In the second part of the paper, a geometric model of the electron's internal state is proposed, based on the assumption of the existence of two close branches of a single internal trajectory. It is shown that this two-sheet structure is naturally consistent with the internal two-component state described by a new idempotent basis and leads to full periodicity only after a rotation by \(4\pi\). Moreover, the geometric branches are interpreted not as observable spatial orbits, but as internal states mapped into physical space via the Pauli operators.

Based on the constructed model, its physical consequences are investigated: the relationship of internal motion to the classical electron radius and fine structure constant, the operator description of spin and magnetic moment, and the emergence of Zeeman splitting and Larmor precession. Particular attention is paid to separating the geometric structure of the internal motion from the observed quantum states, which allows us to combine the geometric interpretation of the two-sheeted orbit with the modern operator apparatus of quantum mechanics.

In the first part of the paper, a new complex idempotent basis \[ \left\{ \ep,\;i\ep,\;\em,\;i\em \right\} \] was interpreted as the internal state space of the system, not directly coinciding with ordinary spacetime. It was shown that complementary idempotents can be viewed as projectors onto two internal states, and the transition operators between them naturally generate a complete set of Pauli matrices.

The internal two-component state \[ |\Psi\rangle = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} \] is mapped to the three-dimensional direction via the bilinear expression \[ \mathbf n = \langle\Psi| \boldsymbol{\sigma} |\Psi\rangle, \] and then to the local four-dimensional vector and physical space-time.

In this paper, this mathematical construction is applied to a hypothetical model of the internal motion of an electron. It is assumed that the internal charge dynamics consists of two close branches located on opposite sides of the average electromagnetic scale corresponding to the classical electron radius. The complete internal cycle involves sequentially passing through both branches and is completed only after a rotation of \(4\pi\).

The main goal of this work is not to identify the electron with a classical charged point moving along a conventional spatial circle, but to construct a geometric model of the internal state. The two orbits are considered as two sheets of a single internal trajectory, and their orientation in the observed three-dimensional space is determined by the mapping of the internal two-component state.

Consider the internal state \[ \tag{1} |\Psi(\tau)\rangle = \begin{pmatrix} \psi_+(\tau) \\ \psi_-(\tau) \end{pmatrix}, \qquad \langle\Psi|\Psi\rangle=1, \] where \(\tau\) is the internal evolution parameter.

Idempotents \[ \tag{2} \ep = \frac{I+\sigma_z}{2}, \qquad \em = \frac{I-\sigma_z}{2} \] distinguish two internal components of state.

The internal state can also be written in algebraic form: \[ \tag{3} \Psi(\tau) = \psi_+(\tau)\ep + \psi_-(\tau)\em. \]

State (1) corresponds to the unit spatial vector \[ \tag{4} \mathbf n(\tau) = \langle\Psi(\tau)| \boldsymbol{\sigma} |\Psi(\tau)\rangle, \] where \[ \tag{5} \boldsymbol{\sigma} = \left( \sigma_x,\sigma_y,\sigma_z \right). \]

The vector \(\mathbf n\) defines the orientation of the internal trajectory in the observed three-dimensional space. Thus, the idempotent components themselves are not spatial coordinates, but define an internal state, which is then mapped to a physical direction.

Let the characteristic mean radius of the internal motion be equal to the classical radius of the electron: \[ \tag{6} r_e = \frac{1}{4\pi\varepsilon_0} \frac{e^2}{m_ec^2} = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec}. \]

We introduce two close branches: \[ \tag{7} r_+ = r_e+\Delta r, \qquad r_- = r_e-\Delta r, \] where \[ \tag{8} \Delta r\ll r_e. \]

The average radius satisfies \[ \tag{9} r_e = \frac{r_++r_-}{2}, \] and the distance between the branches is \[ \tag{10} r_+-r_- = 2\Delta r. \]

The branches are assumed to be located in a single local plane, the normal to which is determined by the vector \(\mathbf n\). Therefore, when passing through both branches, a single geometric axis is maintained internally.frictional motion.

The two circles should not be considered two independent classical orbits. They are two sections of a single internal trajectory: \[ \tag{11} r_+ \longrightarrow r_- \longrightarrow r_+. \]

After traversing the first branch, the system moves to the second sheet of the internal state, and returns to the original configuration only after traversing the second branch.

