2026-08-16
Multilevel electron splitting
Origin of spin 1/2 and transition to photon
The conventional Wave Electricity operator splits the state into two mutually complementary idempotent components. This is sufficient to describe the fundamental internal phase and external motion, but insufficient to capture the entire multilevel structure of the electron. In particular, one revolution of the fundamental wave returns the final operator to the initial state, whereas a complete electron cycle must distinguish two successive traversals of the internal circuit.
To achieve this, a deep splitting is introduced. Its first level creates a two-sheet factor and a half-phase, and its continuation forms a recursive chain of orthogonal projections \(P_n\). After metric mapping, this chain forms a self-similar state \(\boldsymbol\Gamma_\beta\), the norm of which corresponds to the Lorentz factor. These constructions have a common algebraic foundation, but they perform different physical tasks.
A closed electron wave and a free propagating wave belong to different classes of intrinsic geometry. Their difference is characterized by an integer closure index. Therefore, an electron wave cannot spontaneously open, and a free wave cannot spontaneously close, as long as the evolution remains continuous and normalized.
\[\tag{1} \boxed{ J(a,b) \longrightarrow \{P_n\} \longrightarrow \begin{cases} Q_s(a),&\text{two-sheetedness and half-phase},\\ \boldsymbol\Gamma_\beta,&\text{metric of motion and }\gamma(\beta). \end{cases} } \] 1. Finite State Operator
The first level of the algebra is defined by two mutually complementary idempotents:
\[\tag{2} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] Their difference defines the hyperbolic unit \(\j=\ep-\em\), for which \(\j^2=1\). The finite two-parameter operator has the form
\[\tag{3} \boxed{ J(a,b)=\j^a(-\j)^b =\ep e^{i\pi b}+\em e^{i\pi a}. } \] Parameter \(a\) describes the internal periodicity, and parameter \(b\) describes the external motion of the center of state:
\[\tag{4} a=\varpi t, \qquad \pi\varpi=\omega_{\mathrm{int}}, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac vc. \] The operator is normalized:
\[\tag{5} J(a,b)\overline{J(a,b)}=1. \] This equality expresses the conservation of the complete final state during phase redistribution between orthogonal channels. It does not imply that the norm \(J\) is an energy, and does not determine the metric depth of the state.
2. Two Periodicities of the Electronic State
In the rest frame, \(b=0\). The fundamental internal phase repeats as \(a\to a+2\):
\[\tag{6} J(a+2,0)=J(a,0). \] However, the two-sheet model requires that after one rotation, the fundamental phase is already restored, while the number of the inner sheet is still different from the initial one. The complete state of the electron must satisfy the condition
\[\tag{7} \boxed{ J_e(a+2)\ne J_e(a), \qquad J_e(a+4)=J_e(a). } \] Thus, the fundamental frequency of the wave does not decrease. Only the recovery period of the entire internal configuration doubles. A single variable \(a\) can describe both periodicities by adding a factor whose phase changes twice as slowly as the fundamental phase.
3. The first step of recursive splitting
Let the first deep level correspond to an independent commuting hyperbolic unit \(\j_1\). It creates a pair of projectors.
\[\tag{8} p_1^+=\frac{1+\j_1}{2}, \qquad p_1^-=\frac{1-\j_1}{2}. \] We denote the first dedicated channel and the first residual branch as
\[\tag{9} P_0=p_1^-, \qquad R_1=p_1^+, \qquad 1=P_0+R_1, \qquad P_0R_1=0. \] The two-sheet factor is constructed on this first slice of the recursive structure:
\[\tag{10} \boxed{ Q_s^{(\sigma)}(a) =P_0+R_1e^{i\sigma\pi a/2}, \qquad \sigma=\pm1. } \] In the notation of the second idempotent pair, we can set
\[\tag{11} P_0\equiv\pmp, \qquad R_1\equiv\pme, \qquad \jp=\pmp-\pme. \] Then one orientation of the deep factor is written as
\[\tag{12} Q_s^{(+)}(a) =\jp^{a/2} =\pmp+\pme e^{i\pi a/2}, \] and the opposite is the conjugate and inverse:
\[\tag{13} Q_s^{(-)}(a) =\jp^{-a/2} =\pmp+\pme e^{-i\pi a/2}, \qquad Q_s^{(+)}Q_s^{(-)}=1. \] The mathematics of such a splitting is presented here.
