2026-08-28
Temporal Field Projections
Origin of Electric, Magnetic, and Wave Fields
In the first part, the primary temporal field was introduced as the directed change of the global state \(J\) along time. Space was treated not as a pre-existing container, but as a system of independent directions arising from orthogonal splitting of this field. In this approach, a spatial coordinate is the accumulated value of the corresponding projection.
The next step is to determine how different organizations of the temporal field give rise to known physical fields. It is not sufficient to introduce the three-dimensional vectors \(\mathbf E\) and \(\mathbf B\) at once. The corresponding components must first be obtained in the geometry of the global operator \(J\), then mapped into space, and only afterward identified with observable quantities.
\[\tag{1}\boxed{\text{temporal field}\;\longrightarrow\;\text{charge difference}\;\longrightarrow\;\text{splitting of motion}\;\longrightarrow\;\gamma,\; E,\; B\;\longrightarrow\;\text{spatial field}}\] The main hypothesis of the present work is that electric, magnetic and wave fields are not independent primary entities. They arise as different spatial projections of one temporal field, and the type of projection is determined by the internal dynamics, number of leaves, external motion, orientation and closure of the wave structure.
The central principle of this part is as follows. External motion is not added to an already formed field as an independent property; it activates recursive splitting of the external-state channel. The total norm of this splitting gives the Lorentz factor \(\gamma\). The first division of the normalized state yields the initial branch \(1/\gamma\) and the complete orthogonal remainder with norm \(\beta=v/c\). For a two-leaf charge component, these branches manifest, after physical mapping, as the electric and magnetic components.
1. The initial temporal field
The global state of Wave Electricity is written as
\[\tag{2}J(a,b)=\j^a(-\j)^b,\] where \(a\) describes the internal state and \(b\) describes the external motion. For internal evolution
\[\tag{3}a=\varpi\tau,\qquad \omega=\pi\varpi\] The temporal field is determined by the derivative
\[\tag{4}\boxed{\mathcal T_J=\frac{dJ}{d\tau}}.\] At \(b=0\) and
\[\tag{5}J(\tau)=\j^{\varpi\tau}=\ep+\em e^{i\omega\tau}\] it appears
\[\tag{6}\boxed{\mathcal T_J=i\omega\em e^{i\omega\tau}}.\] This field is primary with respect to its spatial projections. Its magnitude is determined by the internal frequency, while its direction changes continuously in the phase plane:
\[\tag{7}|\mathcal T_J|=\omega,\qquad J\cdot\mathcal T_J=0.\] Further, the same field will be considered for different geometric features:
\[\tag{8}\boxed{\mathcal F_k=\Pi_k\!\left[\mathcal T_J\right],\qquad \Pi_k=\Pi_k(\dot a,N,b,\chi,\Omega,\dot b,\ldots)}\] Here \(N\) is the number of leaves, \(\chi\) is the degree of opening, \(\Omega\) is the orientation of the inner plane, and \(\dot b\) characterizes the change in external motion.
2. Mean and difference fields of a multi-leaf structure
Let the global structure contain \(N\) leaves. Each leaf state corresponds to a temporal field.
\[\tag{9}\mathcal T_{J,k}=\frac{dJ_k}{d\tau},\qquad k=1,2,\ldots,N.\] The average component is
\[\tag{10}\boxed{\overline{\mathcal T}_J=\frac1N\sum_{k=1}^{N}\mathcal T_{J,k}}.\] The deviation of each leaf from the average state sets the difference component:
\[\tag{11}\boxed{\delta\mathcal T_{J,k}=\mathcal T_{J,k}-\overline{\mathcal T}_J}.\] By definition, their amount is compensated:
\[\tag{12}\sum_{k=1}^{N}\delta\mathcal T_{J,k}=0.\] The mean part characterizes the overall internal motion of the structure and the energy scale associated with it. Difference parts characterize the differences of leaves. With full internal compensation, they do not change the total energy, but they can produce non-zero external projections.