To describe two close branches, we introduce the internal angular parameter \[ \tag{12} \chi\in[0.4\pi). \]

We define the radial coordinate by the expression \[ \tag{13} r(\chi) = r_e + \Delta r \cos\frac{\chi}{2}. \]

Then \[ \tag{14} r(0) = r_e+\Delta r = r_+, \] \[ \tag{15} r(2\pi) = r_e-\Delta r = r_-, \] \[ \tag{16} r(4\pi) = r_e+\Delta r = r_+. \]

Thus, after changing the parameter by \(2\pi\), the system switches from one branch to another, and the full radial cycle is completed only after changing the parameter by \(4\pi\).

In the local plane orthogonal to the vector \(\mathbf n\), we introduce two unit vectors \[ \tag{17} \mathbf e_1, \qquad \mathbf e_2, \qquad \mathbf e_1\times\mathbf e_2 = \mathbf n. \]

Then the inner trajectory can be written as \[ \tag{18} \mathbf R_{\mathrm{int}}(\chi) = r(\chi) \left( \mathbf e_1\cos\chi + \mathbf e_2\sin\chi \right). \]

Expression (18) describes a continuous curve with a slow transition between two close radial branches. At \(\Delta r\rightarrow0\), it transforms into a regular circle of radius \(r_e\).

Since the radial coordinate contains a half-angle of \(\chi/2\), the geometric state of the trajectory has a period of \(4\pi\), although its azimuthal part contains the usual angle of \(\chi\).

We associate the first branch of the internal trajectory with the state of \(\ep\), and the second with the state of \(\em\): \[ \tag{19} \ep \xrightarrow{\;2\pi\;} \em \xrightarrow{\;2\pi\;} \ep. \]

In this case, \(\ep\) and \(\em\) denote not two different spatial planes, but two sheets of the same internal geometry.

The operator \[ \tag{20} \sigma_z = \ep-\em \] distinguishes these sheets, while \[ \tag{21} \sigma_x = S_++S_- \] permutes them.

The operator \[ \tag{22} \sigma_y = -i(S_+-S_-) \] describes the transition between branches with an additional complex phase shift.

Thus, the geometric transition between two branches and the operator transition between two idempotent components are two descriptions of the same internal two-sheet structure.

The internal two-component state is transformed by the rotation operator \[ \tag{23} U(\mathbf n,\theta) = \exp \left[ -\frac{i\theta}{2} \mathbf n\cdot\boldsymbol{\sigma} \right]. \]

Using the identity \[ \tag{24} \left( \mathbf n\cdot\boldsymbol{\sigma} \right)^2 = I, \] we get \[ \tag{25} U(\mathbf n,\theta) = I\cos\frac{\theta}{2} - i \left( \mathbf n\cdot\boldsymbol{\sigma} \right) \sin\frac{\theta}{2}. \]

After rotation by \(2\pi\) \[ \tag{26} U(\mathbf n,2\pi) = -I, \] therefore \[ \tag{27} |\Psi(2\pi)\rangle = -|\Psi(0)\rangle. \]

After rotation by \(4\pi\) \[ \tag{28} U(\mathbf n,4\pi) = I, \] and \[ \tag{29} |\Psi(4\pi)\rangle = |\Psi(0)\rangle. \]

Thus, the two-sheet geometry of the trajectory is consistent with spinor periodicity: after the first rotation, the system transitions to the conjugate sheet and acquires a common phase factor of \(-1\), and after the second rotation, it returns to its original state.

It should be emphasized that the sign of the wave function itself does not change the bilinear spatial vector: \[ \tag{30} \langle-\Psi| \boldsymbol{\sigma} |-\Psi\rangle = \langle\Psi| \boldsymbol{\sigma} |\Psi\rangle. \] Therefore, after a rotation of \(2\pi\), the observed direction of \(\mathbf n\) remains the same, although the internal state changes sign.

Let one geometric revolution along one branch take time \[ \tag{31} T_{\mathrm{orb}} = \frac{2\pi}{\omega_{\mathrm{orb}}}. \]

A complete two-sheet cycle consists of two revolutions: \[ \tag{32} T_{4\pi} = 2T_{\mathrm{orb}}. \]

Therefore, the frequency of the full state is equal to \[ \tag{33} \nu_{4\pi} = \frac{1}{T_{4\pi}} = \frac{\nu_{\mathrm{orb}}}{2}. \]

To avoid ambiguity, we will further distinguish between: \[ \tag{34} \omega_{\mathrm{orb}} \] — the angular frequency of one branch, and \[ \tag{35} \omega_{4\pi} = \frac{\omega_{\mathrm{orb}}}{2} \] — the angular frequency of the complete internal state.

