This paper develops a previously constructed algebra based on complementary idempotents and complex numbers, leading to a four-dimensional real basis \(\{\ep,i\ep,\em,i\em\}\). A new interpretation of this basis is proposed: it is viewed not as an alternative space-time coordinate system, but as an internal state space describing the one-dimensional phase evolution of the system. It is shown that the transition to the observable four-dimensional world is realized through a sequential mapping of the internal two-component state onto a local four-dimensional vector and then onto ordinary space-time. In this construction, complementary idempotents are naturally interpreted as projectors, and the transition operators between them give rise to a complete set of Pauli matrices.

Based on the resulting mapping, a geometric model of the internal structure of the electron is proposed, consisting of two close branches of a single internal trajectory. This construction leads to a natural (4π)-periodicity of the internal state, allows us to separate the geometric double-sheetedness from the quantum spin states, and relate the internal dynamics to the operator description of the magnetic moment, Zeeman splitting, and Larmor precession. This clearly separates the mathematical results following from the properties of the new basis from the physical hypotheses related to the electron model, which determines the direction of further development of the proposed theory.

Previous works introduced a new complex idempotent basis \[ \tag{1} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\}, \] built on two mutually complementary idempotents \[ \tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] This basis allows us to represent the system as a set of two complex components, each of which forms its own real plane.

This basis has previously been used to describe internal periodic motion. However, it should not be directly identified with the coordinates of ordinary spacetime. In this paper, we assume that the basis \[\tag{3} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\} \] describes the internal one-dimensional state of the system, while the observable four-dimensional space arises after a special mapping of this state onto the ordinary Cartesian basis of spacetime.

The main objective of this work is to construct a sequential transition During this transition, projectors, transition operators between idempotent components, and Pauli matrices naturally arise.

Consider the internal vector \[ \tag{4} J(\tau) = \ep + \em e^{-i\omega\tau}, \] where \(\tau\) is the internal evolution parameter, which can later be related to the system's proper time.

Expanding the complex exponential gives \[ \tag{5} J(\tau) = \ep + \em\cos\omega\tau - i\em\sin\omega\tau. \] The first component \(\ep\) remains constant, while the second component rotates in the complex plane. \[ \left\{ \em,\;i\em \right\}. \]

Despite the presence of four real basis elements, the motion remains one-dimensional in the sense that it is determined by a single parameter \(\tau\). All coordinates of the internal state are functions of a single variable: \[ \tag{6} J=J(\tau). \] Thus, the four-dimensionality of the basis does not imply the presence of four independent spatial directions. It arises as a real representation of two complex components.

In other words, the new basis describes not the ordinary trajectory \[ x(\tau),\;y(\tau),\;z(\tau), \] but the internal phase evolution of the system. To transition to observable motion, it is necessary to separately introduce ordinary space-time and a rule for mapping the internal state into this space.

We write the general internal state as \[ \tag{7} \Psi = \psi_+\ep + \psi_-\em, \] where \[ \psi_+,\psi_-\in\mathbb C. \]

The space of such states has the form \[ \tag{8} \mathcal H_{\mathrm{int}} = \mathbb C\ep \oplus \mathbb C\em. \] It is twoblack over the field of complex numbers and four-dimensional over the field of real numbers: \[ \tag{9} \dim_{\mathbb C}\mathcal H_{\mathrm{int}}=2, \qquad \dim_{\mathbb R}\mathcal H_{\mathrm{int}}=4. \]

State (7) can be associated with a complex column \[ \tag{10} |\Psi\rangle = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}. \] In this case, the idempotent components \(\ep\) and \(\em\) play the role of two internal basis states.

It is important to emphasize that the quantities \(\psi_+\) and \(\psi_-\) are not spatial coordinates. They define the amplitudes of two internal components, and their physical manifestation is determined only after mapping into external space.

