2026-08-05
Geometric origin of electric and magnetic fields
Why does the same charge create an electric field when at rest, yet a magnetic field is also detected nearby when it is in motion? Typically, electric and magnetic fields are introduced as two coupled components of a unified electromagnetic field. However, a geometric question remains: why does the motion of the source reveal a new field direction at all, and why is its magnitude to a first approximation proportional to the ratio \(v/c\)?
In the proposed model, the answer is linked to the multi-level idempotent splitting of space. Upon movement, the initial splitting of the source's external state channel into an original branch and an orthogonal branch is activated. Following physical manifestation, the original branch appears as the electric component, while the deep orthogonal residue manifests as the magnetic component.
This refines the previous interpretation. The magnetic field corresponds neither to a single additional direction nor merely to the imaginary part of a complex exponential. It is the observable manifestation of the entire deep orthogonal remainder that emerges after the first branch of space is isolated. The complex notation \(E+i\,cB\) is derived later as a final, compact representation of this deeper structure.
A single geometric mechanism yields two results simultaneously. The total unnormalized splitting depth has a norm of \(\gamma\) that is, it generates the Lorentz factor. After normalization, the first branch has an amplitude of \(1/\gamma\), while the entire orthogonal remainder has an amplitude of \(\beta=v/c\).
Therefore, \[\tag{1} \boxed{ \text{total splitting depth}\longrightarrow\gamma, \qquad \text{first observable separation}\longrightarrow E,\;B. } \] Below, we first determine the initial scale of the rest field \(E_0\) and construct a normalized deep state; subsequently, its first idempotent splitting is physically mapped to the electric and magnetic components. The direction of the magnetic field emerges only at the final stage upon mapping the abstract orthogonal branch into three-dimensional space.
The fundamental equality of norms is derived for the total transverse splitting corresponding to the geometry \(\mathbf v\perp\mathbf R\). For an arbitrary angle, the observed magnetic field is merely a three-dimensional projection of the deep branch. This distinction will be addressed separately.
1. The rest field as the initial scale
Consider two elementary charges separated by a distance \(R\). In a previous work, the interaction energy for these charges was found to be
\[\tag{2} U(R)=\frac{\alpha_{\mathrm{fs}}\hbar c}{R}. \] The sign of the energy is determined by the relative orientation of the charges. To construct the field magnitude, we consider the absolute value of the force, which equals the negative gradient of the potential energy:
\[\tag{3} F(R)=-\frac{dU}{dR}, \qquad |F(R)|=\frac{\alpha_{\mathrm{fs}}\hbar c}{R^2}. \] The electric field of the source is defined as the force per unit test charge. For a test charge with magnitude (e), we obtain \[\tag{4} E_0(R)=\frac{|F(R)|}{e} =\frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2}. \] Using the definition of the fine-structure constant \[\tag{5} \alpha_{\mathrm{fs}} =\frac{e^2}{4\pi\varepsilon_0\hbar c}, \] we obtain the standard value of the field of an elementary charge at rest: \[\tag{6} \boxed{ E_0(R) =\frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2} =\frac{e}{4\pi\varepsilon_0R^2} }. \] For an arbitrary point charge (q), the field is proportional to the ratio (q/e). Upon switching to a three-dimensional notation, we have \[\tag{7} \boxed{ \mathbf E_0(\mathbf R) =\frac{q}{4\pi\varepsilon_0R^2}\widehat{\mathbf R}, \qquad \widehat{\mathbf R}=\frac{\mathbf R}{R}. } \] Each direction (\widehat{\mathbf R}) corresponds to a single one-dimensional field line of the model. The set of all equivalent radial directions forms the spherically symmetric field of a point charge. The quantity (E_0(R)) has a dual meaning. For a stationary source, it is directly equal to the observed electric field. For a moving source, it serves as the initial, unified scale of the external state prior to its separation into electric and magnetic components: \[\tag{8} \beta=0: \qquad E=E_0, \qquad B=0. \]
2. Why motion is linked to splitting
The mere existence of multiple mathematical channels does not in itself determine their physical meaning. Idempotent algebra answers the question of which independent parts can exist, but it does not assign to any single part...one being electrical and the other magnetic in nature. Such an assignment requires a distinct physical representation.