In the proposed model, the connection with the electron mass is supposed to be established through the frequency of the complete state: \[ \tag{36} m_ec^2 = \hbar\omega_{4\pi}. \]

Then the frequency of one revolution is \[ \tag{37} \omega_{\mathrm{orb}} = 2\omega_{4\pi} = \frac{2m_ec^2}{\hbar}. \]

Expression (36) is a postulate of the model. It associates the rest energy not with the individual passage of one branch, but with the complete return of the internal state after a period of \(4\pi\).

Previously, the fine structure constant was interpreted as the reciprocal of the internal Lorentz factor: \[ \tag{38} \alpha_{\mathrm{fs}} = \frac{1}{\gamma_{\mathrm{int}}}. \]

Then \[ \tag{39} \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}, \] And \[ \tag{40} \beta_{\mathrm{int}} = \sqrt{1-\alpha_{\mathrm{fs}}^2}. \]

The internal speed has the form \[ \tag{41} v_{\mathrm{int}} = c\sqrt{1-\alpha_{\mathrm{fs}}^2}. \]

Numerically \[ \tag{42} v_{\mathrm{int}} \approx 0.99997337c. \]

If the local angular velocity changes with the radius, then the condition of constant linear velocity can be written as \[ \tag{43} \dot\chi(\tau) = \frac{v_{\mathrm{int}}} {r(\chi)}. \]

On the outer branch, the angular velocity is slightly smaller, and on the inner branch, it is slightly larger: \[ \tag{44} \dot\chi_+ = \frac{v_{\mathrm{int}}}{r_+}, \qquad \dot\chi_- = \frac{v_{\mathrm{int}}}{r_-}. \]

This choice allows us to maintain the same intrinsic velocity on both close branches and avoid superluminal motion on the outer branch.

The branch travel times are \[ \tag{45} T_+ = \frac{2\pi r_+}{v_{\mathrm{int}}}, \qquad T_- = \frac{2\pi r_-}{v_{\mathrm{int}}}. \]

The total period is \[ \tag{46} T_{4\pi} = T_++T_-. \]

Using \[ r_++r_-=2r_e, \] we get \[ \tag{47} T_{4\pi} = \frac{4\pi r_e} {v_{\mathrm{int}}}. \]

Therefore, the frequency of the complete internal state is equal to \[ \tag{48} \nu_{4\pi} = \frac{v_{\mathrm{int}}} {4\pi r_e}, \] and angular frequency \[ \tag{49} \omega_{4\pi} = 2\pi\nu_{4\pi} = \frac{v_{\mathrm{int}}} {2r_e}. \]

The average angular frequency of one geometric revolution is \[ \tag{50} \omega_{\mathrm{orb}} = 2\omega_{4\pi} = \frac{v_{\mathrm{int}}}{r_e}. \]

Thus, the symmetrical arrangement of the two branches with respect to \(r_e\) results in the total period being independent of \(\Delta r\) to a first approximation. It is determined by the average radius and the internal velocity.

Substituting the expression \[ r_e = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec} \] and the velocity \[ v_{\mathrm{int}} = c\sqrt{1-\alpha_{\mathrm{fs}}^2} \] into formula (49), we obtain \[ \tag{51} \omega_{4\pi} = \frac{m_ec^2} {2\alpha_{\mathrm{fs}}\hbar} \sqrt{1-\alpha_{\mathrm{fs}}^2}. \]

Accordingly, \[ \tag{52} \hbar\omega_{4\pi} = \frac{m_ec^2} {2\alpha_{\mathrm{fs}}} \sqrt{1-\alpha_{\mathrm{fs}}^2}. \]

This expression does not coincide with the postulate \[ m_ec^2 = \hbar\omega_{4\pi}. \] Therefore, the classical radius, the intrinsic velocity defined by \(\alpha_{\mathrm{fs}}\), and the identification of the rest energy with the total \(4\pi\)-frequency cannot be simultaneously independent assumptions.

To align the model, two initial relations must be selected, and the third must be considered as a consequence or an approximate relationship. In this paper, geometric periodicity is separated from the energy hypothesis so as not to obscure the additional coefficient that arises.

