This second part examines the geometric hypothesis of two nearby branches of a single internal contour with a mean scale of \(r_e\). The internal angular cycle has a period of \(4\pi\). Physical spin is described by a separately adopted spinor representation; matching periods do not prove that the two constructions share a common mechanism.

We consider the internal metric norm, its conditional connection to internal speed, the distinction between geometric and energy frequencies, and the operator description of magnetic moment and precession. Sheet geometry, spin space, and external motion have different physical roles.

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

For a normalized pure spin state \[ |\Psi\rangle = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} \] the bilinear expression \[ \mathbf n = \langle\Psi| \boldsymbol{\sigma} |\Psi\rangle, \] defines a unit polarization vector. Its correspondence to the orientation of the internal contour is adopted as a separate mapping. It does not by itself construct spacetime coordinates.

Here, we retain an exploratory variant with mean radius \(r_e\). It differs from the Compton-scale charge-carrying contour with mean radius \(R=\hbar/(m_ec)\) discussed in the article on magnetic moment. Relation \(r_e=\alpha_{\mathrm{fs}}R\) connects the scales but proves neither the existence of two physical contours nor a mapping between them. Formulas from the two variants cannot be transferred between them without such a mapping.

The main aim is to investigate an internal geometric model, rather than the motion of a classical charged point. The branches belong to the internal contour; its orientation can be related to spin polarization only through an additional physical rule.

Consider a normalized pure state in a separately introduced spin space: \[ \tag{1} |\Psi(\tau)\rangle = \begin{pmatrix} \psi_+(\tau) \\ \psi_-(\tau) \end{pmatrix}, \qquad \langle\Psi|\Psi\rangle=1, \] Below, the parameter \(\tau\) denotes time in the rest frame of the electron's center. It is not automatically the proper time of a hypothetical carrier of internal motion.

In spin space, the projectors \[ \tag{2} \ep = \frac{I+\sigma_z}{2}, \qquad \em = \frac{I-\sigma_z}{2} \] select two components relative to a chosen axis. They must not be identified with the geometric sheets of the contour.

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

State (1) corresponds to a 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 \(\mathbf n\) describes spin polarization. The correspondence between this polarization and the normal to the internal contour is a hypothesis. For a mixed spin state, the polarization vector can have a length smaller than unity; the geometric construction below pertains to a pure, fully polarized state.

In this exploratory variant, let us adopt the classical electromagnetic scale as the mean radius: \[ \tag{6} r_e = \frac{1}{4\pi\varepsilon_0} \frac{e^2}{m_ec^2} = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec}. \] This is an assignment of a radius, rather than a measured electron size.

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

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

The branches are assumed to lie in a single local plane whose normal is determined by the vector \(\mathbf n\). A common geometric axis of internal motion is therefore maintained during traversal of both branches.

The two circles must not be regarded as two independent classical orbits. They are two segments 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. It returns to the initial configuration only after traversing the second branch.

To describe the two nearby branches, introduce the internal angular parameter \[ \tag{12} \chi\in[0,4\pi). \]

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

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, when the parameter changes by \(2\pi\), the system moves from one branch to the other, while the full radial cycle is completed only after a change of \(4\pi\).

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

The internal trajectory can then 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) defines a smoothly modulated radial curve, rather than two exact circles with separate radial connections. As \(\Delta r\to0\), its spatial image becomes a circle of radius \(r_e\), traversed twice over the interval \(4\pi\). Distinguishing the internal sheets in this limit requires an additional state label.

For nonzero \(\Delta r\), the radial function has period \(4\pi\), while the azimuth has period \(2\pi\). This is the periodicity of the chosen parametrization. Possible coincidences of spatial points do not imply coincidence of the complete internal state and do not establish the particle's topological stability.

