energy center) arises from the structure of the relativistic wave equation and the interference of different energy components. In the model under consideration, internal rotation is directly embedded in the geometry of the idempotent vector.

If the rotating component truly corresponds to the motion of an electric charge, it forms a closed current. For a charge \(e\) rotating with angular frequency \(\omega_e\) around a circle of radius \(r\), the classical magnetic moment is: \[ \tag{37} \mu_{\mathrm{circ}} = \frac{e\omega_e r^2}{2}. \]

The model introduces an additional relationship between the observed magnetic moment and the internal Lorentz factor: \[ \tag{38} \mu_e^{(0)} = \gamma_{\mathrm{int}} \mu_{\mathrm{circ}}. \]

At \[ \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}} \] it turns out: \[ \tag{39} \mu_e^{(0)} = \frac{e\omega_e r^2} {2\alpha_{\mathrm{fs}}}. \]

Using the geometric relation \[ \omega_e r_e^{(J)}=c \] and the identity \[ r_e = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec}, \] we can obtain the fundamental scale of the magnetic moment: \[ \tag{40} \mu_e^{(0)} \approx \frac{e\hbar}{2m_e} = \mu_B. \]

It should be emphasized here that the factor \(\gamma_{\mathrm{int}}\) is a separate physical assumption of the model. Therefore, obtaining the Bohr magneton does not follow solely from the classical circular current formula.

The construction of the internal state, the transition to Pauli matrices, and the mapping of the model into familiar space are discussed in the first part of the work: "Geometric Model of the Internal Structure of the Electron. Part 1".

The double orbit, natural frequency, classical radius, energy interpretation, and magnetic moment are discussed in detail in the second part: "Geometric Model of the Internal Structure of the Electron. Part 2". The relationship between the internal motion and the electron's second magnetic field was previously discussed in the article: "The Second Magnetic Field of the Electron. A Mathematical Model".

The fine structure constant \[ \tag{41} \alpha_{\mathrm{fs}} \approx \frac{1}{137.036} \] is one of the most important dimensionless constants in physics. It determines the strength of the electromagnetic interaction and is included in many characteristics of atoms and elementary particles.

The model under consideration offers an additional geometric interpretation: \[ \tag{42} \alpha_{\mathrm{fs}} = \frac{1}{\gamma_{\mathrm{int}}}, \] where \(\gamma_{\mathrm{int}}\) is the hypothetical Lorentz factor of the internal motion.

Then: \[ \tag{43} \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}} \approx 137.036. \]

If we use the standard relationship between the Lorentz factor and velocity, \[ \tag{44} \gamma_{\mathrm{int}} = \frac{1} {\sqrt{1-\beta_{\mathrm{int}}^2}}, \] then we get: \[ \tag{45} \beta_{\mathrm{int}} = \sqrt{1-\alpha_{\mathrm{fs}}^2}. \]

Therefore, the estimated velocity of the internal motion is: \[ \tag{46} v_{\mathrm{int}} = c\sqrt{1-\alpha_{\mathrm{fs}}^2} \approx 0.99997337c. \]

In the energy part of the model, the total energy of the internal process is defined as \[ \tag{47} W_e=h\nu_e. \] The rest energy is considered as its projection: \[ \tag{48} m_ec^2 = \alpha_{\mathrm{fs}}W_e = \frac{W_e}{\gamma_{\mathrm{int}}}. \]

Hence: \[ \tag{49} W_e = \gamma_{\mathrm{int}}m_ec^2. \] The right-hand side has the form of the total relativistic energy.

If we introduce the angle \(\vartheta\), for which \[ \sin\vartheta=\beta_{\mathrm{int}}, \] then: \[ \tag{50} \cos\vartheta = \sqrt{1-\beta_{\mathrm{int}}^2} = \alpha_{\mathrm{fs}}. \]

Thus, the fine structure constant receives several interconnected interpretations in the model:

1. as the reciprocal of the internal Lorentz factor;

2. as the coefficient of projection of the total internal energy onto the rest energy;

3. as a geometric coefficient relating the reduced Compton length to the classical electron radius;

4. as a factor involved in the transition from the magnetic moment of the internal circular current to the Bohr magneton scale.

These interpretations form a consistent system within the model, but do not yet independently derive the numerical value of \(\alpha_{\mathrm{fs}}\). The value of the fine structure constant is taken from known physics, after which its possible geometric meaning is explored.