\[\tag{13}\boxed{\overline{\mathcal T}_J\;\longrightarrow\;\text{total energy and motion},\qquad \delta\mathcal T_{J,k}\;\longrightarrow\;\text{fields of leaf differences}}\] 3. A closed single-leaf structure
Let us first consider the simplest case:
\[\tag{14}N=1,\qquad b=0,\qquad \chi=0.\] There is only one \(\mathcal T_J\) field, so the average field coincides with the full field, and there is no difference component:
\[\tag{15}\overline{\mathcal T}_J=\mathcal T_J,\qquad \delta\mathcal T_J=0.\] If the internal rotation is symmetrically closed, its spatial projections over the full period are compensated by:
\[\tag{16}\int_{\tau}^{\tau+T}\mathcal T_J(\tau')\,d\tau'=0.\] The local temporal field does not disappear. It specifies frequency and energy scale.
\[\tag{17}\mathscr E=\hbar\omega.\] Thus, a single-leaf closed structure at a fixed center has a temporal field, but does not have an external field derived from the difference of leaves.
Spatial projection is not necessarily a force field. Without leaf difference, external motion can manifest as an impulse of the structure itself, but not as an electrical or magnetic interaction.
4. External motion of a single-leaf structure
Let the center of the single-leaf structure begin to move:
\[\tag{18}b>0,\qquad b=\frac{\arcsin\beta}{\pi},\qquad \beta=\frac vc.\] Then
\[\tag{19}\Sin(\pi b)=\beta,\qquad \Cos(\pi b)=\sqrt{1-\beta^2}=\frac1\gamma.\] The accumulation of external spatial coordinates is determined by the projection of temporal motion:
\[\tag{20}\frac{dx}{dt}=c\Sin(\pi b)=v,\qquad x(t)-x(t_0)=\int_{t_0}^{t}v(t')\,dt'.\] The average component of the temporal field acquires a spatial manifestation associated with the movement of the center. After physical normalization, it corresponds to energy and momentum:
\[\tag{21}\mathscr E=\gamma mc^2,\qquad \mathbf p=\gamma m\mathbf v.\] But since \(N=1\), the difference is still missing. Therefore, external motion by itself does not create a charge field:
\[\tag{22}\boxed{N=1,\, b>0\quad\longrightarrow\quad\text{energy and momentum, but not the charge field}}\] 5. Two-leaf electron temporal field
For an electron, a single wave successively passes two close internal branches. In deep idempotent splitting, such a state can be represented by the operator.
\[\tag{23}J_e(\tau)=\j^a\ast\j_{\prp}^{a/2}.\] Denote the two leaf states by \(J_+\) and \(J_-\). They match the fields.
\[\tag{24}\mathcal T_+=\frac{dJ_+}{d\tau},\qquad \mathcal T_-=\frac{dJ_-}{d\tau}.\] Let us introduce the average and difference components:
\[\tag{25}\overline{\mathcal T}_J=\frac{\mathcal T_++\mathcal T_-}{2},\qquad \boxed{\delta\mathcal T_J=\frac{\mathcal T_+-\mathcal T_-}{2}}.\] The average field characterizes the overall internal dynamics and does not depend on which leaf the wave passes. The difference field exists only due to two-leaf structure. It is not yet a ready-made electric field in three-dimensional space, but serves as its geometric predecessor:
\[\tag{26}\boxed{\mathcal E_J\equiv\delta\mathcal T_J}.\] It is the \(\mathcal E_J\) that contains information about the difference between the two branches and the direction of their deep traversal. To obtain the observed field, it is necessary to determine how this difference is projected onto the external object.
6. Spatial projection of two leaves
The geometric derivation of the differential spatial projection is discussed in detail in “Geometric Origin of the Electric Force”. Here we reproduce only its principal steps and connect them with the temporal field.