It is necessary to clarify here exactly what undergoes splitting. We are not referring to the literal division of pre-existing physical space. Motion triggers the splitting of the external-state channel of the wave structure. The resulting independent orthogonal channels are then manifested as spatial directions and their associated observable field components. Therefore, the term "splitting of space" is used hereafter in a shorthand sense to denote the formation of a system of independent channels from which the spatial structure emerges.
In this model, the following correspondence is adopted. While the wave center is at rest, the transition from the initial external direction to the deep orthogonal branch has zero amplitude. Therefore, the entire available scale \(E_0\) manifests as an electric field. Upon motion with a dimensionless velocity
\[\tag{9} \beta=\frac{v}{c}, \qquad |\beta|<1, \] a transition into the orthogonal remainder of the space emerges. Its relative amplitude is multiplied by \(\beta\) again at each successive level. Thus, the space reproduces itself after each splitting.
It is precisely this self-similarity that distinguishes the new interpretation from a simple two-dimensional rotation. Concealed behind the observed magnetic component is not merely a single axis, but an entire sequence of nested orthogonal directions.
3. Deep state of motion
Let us introduce the orthonormal directions of deep space
\[\tag{10} \boldsymbol{\xi}_n\boldsymbol{\cdot}\boldsymbol{\xi}_m =\delta_{nm}, \qquad n,m=0,1,2,\ldots \] and a shift operator \(S\) that moves the state to the next orthogonal level:
\[\tag{11} S\boldsymbol{\xi}_n=\boldsymbol{\xi}_{n+1}. \] If each transition is accompanied by the same coefficient \(\beta\), the natural deep state takes the form
\[\tag{12} \boxed{ \GF =\boldsymbol{\xi}_0 +\beta\boldsymbol{\xi}_1 +\beta^2\boldsymbol{\xi}_2 +\beta^3\boldsymbol{\xi}_3+\cdots }. \] The same state can be written recursively:
\[\tag{13} \boxed{ \GF =\boldsymbol{\xi}_0+\beta S\GF. } \] After isolating the initial direction \(\boldsymbol{\xi}_0\), a shifted copy of the entire original state remains. Therefore, the second term in formula (13) represents not just a single channel, but the entire residual space.
Due to the orthogonality of the directions, the squared norm is equal to
\[\tag{14} \begin{aligned} \|\GF\|^2 &=1+\beta^2+\beta^4+\beta^6+\cdots\\ &=\frac{1}{1-\beta^2}. \end{aligned} \] Consequently, as shown in detail in the article “Geometric Origin of the Lorentz Factor and Energy Invariant,”
\[\tag{15} \boxed{ \|\GF\| =\frac{1}{\sqrt{1-\beta^2}} =\gamma. } \] Here, the Lorentz factor emerges as the metric length of the entire unnormalized deep state. Achieving this result requires not only idempotents but also the orthogonality of the deep directions, their metric, the self-similarity of the residual space, and a unified transition coefficient \(\beta\).
4. Normalized state
To facilitate comparison with the finite operator and with fields, we normalize the deep state:
\[\tag{16} \boxed{ \NF=\frac{\GF}{\gamma}, \qquad \|\NF\|=1. } \] In expanded form:
\[\tag{17} \NF =\frac1\gamma\boldsymbol{\xi}_0 +\frac\beta\gamma\boldsymbol{\xi}_1 +\frac{\beta^2}{\gamma}\boldsymbol{\xi}_2 +\frac{\beta^3}{\gamma}\boldsymbol{\xi}_3+\cdots. \] Let us partition the expression not into individual terms, but into the initial branch and the entire subsequent remainder:
\[\tag{18} \boxed{ \NF =\frac1\gamma\boldsymbol{\xi}_0 +\beta S\NF. } \] The first part lies in the direction of \(\boldsymbol{\xi}_0\). The second part lies in the orthogonal subspace spanned by \(\boldsymbol{\xi}_1, \boldsymbol{\xi}_2, \ldots\). This constitutes the first level of the decomposition of the complete normalized state.
