2026-08-16
Multilevel idempotent electron splitting
Origin of spin 1/2 and transition to photon
Multilevel Idempotent Splitting of the Electron
In the conventional wave operator notation, one pair of complementary idempotents splits the state into two orthogonal phase components. This is sufficient to define the fundamental wave cycle, but insufficient to describe the electron, whose full internal state must be restored after two successive revolutions, rather than one.
Previously, the two-turn splitting was introduced primarily geometrically: it was assumed that the wave sequentially passes through two close sheets or two split orbits. In this paper, this geometry is given an independent algebraic notation. For this purpose, a second idempotent level is introduced, embedded within the general state operator. The mathematics of this splitting is presented here.
The main result of the paper is that an electron can be represented as the product of a conventional wave operator and an additional two-sheet factor. The fundamental frequency of the wave does not decrease, but the recovery period for the full state doubles.
\[\tag{1} \boxed{ J_e(x)=\j^{a}\jp^{a/2}. } \] When the electron transitions to a free wave, the additional internal factor cancels out along with its inverse operator. As a result, the original wave phase remains:
\[\tag{2} \boxed{ J_\gamma(x) =J_e(x)\jp^{-a/2} =\j^{a}. } \] 1. First level of idempotent splitting
The original algebra is built on two mutually complementary idempotents \(\ep\) and \(\em\):
\[\tag{3} \ep^{2}=\ep, \qquad \em^{2}=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] Their difference defines the hyperbolic unit:
\[\tag{4} \boxed{ \j=\ep-\em, \qquad \j^{2}=1. } \] After complex expansion, any real power \(\j\) is decomposed into two idempotent planes:
\[\tag{5} \boxed{ \j^{a} =\ep+\em e^{i\pi x}. } \] The plane \(\ep\) in this notation remains unchanged, and the complex phase \(e^{i\pi x}\) develops in the plane \(\em\). If the parameter \(x\) varies from zero to two, the phase angle varies from zero to \(2\pi\), that is, the wave makes one complete revolution:
\[\tag{6} x:0\longrightarrow2 \qquad\Longleftrightarrow\qquad \pi x:0\longrightarrow2\pi. \] Therefore, the normal operator has a period of two in the parameter \(x\):
\[\tag{7} \j^{x+2} =\ep+\em e^{i\pi(x+2)} =\ep+\em e^{i\pi x} =\j^{a}. \] This is the normal period of a free wave. But if the same operator without additional structure is used to describe an electron, the full state will also return after one rotation. Such a notation does not contain a separate feature that allows one to distinguish the first and second sheets of the inner contour.
2. Why a deeper level is required
In a two-sheet model, the spatial position of the wave can be repeated after the first rotation, but the inner sheet does not necessarily coincide with the initial one. After the first rotation, the wave must find itself in a conjugate internal state, and after the second, it must return to the original one.
\[\tag{8} \boxed{ J_e(x+2)\ne J_e(x), \qquad J_e(x+4)=J_e(x). } \] It is important that this requirement does not imply a reduction in the frequency of the wave itself. The phase \(e^{i\pi x}\) must still complete a full cycle at \(x:0\to2\). Only the recovery period of the entire internal configuration doubles. Therefore, the operator must simultaneously store two different periodicities:
\[\tag{9} \boxed{ \begin{aligned} \text{fundamental wave phase:}&\quad x\sim x+2,\ \text{complete electron state:}&\quad x\sim x+4. \end{aligned} } \] A single variable can describe both periodicities by adding an independent internal factor to the basic operator, whose phase changes twice as slowly.
3. The Second Pair of Idempotents
We introduce a second level of splitting, formed by the idempotents \(\pmp\) and \(\pme\):
\[\tag{10} \pmp^{2}=\pmp, \qquad \pme^{2}=\pme, \qquad \pmp\pme=0, \qquad \pmp+\pme=1. \] This pair is not a repeat of \(\ep,\em\). The first pair specifies the initial splitting of the full state, and the second describes additional internal structure that may be present within each first-level state. Therefore, it is convenient to assume that elements of two levels commute:
\[\tag{11} \ep\pmp=\pmp\ep, \qquad \ep\pme=\pme\ep, \qquad \em\pmp=\pmp\em, \qquad \em\pme=\pme\em. \] The combined action of the two splittings creates four deep components:
\[\tag{12} \boxed{ \ep\pmp, \qquad \ep\pme, \qquad \em\pmp, \qquad \em\pme. } \] These are not four independent particles or four independent waves. They are four joint projections of a single state onto two successive idempotent levels. Just as two coordinates can define four combinations of basic directions, two pairs of idempotents create four joint sectors.