The energetic interpretation of the fine structure constant, the internal Lorentz factor, and its relationship to the radius and magnetic moment of the electron are discussed in the paper "Geometric Model of the Internal Structure of the Electron. Part 2".

The general origin of the Lorentz factor from mutually inverse idempotent components is explored in the paper "Origin of the Lorentz Factor and the Energy Invariant from the Hyperbolic Unit".

Now the general logic of the loop can be represented as a sequence of several levels.

The first level is algebraic. Two mutually complementary idempotents are derived from the hyperbolic unit: \[ \tag{51} \ep=\frac{1+\j}{2}, \qquad \em=\frac{1-\j}{2}. \] Their complex expansion creates a four-dimensional real basis: \[ \tag{52} \{\ep,i\ep,\em,i\em\}. \]

The second level is geometric. Vector \[ \tag{53} J(t) = \ep+\em e^{-i\omega t} \] separates the stationary and rotating components. Conservation of the modulus of the complex phase leads to circular motion and the orthogonality of the vector to its derivative.

The third level is kinematic. After multiplying by the velocity and integrating, the translational and rotational parts of the trajectory appear: \[ \tag{54} R(t) = ut\ep + \frac{iu}{\omega} \em e^{-i\omega t} + R_0. \]

The fourth level is quantum energy. Differentiating the internal component and multiplying by \(i\hbar\) yields: \[ \tag{55} i\hbar \frac{\partial\Psi}{\partial t} = \hbar\omega\Psi. \] This relates the internal frequency to energy and time-phase evolution.

The fifth level is mass. When identifying the energy of the internal mode with the rest energy: \[ \tag{56} \hbar\omega_0=mc^2 \] mass is interpreted as an energy measure of the natural frequency.

The sixth level is relativistic. Mutually inverseThe idempotent coefficients of two idempotent components lead to a hyperbolic invariant and a Lorentz factor: \[ \tag{57} \gamma = \frac{1} {\sqrt{1-\beta^2}}. \]

After multiplying the hyperbolic state by \(mc^2\), an object emerges: \[ \tag{58} \mathcal E = E+\j pc. \] Its norm yields an energy invariant: \[ \tag{59} E^2-p^2c^2=m^2c^4. \]

The seventh level is the electron model. The internal frequency is related to the geometric radius, charge motion, magnetic moment, and fine structure constant.

The entire sequence can be represented in compact form:

The main stages of this chain are presented in the following works:

"A New Cartesian Basis from Two Idempotents" — construction of the initial algebra and basis.

"Supplement to the New Cartesian Basis" — clarification of the properties, operations, and geometric meaning of the basis.

"A New Cartesian Basis." Part 2" — development of the geometry of motion and complex components.

"New Cartesian Basis. Part 3" — further construction of the motion vector and its representation.

"Rotation as a consequence of norm conservation" — orthogonality of the vector and its derivative.

"Origin of the Lorentz Factor and the Energy Invariant" — idempotent imbalance, the Lorentz factor, and the energy-momentum norm.

"The Schrödinger Equation and the Origin of Mass in a New Idempotent Basis" — the time-dependent Schrödinger equation and the relationship of mass to eigenfrequency.

"Geometric Model of the Internal Structure of the Electron. Part 1" — internal state, matrix representation, and mapping of the model into observable space.

"Geometric Model of the Internal Structure of the Electron. Part 2" — double orbit, eigenfrequency, energy, radius, fine structure, and magnetic moment.

To correctly evaluate the results, it is necessary to distinguish between three levels of assertions: mathematically proven properties of algebra, conclusions valid within the accepted model, and physical hypotheses requiring additional substantiation or experimental verification.

The mathematical part includes the properties of idempotents: \[ \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \] the transition between the bases \(\{1,\j\}\) and \(\{\ep,\em\}\), as well as the complex extension to four real directions.

This also includes the decomposition of the vector \[ J(t) = \ep+\em e^{-i\omega t}, \] its differentiation, the constancy of the modulus of the rotational component, and the orthogonality of the radial and tangential directions.

The mutually inverse idempotent coefficients, the hyperbolic identity, and the formula \[ \gamma = \frac{1}{\sqrt{1-\beta^2}}, \] are mathematically derived if \(\beta\) is defined as the relative imbalance of the components.