Let's introduce two different frequencies: \[ \tag{53} \omega_{\mathrm{geom}} = \frac{v_{\mathrm{int}}} {2r_e} \] — the frequency of the geometric \(4\pi\)-cycle, and \[ \tag{54} \omega_E = \frac{m_ec^2}{\hbar} \] — the resting energy frequency of the electron.

Their ratio is \[ \tag{55} \frac{\omega_{\mathrm{geom}}}{\omega_E} = \frac{ \sqrt{1-\alpha_{\mathrm{fs}}^2} }{ 2\alpha_{\mathrm{fs}} }. \]

Therefore, for direct identification of these frequencies,This would require changing the characteristic radius or the relationship between the geometric period and energy.

Therefore, in what follows, the geometric two-sheet cycle is used primarily to describe spinor periodicity, while the relationship between mass and internal frequency is retained as a separate dynamic hypothesis.

The motion of a charge along a closed trajectory creates a circular current. For one complete geometric revolution around a circle of radius \(r\), the classical magnetic moment has the form \[ \tag{56} \mu_{\mathrm{circ}} = \frac{e\omega r^2}{2}. \]

For a negative charge, the direction of the magnetic moment is opposite to the direction of the mechanical angular momentum: \[ \tag{57} \boldsymbol{\mu} = -\mu\mathbf n. \]

Since both branches lie in the same local plane and extend in the same direction, their magnetic moments are directed along the same axis. They differ only in magnitude: \[ \tag{58} \mu_+ = \frac{e\omega_+r_+^2}{2}, \qquad \mu_- = \frac{e\omega_-r_-^2}{2}. \]

At constant linear velocity \[ \omega_pm = \frac{v_{\mathrm{int}}}{r_pm}, \] therefore \[ \tag{59} \mu_pm = \frac{ev_{\mathrm{int}}r_pm}{2}. \]

The average magnetic moment is \[ \tag{60} \overline{\mu} = \frac{\mu_++\mu_-}{2} = \frac{ev_{\mathrm{int}}}{4} \left( r_++r_- \right). \]

Using \[ r_++r_-=2r_e, \] we get \[ \tag{61} \overline{\mu} = \frac{ev_{\mathrm{int}}r_e}{2}. \]

Substitution \[ v_{\mathrm{int}} = c\sqrt{1-\alpha_{\mathrm{fs}}^2} \] And \[ r_e = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec} \] gives \[ \tag{62} \overline{\mu} = \alpha_{\mathrm{fs}} \sqrt{1-\alpha_{\mathrm{fs}}^2} \frac{e\hbar}{2m_e}. \]

Therefore, \[ \tag{63} \overline{\mu} = \alpha_{\mathrm{fs}} \sqrt{1-\alpha_{\mathrm{fs}}^2} \mu_B. \]

The classical circular current along two close branches does not by itself reproduce the Bohr magneton. To obtain the scale \[ \mu_B = \frac{e\hbar}{2m_e} \] an additional internal coefficient is needed.

In the previous model, this coefficient was the internal Lorentz factor: \[ \tag{64} \gamma_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}. \]

After taking it into account, we obtain \[ \tag{65} \mu^{(0)} = \gamma_{\mathrm{int}} \overline{\mu} = \sqrt{1-\alpha_{\mathrm{fs}}^2}\mu_B. \]

Since \[ \sqrt{1-\alpha_{\mathrm{fs}}^2} \approx 0.99997337, \] the result is close to the Bohr magneton, but does not coincide with it exactly: \[ \tag{66} \mu^{(0)} \approx 0.99997337\mu_B. \]

Thus, using a physical velocity slightly smaller than \(c\) leads to a small geometric correction. An accurate derivation of the Bohr magneton requires either a different normalization of the magnetic moment or an additional relativistic transformation rule for the internal current.

The geometric periodicity \(4\pi\) by itself does not determine the magnitude of angular momentum. Therefore, the spin operator is introduced on the internal state space: \[ \tag{67} \hat{\mathbf S} = \frac{\hbar}{2} \boldsymbol{\sigma}. \]

Its components satisfy \[ \tag{68} \left[ \hat S_i,\hat S_j \right] = i\hbar \varepsilon_{ijk} \hat S_k. \]

The square of the spin operator is \[ \tag{69} \hat{\mathbf S}^2 = \frac{3}{4}\hbar^2I. \]

The eigenvalues ​​of the projection onto the chosen direction: \[ \tag{70} S_{\mathbf n} = \pm\frac{\hbar}{2}. \]

In the proposed model, the Pauli matrices arise as transformation operators of two idempotent components. However, the coefficient \(\hbar/2\) remains a quantum normalization of the spin operator and is not yet directly derivable from the sizes of the two orbits.