We will describe the observed motion in ordinary four-dimensional spacetime with coordinates \[ \tag{11} X^\mu = \left( ct,x,y,z \right), \qquad \mu=0,1,2,3. \]

We introduce a local Cartesian basis \[ \tag{12} \left\{ e_0,e_1,e_2,e_3 \right\}, \] where \(e_0\) corresponds to the temporal direction, and \[ e_1,e_2,e_3 \] corresponds to the three spatial directions.

This basis is fundamentally different from the internal basis \[ \left\{ \ep,i\ep,\em,i\em \right\}. \] The former describes physical space-time, while the latter describes the internal state of the system. No direct component-by-component identification is assumed between them.

Therefore, the mapping required is \[ \tag{13} \mathcal M: \mathcal H_{\mathrm{int}} \longrightarrow T_XM, \] where \(T_XM\) is the local tangent space of spacetime at the point \(X\).

In matrix representation, complementary idempotents can be written as \[ \tag{14} \ep = \begin{pmatrix} 1&0 \\ 0&0 \end{pmatrix}, \qquad \em = \begin{pmatrix} 0&0 \\ 0&1 \end{pmatrix}. \]

These matrices directly imply the properties \[ \tag{15} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=I. \]

The difference between two projectors is \[ \tag{16} \ep-\em = \begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}. \] The right-hand side coincides with the third Pauli matrix: \[ \tag{17} \sigma_z = \begin{pmatrix} 1&0 \\ 0&-1 \end{pmatrix}. \]

Therefore, \[ \tag{18} I=\ep+\em, \qquad \sigma_z=\ep-\em. \] The inverse relations are of the form \[ \tag{19} \ep = \frac{I+\sigma_z}{2}, \qquad \em = \frac{I-\sigma_z}{2}. \]

Thus, two complementary idempotents are naturally spectral projections of the operator \(\sigma_z\). They identify two internal states with eigenvalues \[ \tag{20} \sigma_z\ep=+\ep, \qquad \sigma_z\em=-\em. \]

Diagonal projections alone are insufficient to describe transitions between two states. Therefore, we introduce the operators \[ \tag{21} S_+ = |\ep\rangle\langle\em|, \qquad S_- = |\em\rangle\langle\ep|. \]

In matrix form \[ \tag{22} S_+ = \begin{pmatrix} 0&1 \\ 0&0 \end{pmatrix}, \qquad S_- = \begin{pmatrix} 0&0 \\ 1&0 \end{pmatrix}. \]

They perform the transitions \[ \tag{23} S_+|\em\rangle = |\ep\rangle, \qquad S_-|\ep\rangle = |\em\rangle, \] and also satisfy \[ \tag{24} S_+|\ep\rangle=0, \qquad S_-|\em\rangle=0. \]

The products of the transition operators yield the original idempotents: \[ \tag{25} S_+S_-=\ep, \qquad S_-S_+=\em. \] Furthermore, \[ \tag{26} S_+^2=0, \qquad S_-^2=0. \]

The two remaining Pauli matrices are constructed from the transition operators: \[ \tag{27} \sigma_x = S_++S_-, \] \[ \tag{28} \sigma_y = -i\left(S_+-S_-\right). \]

Indeed, \[ \tag{29} \sigma_x = \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}, \qquad \sigma_y = \begin{pmatrix} 0&-i \\ i&0 \end{pmatrix}. \]

Thus, the complete set of Pauli matrices arises as an algebra of transformations of two idempotent components: \[ \tag{30} I=\ep+\em, \qquad \sigma_z=\ep-\em, \qquad \sigma_x=S_++S_-, \qquad \sigma_y=-i(S_+-S_-). \]

Matrix \(\sigma_z\) distinguishes between two internal states, \(\sigma_x\) swaps them without an additional phase shift, and \(\sigma_y\) performs the same transition with a complex phase factor.