4. General Recursive Chain of Projectors
The first binary slice need not be the last. Let each subsequent level correspond to an independent commuting unit \(\j_n\) and projectors.
\[\tag{14} p_n^+=\frac{1+\j_n}{2}, \qquad p_n^-=\frac{1-\j_n}{2}. \] In the recursive residual branch, channels are allocated
\[\tag{15} \boxed{ P_n =\left(\prod_{k=1}^{n}p_k^+\right)p_{n+1}^-, \qquad P_nP_m=\delta_{nm}P_n. } \] After \(N+1\) steps, the unit factorization is as
\[\tag{16} 1=\sum_{n=0}^{N}P_n+R_{N+1}, \qquad R_{N+1}=\prod_{k=1}^{N+1}p_k^+. \] The recursive continuation of the first remainder is as follows
\[\tag{17} R_1=P_1+R_2, \qquad R_2=P_2+R_3, \qquad\ldots \] Therefore, the pair \(\pmp,\pme\) is not a construct separate from the general concept. It represents the first physically isolated slice \(P_0+R_1\). Further expansion of the residual branch results in channels \(P_1,P_2,\ldots\).
Algebra determines the orthogonality of the channels and the preservation of the whole. The physical meaning of each \(P_n\)—an internal sheet, a spatial direction, or some other characteristic—appears only after additional mapping.
5. The Complete Electron Operator
A two-sheet multiplier must be distinguished from a finite dynamic operator and from a charge structure. The complete state can be conveniently written as a product of independent levels:
\[\tag{18} \boxed{ J_e^{(\sigma)}(a,b) =J_{\mathrm{dyn}}^{(\sigma)}(a,b) Q_s^{(\sigma)}(a) J_q\cdots. } \] The two orientations of the final dynamic state have the form
\[\tag{19} \begin{aligned} J_{\mathrm{dyn}}^{(+)}(a,b) &=\ep e^{i\pi b}+\em e^{i\pi a},\\ J_{\mathrm{dyn}}^{(-)}(a,b) &=\ep e^{i\pi a}+\em e^{i\pi b}. \end{aligned} \] In both cases, the physically labeled parameter \(b\) describes the same velocity of the center:
\[\tag{20} \mathcal P_{\mathrm{ext}}[J_{\mathrm{dyn}}^{(+)}] =\mathcal P_{\mathrm{ext}}[J_{\mathrm{dyn}}^{(-)}] =\beta. \] The factor \(Q_s\) creates a two-sheet period but does not determine the electric charge. The sign of the charge must belong to a separate operator \(J_q\). This allows the external motion, spin orientation, two-sheet closure, and charge state to be specified independently.
6. Geometric mapping of two sheets
The first deep slice can be mapped into two close branches of a single closed contour:
\[\tag{21} P_0\longleftrightarrow C_+, \qquad R_1\longleftrightarrow C_-, \qquad R_\pm=R\pm\frac{\Delta R}{2}. \] One wave passes through both branches sequentially. To the leading approximation, each branch has length \(2\pi R\), so the length of the complete two-sheet path is
\[\tag{22} L_1=L_2=2\pi R, \qquad \boxed{L_J=L_1+L_2=4\pi R.} \] The fundamental phase \(2\pi\) is distributed over the complete path, and therefore
\[\tag{23} k_J=\frac{2\pi}{L_J}=\frac1{2R}, \qquad \boxed{\chi=k_Js=\frac{\theta}{2}.} \] The two-turnover, not the radial distance between branches, creates a coefficient of \(1/2\). The magnitude of \(\Delta R\), the shape of the transitions between branches, and their precise physical arrangement are separate geometric hypotheses.