5. Idempotent projectors for the electric and magnetic branches
To express this initial splitting rigorously, we introduce two orthogonal linear projectors. The projector $PE$ picks out the initial direction $oldsymbol{ i}_0$, while $PB$ picks out its entire orthogonal complement: \[\tag{19} \PE=|\boldsymbol{\xi}_0\rangle\langle\boldsymbol{\xi}_0|, \qquad \PB=I-\PE. \]
These projectors are idempotent, mutually annihilating, and sum to the identity operator on the space:
\[\tag{20} \boxed{ \PE^2=\PE, \qquad \PB^2=\PB, \qquad \PE\PB=\PB\PE=0, \qquad \PE+\PB=I. } \] Consequently, the same structural principle applies here as in the standard idempotent decomposition. Any state can be uniquely split into two independent parts:
\[\tag{21} \NF=\PE\NF+\PB\NF. \] For state (18), these parts are
\[\tag{22} \boxed{ \PE\NF=\frac1\gamma\boldsymbol{\xi}_0, \qquad \PB\NF=\beta S\NF. } \] If desired, an involution can be associated with the two projectors:
\[\tag{23} \j_F=\PE-\PB, \qquad \j_F^2=I, \] in which case
\[\tag{24} \PE=\frac{I+\j_F}{2}, \qquad \PB=\frac{I-\j_F}{2}. \] Formulas (23)–(24) demonstrate an exact correspondence with the hyperbolic unit and a pair of mutually complementary idempotents. However, \(\PE\) and \(\PB\) are projectors of the deep metric space; they should not be automatically identified with the original algebraic idempotents \(\ep\) and \(\em\) of the finite operator \(J\). These constructions belong to different levels of description.
6. Norm of the deep remainder
The main property of the second projector is that it isolates the entire remainder:
\[\tag{25} \PB\NF =\frac\beta\gamma\boldsymbol{\xi}_1 +\frac{\beta^2}{\gamma}\boldsymbol{\xi}_2 +\frac{\beta^3}{\gamma}\boldsymbol{\xi}_3+\cdots. \] Let us calculate its norm:
\[\tag{26} \begin{aligned} \|\PB\NF\|^2 &=\frac{\beta^2}{\gamma^2} \left(1+\beta^2+\beta^4+\cdots\right)\\ &=\frac{\beta^2}{\gamma^2}\,\gamma^2 =\beta^2. \end{aligned} \] Therefore,
\[\tag{27} \boxed{ \|\PE\NF\|=\frac1\gamma, \qquad \|\PB\NF\|=\beta. } \] This is a fundamental result.
The deep branch begins with the term \((\beta/\gamma)\boldsymbol{\xi}_1\) but is not limited to it. If only the first additional channel were associated with the magnetic component, its relative magnitude would be \(\beta/\gamma\). The total norm of the entire infinite residual branch is exactly \(\beta\). Since the two parts are orthogonal, the normalization of the full state yields
\[\tag{28} \boxed{ \|\PE\NF\|^2+\|\PB\NF\|^2 =\frac1{\gamma^2}+\beta^2=1. } \] Thus, the final identity \(1/\gamma^2+\beta^2=1\) expresses the norm of the first splitting, whereas the value \(\gamma\) itself arises from the full infinite depth of the state.
7. Physical mapping to electric and magnetic components
We now introduce a physical correspondence between the two geometric branches and the field. We associate the normalized deep state with the initial scale \(E_0(R)\)
We associate the electric component with the distinguished branch \(\PE\NF\), and the full magnetic component with the entire orthogonal remainder \(\PB\NF\):
\[\tag{30} \boxed{ \frac{E}{E_0} \;\longleftrightarrow\; \|\PE\NF\|, \qquad \frac{cB_{\mathrm{full}}}{E_0} \;\longleftrightarrow\; \|\PB\NF\|. } \] The factor \(c\) converts the magnetic induction to the same dimension as the electric field strength. Using formula (27), we obtain \[\tag{31} \boxed{ E(R,\beta)=\frac{E_0(R)}{\gamma}, \qquad cB_{\mathrm{full}}(R,\beta)=E_0(R)\beta. } \]
Here, \(B_{\mathrm{full}}\) denotes the full magnitude of the deep magnetic branch prior to the selection of a specific observation direction in three-dimensional space. In the case of full transverse geometry, it coincides with the observed magnetic field.
The normalization equality follows directly from the orthogonality of the projectors:
\[\tag{32} \boxed{ E^2+c^2B_{\mathrm{full}}^2=E_0^2. } \] Formula (30) represents the physical manifestation of the model. It does not follow solely from the idempotency of the projectors. The geometry uniquely determines the norms \(1/\gamma\) and \(\beta\), whereupon the model assigns the normalized electric and magnetic components to them.