Pauli matrices satisfy the well-known multiplication rule \[ \tag{31} \sigma_i\sigma_j = \delta_{ij}I + i\varepsilon_{ijk}\sigma_k. \]

From this follow the commutators \[ \tag{32} \left[ \sigma_i,\sigma_j \right] = 2i\varepsilon_{ijk}\sigma_k \] and anticommutators \[ \tag{33} \left\{ \sigma_i,\sigma_j \right\} = 2\delta_{ij}I. \]

Thus, the original idempotents define projections onto two states, and the natural transition operators between these states generate a complete noncommutative Pauli algebra.

This does not mean that the matrices \(\sigma_x\) and \(\sigma_y\) arise only from the inner multiplication of commutative elements \(\ep\) and \(\em\). They arise in a larger space of linear operators acting on \[ \mathcal H_{\mathrm{int}} = \mathbb C\ep\oplus\mathbb C\em. \]

After constructing the Pauli matrices, we can define the rotation operator of the internal state around the unit direction \(\mathbf n\): \[ \tag{34} U(\mathbf n,\theta) = \exp \left[ -\frac{i\theta}{2} \mathbf n\cdot\boldsymbol{\sigma} \right], \] where \[ \tag{35} \mathbf n\cdot\boldsymbol{\sigma} = n_x\sigma_x+n_y\sigma_y+n_z\sigma_z. \]

Since \[ \tag{36} \left( \mathbf n\cdot\boldsymbol{\sigma} \right)^2 = I, \] the exponential can be represented as \[ \tag{37} U(\mathbf n,\theta) = I\cos\frac{\theta}{2} - i \left( \mathbf n\cdot\boldsymbol{\sigma} \right) \sin\frac{\theta}{2}. \]

When rotating by the full angle \(2\pi\), we get \[ \tag{38} U(\mathbf n,2\pi) = -I, \] and when rotating by \(4\pi\), \[ \tag{39} U(\mathbf n,4\pi) = I. \]

Therefore, the internal two-component state transforms according to the law \[ \tag{40} |\Psi(2\pi)\rangle = -|\Psi(0)\rangle, \qquad |\Psi(4\pi)\rangle = |\Psi(0)\rangle. \] This periodicity arises not from the ordinary motion of a point along a circle, but from the structure of operators acting on a two-component complex state.

Consider the normalized internal state \[ \tag{41} |\Psi\rangle = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix}, \qquad \langle\Psi|\Psi\rangle=1. \]

Assign three real quantities to it \[ \tag{42} n_x = \langle\Psi|\sigma_x|\Psi\rangle, \] \[ \tag{43} n_y = \langle\Psi|\sigma_y|\Psi\rangle, \] \[ \tag{44} n_z = \langle\Psi|\sigma_z|\Psi\rangle. \]

Explicitly, \[ \tag{45} n_x = \psi_+^*\psi_- + \psi_-^*\psi_+, \] \[ \tag{46} n_y = -i\psi_+^*\psi_- + i\psi_-^*\psi_+, \] \[ \tag{47} n_z = |\psi_+|^2-|\psi_-|^2. \]

For the normalized pure state, we have \[ \tag{48} n_x^2+n_y^2+n_z^2=1. \] Therefore, the internal two-component state naturally corresponds to a unit vector \[ \tag{49} \mathbf n = \left( n_x,n_y,n_z \right) \] in ordinary three-dimensional space.

This mapping shows how two complex amplitudes of the internal state determine three real spatial components. The transition is bilinear and therefore cannot be reduced to simply identifying the elements of the internal basis with the axes \(x,y,z\).

The general normalized two-component state can be written as \[ \tag{50} |\Psi\rangle = e^{i\chi} \begin{pmatrix} \cos\dfrac{\vartheta}{2} \\ e^{i\varphi} \sin\dfrac{\vartheta}{2} \end{pmatrix}, \] where \(\chi\) is the overall phase, and \(\vartheta\) and \(\varphi\) determine the direction of the mapped vector.