After defining the hyperbolic energy state, the norm \[ E^2-p^2c^2 \] Its identification with \(m^2c^4\) follows from the chosen normalization of the state \(mc^2\j^\alpha\).

Within the accepted geometric interpretation, the component \(\ep\) can be associated with the translational direction, and the component \(\em e^{-i\omega t}\) with the internal rotation.

Differentiation of the internal mode leads to the time-dependent form of the stationary Schrödinger equation. This result is exact for the chosen phase dependence, but does not yet represent a complete derivation of the spatial quantum dynamics.

With physical identification \[ \beta=\frac{v}{c} \] the mathematical normalization coefficient becomes the Lorentz factor. When multipliedstates on \(mc^2\), its components are interpreted as energy and momentum.

When identifying \[ \hbar\omega_0=mc^2 \] mass becomes a measure of the natural frequency of internal motion.

The hypothesis is the assertion that the rotational component describes the real internal motion of the electron's charge, and not just its mathematical phase evolution.

The identification of the geometric radius \[ \frac{c}{\omega_e} \] with the classical radius of the electron remains a hypothesis.

The interpretation of the fine structure constant as the inverse of the internal Lorentz factor requires additional physical justification: \[ \alpha_{\mathrm{fs}} = \frac{1}{\gamma_{\mathrm{int}}}. \]

A separate postulate is the amplification of the magnetic moment of the internal circular current by a factor of \(\gamma_{\mathrm{int}}\). Without this assumption, the classical circular motion of a charge does not directly lead to the Bohr magneton.

It is also necessary to demonstrate how the model reproduces the spinor properties of the electron, the statistics of fermions, the full spectrum of interactions with an external electromagnetic field, the anomalous magnetic moment, and the experimental results of quantum electrodynamics.

The first direction is to construct a complete differential operator acting only in physically defined directions of the new basis. This would allow us to transition from the time-internal mode to the space-time wave equation.

The second direction is related to transformations between different observers and different idempotent bases. It is necessary to show whether the composition of such transformations can naturally reproduce the relativistic law of velocity addition.

The third direction is to construct the Lagrangian and Hamiltonian of the model. This step would allow us to determine the conservation laws, symmetries, and possible interactions in the standard form of theoretical physics.

The fourth direction is the search for distinguishable experimental consequences. A theory acquires independent physical value only when it allows for the calculation of a result that differs from existing models or explains an observed value without introducing an additional parameter.

The fifth direction is concerned with deriving the fine-structure constant. In the current model, its known value receives a geometric interpretation. A more powerful result would be an independent derivation of \(\alpha_{\mathrm{fs}}\) from the structure of the algebra, boundary conditions, or the stability of the inner orbit.

The series of papers in the "Wave Electricity" section begins with a simple algebraic idea: two complementary idempotents. After complex expansion, they form a four-dimensional real basis. \[ \{\ep,i\ep,\em,i\em\}. \] This structure naturally separates the two complex planes and allows one component to remain constant while the other undergoes an internal rotation.

Preserving the norm of the rotational component leads to orthogonality of the vector and its derivative. Differentiation of the internal phase generates the factor \(\omega\), and multiplication by \(\hbar\) relates frequency to energy. As a result, the time-varying Schrödinger equation receives a geometric interpretation as the law of evolution of the internal rotational mode.

When identifying the energy of this mode with the rest energy, the relation arises \[ \hbar\omega_0=mc^2, \] and mass can be considered as an energy measure of the natural frequency.

Another branch of the model's development is related to hyperbolic algebra. Mutually inverse idempotent components lead to an internal invariant, the Lorentz factor, and the energy state \[ \mathcal E=E+\j pc, \] whose norm reproduces the relativistic relation between energy and momentum.

Applying this geometry to the electron allows us to relate the internal frequency, characteristic radius, rest energy, fine structure constant, and fundamental scale of the magnetic moment in a single framework. Moreover, some of the resulting relationships are mathematical consequences of the chosen algebra, some are the result of the adopted geometric interpretation, and the strongest physical statements remain hypotheses.

The main advantage of the proposed approach is not the definitive solution to the problem of the electron's internal structure, but the creation of a compact language in which several known physical constructs can be represented as manifestations of a single internal motion.

In this language, rotation arises from conservationI am the norm, energy is derived from frequency, mass is derived from the energy of the eigenmode, and the electron's characteristics are considered as different projections of a single idempotent state. The next challenge is to transform this geometrically consistent picture into a testable physical theory.