To describe the sheets, introduce a separate space \(\mathcal H_\Gamma\simeq\mathbb C^2\) with basis \(|\Gamma_+\rangle,|\Gamma_-\rangle\) and projectors \(P_{\Gamma,\pm}=|\Gamma_\pm\rangle\langle\Gamma_\pm|\). The schematic traversal sequence is written as \[ \tag{19} P_{\Gamma,+}\xrightarrow{\;2\pi\;}P_{\Gamma,-}\xrightarrow{\;2\pi\;}P_{\Gamma,+}. \]

The projectors \(P_{\Gamma,\pm}\) label the sheets. The spin projectors \(\ep,\em\) in (2) act in a different space \(\mathcal H_s\). The two-sheeted extension can be described by a factor \(Q_\Gamma\) consistent with the article on deep splitting; charge is specified by an independent sector.

In sheet space, the matrices \(\boldsymbol\tau\), algebraically analogous to Pauli matrices, define \[ \tag{20} \tau_z=P_{\Gamma,+}-P_{\Gamma,-}, \] and \[ \tag{21} \tau_x=S_{\Gamma,+}+S_{\Gamma,-}, \] where \(S_{\Gamma,+}=|\Gamma_+\rangle\langle\Gamma_-|\) and \(S_{\Gamma,-}=S_{\Gamma,+}^{\dagger}\).

The operator \[ \tag{22} \tau_y=-i(S_{\Gamma,+}-S_{\Gamma,-}). \] describes a transition with an additional phase factor. It is not a component of physical spin.

When sheets and spin are described independently, the space is \(\mathcal H_\Gamma\otimes\mathcal H_s\). These two binary factors are not identified with each other. Any connection must be specified by a mapping or an interaction operator.

For physical rotations of the spin state, adopt the standard law \[ \tag{23} U(\mathbf n,\theta) = \exp \left[ -\frac{i\theta}{2} \mathbf n\cdot\boldsymbol{\sigma} \right]. \] The angle \(\theta\) describes a rotation relative to a laboratory axis and differs from the internal parameter \(\chi\).

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

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

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

The adopted law (23) implies the overall spinor sign after a physical rotation through \(2\pi\). Geometry (13) implies a different result: restoration of the radial state after an internal cycle of \(4\pi\). Their correspondence requires a separate proof; a transition between sheets does not by itself produce an overall factor of \(-1\).

The sign of the wave function does not by itself change the bilinear spatial vector: \[ \tag{30} \langle-\Psi| \boldsymbol{\sigma} |-\Psi\rangle = \langle\Psi| \boldsymbol{\sigma} |\Psi\rangle. \] Thus, after a rotation through \(2\pi\), the observable direction \(\mathbf n\) remains unchanged, although the internal state changes sign.

In the circular approximation, define the mean time for one revolution: \[ \tag{31} T_{\mathrm{orb}} = \frac{2\pi}{\omega_{\mathrm{orb}}}. \]

Excluding the time spent in connecting transitions, the full cycle contains two mean revolutions: \[ \tag{32} T_{4\pi} = 2T_{\mathrm{orb}}. \]

The frequency of the complete state is therefore \[ \tag{33} \nu_{4\pi} = \frac{1}{T_{4\pi}} = \frac{\nu_{\mathrm{orb}}}{2}. \]

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

Define the energy frequency independently of the contour period: \[ \tag{36} m_ec^2=\hbar\omega_E,\qquad\omega_E=\frac{m_ec^2}{\hbar}. \]

The geometric frequency of the full cycle and the mean revolution frequency are related by \[ \tag{37} \omega_{\mathrm{orb}}=2\omega_{4\pi},\qquad\omega_{4\pi}\not\equiv\omega_E. \] Their connection to \(\omega_E\) will be considered separately.

Equality (36) defines \(\omega_E\) through the rest energy. The previous identification \(\omega_E=\omega_{4\pi}\) is not adopted here. Internal geometric periodicity does not by itself determine mass.

In the works on infinite splitting and the energy invariant, the norm of the self-similar state is obtained after specifying a metric and a transition coefficient. For the internal level, introduce an independent coefficient \(\rho_{\mathrm{int}}\): \(\boldsymbol\Gamma_{\rho}=\sum_{n\ge0}\rho_{\mathrm{int}}^n\boldsymbol\xi_n\), \(\|\boldsymbol\Gamma_{\rho}\|=(1-\rho_{\mathrm{int}}^2)^{-1/2}=\kappa_{\mathrm{int}}\). Separately, adopt the metric hypothesis \[ \tag{38} \alpha_{\mathrm{fs}} = \frac{1}{\kappa_{\mathrm{int}}}. \]

This hypothesis gives \[ \tag{39} \kappa_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}, \] and \[ \tag{40} \rho_{\mathrm{int}} = \sqrt{1-\alpha_{\mathrm{fs}}^2}. \] The value \(\kappa_{\mathrm{int}}=1/\alpha_{\mathrm{fs}}\) is not selected by the splitting theorem itself: it is specified additionally.