The two branches explain the bivalent and periodic nature of the internal state, while the spin value is determined by the operator structure of the state space.

The magnetic moment of an electron is related to its spin by the expression \[ \tag{71} \hat{\boldsymbol{\mu}} = -g_e \frac{e}{2m_e} \hat{\mathbf S}. \]

Taking into account \[ \hat{\mathbf S} = \frac{\hbar}{2} \boldsymbol{\sigma} \] we obtain \[ \tag{72} \hat{\boldsymbol{\mu}} = -\frac{g_e}{2} \mu_B \boldsymbol{\sigma}. \]

For the Dirac value \[ \tag{73} g_e=2 \] we have \[ \tag{74} \hat{\boldsymbol{\mu}} = -\mu_B \boldsymbol{\sigma}. \]

In this case, the projections of the magnetic moment onto the direction \(\mathbf n\) are equal to \[ \tag{75} \mu_{\mathbf n} = \mp\frac{g_e}{2}\mu_B. \]

A distinction should be made between the classical magnetic moment of the circular current and the quantum magnetic moment operator. The former is associated with the geometry of the internal trajectory, while the latter acts on the two-component state and determines the observed magnetic projections.

In this model, it is assumed that the classical internal trajectory defines the geometric basis of the magnetic moment, while its exact value and orientation are determined by operator (71).

In an external magnetic field \(\mathbf B\), the interaction Hamiltonian has the form \[ \tag{76} \hat H_B = -\hat{\boldsymbol{\mu}} \cdot \mathbf B. \]

Substituting expression (72), we obtain \[ \tag{77} \hat H_B = \frac{g_e\mu_B}{2} \boldsymbol{\sigma} \cdot \mathbf B. \]

The sign of the Hamiltonian depends on the adopted notation the electron charge and the direction of the magnetic moment operator. The difference in self-energies is physically significant.

For a field directed along the z-axis, \[ \tag{78} \mathbf B = B\mathbf e_z, \] the Hamiltonian takes the form \[ \tag{79} \hat H_B = \frac{g_e\mu_BB}{2} \sigma_z. \]

Its eigenvalues: \[ \tag{80} E_+ = +\frac{g_e\mu_BB}{2}, \qquad E_- = -\frac{g_e\mu_BB}{2}. \]

The level splitting is \[ \tag{81} \Delta E = E_+-E_- = g_e\mu_BB. \]

Thus, the external magnetic field splits not two geometric circles as such, but two eigenstates of the operator \[ \boldsymbol{\sigma}\cdot\mathbf B. \]

The geometric branches \(r_+\) and \(r_-\) define a two-sheeted internal structure, while the states with energies \(E_+\) and \(E_-\) are quantum projections of this structure relative to the external field.

Let the initial state be a superposition of the eigenstates of the magnetic Hamiltonian: \[ \tag{82} |\Psi(0)\rangle = a|+\rangle + b|-\rangle. \]

Its time evolution is \[ \tag{83} |\Psi(t)\rangle = a e^{-iE_+t/\hbar} |+\rangle + b e^{-iE_-t/\hbar} |-\rangle. \]

The phase difference between the two components is \[ \tag{84} \Delta\phi(t) = \frac{E_+-E_-}{\hbar}t. \]

Using expression (81), we obtain \[ \tag{85} \Delta\phi(t) = \frac{g_e\mu_BB}{\hbar}t. \]

Therefore, the angular frequency of precession is \[ \tag{86} \omega_{\mathrm{prec}} = \frac{g_e\mu_BB}{\hbar}. \]

Since \[ \mu_B = \frac{e\hbar}{2m_e}, \] we obtain \[ \tag{87} \omega_{\mathrm{prec}} = \frac{g_ee}{2m_e}B. \]

For the Dirac value of \(g_e=2\) \[ \tag{88} \omega_{\mathrm{prec}} = \frac{eB}{m_e}. \]

It is not necessarily the inner circle itself that precesses as a solid geometric object. The mapped vector precesses \[ \tag{89} \mathbf n(t) = \langle\Psi(t)| \boldsymbol{\sigma} |\Psi(t)\rangle. \]

Its evolution satisfies the equation \[ \tag{90} \frac{d\mathbf n}{dt} = \frac{g_e\mu_B}{\hbar} \mathbf n\times\mathbf B. \]

Thus, slow spatial precession arises as a consequence of the accumulation of relative phase between the two internal components of the state.