8. State of rest
For a stationary source,
\[\tag{33} \beta=0, \qquad \gamma=1. \] The deep state degenerates into the initial direction:
\[\tag{34} \mathbf N_0=\boldsymbol{\xi}_0, \qquad \PE\mathbf N_0=\boldsymbol{\xi}_0, \qquad \PB\mathbf N_0=0. \] Therefore, the entire initial scale of the field resides in the electric branch:
\[\tag{35} \boxed{ \beta=0: \qquad E=E_0, \qquad B=0. } \] The magnetic field is absent not because the orthogonal subspace is fundamentally impossible. At rest, the transition amplitude into it is zero. Movement does not recreate a pre-existing space but activates the orthogonal remainder of the external channel; in spatial mapping, this remainder becomes a magnetic branch of the field.
9. Motion and the emergence of the deep branch
For \(\beta>0\), the normalized state acquires two non-zero components:
\[\tag{36} \NF =\underbrace{\frac1\gamma\boldsymbol{\xi}_0}_{\text{electric branch}} +\underbrace{\beta S\NF}_{\text{deep magnetic branch}}. \] As velocity increases, the norm of the first part decreases as
\[ \frac1\gamma=\sqrt{1-\beta^2}, \] while the norm of the second increases as \(\beta\). At low velocity,
\[\tag{37} \frac1\gamma =1-\frac{\beta^2}{2}+O(\beta^4). \] Therefore, the electric component changes only at the second order of velocity, whereas the magnetic one appears already at the first...where:
\[\tag{38} E\approx E_0\left(1-\frac{\beta^2}{2}\right), \qquad cB_{\mathrm{full}}\approx E_0\beta. \] This is precisely why, in the non-relativistic regime, the electric field may differ hardly at all from the field of a stationary charge, even though a weak magnetic field already exists.
10. The finite operator \(J\) as a squeezed representation
The complete phase state of the particle in the finite split-geometry is described by the operator
\[\tag{39} \boxed{ J(a,b)=\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a}. } \] The parameter \(a\) describes the internal periodic state, while the parameter \(b\) describes the external motion:
\[\tag{40} a=\varpi t, \qquad \pi\varpi=\omega, \qquad b=\frac{\arcsin\beta}{\pi}. \] The external branch of the operator is equal to
\[\tag{41} J_{\mathrm{ext}}(b)=\ep e^{i\pi b}. \] From the connection between the finite operator and the deep state, it follows that
\[\tag{42} \sin(\pi b)=\beta, \qquad \cos(\pi b)=\frac1\gamma. \] Therefore,
\[\tag{43} \boxed{ J_{\mathrm{ext}}(b) =\ep\left(\frac1\gamma+i\beta\right). } \] The meaning of the complex decomposition now becomes deeper. The finite operator does not show each channel \(\boldsymbol{\xi}_n\) separately. It compresses the infinite state into two quantities: the norm of the distinguished branch \(1/\gamma\) and the norm of the entire orthogonal remainder \(\beta\). Therefore, the correspondence can be written as
\[\tag{44} \boxed{ \NF =\frac1\gamma\boldsymbol{\xi}_0+\beta S\NF \quad\longrightarrow\quad e^{i\pi b}=\frac1\gamma+i\beta. } \] After physical mapping onto the field, this yields a single finite expression
\[\tag{45} \boxed{ E+i\,cB_{\mathrm{full}} =E_0e^{i\pi b}. } \] The real and imaginary parts of formula (45) are not idempotents in themselves. They represent two finite quadrature projections of deep idempotent branches. Therefore, the complex formula is a representation of the splitting, rather than a replacement for its mathematical mechanism.
11. Direction of the magnetic field in three-dimensional space
Deep splitting determines the relative modulus of the second branch, but by itself, it does not yet specify the direction of the magnetic field in our three-dimensional reality. For this, two observable vectors are required:
\[\tag{46} \boldsymbol{\beta}=\frac{\mathbf v}{c}, \qquad \mathbf E_0(\mathbf R)=E_0(R)\widehat{\mathbf R}. \] The source velocity and the radial direction of the electric field line form an oriented plane. Its normal is defined by the vector product \[ \boldsymbol{\beta}\times\widehat{\mathbf R}. \]
Therefore, the spatial mapping of the magnetic branch is determined by the formula
\[\tag{47} \boxed{ \mathbf B(\mathbf R) =\frac1c\boldsymbol{\beta}\times\mathbf E_0(\mathbf R) =\frac1{c^2}\mathbf v\times\mathbf E_0(\mathbf R). } \] For a point charge, substituting formula (7) yields
\[\tag{48} \mathbf B(\mathbf R) =\frac{q}{4\pi\varepsilon_0c^2} \frac{\mathbf v\times\widehat{\mathbf R}}{R^2}. \] Taking into account \(\mu_0\varepsilon_0c^2=1\), we obtain
\[\tag{49} \boxed{ \mathbf B(\mathbf R) =\frac{\mu_0q}{4\pi} \frac{\mathbf v\times\widehat{\mathbf R}}{R^2}. } \] This expression coincides with the non-relativistic form of the magnetic field of a uniformly moving point charge. Thus, the deep splitting determines the coefficient \(\beta\), while the three-dimensional mapping imparts direction to the magnetic branch.