7. Two Sheets and Two Spin Orientations
The projectors \(P_0\) and \(R_1\) are not spin-up and spin-down states. They denote two consecutive sheets. Each of the two spin orientations uses the entire \(C_+\cup C_-\) contour, but has opposite directions of deep relative phase accumulation:
\[\tag{24} Q_s^{(\pm)}(\theta) =P_0+R_1e^{\pm i\theta/2}, \qquad \theta=\pi a. \] Therefore, after a rotation of \(2\pi\), the phase representative changes sign, and after \(4\pi\), it is restored:
\[\tag{25} \Psi(2\pi)=-\Psi(0), \qquad \Psi(4\pi)=\Psi(0). \] If the generator of rotation about the chosen axis is defined as
\[\tag{26} \widehat S_{\mathbf n} =i\hbar\frac{\partial}{\partial\theta}, \] then the action on the conjugate deep phases yields
\[\tag{27} \boxed{ \widehat S_{\mathbf n}e^{-i\theta/2} =+\frac{\hbar}{2}e^{-i\theta/2}, \qquad \widehat S_{\mathbf n}e^{+i\theta/2} =-\frac{\hbar}{2}e^{+i\theta/2}. } \] The deep factor algebra creates a half phase factor. The scale \(\hbar\), its relation to the measured angular momentum, and the probability rule are related to the physical mapping. A detailed spinor description is given in the article "Electron Spin as a Consequence of a Two-Sheet Cycle of a Closed Wave".
8. Metric mapping of a deep chain
To transition from algebraic projections to metric depth, we associate an orthonormal vector with each channel:
\[\tag{28} P_n\longleftrightarrow\boldsymbol\xi_n, \qquad \boldsymbol\xi_n\boldsymbol\cdot\boldsymbol\xi_m =\delta_{nm}. \] This is how a countable-dimensional space of internal channels arises.
\[\tag{29} \mathcal H_J =\operatorname{span} \{\boldsymbol\xi_0,\boldsymbol\xi_1, \boldsymbol\xi_2,\ldots\}. \] Vectors \(\boldsymbol\xi_n\) are not additional coordinates of the observed three-dimensional space. They serve as metric representatives of orthogonal levels of the internal state.
9. Self-similar state \(\boldsymbol\Gamma_\beta\)
Let the transition operator \(S\boldsymbol\xi_n=\boldsymbol\xi_{n+1}\) transition the state to the next deepest level. A self-similar state with a single transition coefficient \(\beta\) satisfies the equation
\[\tag{30} \boxed{ \boldsymbol\Gamma_\beta =\boldsymbol\xi_0 +\beta S\boldsymbol\Gamma_\beta. } \] When \(|\beta|<1\), it expands into a row
\[\tag{31} \boldsymbol\Gamma_\beta =\sum_{n=0}^{\infty}\beta^n\boldsymbol\xi_n. \] Due to the orthogonality of the channels
\[\tag{32} \|\boldsymbol\Gamma_\beta\|^2 =1+\beta^2\|\boldsymbol\Gamma_\beta\|^2, \] from where
\[\tag{33} \boxed{ \|\boldsymbol\Gamma_\beta\| =\frac1{\sqrt{1-\beta^2}} =\gamma(\beta). } \] The relationship with the external exponent of the finite operator is defined by the equalities
\[\tag{34} \beta=\Sin(\pi b), \qquad \Cos(\pi b)=\sqrt{1-\beta^2}, \qquad \gamma=\frac1{\Cos(\pi b)}. \] 10. Why are \(Q_s\) and \(\boldsymbol\Gamma_\beta\) different objects?
The two-sheet multiplier and the self-similar state are built on the same deep channel architecture, but they cannot be identified:
\[\tag{35} \boxed{ Q_s(a)\ne\boldsymbol\Gamma_\beta. } \] The \(Q_s\) multiplier uses the first slice \(P_0+R_1\), contains the phase \(e^{\pm i\theta/2}\), and describes a two-sheet periodicity. The state \(\boldsymbol\Gamma_\beta\) uses the entire recursive chain, contains metric weights \(\beta^n\), and describes the depth of the external motion.