After substituting into expressions (42)–(44), we obtain \[ \tag{51} n_x = \sin\vartheta\cos\varphi, \] \[ \tag{52} n_y = \sin\vartheta\sin\varphi, \] \[ \tag{53} n_z = \cos\vartheta. \]

Thus, the internal complex phase and the ratio of the amplitudes of two idempotent components determine the orientation of the unit vector in ordinary three-dimensional space.

The overall phase \(e^{i\chi}\) does not change the vector \(\mathbf n\), since it cancels out in bilinear expressions \[ \langle\Psi|\sigma_i|\Psi\rangle. \] Therefore, the set of internal states contains more information than the observed direction in three-dimensional space.

To include the time component, we introduce a set of matrices \[ \tag{54} \sigma^a = \left( I,\sigma_x,\sigma_y,\sigma_z \right), \qquad a=0,1,2,3. \]

The internal state is associated with a local four-dimensional vector \[ \tag{55} v^a = \langle\Psi|\sigma^a|\Psi\rangle. \]

Its components are \[ \tag{56} v^0 = \langle\Psi|I|\Psi\rangle, \] \[ \tag{57} v^1=n_x, \qquad v^2=n_y, \qquad v^3=n_z. \]

For normalstate \[ \tag{58} v^0=1, \qquad \left(v^1\right)^2+ \left(v^2\right)^2+ \left(v^3\right)^2=1. \] Therefore, with the metric \[ \eta_{ab} = \operatorname{diag} \left( 1,-1,-1,-1 \right) \] we obtain \[ \tag{59} v^av_a = \left(v^0\right)^2 - \left(v^1\right)^2 - \left(v^2\right)^2 - \left(v^3\right)^2 = 0. \]

Thus, the normalized pure two-component state is naturally mapped onto a local light-like four-vector.

If we multiply it by the speed of light, we get \[ \tag{60} k^a = c\,v^a = c \left( 1,\mathbf n \right), \] for which \[ \tag{61} k^ak_a=0. \]

Such a vector should not be directly identified with the four-velocity of the center of a massive particle. It can describe the internal direction of motion or propagation, whereas the observed timelike motion of the center of the system arises after averaging out the internal dynamics.

To transition from the local four-dimensional vector \(v^a\) to the coordinate four-vector of space-time, we introduce the tetrad \[ \tag{62} e_a^{\;\mu}. \]

Then the physical vector is defined by the expression \[ \tag{63} V^\mu = e_a^{\;\mu}v^a. \]

The full mapping is of the form \[ \tag{64} |\Psi\rangle \longrightarrow v^a = \langle\Psi|\sigma^a|\Psi\rangle \longrightarrow V^\mu = e_a^{\;\mu}v^a. \]

The first part of the mapping transforms the internal two-component state into a local four-dimensional vector. The second part orients this vector relative to the chosen Cartesian space-time system.

In flat space and an inertial frame of reference, one can choose \[ \tag{65} e_a^{\;\mu} = \delta_a^{\mu}, \] and then \[ \tag{66} V^\mu=v^\mu. \] In general, the tetrad allows for arbitrary orientation of the local basis and the transition between different coordinate systems.

Within the proposed design, it is necessary to distinguish between the internal state \[ |\Psi(\tau)\rangle \] and the position of the system's center \[ X^\mu(\tau). \]

The internal state determines the local direction \[ \tag{67} v^a(\tau) = \langle\Psi(\tau)|\sigma^a|\Psi(\tau)\rangle, \] and the external motion of the center can be specified by a separate equation \[ \tag{68} \frac{dX^\mu}{d\tau} = U^\mu. \]

In this case, the internal vector can be light-like: \[ \tag{69} v^av_a=0, \] while the observed four-velocity of the center of a massive system should be timelike: \[ \tag{70} U^\mu U_\mu=c^2. \]

One possible way to relate these levels is averaging: \[ \tag{71} U^\mu = c \left\langle e_a^{\;\mu}v^a \right\rangle. \] Then the rapidly changing internal direction can produce a stable timelike motion of the center of the system.