12. The complete branch and its observable projection
Let \(\theta\) be the angle between \(\mathbf v\) and \(\mathbf R\). It follows from formula (47) that
\[\tag{50} B(R,\beta,\theta) =\frac{E_0(R)}{c}\,\beta\sin\theta. \] It is necessary to distinguish between the full magnitude of the deep magnetic branch and its observed spatial projection:
\[\tag{51} \boxed{ cB_{\mathrm{full}}=E_0\beta, \qquad cB(\theta)=E_0\beta\sin\theta. } \] In the transverse geometry
\[\tag{52} \theta=\frac\pi2 \qquad\Longrightarrow\qquad B=B_{\mathrm{full}}. \] It is precisely in this case that both branches of the first splitting are fully mapped onto the observed \(E\) and \(B\); therefore,
\[\tag{53} \boxed{ \mathbf v\perp\mathbf R: \qquad E^2+c^2B^2=E_0^2. } \] At an arbitrary angle, only the transverse part of the deep branch is mapped onto the magnetic induction. Therefore, for the quantities directly observed at a given point,
\[\tag{54} E^2+c^2B^2(\theta) =E_0^2\left(1-\beta^2\cos^2\theta\right). \] This does not imply a violation of the normalization of the full deep state. The difference is associated with an additional three-dimensional projection: when \(\theta\ne\pi/2\), the magnetic com...representation does not utilize the entire orthogonal residual module. In particular, when \(\mathbf v\parallel\mathbf R\), the vector product vanishes, even though the deep state of motion itself continues to depend on \(\beta\).
13. The meaning of the normalization equality
The equality
\[\tag{55} E^2+c^2B_{\mathrm{full}}^2=E_0^2 \] expresses the conservation of the Euclidean norm of the two orthogonal branches of the internal geometric state. It can be termed the normalization invariant of the split field state.
This result should not be identified with the standard Lorentz invariant of the electromagnetic tensor, which involves the difference \(E^2-c^2B^2\). The two expressions pertain to different mathematical objects. The sum of squares in formula (55) describes the internal norm associated with the splitting adopted in the model, whereas the standard difference characterizes transformations of the physical electromagnetic tensor between inertial reference frames.
Furthermore, the construct under consideration does not describe a free traveling electromagnetic wave. In a traveling wave, the \(\mathbf E\) and \(\mathbf B\) fields vary in phase and are related by \(\mathbf B=\widehat{\mathbf k}\times\mathbf E/c\). Here, however, we investigate a localized particle regime and the geometric separation of the initial scale of its external state.
14. Common origin of the magnetic field and the Lorentz factor
The relationship between the two results can now be written out in full. The unnormalized state
\[\tag{56} \GF=\boldsymbol{\xi}_0+\beta S\GF \] has a length of
\[\tag{57} \|\GF\|=\gamma. \] After normalization, the same state takes the form
\[\tag{58} \NF =\frac1\gamma\boldsymbol{\xi}_0 +\beta S\NF. \] From this, three related yet distinct quantities follow:
\[\tag{59} \boxed{ \begin{aligned} \gamma &=\text{norm of the full unnormalized deep state},\\ \frac1\gamma &=\text{norm of the distinguished initial branch},\\ \beta &=\text{norm of the entire deep orthogonal remainder}. \end{aligned} } \] Following the adopted physical mapping, the initial branch manifests as the electric component, while the deep remainder manifests as the magnetic component. Therefore, the Lorentz factor and the magnetic field are not derived from one another. They arise in parallel from a single recursive geometry:
\[\tag{60} \boxed{ \begin{gathered} \GF=\{1,\beta,\beta^2,\ldots\}, \qquad \|\GF\|=\gamma,\\ \NF=\frac1\gamma\boldsymbol{\xi}_0+\beta S\NF,\\ \frac{E}{E_0}=\frac1\gamma, \qquad \frac{cB_{\mathrm{full}}}{E_0}=\beta. \end{gathered} } \] 15.