Let the effective radii of the two branches be equal
\[\tag{27}r_+=r_0+\frac{\Delta r}{2},\qquad r_-=r_0-\frac{\Delta r}{2}.\] For an external point located at the distance of \(\ell\), two projection coefficients arise:
\[\tag{28}P_\pm(\ell)=\frac{r_\pm}{\sqrt{\ell^2+r_\pm^2}}.\] The difference between the two projections is
\[\tag{29}\boxed{\Delta P(\ell)=P_+(\ell)-P_-(\ell)}.\] In the far-field region,
\[\tag{30}\ell\gg r_+,r_-\] we obtain
\[\tag{31}\boxed{\Delta P(\ell)\approx\frac{\Delta r}{\ell}}.\] The temporal field specifies the directional energy scale of the \(E_*\), and the \(\Delta P\) specifies its uncompensated spatial part:
\[\tag{32}\Delta E_{\mathrm{proj}}(\ell)=E_*\Delta P(\ell).\] For a correlated state, the potential energy is taken to be
\[\tag{33}U(\ell)=-E_*\Delta P(\ell)\approx-E_*\frac{\Delta r}{\ell}.\] Its outer spatial gradient creates a force:
\[\tag{34}\boxed{F_\ell=-\frac{dU}{d\ell}\approx-E_*\frac{\Delta r}{\ell^2}}.\] Thus, there are two different mechanisms in succession: the internal leaf difference creates an uncompensated projection, and the derivative of this projection over the external distance creates a force field.
\[\tag{35}\boxed{\delta\mathcal T_J\;\longrightarrow\;\Delta P(\ell)\;\longrightarrow\;U(\ell)\;\longrightarrow\;-\frac{dU}{d\ell}}\] 7. From Difference Projection to Electric Field
For an electron, the observed energy scale is
\[\tag{36}E_*=m_ec^2.\] If the effective separation of branches is equal to the classical radius of the electron,
\[\tag{37}\Delta r=r_e,\] The coefficient of spatial force becomes
\[\tag{38}E_*\Delta r=m_ec^2r_e=\alpha_{\mathrm{fs}}\hbar c.\] From the definition of the fine structure constant:
\[\tag{39}\alpha_{\mathrm{fs}}\hbar c=\frac{e^2}{4\pi\varepsilon_0}.\] Therefore, the force magnitude in the far-field region coincides with the Coulomb force:
\[\tag{40}\boxed{|F_\ell|=\frac{e^2}{4\pi\varepsilon_0\ell^2}}.\] Only after obtaining the \(1/\ell^2\) dependence and the correct coefficient can the geometric component of \(\mathcal E_J\) be compared with the electric field:
\[\tag{41}\boxed{\mathcal E_J\overset{\Pi_{\mathrm E}}{\longrightarrow}\mathbf E}.\] The electric field is a spatial difference projection of a two-leaf temporal field. Two-leaf structure creates a source of difference, and dependence on external distance turns this difference into a field of interaction.
The \(\Delta r=r_e\) condition, as well as the agreed energy sign, remain the physical principles of the model. They have yet to be obtained directly from the algebra of the global operator.
8. Sign of the leaf traversal
Opposite directions of traversal through the deep splitting can be represented as
\[\tag{42}J_{e^-}=\j^a\ast\j_{\prp}^{a/2},\qquad J_{e^+}=\j^a\ast\j_{\prp}^{-a/2}.\] The mean component is preserved, and the difference changes orientation:
\[\tag{43}\overline{\mathcal T}_{e^-}=\overline{\mathcal T}_{e^+},\qquad \delta\mathcal T_{e^+}=-\delta\mathcal T_{e^-}.\] After spatial mapping, this should lead to opposite directions of the electric field:
\[\tag{44}\mathbf E_{e^+}=-\mathbf E_{e^-}.\] Thus, the equal masses of the electron and positron are associated with the mean part of the temporal field, whereas their opposite charges are associated with the orientation of its deep difference component. A rigorous derivation of charge sign from the conjugation rule for \(J\) remains a separate task.