In this paper, expression (71) is considered as a possible mapping scheme, not as a definitively derived dynamic equation. Its rigorous justification requires the specification of an internal evolution law and an averaging procedure.

In the proposed construction, the Pauli matrices are not additional spatial directions within the original idempotent basis.

They perform two related functions:

the first is to describe all linear transitions between states \(\ep\) and \(\em\);

the second is to define a bilinear mapping of the internal two-component state onto a three-dimensional spatial direction.

Therefore, the sequence of occurrence of the Pauli matrices can be represented as \[ \tag{72} \left\{ \ep,\em \right\} \longrightarrow \left\{ S_+,S_- \right\} \longrightarrow \left\{ I,\sigma_x,\sigma_y,\sigma_z \right\}. \]

It is the complete set of operators \[ \left\{ I,\sigma_x,\sigma_y,\sigma_z \right\} \] that allows us to transition from an internal two-component state to a local four-dimensional structure.

The resulting structure can be represented as the following sequence: \[ \tag{73} \boxed{ \begin{array}{c} \text{one-dimensional internal motion} \\[4pt] J(\tau) \\[6pt] \downarrow \\[6pt] \text{two-component state} \\[4pt] |\Psi\rangle = \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} \\[8pt] \downarrow \\[6pt] \text{projectors and transitions} \\[4pt] \ep,\em,S_+,S_- \\[8pt] \downarrow \\[6pt] \text{Pauli matrices} \\[4pt] I,\sigma_x,\sigma_y,\sigma_z \\[8pt] \downarrow \\[6pt] \text{local four-vector} \\[4pt] v^a = \langle\Psi|\sigma^a|\Psi\rangle \\[8pt] \downarrow \\[6pt] \text{physical space-time} \\[4pt] V^\mu = e_a^{\;\mu}v^a \end{array} } \]

This scheme allows us to preserve the original postulate of one-dimensional internal motion and simultaneously obtain a three-dimensional spatial orientation and time component without directly identifying the internal basis with space-time coordinates.

The proposed mapping demonstrates a natural mathematical relationship between two complementary idempotents, a two-component complex state, and a local four-dimensional vector.

However, this paper does not claim that ordinary spacetime is completely derived from the internal basis. The Cartesian system \[ \left( ct,x,y,z \right) \] is introduced as an external geometric structure, and the new basis defines the internal state mapped onto this structure.

Also, the final dynamic equation governing the evolution of the tetrad, the internal state, and the external four-velocity is not yet specified. These issues should be addressed at the next stage of the model's development.

This paper proposes separating the internal and external geometry of the system. The new idempotent basis \[ \left\{ \ep,i\ep,\em,i\em \right\} \] is interpreted as a real representation of a two-component complex internal state depending on a single evolution parameter.

Two mutually complementary idempotents are naturally represented as spectral projectors \[ \ep = \frac{I+\sigma_z}{2}, \qquad \em = \frac{I-\sigma_z}{2}. \] After introducing the transition operators between components, a complete set of Pauli matrices arises: \[ I,\sigma_x,\sigma_y,\sigma_z. \]

The Pauli matrices act as internal transformation operators and simultaneously define a way to map a two-component state onto a three-dimensional direction: \[ \mathbf n = \langle\Psi|\boldsymbol{\sigma}|\Psi\rangle. \] After adding the identity matrix, this mapping is extended to a local four-dimensional vector: \[ v^a = \langle\Psi|\sigma^a|\Psi\rangle. \]

The final transition to physical space-time is accomplished through a tetrad: \[ V^\mu = e_a^{\;\mu}v^a. \] Thus, the new basis does not replace ordinary space-time, but describes the system's internal degree of freedom, which manifests itself in the observable world through a consistent operator mapping.

The constructed scheme creates a mathematical foundation for further consideration of the electron's internal structure. In the second part of the paper, this construction will be applied to the model of two close branches of the internal orbit, the full periodicity of \(4\pi\), the magnetic moment, energy splitting, and precession.