For the following kinematic estimate, additionally adopt the mapping \(v_{\mathrm{int}}=c\rho_{\mathrm{int}}\): \[ \tag{41} v_{\mathrm{int}} = c\sqrt{1-\alpha_{\mathrm{fs}}^2}. \] It does not follow from the metric norm. Here, \(v_{\mathrm{int}}\) is the proposed speed along the internal contour with respect to time \(\tau\).

Under the adopted hypothesis, numerically \[ \tag{42} v_{\mathrm{int}} \approx 0.99997337c. \] The external factor \(\gamma_{\mathrm{ext}}=(1-v_{\mathrm{center}}^2/c^2)^{-1/2}\) pertains to the motion of the electron as a whole and equals unity in the rest frame of its center. It is not identified with \(\kappa_{\mathrm{int}}\).

For the smooth curve (18) in a fixed plane, \(|d\mathbf R_{\mathrm{int}}/d\chi|=\sqrt{r^2+(dr/d\chi)^2}\). A constant speed along the curve therefore requires \[ \tag{43} \dot\chi(\tau)=\frac{v_{\mathrm{int}}}{\sqrt{r(\chi)^2+[r^{\prime}(\chi)]^2}}. \]

On exact circular branches, the radial derivative vanishes, giving \[ \tag{44} \dot\chi_+ = \frac{v_{\mathrm{int}}}{r_+}, \qquad \dot\chi_- = \frac{v_{\mathrm{int}}}{r_-}. \] These expressions are the circular approximation, rather than an exact law for (18).

Including the radial component ensures the specified total speed \(v_{\mathrm{int}}\) along (18). For a slowly varying plane, derivatives of the basis vectors introduce additional terms; below, these are neglected under the assumption of separated timescales.

For two circular parts, excluding the connecting transitions, the times are \[ \tag{45} T_+ = \frac{2\pi r_+}{v_{\mathrm{int}}}, \qquad T_- = \frac{2\pi r_-}{v_{\mathrm{int}}}. \]

For the smooth curve, the exact period at constant speed is specified by an integral; the sum of the circular traversal times gives its leading approximation: \[ \tag{46} T_{4\pi}=\frac1{v_{\mathrm{int}}}\int_0^{4\pi}\sqrt{r(\chi)^2+[r^{\prime}(\chi)]^2}\,d\chi\simeq T_++T_-. \]

Using \(r_++r_-=2r_e\), in the circular approximation we obtain \[ \tag{47} T_{4\pi} \simeq \frac{4\pi r_e} {v_{\mathrm{int}}}. \]

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

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

For the smooth curve, the length correction starts at order \((\Delta r/r_e)^2\). For two exact circles, the sum of their lengths is independent of the separation, but connecting segments can add a first-order term. These are different geometries; formulas for the radial transitions of the Compton contour do not automatically apply here.

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

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

Ratio (55) shows that the geometric frequency does not equal the energy frequency. In the present formulation, the postulate \(m_ec^2=\hbar\omega_{4\pi}\) is omitted; (36) defines only \(\omega_E\).

For the adopted radius and speed, one can write \(\omega_E\simeq[2\alpha_{\mathrm{fs}}/\sqrt{1-\alpha_{\mathrm{fs}}^2}]\omega_{4\pi}\). This follows from the definitions in the circular approximation, rather than being a dynamical derivation of mass. A possible transverse-Doppler interpretation requires specification of the source, observer, and times being compared.

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

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

Directly identifying these frequencies would therefore require changing the characteristic radius or the rule relating the geometric period to energy.

The geometric cycle and energy frequency retain different definitions. Its connection to spinor rotations is also not established merely by matching periods.