9. Internal circulation and magnetic moment
The resting center of the electron has a \(b=0\), but the charge wave continues to circulate along an internal closed trajectory. Therefore, it is necessary to distinguish between the immobility of the center and the absence of internal motion:
\[\tag{45}b=0,\qquad \dot a\ne0.\] The oriented circulation of the difference field creates an internal moment component:
\[\tag{46}\boxed{\mathcal E_J+\text{internal circulation}\;\longrightarrow\;\mathcal M_J^{\mathrm{int}}}.\] After mapping into three-dimensional space, it manifests as a magnetic moment:
\[\tag{47}\mathcal M_J^{\mathrm{int}}\overset{\Pi_\mu}{\longrightarrow}\boldsymbol\mu.\] The geometry of the normal magnetic moment is discussed in “Electron Spin”, and the effect of additional transitions between the two branches is discussed in “The Anomalous Magnetic Moment of the Electron”.
It is important to distinguish the intrinsic magnetic moment from the magnetic field of a translationally moving charge:
\[\tag{48}\boxed{\boldsymbol\mu\;\longleftarrow\;\text{internal circulation},\qquad \mathbf B_{\mathrm{move}}\;\longleftarrow\;\text{external motion of the center}}\] 10. External motion of the two-leaf field
Now let's start moving the center of the two-leaf structure. The external operator does not act on the ready-made three-dimensional vector \(\mathbf E\), but on its internal source - the difference component \(\mathcal E_J=\delta\mathcal T_J\):
\[\tag{49}\mathcal E_J(0)\;\longrightarrow\;\mathcal E_J(0)(-\j)^b,\qquad \beta=\Sin(\pi b)=\frac vc.\] Here it is necessary to clarify what is being split. This is not a literal division of already existing physical space. Motion activates recursive splitting of the external-state channel of the charge structure. The resulting orthogonal channels are subsequently mapped to spatial directions and field components.
Introduce a deep state of motion
\[\tag{50}\boxed{\GF=\boldsymbol\xi_0+\beta S\GF=\boldsymbol\xi_0+\beta\boldsymbol\xi_1+\beta^2\boldsymbol\xi_2+\cdots}.\] Because the initial direction is orthogonal to the shifted copy of the entire remainder space, its norm satisfies
\[\tag{51}\boxed{\|\GF\|^2=1+\beta^2\|\GF\|^2,\qquad \|\GF\|=\frac1{\sqrt{1-\beta^2}}=\gamma}.\] Thus, the Lorentz factor arises automatically as the total metric depth of the recursive splitting of motion. This result is discussed in detail in “Geometric Origin of the Lorentz Factor and the Energy Invariant”.
11. First splitting and magnetic component
Normalize the deep state:
\[\tag{52}\boxed{\NF=\frac{\GF}{\gamma}=\frac1\gamma\boldsymbol\xi_0+\beta S\NF,\qquad \|\NF\|=1}.\] Introduce the projector \(\PE\), which selects the initial direction \(\boldsymbol\xi_0\), and the projector \(\PB=I-\PE\), which selects the complete deep orthogonal remainder:
\[\tag{53}\PE^2=\PE,\qquad \PB^2=\PB,\qquad \PE\PB=0,\qquad \PE+\PB=I.\] The norms of the two branches are
\[\tag{54}\boxed{\|\PE\NF\|=\frac1\gamma,\qquad \|\PB\NF\|=\beta}.\] Therefore, the first division of the normalized state retains the unit norm:
\[\tag{55}\boxed{\frac1{\gamma^2}+\beta^2=1}.\] Equations (50)–(55) constitute a geometric result. Connecting this result with a physical field requires a separate mapping. For a two-leaf charge component, the initial branch is mapped to the electric component, while the complete orthogonal remainder is mapped to the full magnetic component:
\[\tag{56}\boxed{\frac{E}{E_0}\;\longleftrightarrow\;\|\PE\NF\|,\qquad \frac{cB_{\mathrm{full}}}{E_0}\;\longleftrightarrow\;\|\PB\NF\|}.\] Hence,
\[\tag{57}\boxed{E=\frac{E_0}{\gamma},\qquad cB_{\mathrm{full}}=E_0\beta}.\] The magnetic component does not arise only after taking the curl of an already formed potential. It appears earlier, as the first orthogonal remainder in the splitting of the charge component’s motion. The Lorentz factor and the magnetic branch are not derived from one another: \(\gamma\) is the norm of the complete deep state, whereas \(\beta\) is the norm of its first orthogonal remainder.