Let the local plane of the inner trajectory be defined by the vectors \[ \mathbf e_1(t), \qquad \mathbf e_2(t), \] and its normal be equal to \[ \tag{91} \mathbf n(t) = \mathbf e_1(t) \times \mathbf e_2(t). \]

Then the complete internal trajectory can be written as \[ \tag{92} \mathbf R_{\mathrm{int}}(\chi,t) = r(\chi) \left[ \mathbf e_1(t)\cos\chi + \mathbf e_2(t)\sin\chi \right]. \]

The parameter \(\chi\) describes the rapid internal motion along the two branches, and the time dependence of the basis vectors describes the slow change in the orientation of the entire internal plane.

Thus, the model features two different time scales: \[ \tag{93} \omega_{\mathrm{int}} \gg \omega_{\mathrm{prec}}. \]

The fast motion is associated with the internal passage of the two-sheet trajectory, while the slow motion is associated with the precession of its mapping in the external magnetic field.

It is important not to directly identify: \[ r_+ \quad\text{and}\quad r_- \] with the states \[ |+\rangle \quad\text{and}\quad |-\rangle. \]

The first pair refers to the geometric sheets of the internal trajectory, and the second to the eigenstates of the operatormagnetic projection.

A possible relationship between them is defined by the mapping: \[ \tag{94} \left( r_+,r_- \right) \longrightarrow |\Psi\rangle \longrightarrow \mathbf n \longrightarrow \mu_{\mathbf n}. \]

That is, the geometric bivalent structure forms a two-component internal space, and the direction of the external field selects two proper quantum states in this space.

When the magnetic field direction changes, it is not the radii (r_+) and (r_-) that change, but the basis of eigenstates \[ \boldsymbol{\sigma}\cdot\mathbf B. \]

The value of \(\Delta r\) is not yet determined by the operator structure of the model. To derive it physically, it is necessary to specify an internal potential or a geometric stability condition.

For example, one can assume the existence of an effective potential \[ \tag{95} U(r), \] having two close minima: \[ \tag{96} \left. \frac{dU}{dr} \right|_{r=r_+} = 0, \qquad \left. \frac{dU}{dr} \right|_{r=r_-} = 0. \]

Stability requires \[ \tag{97} \left. \frac{d^2U}{dr^2} \right|_{r=r_+} >0, \qquad \left. \frac{d^2U}{dr^2} \right|_{r=r_-} >0. \]

In its simplest symmetric form, such a potential can be written as \[ \tag{98} U(r) = \lambda \left[ (r-r_e)^2-(\Delta r)^2 \right]^2, \] where \(\lambda>0\).

The potential minima are located at the points \[ \tag{99} r=r_e\pm\Delta r. \]

Expression (98) is only a model example. It shows how two close branches can arise as stable states of a single internal radial coordinate.

The transition between two branches can be described either by a continuous trajectory (13) or by the operator dynamics of a two-component state.

In operator form, the internal Hamiltonian can be written as \[ \tag{100} \hat H_{\mathrm{int}} = E_0I + \Delta_0\sigma_z + \kappa\sigma_x, \] where:

\(E_0\) is the average energy of the internal state;

\(Delta_0\) is the possible intrinsic asymmetry of the two branches;

\(\kappa\) is the transition amplitude between them.

If the branches are completely symmetrical, \[ \tag{101} \Delta_0=0, \] then \[ \tag{102} \hat H_{\mathrm{int}} = E_0I+\kappa\sigma_x. \]

Then the eigenstates are the symmetric and antisymmetric combinations: \[ \tag{103} |\Psi_s\rangle = \frac{1}{\sqrt2} \left( |\ep\rangle+|\em\rangle \right), \] \[ \tag{104} |\Psi_a\rangle = \frac{1}{\sqrt2} \left( |\ep\rangle-|\em\rangle \right). \]

Their energies are \[ \tag{105} E_s = E_0+\kappa, \qquad E_a = E_0-\kappa. \]

The internal splitting is \[ \tag{106} \Delta E_{\mathrm{int}} = 2\kappa. \]