What is derived from geometry and what is introduced physicallyTo evaluate the result, it is necessary to distinguish between three levels of construction.
From the geometry of deep space following the introduction of its metric, orthogonality, self-similarity, and the transition coefficient \(\beta\) we obtain
\[\tag{61} \GF=\boldsymbol{\xi}_0+\beta S\GF, \qquad \|\GF\|=\gamma. \] From the first idempotent projection of the normalized state, we obtain
\[\tag{62} \PE\NF=\frac1\gamma\boldsymbol{\xi}_0, \qquad \PB\NF=\beta S\NF, \qquad \frac1{\gamma^2}+\beta^2=1. \] Additional physical mappings of the model include the correspondences
\[\tag{63} \frac{E}{E_0}=\frac1\gamma, \qquad \frac{cB_{\mathrm{full}}}{E_0}=\beta, \qquad \mathbf B=\frac1c\boldsymbol{\beta}\times\mathbf E_0. \] Adopting these mappings leads unambiguously to
\[\tag{64} E=\frac{E_0}{\gamma}, \qquad cB_{\mathrm{full}}=E_0\beta, \qquad E^2+c^2B_{\mathrm{full}}^2=E_0^2. \] This distinction is essential: idempotent algebra creates independent channels, the metric assigns norms to them, and the physical mapping determines which observable quantities correspond to these channels.
16. Limits of applicability
The result obtained is a fundamental geometric construction rather than a complete electrodynamic theory. Therefore, it is necessary to specify its limits.
First, the localized wave mode of a particle is considered, not free electromagnetic radiation. Second, the formula \(E=E_0/\gamma\) is a physical representation of this model and does not, in the general case, coincide with the full standard relativistic transformation of the field of a moving point charge, where the field strength depends also on the observation angle and the reference frame.
Third, the equality \(E^2+c^2B^2=E_0^2\) applies to the complete abstract splitting or to its transverse three-dimensional projection. For an arbitrary angle, the observed magnetic field contains an additional factor of \(\sin\theta\).
Fourth, the mere existence of idempotent projectors does not prove that the corresponding branches must manifest specifically as electric and magnetic fields. Such a correspondence is a physical hypothesis of the model. Its value lies in the fact that, upon projection into 3D, it reproduces the correct non-relativistic dependence \(\mathbf B\propto\mathbf v\times\mathbf E_0\) and links it to the independently derived geometric origin of the Lorentz factor.
Conclusions
In the proposed model, the electric and magnetic components are two observable manifestations of the first level of deep spatial splitting. After normalization of the self-similar state, its initial branch has an amplitude of \(1/\gamma\), while the complete orthogonal remainder has an amplitude of \(\beta\).
The electric component is associated with the initial branch
\[ E=\frac{E_0}{\gamma}, \] and the full scale of the magnetic component is associated with the entire deep remainder
\[ cB_{\mathrm{full}}=E_0\beta. \] Thus, the magnetic field corresponds not to a single additional channel, but to the aggregate of all subsequent splitting levels. The first term of this sequence has an amplitude of \(\beta/\gamma\); however, the norm of the entire infinite branch is equal to \(\beta\).
Prior to normalization, this same recursive series has a length of \(\gamma\).
Thus, the magnetic field and the Lorentz factor acquire a common geometric origin: the Lorentz factor characterizes the total metric depth of the state, while the magnetic field is the physical manifestation of its residual branch following the initial splitting.The final operator \(J(a,b)\) compresses this structure into two external projections:
\[\tag{65} \boxed{ e^{i\pi b}=\frac1\gamma+i\beta. } \] After the physical mapping, we obtain
\[\tag{66} \boxed{ E+i\,cB_{\mathrm{full}}=E_0e^{i\pi b}. } \] Thus, motion does not create a second, independent energetic entity. It opens up an orthogonal branch of the pre-existing spatial state. The underlying geometry determines the magnitude of this branch, while the mapping into three-dimensional space assigns it a direction
\[\tag{67} \boxed{ \mathbf B =\frac1c\boldsymbol{\beta}\times\mathbf E_0 =\frac1{c^2}\mathbf v\times\mathbf E_0. } \] In this sense, the magnetic field can be viewed as the first directly observable manifestation of the multi-level idempotent splitting of external space.