\[\tag{58} \boxed{ \delta\mathcal T_J\;\longrightarrow\;\text{splitting of motion}\;\longrightarrow\;\begin{gathered}\|\GF\|=\gamma,\\ E/E_0=1/\gamma,\\ cB_{\mathrm{full}}/E_0=\beta. \end{gathered}}\] A detailed metric construction of this correspondence is given in “Geometric Origin of the Electric and Magnetic Fields”. Here we establish its place in the general hierarchy of the temporal field.
12. Three-dimensional direction and tensor representationing
Deep splitting determines the relative magnitude of the magnetic branch, but does not yet select its direction in three-dimensional space. Introduce the radial direction of the electric projection and the direction of motion:
\[\tag{59}\mathbf n_\ell=\frac{\mathbf r-\mathbf r_0}{|\mathbf r-\mathbf r_0|},\qquad \mathbf n_v=\frac{\mathbf v}{v},\qquad \mathbf E_0=E_0\mathbf n_\ell.\] The oriented plane formed by \(\mathbf v\) and \(\mathbf E_0\) determines the normal direction of the magnetic projection. Therefore, the three-dimensional mapping of the orthogonal branch has the form
\[\tag{60}\boxed{\mathbf B=\frac1c\boldsymbol\beta\times\mathbf E_0=\frac1{c^2}\mathbf v\times\mathbf E_0}.\] For a uniformly moving point charge in the far non-relativistic region:
\[\tag{61}\boxed{\mathbf B=\frac{\mu_0q}{4\pi}\frac{\mathbf v\times\mathbf n_\ell}{\ell^2}}.\] If \(\vartheta\) is the angle between \(\mathbf v\) and \(\mathbf E_0\), then the observed three-dimensional projection is
\[\tag{62}B(\vartheta)=\frac{E_0}{c}\beta\sin\vartheta.\] Therefore, it is necessary to distinguish between the full magnitude of the deep magnetic branch and its observed spatial projection:
\[\tag{63}\boxed{cB_{\mathrm{full}}=E_0\beta,\qquad cB(\vartheta)=E_0\beta\sin\vartheta}.\] After obtaining the geometric branches and their three-dimensional direction, the electrical and magnetic components can be combined into a standard field representation. Introduce the four-potential
\[\tag{64}A^\mu=\left(\frac{\Phi}{c},\mathbf A\right).\] Its antisymmetric gradient determines the electromagnetic field tensor:
\[\tag{65}\boxed{F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu}.\] Mixed temporal-spatial components correspond to an electric field, and space-spatial components correspond to a magnetic field:
\[\tag{66}F_{0i}\;\longleftrightarrow\;E_i,\qquad F_{ij}\;\longleftrightarrow\;B_k.\] In three-dimensional notation:
\[\tag{67}\mathbf E=-\nabla\Phi-\frac{\partial\mathbf A}{\partial t},\qquad \mathbf B=\nabla\times\mathbf A.\] In the present construction, the antisymmetric gradient is not the primary cause of the magnetic component. It presents the electrical and magnetic projections already obtained in standard differential form and describes their consistent variation in space and time.