Their norms also have different meanings:
\[\tag{36} Q_s\overline{Q_s}=1, \qquad J\overline J=1, \qquad \|\boldsymbol\Gamma_\beta\|=\gamma. \] The first two equalities refer to finite phase operators. The third uses the Euclidean metric of a countable set of deep channels. Therefore, the unit norm of \(J_e\) does not contradict the relativistic factor \(\gamma\).
11. Closure of a Localized Wave
The algebraic factor exhibits internal bivalent properties, but particle localization requires a geometric closure condition. For
\[\tag{37} V(t)=cJ(t) \] Introduce the path functional
\[\tag{38} \mathcal C(T) =\int_0^T V(t)\,dt. \] A closed state corresponds to
\[\tag{39} \boxed{ \mathcal C(T_e)=0, \qquad \text{electron}=\text{stable spatially closed wave}. } \] The internal spatial structure is closed, not the world line of the particle's center. Therefore, the electron can move in the laboratory frame while maintaining an internal closed cycle.
12. Topological index of internal closure
To distinguish a closed two-sheet mode from a free wave, consider the coefficients of the deep multiplier:
\[\tag{40} Q_s^{(\sigma)} =P_0q_0+R_1q_1, \qquad q_0=1, \qquad q_1=e^{i\sigma\chi}. \] Their relative phase is
\[\tag{41} q_{\mathrm{rel}} =\frac{q_1}{q_0} =e^{i\sigma\chi}. \] On a full two-turn \(\Gamma\) circuit, the phase \(\chi\) changes from zero to \(2\pi\). We define the integer closure index:
\[\tag{42} \boxed{ N_\Gamma[Q_s] =\frac1{2\pi i} \oint_\Gamma q_{\mathrm{rel}}^{-1} \,dq_{\mathrm{rel}} \in\mathbb Z. } \] For two directions of deep phase accumulation, we obtain
\[\tag{43} N_\Gamma[Q_s^{(+)}]=+1, \qquad N_\Gamma[Q_s^{(-)}]=-1. \] For a free state without relative deep turnover
\[\tag{44} Q_{\mathrm{free}}=1=P_0+R_1, \qquad q_{\mathrm{rel}}=1, \qquad N_\Gamma[Q_{\mathrm{free}}]=0. \] Therefore, the closed electron wave and the free wave belong to different topological sectors:
\[\tag{45} \boxed{ \text{closed loop: }N_\Gamma=\pm1, \qquad \text{free mode: }N_\Gamma=0. } \] 13. Closure Index Conservation Law
The integer index cannot change under continuous deformation of a non-zero relative phase. Normalization
\[\tag{46} Q_s\overline{Q_s}=1 \] ensures that the phase channel coefficients do not pass through zero. Therefore, the continuous normalized evolution of an isolated state ne cannot transform \(N_\Gamma=\pm1\) into \(N_\Gamma=0\).
\[\tag{47} \boxed{ \text{for an isolated continuous normalized wave} \qquad N_\Gamma=\operatorname{const}. } \] This rule prohibits two spontaneous processes:
\[\tag{48} \begin{aligned} &\text{closed electron wave} \not\longrightarrow \text{free wave},\\ &\text{free wave} \not\longrightarrow \text{a single closed electron state}. \end{aligned} \] The stability of the electron is not due to an external prohibition or to the fact that there is "insufficient energy" for opening. The reason lies in the impossibility of continuously changing the integer class of the internal closure while preserving the norm.
14. The Physical Meaning of the Inverse Operator
Algebraic Equality
\[\tag{49} Q_s^{(+)}Q_s^{(-)}=1 \] is not an equation for the spontaneous transformation of an electron into a photon. The inverse factor carries the opposite index:
\[\tag{50} N_\Gamma[Q_s^{-1}] =-N_\Gamma[Q_s]. \] For the product of nonzero phases, the indices are added:
\[\tag{51} N_\Gamma[Q_AQ_B] =N_\Gamma[Q_A]+N_\Gamma[Q_B]. \] Therefore, canceling the deep multiplier means compensating for two opposing closure classes:
\[\tag{52} \boxed{ N_\Gamma[Q_sQ_s^{-1}] =N_\Gamma[Q_s]+N_\Gamma[Q_s^{-1}] =0. } \] The notation \(J_eQ_s^{-1}=J_{\mathrm{free}}\) only shows the algebraic form of the result of such compensation. It doesn't explain where the inverse factor comes from. In an isolated electron, it cannot arise as a separate cause of its own disconnection; it must be introduced by another participant in the interaction.