If a physical current is additionally specified, the positive magnetic-moment scale for a charge \(-e\) on a circle of radius \(r\) is \[ \tag{56} \mu_{\mathrm{circ}} = \frac{e\omega r^2}{2}. \] Here, \(\omega\) is the charge-transport frequency, rather than an arbitrary phase frequency.

For a negative charge, the moment points opposite to the geometric normal of the traversal: \[ \tag{57} \boldsymbol{\mu} = -\mu\mathbf n. \] The connection of this normal to physical spin is specified separately.

For comparison, define the hypothetical moments of a single charge traversing each branch separately: \[ \tag{58} \mu_+ = \frac{e\omega_+r_+^2}{2}, \qquad \mu_- = \frac{e\omega_-r_-^2}{2}. \] These are not two simultaneously independent electron charges.

At constant linear speed, \[ \omega_\pm = \frac{v_{\mathrm{int}}}{r_\pm}, \] hence \[ \tag{59} \mu_\pm = \frac{ev_{\mathrm{int}}r_\pm}{2}. \]

For a single charge traversing the full path sequentially, physical averaging must account for time. For \(q=-e\), the positive scale of the time-averaged moment in the plane is \[ \tag{60} \overline\mu=\frac{e}{2T_{4\pi}}\int_0^{4\pi}r(\chi)^2\,d\chi\simeq\frac{ev_{\mathrm{int}}(r_+^2+r_-^2)}{2(r_++r_-)}. \] The final expression uses two circular parts without transitions. The arithmetic mean of the separate moments (58) agrees with it only to leading order in the separation.

For \(r_++r_-=2r_e\), the leading magnetic scale is \[ \tag{61} \overline{\mu} \simeq \frac{ev_{\mathrm{int}}r_e}{2}. \]

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

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

The current estimate (63) does not reproduce \(\mu_B\). Consistency requires a physical magnetic mapping, rather than an arbitrary normalization factor. Normalization of the total charge \(-e\) and current conservation must be specified over the entire contour.

In the previous variant, it was proposed to use the internal coefficient \[ \tag{64} \kappa_{\mathrm{int}} = \frac{1}{\alpha_{\mathrm{fs}}}. \]

With the formal choice of magnetic mapping \(\mu^{(0)}=\kappa_{\mathrm{int}}\overline\mu\), one obtains \[ \tag{65} \mu^{(0)} \simeq \kappa_{\mathrm{int}} \overline{\mu} \simeq \sqrt{1-\alpha_{\mathrm{fs}}^2}\mu_B. \] This is an additional assumption that follows neither from the metric-norm theorem nor from a current-transformation law.

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: \[ \tag{66} \mu^{(0)} \approx 0.99997337\mu_B. \]

Even this conditional mapping does not give exactly \(\mu_B\). It is retained here as an exploratory variant. In the Compton construction, a different radius and speed give a different current calculation; the connection between the two variants has not yet been established.

Geometric \(4\pi\) periodicity does not by itself determine the magnitude of angular momentum. The spin operator is therefore introduced in 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 squared-spin operator is \[ \tag{69} \hat{\mathbf S}^2 = \frac{3}{4}\hbar^2I. \]

The projection eigenvalues along a chosen direction are: \[ \tag{70} S_{\mathbf n} = \pm\frac{\hbar}{2}. \]

The matrices \(\boldsymbol\sigma\) act in a separate spin space. The matrices \(\boldsymbol\tau\) act in sheet space. The normalization \(\hbar/2\) and the law of physical rotations are adopted separately; the quantum magnitude of spin is \(\sqrt3\hbar/2\), while the projections are \(\pm\hbar/2\).

The two branches specify internal geometric periodicity. The standard spin representation specifies observable rotations and projections. Their physical correspondence remains a task for the model.

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

Using \[ \hat{\mathbf S} = \frac{\hbar}{2} \boldsymbol{\sigma} \] 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 \[ \tag{74} \hat{\boldsymbol{\mu}} = -\mu_B \boldsymbol{\sigma}. \]

The magnetic-moment projections along the direction \(\mathbf n\) are \[ \tag{75} \mu_{\mathbf n} = \mp\frac{g_e}{2}\mu_B. \]

The classical magnetic moment of a circular current must be distinguished from the quantum magnetic-moment operator. The former is related to the geometry of the internal trajectory, while the latter acts on a two-component state and determines observable magnetic projections.