4. Intrinsic Hyperbolic Unit
For the second pair, we define its own hyperbolic unit:
\[\tag{13} \boxed{ \jp =\pmp-\pme, \qquad \jp^{2}=1. } \] Its power is constructed according to exactly the same rule as the power of the original unit \(\j\):
\[\tag{14} \boxed{ \jp^{y} =\pmp+\pme e^{i\pi y}. } \] For a two-turn structure, the exponent \(y=x/2\) is chosen:
\[\tag{15} \boxed{ \jp^{a/2} =\pmp+\pme e^{i\pi x/2}. } \] As \(x\) changes from zero to two, the fundamental phase \(e^{i\pi x}\) completes a full rotation, while the internal phase \(e^{i\pi x/2}\) completes only half of its cycle. One more external rotation completes the internal cycle.
The exponent \(x/2\) does not replace the fundamental phase of the electron. It refers only to the additional leaf factor. Therefore, the fundamental wave frequency remains the same.
5. The Complete Electron Operator
Now the electron state can be defined as the product of two operators:
\[\tag{16} \boxed{ J_e(x) =\j^{a}\jp^{a/2}. } \] The first factor describes the fundamental wave motion. The second stores the number of the inner sheet and doubles the period of recovery of the complete configuration. Substituting the explicit forms of both factors, we obtain:
\[\tag{17} J_e(x) =\left(\ep+\em e^{i\pi x}\right) \left(\pmp+\pme e^{i\pi x/2}\right). \] Expanding the product reveals all four joint components:
\[\tag{18} \boxed{ \begin{aligned} J_e(x)={}& \ep\pmp +\ep\pme e^{i\pi x/2}\\ &+\em\pmp e^{i\pi x} +\em\pme e^{i3\pi x/2}. \end{aligned} } \] In the last sector, the phases of the two levels add up: \(\pi x+\pi x/2=3\pi x/2\). This does not imply the appearance of a third independent frequency. We are talking about the joint phase of the component that simultaneously belongs to the \(\em\) and \(\pme\) planes.
For a moving electron, this same principle can be attached to the general operator \(J(a,b)\), where \(a\) describes the internal state, and \(b\) the external motion:
\[\tag{19} \boxed{ J_e(a,b) =\left( \ep e^{i\pi b} +\em e^{i\pi a} \right) \jp^{a/2}. } \] At rest, \(b=0\), and the notation \(x=a\) returns formula (16). Thus, deep splitting refers to the particle's internal parameter and does not replace its external velocity.
6. Norm Preservation
Both factors have unit norm. For the internal operator, the conjugate power is
\[\tag{20} \jp^{-a/2} =\pmp+\pme e^{-i\pi x/2}. \] Thanks to idempotency and the disappearance of mixed products:
\[\tag{21} \begin{aligned} \jp^{a/2}\jp^{-a/2} &=\left(\pmp+\pme e^{i\pi x/2}\right) \left(\pmp+\pme e^{-i\pi x/2}\right)\ &=\pmp+\pme=1. \end{aligned} \] Similar to \(\j^{a}\j^{-x}=1\). Therefore, the total electron operator is also normalized:
\[\tag{22} \boxed{ J_e(x)J_e^{-1}(x)=1. } \] Deep splitting does not add a new total norm to the electron and does not double its energy. It redistributes the unit state between additional orthogonal components.