This splitting exists independently of the external magnetic field and is related to the intrinsic transition between the geometric branches. It should not be confused with the Zeeman splitting \[ \Delta E_B = g_e\mu_BB. \]

Taking into account the external magnetic field, the overall Hamiltonian can be written as \[ \tag{107} \hat H = E_0I + \Delta_0\sigma_z + \kappa\sigma_x + \frac{g_e\mu_B}{2} \boldsymbol{\sigma}\cdot\mathbf B. \]

The coefficient vector of the Pauli matrices has the form \[ \tag{108} \mathbf h = \left( \kappa+\frac{g_e\mu_B}{2}B_x, \; \frac{g_e\mu_B}{2}B_y, \; \Delta_0+\frac{g_e\mu_B}{2}B_z \right). \]

Then \[ \tag{109} \hat H = E_0I + \mathbf h\cdot\boldsymbol{\sigma}. \]

Self energies are equal \[ \tag{110} E_\pm = E_0 \pm |\mathbf h|. \]

Complete splitting: \[ \tag{111} \Delta E = 2|\mathbf h|. \]

This formula shows that the internal geometric interaction between the branches and the external magnetic field are summed in a single two-component operator space.

Previously, the relative difference between the physical radius and the light limit was estimated as \[ \tag{112} 1- \sqrt{1-\alpha_{\mathrm{fs}}^2} \approx \frac{\alpha_{\mathrm{fs}}^2}{2}. \]

This allows us to consider the hypothesis \[ \tag{113} \frac{\Delta r}{r_e} = \frac{\alpha_{\mathrm{fs}}^2}{2}. \]

Then \[ \tag{114} \Delta r = \frac{\alpha_{\mathrm{fs}}^2}{2} r_e. \]

Numerically \[ \tag{115} \frac{\Delta r}{r_e} \approx 2.66\cdot10^{-5}. \]

This means that the two branches are practically indistinguishable on the scale of \(r_e\).

However, expression (113) is still a geometric assumption. To justify it, it is necessary to derive \(\Delta r\) from the internal potential, the equation of motion, or the energy minimum condition.

The experimental magnetic moment of the electron differs from the Dirac value by a small anomalous correction: \[ \tag{116} g_e = 2(1+a_e). \]

Within the framework of this geometric model, it is possible to investigate whether the small difference in radii \[ \frac{\Delta r}{r_e}\ll1 \] is related to the value of \(a_e\).

However, a direct identification \[ \tag{117} a_e \sim \frac{\Delta r}{r_e} \] does not follow from the constructed algebra and requires a separate dynamical derivation.

Therefore, in this paper, the double orbital model is used to explain the two-sheeted geometry and the internal transition, but is not considered as an already obtained explanation for the anomalous magnetic moment.

The proposed model shares several qualitative similarities with the zitterbewegung effect arising in Dirac theory.

First, both constructions assume a rapid internal motion, separate from the observed motion of the particle's center.

Second, the internal state is two-component and requires operators that mix the different components.

Third, the observed spatial direction arises as a bilinear expression of the internal state, not as a direct coordinate of the internal basis.

However, the proposed two-orbital geometry is not derived from the Dirac equation and is not a replacement for it. It is considered as a possible geometric interpretation of the internal two-sheet structure of the spinor state.

If the internal trajectory is literally interpreted as the motion of a point charge in ordinary space, the accelerated motion should be accompanied by electromagnetic radiation.

Therefore, within the framework of the present model, the internal orbit is considered primarily as a trajectory in state space, rather than as the classical trajectory of a localized charged point.

The observed electromagnetic field is determined by the averaged internal state and its operator magnetic moment.

This interpretation avoids the direct application of the classical radiation formula to the internal phase coordinate; however, a rigorous mechanism for the absence of radiation must be derived from the full dynamical equation.

In the present model, the quantity \[ r_e = \frac{e^2} {4\pi\varepsilon_0m_ec^2} \] is not considered a fixed spatial boundary for the electron.

It is interpreted as a characteristic internal electromagnetic scale, relative to which two close branches are located: \[ r_-=r_e-\Delta r, \qquad r_+=r_e+\Delta r. \]

This approach maintains compatibility with the representation of the electron as a point object in external space-time: the internal orbit belongs to the state space and manifests itself in the physical world through magnetic moment, orientation, and phase evolution.