Operator (71) is introduced as the standard magnetic description of spin. Using it reproduces Zeeman splitting for a specified \(g_e\), but does not derive \(g_e\) from geometry or the metric factor.

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

Substituting (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 convention for the electron charge and the direction of the magnetic-moment operator. The difference between energy eigenvalues 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 are: \[ \tag{80} E_+ = +\frac{g_e\mu_BB}{2}, \qquad E_- = -\frac{g_e\mu_BB}{2}. \]

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

Thus, the external magnetic field splits the two eigenstates of the operator \[ \boldsymbol{\sigma}\cdot\mathbf B. \] rather than the two geometric circles themselves.

The states with energies \(E_+,E_-\) belong to spin space. The geometric sheets belong to a separate internal factor.

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

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

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

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

The angular precession frequency is therefore \[ \tag{86} \omega_{\mathrm{prec}} = \frac{g_e\mu_BB}{\hbar}. \]

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 \(g_e=2\), \[ \tag{88} \omega_{\mathrm{prec}} = \frac{eB}{m_e}. \]

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

For the adopted positive coefficient in Hamiltonian (77), the polarization evolves according to \[ \tag{90} \frac{d\mathbf n}{dt}=\frac{g_e\mu_B}{\hbar}\mathbf B\times\mathbf n. \]

Thus, slow spatial precession follows from the accumulation of a relative phase between the two internal state components.

Let the local plane of the internal trajectory be specified by the vectors \[ \mathbf e_1(t), \qquad \mathbf e_2(t), \] with normal \[ \tag{91} \mathbf n(t) = \mathbf e_1(t) \times \mathbf e_2(t). \]

The full internal trajectory can then 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 fast internal motion along the two branches, while the time dependence of the basis vectors describes the slow reorientation of the entire internal plane.

The model thus contains two distinct timescales: \[ \tag{93} \omega_{\mathrm{int}} \gg \omega_{\mathrm{prec}}. \]

The fast motion is associated with internal traversal of the two-sheeted trajectory, while the slow motion is associated with 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 sheets, and the second to eigenstates of the spin magnetic projection. These are independent attributes until their dynamical connection is specified.

A possible connection is represented symbolically but is not considered to have been constructed: \[ \tag{94} \text{geometric sheets\athcal H_\Gamma\quad\xrightarrow{\;\text{separate mapping}\;}\quad\text{spin state\athcal H_s\quad\longrightarrow\quad\mathbf n\u_{\mathbf n}. \]

The direction of the external field selects the spin basis, rather than automatically selecting the geometric-sheet basis. The complete state can contain correlations between these factors once an interaction is specified.

Changing the direction of the magnetic field changes the eigenstate basis \[ \boldsymbol{\sigma}\cdot\mathbf B. \] rather than the radii \(r_+\) and \(r_-\) themselves.

The quantity \(\Delta r\) is not yet determined by the model's operator structure. Deriving it physically requires an internal potential or a geometric stability condition.

For example, one can postulate an effective potential \[ \tag{95} U(r), \] with two nearby 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 the 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 occur at \[ \tag{99} r=r_e\pm\Delta r. \]

Expression (98) is only a model example. It illustrates how two nearby branches could arise as stable states of a single internal radial coordinate.

The geometric curve (13) is a kinematic description. An operator transition between sheets is a separate model; matching it to the chosen curve requires a dynamical mapping.

In sheet space, introduce a model Hermitian Hamiltonian \[ \tag{100} \hat H_\Gamma=E_0I_\Gamma+\Delta_0\tau_z+\kappa\tau_x, \] Here, the matrices \(\boldsymbol\tau\) are not physical spin operators.

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

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

\(\kappa\) is a real transition amplitude with the dimensions of energy. It differs from the dimensionless metric norm \(\kappa_{\mathrm{int}}\).