15. Compensation for Closed Interactions
In an interacting system, the local index of an individual participant may change, but the total index of the complete state must be preserved:
\[\tag{53} \boxed{ N_\Gamma^{\mathrm{in}} =\sum_kN_\Gamma[Q_k^{\mathrm{in}}] =\sum_lN_\Gamma[Q_l^{\mathrm{out}}] =N_\Gamma^{\mathrm{out}}. } \] In the simplest compensated channel, two interacting closed states have opposite indices:
\[\tag{54} N_\Gamma[Q_A]=+1, \qquad N_\Gamma[Q_B]=-1, \qquad N_\Gamma[Q_AQ_B]=0. \] Such a complete state can already transform into a set of free waves with a zero total index. For an electron-positron pair, this can be written schematically:
\[\tag{55} \boxed{ J_{e^-}J_{e^+} \longrightarrow J_{\gamma_1}J_{\gamma_2}, \qquad N_\Gamma^{e^-e^+} =N_\Gamma^{\gamma_1\gamma_2}=0. } \] Here, the oppositeness of the indices refers to a specific interacting channel and should not be automatically identified with the sign of the electric charge. The charge is stored in an independent factor \(J_q\) and is compensated separately. Similarly, the total energy, momentum, and angular momentum are redistributed among all participants in the process.
Reverse closure is also only possible in a compensated process:
\[\tag{56} J_{\gamma_1}J_{\gamma_2} \longrightarrow J_{e^-}J_{e^+}, \qquad 0\longrightarrow(+1)+(-1)=0. \] Thus, the pairing of creation and annihilation acquires a geometric meaning: closures arise or disappear not separately, but as mutually compensated changes in the total operator state.
16. Energy and the Closure Index
Conservation of the index explains the stability of the closure class and does not require asserting that the electron does not open solely due to the energy prohibition. Energy relates to another invariant—the generator of time phase shifts:
\[\tag{57} \widehat E=i\hbar\frac{\partial}{\partial t}, \qquad E=\hbar\omega. \] Conservation of the index requires homogeneity of time and invariance of the complete dynamics with respect to time shifts. This does not follow only from the identity \(Q_s\overline{Q_s}=1\). Therefore, in the model, it is useful to distinguish:
\[\tag{58} \begin{aligned} N_\Gamma=\operatorname{const} &\quad\Longleftrightarrow\quad \text{preservation of the closure class},\\ E_{\mathrm{total}}=\operatorname{const} &\quad\Longleftrightarrow\quad \text{homogeneity of the time of the complete system}. \end{aligned} \] During physical interaction, all independent characteristics of the complete state must be simultaneously preserved. However, the prohibition of spontaneous opening within the present design is ensured primarily by the topological index, rather than by the pre-invoked law of conservation of energy.
17. Why isn't a photon obtained by substituting \(\beta=1\) into \(\boldsymbol\Gamma_\beta\)?
The self-similar state \(\boldsymbol\Gamma_\beta\) describes a deep metric of the motion of the formed massive particlepoints. As it approaches the speed of light, its norm diverges:
\[\tag{59} \|\boldsymbol\Gamma_\beta\|^2 =1+\beta^2+\beta^4+\cdots \longrightarrow\infty, \qquad |\beta|\to1. \] Therefore, a photon cannot be obtained by simply substituting \(\beta=1\) into a massive chain. The limit shows the applicability boundary of the \(\boldsymbol\Gamma_\beta\) state, not the photon production mechanism.