Splitting gives the existence and full magnitude of the magnetic branch; mapping through \(\mathbf v\times\mathbf E_0\) determines its three-dimensional direction; the four-potential and the tensor \(F_{\mu\nu}\) provide the standard field representation of the result.
13. Closed and open structures
So far, we have considered a localized structure in which the internal wave returns to the center. Its closure can be characterized by the functional
\[\tag{68}\mathcal C(T)=\int_0^T e^{i\alpha(t)}\j^{\varpi t}\,dt.\] For a closed state
\[\tag{69}\boxed{\mathcal C(T)=0}.\] When closure is broken, an uncompensated displacement appears:
\[\tag{70}\boxed{\mathcal C(T)\ne0}.\] Therefore, closure determines not the presence of a temporal field, but the manner of its spatial manifestation:
\[\tag{71}\boxed{\text{closure}\;\longrightarrow\;\text{localized projection},\qquad \text{opening}\;\longrightarrow\;\text{spatial propagation}}\] The geometry by which closed motion unfolds into a propagating wave is discussed in “Geometric Origin of a Propagating Wave”.
14. Two leaves as two parts of the period
In a closed electron, a single wave successively passes two leaves:
\[\tag{72}J_+\;\longrightarrow\;J_-\;\longrightarrow\;J_+.\] After passing one leaf, the internal rotation is completed, but the full leaf state has not yet been restored. Complete closure occurs after both leaves:
\[\tag{73}T_{\mathrm w}=2T_{\mathrm{leaf}}.\] Upon opening, the leaves neither disappear nor become averaged. Their sequence unfolds along the direction of propagation:
\[\tag{74}\boxed{J_+\;\longrightarrow\;\text{first half of the period},\qquad J_-\;\longrightarrow\;\text{second half of the period}}\] For propagation at speed \(c\), the full period becomes a wavelength:
\[\tag{75}\boxed{\lambda=cT_{\mathrm w}=2cT_{\mathrm{leaf}}}.\] Thus, the internal discrete index of the leaf becomes a phase of the propagating wave:
\[\tag{76}\{+,-\}\;\longrightarrow\;\left\{0\le\varphi<\pi,\quad \pi\le\varphi<2\pi\right\}.\] 15. The electromagnetic wave
Introduce a coordinate along the direction of propagation:
\[\tag{77}\xi=\mathbf n\cdot\mathbf r-ct.\] The difference components become functions of \(\xi\):
\[\tag{78}\mathcal E_J=\mathcal E_J(\xi),\qquad \mathcal B_J=\mathcal B_J(\xi).\] The transition to the former second leaf corresponds to a change in orientation after half the period:
\[\tag{79}\mathcal E_J\!\left(\xi+\frac\lambda2\right)=-\mathcal E_J(\xi),\qquad \mathcal B_J\!\left(\xi+\frac\lambda2\right)=-\mathcal B_J(\xi).\] After a full period, the condition is repeated:
\[\tag{80}\mathcal E_J(\xi+\lambda)=\mathcal E_J(\xi),\qquad \mathcal B_J(\xi+\lambda)=\mathcal B_J(\xi).\] After mapping into three-dimensional space, the propagating components must satisfy
\[\tag{81}\mathbf E\perp\mathbf B,\qquad \mathbf E\perp\mathbf n,\qquad \mathbf B\perp\mathbf n\] and the relation for a free electromagnetic wave
\[\tag{82}\boxed{E=cB}.\] The direction of energy transfer is determined by the Poynting vector:
\[\tag{83}\mathbf S=\frac1{\mu_0}\mathbf E\times\mathbf B\parallel\mathbf n.\] Both fields change the sign at the same time after half the period, so the direction of the \(\mathbf S\) is maintained. The average electric field for a period is zero, but the average energy does not disappear.
\[\tag{84}\langle\mathbf E\rangle_T=0,\qquad \langle E^2\rangle_T\ne0.\] The electromagnetic wave is a spatially unfolded two-leaf projection of the temporal field. Two leaves of a closed structure become two consecutive halves of a full period.