If the branches are fully symmetric, \[ \tag{101} \Delta_0=0, \] then \[ \tag{102} \hat H_\Gamma=E_0I_\Gamma+\kappa\tau_x. \]

The eigenstates are then the symmetric and antisymmetric combinations: \[ \tag{103} |\Phi_s\rangle=\frac{|\Gamma_+\rangle+|\Gamma_-\rangle}{\sqrt2}, \] \[ \tag{104} |\Phi_a\rangle=\frac{|\Gamma_+\rangle-|\Gamma_-\rangle}{\sqrt2}. \]

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

The positive magnitude of the sheet splitting is \[ \tag{106} \Delta E_\Gamma=2|\kappa|. \]

This splitting pertains to internal sheet modes, rather than the two projections of free spin. Its observability, mode populations, and energy normalization require separate justification.

In the minimal variant without coupling between sheets and spin, the full Hamiltonian acts on the tensor product: \[ \tag{107} \hat H=\hat H_\Gamma\otimes I_s+I_\Gamma\otimes\hat H_B. \] Interaction between the factors requires an additional Hermitian operator \(V_{\Gamma s}\), which has not yet been specified.

Define two independent coefficient vectors: \[ \tag{108} \mathbf h_\Gamma=(\kappa,0,\Delta_0),\qquad\mathbf h_B=\frac{g_e\mu_B}{2}\mathbf B. \]

Then \[ \tag{109} \hat H=E_0I_\Gamma\otimes I_s+(\mathbf h_\Gamma\cdot\boldsymbol\tau)\otimes I_s+I_\Gamma\otimes(\mathbf h_B\cdot\boldsymbol\sigma). \]

For \(V_{\Gamma s}=0\), the energy eigenvalues have two independent indices: \[ \tag{110} E_{r,s}=E_0+r\sqrt{\Delta_0^2+\kappa^2}+s\frac{g_e\mu_BB}{2},\qquad r,s=\pm1,\quad B=|\mathbf B|. \]

The corresponding level separations are: \[ \tag{111} \Delta E_\Gamma=2\sqrt{\Delta_0^2+\kappa^2},\qquad\Delta E_B=g_e\mu_BB. \]

This Hamiltonian preserves the degeneracy of spin projections at \(B=0\) even if the internal sheet modes are split. Adding their coefficients within a single Pauli vector would identify the factors differently and could create unjustified spin splitting in the absence of a field.

Under the adopted speed hypothesis, the relative departure from \(c\) is \[ \tag{112} 1- \sqrt{1-\alpha_{\mathrm{fs}}^2} \approx \frac{\alpha_{\mathrm{fs}}^2}{2}. \] This is a ratio of speeds, rather than automatically a ratio of radii.

Independently of this, an additional geometric hypothesis can be investigated: \[ \tag{113} \frac{\Delta r}{r_e} = \frac{\alpha_{\mathrm{fs}}^2}{2}. \]

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 nearly indistinguishable on the scale \(r_e\).

However, expression (113) remains a geometric assumption. Justifying it requires deriving \(\Delta r\) from an internal potential, an equation of motion, or an energy-minimum condition.

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

Within this geometric model, one can investigate whether the small difference between radii \[ \frac{\Delta r}{r_e}\ll1 \] is related to \(a_e\).

However, the 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.

The splitting of this contour is not considered an established mechanism for the anomaly. It is also not identified with the separation \(r_e\) between the branches of the Compton contour: here, the separation is \(2\Delta r\).

The proposed model has several qualitative features in common with zitterbewegung in Dirac theory.

First, both constructions involve fast internal motion separated from the observable motion of the particle's center.

The two-component spin description is used as a separate sector. The full Dirac spinor has four complex components; zitterbewegung cannot be derived merely from the two-component nature of sheet space.

Third, the observable spatial direction arises as a bilinear expression of the internal state, rather than as a direct coordinate of the internal basis.

The qualitative comparison does not establish equivalence with the Dirac equation. In particular, the geometric frequency of a contour of radius \(r_e\) does not automatically coincide with the frequency \(2m_ec^2/\hbar\) characteristic of the standard discussion of zitterbewegung for a free particle in the rest frame.