The resulting structure can be represented as a sequence: \[ \tag{118} \boxed{ \begin{array}{c} \text{two idempotent states} \\[4pt] \ep,\em \\[8pt] \downarrow \\[8pt] \text{two close internal branches} \\[4pt] r_+,r_- \\[8pt] \downarrow \\[8pt] \text{full cycle} \\[4pt] \ep \xrightarrow{2\pi} \em \xrightarrow{2\pi} \ep \\[8pt] \downarrow \\[8pt] \text{spin periodicity} \\[4pt] |\Psi(2\pi)\rangle = -|\Psi(0)\rangle \\[4pt] |\Psi(4\pi)\rangle = |\Psi(0)\rangle \\[8pt] \downarrow \\[8pt] \text{spin operator} \\[4pt] \hat{\mathbf S} = \frac{\hbar}{2} \boldsymbol{\sigma} \\[8pt] \downarrow \\[8pt] \text{magnetic moment} \\[4pt] \hat{\boldsymbol{\mu}} = -g_e \frac{e}{2m_e} \hat{\mathbf S} \\[8pt] \downarrow \\[8pt] \text{Zeeman splitting} \\[4pt] \Delta E = g_e\mu_BB \\[8pt] \downarrow \\[8pt] \text{precession} \\[4pt] \omega_{\mathrm{prec}} = \frac{g_e\mu_BB}{\hbar} \end{array} } \]

In this sequence, geometric bivalent and operator dynamics perform different but coordinated functions. The double orbit defines the internal topology of the state, and the Pauli matrices describe its transformations and mapping to observables.It is a three-dimensional space.

The constructed scheme allows for a consistent description of the two-sheet internal geometry, the 4-pi periodicity, the two-component state, the spin operator, magnetic splitting, and precession.

However, questions remain that are not considered resolved in this paper.

First, the distance (Delta r) between the two branches has not yet been derived from the fundamental equation.

Second, the coefficient (hbar/2) in the spin operator is introduced as a quantum normalization and is not directly obtained from the dimensions of the internal trajectory.

Third, the exact value of the magnetic moment and the coefficient \(g_e\) is determined by the operator law and has not yet been fully derived from the classical circular current.

Fourth, the geometric frequency of the two-sheet cycle and the electron's resting energy frequency do not automatically coincide. Their relationship requires an additional dynamic equation.

Fifth, the internal orbit should be considered as an element of the state space, and not as a directly observable classical charge trajectory.

This paper proposes a geometric model of the internal state of the electron based on two close branches located on opposite sides of the average electromagnetic scale \(r_e\). The branches are described by a single continuous trajectory \[ r(\chi) = r_e + \Delta r \cos\frac{\chi}{2}, \] which returns to the original radial state only after changing the parameter to \(4\pi\).

The two branches are associated with mutually complementary idempotent states \[ \ep \quad\text{and}\quad \em. \] Transition \[ \ep \xrightarrow{2\pi} \em \xrightarrow{2\pi} \ep \] defines a two-sheeted intrinsic geometry and is consistent with the spinor transformation \[ |\Psi(2\pi)\rangle = -|\Psi(0)\rangle, \qquad |\Psi(4\pi)\rangle = |\Psi(0)\rangle. \]

The spatial orientation of the intrinsic trajectory is determined not by the radii themselves, but by the mapping of the two-component state: \[ \mathbf n = \langle\Psi| \boldsymbol{\sigma} |\Psi\rangle. \] Therefore, internal movement can remain one-dimensional and two-sheeted, while its manifestation in the observable world has an arbitrary orientation in three-dimensional space.

The spin operator \[ \hat{\mathbf S} = \frac{\hbar}{2} \boldsymbol{\sigma} \] and the magnetic moment operator \[ \hat{\boldsymbol{\mu}} = -g_e \frac{e}{2m_e} \hat{\mathbf S} \] allow us to obtain two magnetic states, the Zeeman splitting \[ \Delta E = g_e\mu_BB \] and the precession frequency \[ \omega_{\mathrm{prec}} = \frac{g_e\mu_BB}{\hbar}. \]

However, the two geometric branches are not directly identified with the spin-up and spin-down states. They form an internal two-component space in which the direction of the external field selects the eigenstates of the magnetic projection.

The model shows that the double orbit can serve as a geometric image of a two-sheeted spinor structure, and the Pauli matrices can serve as a natural apparatus for its transformation and mapping into the physical world. However, final determination of the distance between the branches, the magnetic moment, the coefficient \(g_e\), the energy frequency, and the stability of the internal state requires further dynamic development of the model.