16. Spin and polarization – an independent feature
The conversion of two leaves into two parts of the period should not be confused with the orientation of the inner plane. Conditions
\[\tag{85}J(a,0),\qquad J(0,a)\] may specify two spin orientations of a closed particle. Upon opening, the orientation of the plane should become the orientation of the transverse wave projection:
\[\tag{86}\boxed{\text{spin orientation}\;\longrightarrow\;\text{wave polarization}}\] Consequently, the two features perform different functions:
\[\tag{87}\boxed{\text{two-leaf structure}\;\longrightarrow\;\text{period structure},\qquad \text{plane orientation}\;\longrightarrow\;\text{polarization}}\] This transition is discussed in detail in “Transition of Electron Spin into Photon Polarization”.
17. Partial opening and acceleration
Between a fully closed particle and a free wave, a degree of opening can be formally introduced.
\[\tag{88}0\le\chi\le1.\] Then
\[\tag{89}\chi=0\;\longrightarrow\;\text{closed structure},\qquad 0<\chi<1\;\longrightarrow\;\text{transitional state},\qquad \chi=1\;\longrightarrow\;\text{free wave}.\] If the state parameters change, the total derivative acquires additional terms:
\[\tag{90}\boxed{\frac{dJ}{dt}=\dot a\frac{\partial J}{\partial a}+\dot b\frac{\partial J}{\partial b}+\dot\chi\frac{\partial J}{\partial\chi}+\ldots}.\] The first term describes internal evolution, the second describes a change in external orientation, and the third describes the transition between closed and open modes. At acceleration
\[\tag{91}\dot b\ne0\] A variable uncompensated component associated with radiation may occur. However, for physical identification it is necessary to deduce the power of radiation and its dependence on acceleration. Therefore, the relation of \(\dot b\ne0\) to radiation remains a hypothesis.
18. Deeper splits
Two-leaf structure is only the first non-trivial level. For the \(N>2\), several independent difference components arise:
\[\tag{92}\delta\mathcal T_{J,1},\quad\delta\mathcal T_{J,2},\quad\ldots,\quad\delta\mathcal T_{J,N-1}.\] Each of them can have its own spatial mapping rule:
\[\tag{93}\mathcal F_k=\Pi_k\!\left[\delta\mathcal T_{J,k}\right].\] When closed, such components can manifest as additional internal properties, and when opened, as a more complex multiphase or modulated wave structure. As long as geometry does not reproduce a particular observable law and its coefficient, such projections cannot be identified with known or new physical interactions.
19. General classification
The resulting system can be reduced to the following main modes.
Single-leaf structure, \(b=0\), closure.There is a temporal field associated with internal frequency and energy; there is no external difference projection.
Single-leaf structure, \(b>0\), closure.The spatial projection of the mean field manifests itself as the energy and momentum of the center.
Two-leaf structure, \(b=0\), closure.The difference of the leaves creates an electrical projection, and the internal circulation creates its own magnetic moment.
Two-leaf structure, \(b>0\), closure.The motion activates the recursive splitting of the charge component. The total depth of splitting gives the Lorentz factor, the original normalized branch retains the electrical manifestation, and the first orthogonal remainder becomes the magnetic branch of the field of the moving charge.
Two-leaf open structure.The two leaves become two parts of the period, and the electrical and magnetic components are transferred as an electromagnetic wave.
Changing external motion and partial opening.There is a possible radiation regime that requires a separate quantitative inference.