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

In this model, the internal orbit is therefore regarded primarily as a trajectory in state space, rather than a classical trajectory of a localized charged point.

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

This interpretation avoids directly applying 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 regarded as an established spatial boundary of the electron.

It is interpreted as a characteristic internal electromagnetic scale around which the two nearby branches are situated: \[ r_-=r_e-\Delta r, \qquad r_+=r_e+\Delta r. \]

Interpreting \(r_e\) as an internal scale does not by itself prove consistency with experimental constraints on electron structure. This requires calculation of the external currents, form factors, and response generated by the adopted internal dynamics.

The distinction between the constructions can be represented schematically: \[ \tag{118} \boxed{\begin{gathered} \text{sheet geometry:}\quad r_\pm=r_e\pm\Delta r,\quad\chi\in[0,4\pi),\quad\mathcal H_\Gamma,\\ \text{internal metric:}\quad\|\boldsymbol\Gamma_\rho\|=\kappa_{\mathrm{int}},\quad\kappa_{\mathrm{int}}=1/\alpha_{\mathrm{fs}}\, \text{(hypothesis)},\\ \text{separate spin representation:}\quad\mathcal H_s,\quad U(2\pi)=-I,\quad U(4\pi)=I,\\ \hat{\mathbf S}=\frac{\hbar}{2}\boldsymbol\sigma,\qquad\hat{\boldsymbol\mu}=-g_e\frac{e}{2m_e}\hat{\mathbf S},\\ \Delta E_B=g_e\mu_BB,\qquad\omega_{\mathrm{prec}}=\frac{g_e\mu_BB}{\hbar}. \end{gathered}} \]

The geometric cycle, internal metric, and spin dynamics are separate constructions. Arrows indicating physical correspondence require justification and do not replace it.

The model describes the chosen internal geometry and a separately adopted magnetic spin response. It does not automatically derive spin or magnetic moment from the two-sheeted structure.

Several questions are not considered resolved in this work.

First, the half-difference between radii \(\Delta r\) and the full separation \(2\Delta r\) have not been derived from fundamental dynamics.

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

Third, the exact magnetic moment and the coefficient \(g_e\) are specified by the operator law and have not yet been fully derived from the classical circular current.

Fourth, the geometric frequency of the two-sheeted cycle and the electron's rest-energy frequency do not automatically coincide. Their connection requires an additional dynamical equation.

Fifth, the metric choice \(\kappa_{\mathrm{int}}=1/\alpha_{\mathrm{fs}}\), its possible connection to speed, current normalization, stability, and the physical mapping between variants with radii \(r_e\) and \(R\) must be justified.

This work proposes a geometric model of the electron's internal state based on two nearby branches situated on either side of the mean electromagnetic scale \(r_e\). The branches are described by a single continuous trajectory \[ r(\chi) = r_e + \Delta r \cos\frac{\chi}{2}, \] that returns to its initial radial state only after the parameter changes by \(4\pi\).

The sheets are described by an independent binary factor with projectors \(P_{\Gamma,\pm}\), while spin is described by the state \(|\Psi\rangle\) and matrices \(\boldsymbol\sigma\). Internal \(4\pi\) periodicity and the overall spinor sign under a physical rotation have different definitions. Their correspondence has not yet been proved.

The spatial orientation of the internal trajectory is determined by the mapping of the two-component state, rather than by the radii themselves: \[ \mathbf n = \langle\Psi| \boldsymbol{\sigma} |\Psi\rangle. \] Internal motion can therefore remain parametrized by a single variable and two-sheeted, while its manifestation in the observable world can have an arbitrary orientation in three-dimensional space.

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

The geometric sheets are not spin-up and spin-down. For an independent description, \(\mathcal H_\Gamma\otimes\mathcal H_s\) is used; the external field selects eigenstates of the spin projection.

The exploratory variant with radius \(r_e\) retains the internal metric coefficient \(\kappa_{\mathrm{int}}\) separately from the external Lorentz factor. It requires further justification of the speed, frequency and magnetic mappings, connection to the Compton contour, and stable dynamics. The standard description of spin and precession is adopted here as a consistent observable sector, rather than derived solely from the dimensions of the internal branches.