Electron spin cannot be understood as the mechanical rotation of a charged sphere. In the Wave Electricity model, the electron is viewed as a stable, localized mode of a unified normalized wave, while spin kinematics is linked to the two-sheeted closure of its internal circuit.

In this construction, it is necessary to distinguish three separate structures from the outset. The two sheets are successive segments of a single internal cycle, generating a half-phase and a spin magnitude of \(1/2\). The two internal orientations are determined by the permutation of the fundamental idempotent channels: the operator can be realized as either \(J(a,b)\) or \(J(b,a)\). Finally, electric charge pertains to a distinct operator level and is derived neither from the choice of sheet nor from the permutation of channels.

For a particle with spin \(1/2\), measuring the projection onto an arbitrarily chosen axis \(\mathbf n\) yields two results:

When the analyzer axis is rotated by an angle \(\theta\), the probabilities of these results depend on the half-angle, and the state representative changes sign after a rotation of \(2\pi\) and is restored after \(4\pi\):

The task of the model is not to replicate these properties using pre-defined Pauli matrices, but to solve two different problems sequentially. First, the two-sheeted geometry must generate the factor of \(1/2\) and, thereby, the geometric spin magnitude. Then, the permutation of the fundamental channels must determine the two possible internal orientations of the already formed two-sheeted cycle. Normalization of the result via \(\hbar\), the probability rule, and the measurement dynamics remain separate physical correspondences in this process.

At the final operator level, two mutually complementary idempotents are used:

The state operator has the form

Here, the parameters play different physical roles:

The parameter \(a\) describes the internal state of the localized wave, while \(b\) describes the external motion of its center. The case \(b=0\) corresponds to the rest frame. The equality \(J\bar J=1\) expresses operator normalization but represents neither energy nor probability in itself.

The resulting operator \(J(a,b)\) defines the state at the level of two fundamental idempotent channels. Describing the two-sheeted closure requires a deeper splitting of the internal channel. Let us introduce an additional hyperbolic unit and its idempotents:

Let us denote the minimal internal factor distinguishing two consecutive sheets by \(Q_\Gamma\):

The subscript \(\Gamma\) emphasizes that this factor describes the topology of the closure of the internal loop. It does not yet select one of the two spin orientations or determine the sign of the electric charge. Then, the extended electron rest-state operator for a single chosen realization of the principal channels can be represented in the form

The shift \(a\to a+2\) returns the finite part \(J(a,0)\) to the same value but maps the deep component to a different representative. Only the shift \(a\to a+4\) restores the entire extended operator:

In a physical representation, the two deep channels can be depicted as two closely spaced branches of a single closed loop:

The branches may lie in the same physical plane, differing only by a small radial displacement. There is no need to place them in mutually perpendicular spatial or phase planes. The same wave traverses them sequentially:

To leading order, the lengths of the branches are equal:

Radial splitting affects small corrections and potential internal dynamics but does not generate the factor of \(1/2\). The half-angle arises from the sequential traversal of two sheets and the doubled length of the closed loop.

The fundamental mode on the full loop must accumulate a phase of \(2\pi\) over the length \(L_J=4\pi R\). Therefore, its internal wave number is

If \(s\) is the distance traveled and \(\theta=s/R\) is the corresponding geometric angle, then

After the first revolution, the wave is on the second sheet, and its phase differs in sign. After the second revolution, both the sheet and the phase are restored:

The overall sign does not alter the probability of an individual measurement; however, the relative sign is observable through interference with a reference state. Therefore, the \(4\pi\)-periodicity applies to the state amplitude, not to the ordinary spatial trajectory of a material point. \[\tag{18} \mathsf S_\Pi:\quad \ep\longleftrightarrow\em, \qquad \mathsf S_\Pi^2=I. \] \[\tag{19} \boxed{ \begin{aligned} J_1(a,b)&=J(a,b) =\ep e^{i\pi b}+\em e^{i\pi a},\\ J_2(a,b)&=\mathsf S_\Pi J_1(a,b) =J(b,a) =\ep e^{i\pi a}+\em e^{i\pi b}. \end{aligned} } \] \[\tag{20} J_1(a,0)=J(a,0), \qquad J_2(a,0)=J(0,a). \] \[\tag{21} \mathcal J_{e,1}(a,b) =J_1(a,b)Q_\Gamma(a), \qquad \mathcal J_{e,2}(a,b) =J_2(a,b)Q_\Gamma(a). \] \[\tag{22} \Psi_1(\theta)=e^{-i\theta/2}|J_1\rangle, \qquad \Psi_2(\theta)=e^{+i\theta/2}|J_2\rangle. \] \[\tag{23} \widehat S_{\mathbf n} =i\hbar\frac{\partial}{\partial\theta} \] \[\tag{24} \widehat S_{\mathbf n}\Psi_1 =+\frac{\hbar}{2}\Psi_1, \qquad \widehat S_{\mathbf n}\Psi_2 =-\frac{\hbar}{2}\Psi_2. \] \[\tag{25} |+\rangle= \begin{pmatrix}1\\ 0\end{pmatrix}, \qquad |-\rangle= \begin{pmatrix}0\\ 1\end{pmatrix}. \] \[\tag{26} |+\mathbf n\rangle = \begin{pmatrix} \cos(\theta/2)\\ e^{i\varphi}\sin(\theta/2) \end{pmatrix}. \] \[\tag{27} |-\mathbf n\rangle = \begin{pmatrix} -e^{-i\varphi}\sin(\theta/2)\\ \cos(\theta/2) \end{pmatrix}. \] \[\tag{28} U_{\mathbf n}(\theta) =\exp\left(-\frac{i\theta}{2} \boldsymbol{\sigma}\cdot\mathbf n\right), \qquad \widehat S_{\mathbf n} =\frac{\hbar}{2}\boldsymbol{\sigma}\cdot\mathbf n. \] \[\tag{29} U_{\mathbf n}(2\pi)=-I, \qquad U_{\mathbf n}(4\pi)=I. \] \[\tag{30} \boxed{ P_{+\to+}(\theta)=\cos^2\frac{\theta}{2}, \qquad P_{+\to-}(\theta)=\sin^2\frac{\theta}{2}. } \] \[\tag{31} P_{+\to+}+P_{+\to-}=1, \qquad \langle S_{\mathbf n}\rangle =\frac{\hbar}{2}\cos\theta. \] \[\tag{40} \boxed{ C_+\to C_-\to C_+ \quad\Longrightarrow\quad L_J=4\pi R \quad\Longrightarrow\quad \chi=\frac{\theta}{2} \quad\Longrightarrow\quad \Psi(4\pi)=\Psi(0). } \]