Multi-leaf structure.Additional difference fields appear, the physical meaning of which must be determined only after the observable laws are obtained.
\[\tag{94} \boxed{ \begin{aligned}\overline{\mathcal T}_J&\;\longrightarrow\;\text{energy and momentum},\\ \delta\mathcal T_J&\;\longrightarrow\;\text{electric projection},\\ \delta\mathcal T_J,\, b>0&\;\longrightarrow\;\gamma,\, E,\, B_{\mathrm{full}},\\ \delta\mathcal T_J,\, \chi>0&\;\longrightarrow\;\text{electromagnetic wave}. \end{aligned}}\] 20. What has been obtained and what remains hypothetical
From the geometry of two close branches, the difference of the coefficients \(\Delta P(\ell)\) is directly obtained. In the far region, it has a dependence of \(1/\ell\), and the outer gradient of the corresponding potential energy is the dependence of \(1/\ell^2\). Under the condition \(\Delta r=r_e\) coefficient coincides with Coulomb.
From recursive splitting of motion at \(b>0\), after the introduction of orthogonal metrics and self-similarity, a deep state of \(\GF\) with a norm of \(\gamma\) is obtained. Its first normalized division has branches with norms \(1/\gamma\) and \(\beta\). Their comparison with the electrical and magnetic components is a physical representation of the model, not a consequence of idempotence alone.
In the far-field nonrelativistic regime, the three-dimensional mapping of the magnetic branch leads to the relation \(\mathbf B=\mathbf v\times\mathbf E_0/c^2\). The four-potential and the antisymmetric gradient are then introduced as the standard tensor representation of the resulting field components, rather than as the primary cause of the magnetic channel.
The following points remain hypotheses:
— rigorous derivation of charge sign from the direction of deep traversal;
— rigorous derivation of the physical mapping \(E/E_0=1/\gamma\) and \(cB_{\mathrm{full}}/E_0=\beta\) directly from the operator \((-\j)^b\);
— consistency of the field mapping at relativistic speeds and arbitrary observation angles;
— a quantitative relation between partial opening and radiated power;
— derivation of Maxwell’s equations from the geometry of \(J\);
— the physical meaning of deeper splitting levels.
— rigorous derivation of the physical mapping \(E/E_0=1/\gamma\) and \(cB_{\mathrm{full}}/E_0=\beta\) directly from the operator \((-\j)^b\);
— consistency of the field mapping at relativistic speeds and arbitrary observation angles;
— a quantitative relation between partial opening and radiated power;
— derivation of Maxwell’s equations from the geometry of \(J\);
— the physical meaning of deeper splitting levels.
This distinction is necessary: geometric classification specifies a possible mechanism for the origin of fields, but each physical identification requires the reproduction of the form of law, dimensionality, coefficient, and observable orientation.
Conclusion
The temporal field \(\mathcal T_J=dJ/d\tau\) is the common source of all spatial manifestations considered. For a single-leaf structure, its mean part sets the internal energy, and for external motion, the energy and momentum of the center.
The two-leaf structure creates a difference temporal field. The difference between the spatial projections of the two branches produces a \(1/\ell\) potential and a \(1/\ell^2\) field, which, with the required geometric normalization, coincides with the electric field. When the center moves, the external-state channel of this charge component splits. The total depth of the splitting has norm \(\gamma\), while its first orthogonal remainder, with norm \(\beta\), manifests after physical and three-dimensional mapping as the magnetic field.
Upon opening, the two-leaf structure does not disappear; it unfolds into the period of a free wave. The two leaves become two successive halves of the period, the orientation of the internal plane becomes polarization, and the related electric and magnetic projections carry energy through space.
\[\tag{95} \boxed{ \text{temporal field}\;\longrightarrow\;\text{leaf difference}\;\longrightarrow\;\text{splitting of motion}\;\longrightarrow\;\gamma,\, E,\, B\;\longrightarrow\;\text{wave unfolding} }\] Thus, the mass, the electric field, the magnetic moment, the Lorentz factor, the magnetic field of motion and the electromagnetic wave do not form a set of independent constructions, but successive manifestations of one geometry. Internal dynamics creates a temporal field, two-leaf structure creates a charge difference, external motion creates its recursive splitting, and opening creates the transfer of bound field components in the form of a free wave.