The free photon regime belongs to a sector without internal spatial closure:
\[\tag{60} N_\Gamma[J_\gamma]=0, \qquad \beta_\gamma=1, \qquad b_\gamma=\frac12. \] It arises after the rearrangement of the complete interacting state, and not as an infinitely accelerated electron. In this case, the fundamental wave phase can be preserved and continue in the free regime, but its deep topology and metric organization are already different.
18. Possible Physical Consequences of Deep Splitting
Mapping the first level into two close branches creates a basis for studying small corrections to the internal path length, magnetic moment, and external energy gradient. However, these quantities do not follow solely from the existence of the projectors \(P_0\) and \(R_1\).
For the anomalous magnetic moment, it is necessary to further substantiate the transition lengths between branches and the relationship between the change in the total path and the magnetic frequency. This is discussed in the article "Geometric Origin of the Electron's Anomalous Magnetic Moment".
The Coulomb law requires a separate mapping of the internal difference to the external potential energy \(U(R)\). The phase operator itself does not contain an external distance and cannot generate the dependence \(1/R\) without an additional rule. This problem is considered in the article "Geometric Origin of Electric Force".
19. What Follows from the Construction, and What Remains a Hypothesis
The following follow directly from algebra: orthogonality of the projectors \(P_n\); conservation of unity under recursive expansion; existence of a residual branch \(R_n\); unit norm of phase factors; invertibility of \(Q_s^{-1}\); additivity of phase indices under multiplication; difference in the periods of the fundamental and deep phases.
From the adopted metric mapping follow: orthonormal space \(\boldsymbol\xi_n\); Self-similar series \(\boldsymbol\Gamma_\beta\); its norm \(\gamma(\beta)\); the difference between the finite operator norm and the metric norm of the deep state.
Topological result: if the relative deep phase remains continuous and nonzero, its integer index does not change. This implies the stability of the closed and free classes under isolated normalized evolution.
The physical hypotheses remain: mapping the first splitting into specific radial branches; the value of \(\Delta R\); identification of the half-deep phase with the electron spin; the structure of the charge operator; the dynamics of the interaction that compensates the index; Precise transformation of internal energy and information into a photonic state.
Conclusion
Multilevel idempotent splitting links several levels of electron description into a single architecture. The final operator \(J(a,b)\) defines the internal phase and external motion. The first deep slice \(P_0+R_1\) forms a two-sheet factor \(Q_s\), creating a half-phase and a recovery period \(4\pi\). The continuation of the residual branch forms a chain \(P_n\), the metric mapping of which yields the state \(\boldsymbol\Gamma_\beta\) and the Lorentz factor.
\[\tag{61} \boxed{ 1=P_0+R_1 \longrightarrow \begin{cases} Q_s^{(\pm)}=P_0+R_1e^{\pm i\theta/2},\\ R_1=P_1+R_2\longrightarrow\{P_n\} \longrightarrow\boldsymbol\Gamma_\beta. \end{cases} } \] The two-sheet factor and \(\boldsymbol\Gamma_\beta\) use the same deep architecture, but are not interchangeable. The former describes the internal closure and spin periodicity, while the latter describes the metric depth of the external motion.
The main new result is the conservation law for the closure index:
\[\tag{62} \boxed{ N_\Gamma =\frac1{2\pi i} \oint_\Gamma q_{\mathrm{rel}}^{-1}dq_{\mathrm{rel}} =\operatorname{const}. } \] It explains why a normalized closed electron wave cannot spontaneously open, while a free wave cannot spontaneously close. The inverse operator does not destroy the closure without cause, but carries the opposite index. Therefore, the removal or formation of a closure is possible only within a complete interaction, where changes in indices mutually compensate.are:
\[\tag{63} \boxed{ \Delta N_\Gamma^{\mathrm{total}}=0, \qquad (+1)+(-1)\longleftrightarrow0. } \] Thus, the electron and free radiation are not two values of a single continuously variable parameter, but different topological regimes of a single wave. Their mutual transformation requires a restructuring of the entire joint state, and not a simple spontaneous removal of an internal factor.