7. Algebraic Proof of Two-Turn Transformation
Consider several successive values of the parameter. At the starting point:
\[\tag{23} \j^{0}=1, \qquad \jp^{0}=1, \qquad J_e(0)=1. \] After one full rotation of the main wave, the parameter becomes equal to \(x=2\). The first factor has already returned to its initial state:
\[\tag{24} \j^{2} =\ep+\em e^{i2\pi} =1. \] But the inner factor is only in the middle of its cycle:
\[\tag{25} \jp^{1} =\pmp+\pme e^{i\pi} =\pmp-\pme =\jp. \] Therefore, after the first rotation, the full state is
\[\tag{26} \boxed{ J_e(2)=\jp\ne J_e(0). } \] After the second rotation (x=4). Now both factors return to one simultaneously:
\[\tag{27} \j^{4}=1, \qquad \jp^{2}=1, \qquad \boxed{J_e(4)=J_e(0)}. \] Thus, the sequence of states is
\[\tag{28} \boxed{ 1\xrightarrow{\;2\pi\;} \jp \xrightarrow{\;2\pi\;} 1. } \] The spatial phase repeats after each revolution, but an additional idempotent index distinguishes two successive passes. This is the algebraic analog of a two-sheeted circle covering.
8. Wave frequency and state period are not the same thing
The presence of the exponent \(x/2\) may create the impression that the electron frequency has halved. This would be true if \(\jp^{a/2}\) replaced the fundamental operator. But in formula (16), it does not replace, but complements \(\j^{a}\).
The fundamental wave phase remains equal to
\[\tag{29} \phi(x)=\pi x. \] If \(x=x(t)\), the instantaneous angular frequency of the fundamental wave is
\[\tag{30} \boxed{ \omega =\frac{d\phi}{dt} =\pi\frac{dx}{dt}. } \] The deep phase changes as \(\phi_{\pmp}=\pi x/2\). It determines not the frequency of the wave's passage through the main orbit, but the rate at which the internal sheet state changes:
\[\tag{31} \omega_{\pmp} =\frac{d\phi_{\pmp}}{dt} =\frac{\omega}{2}. \] Therefore, the wave continues to make its usual full rotation with a frequency of \(\omega\), but the entire system requires two such rotations to restore the internal phase. This difference can be expressed briefly:
\[\tag{32} \boxed{ \text{wave motion frequency} \ne \text{full electron state repetition frequency}. } \] 9. The geometric meaning of deep planes
Algebra itself defines two internal sheets, but does not yet determine their specific form in physical space. For a geometric mapping, we can associate two close branches of a single closed contour with the idempotents \(\pmp\) and \(\pme\):
\[\tag{33} \pmp\longmapsto C_{+}, \qquad \pme\longmapsto C_{-}. \] In the radial representation, these branches correspond
\[\tag{34} R_{+}=R+\frac{\Delta r}{2}, \qquad R_{-}=R-\frac{\Delta r}{2}. \] The first rotation transforms the complete operator from the state \(1\) to the state \(\jp=\pmp-\pme\). The second rotation returns the relative phase of the internal branches to the original. Therefore, the geometric statement about successive traversal of two sheets now has an algebraic support.
It is necessary to distinguish between idempotent phase planes and their physical representation. The equalities for \(\pmp,\pme\) are algebraic. The specific radii \(R_{+},R_{-}\), the distance between branches, and the shape of the transitions constitute an additional physical model.
10. Two-sheetedness and two spin orientations
The deep planes \(\pmp\) and \(\pme\) should not be directly referred to as "spin-up" and "spin-down" states. They represent the two sheets of the complete inner loop and the state recovery after two rotations.
The two observed spin orientations correspond to two modes of the complete operator:
\[\tag{35} \boxed{ J(a,0) \longleftrightarrow J(0,a). } \] Now the relationship between the deep factor and spin can be demonstrated directly. We denote the physical rotation angle by \(\theta\). Since the fundamental phase is \(\pi x\), we have
\[\tag{36} \boxed{ \theta=\pi x. } \] Then the two opposite orientations of the deep cycle are described by conjugate powers of the inner operator:
\[\tag{37} \boxed{ \jp^{\pm x/2} =\pmp+\pme e^{\pm i\theta/2}. } \] The idempotents \(\pmp\) and \(\pme\) still denote the two inner sheets. The spin information is carried not by each idempotent separately, but by the relative phase between them. For two opposing modes, this phase is equal to
\[\tag{38} \chi_{\uparrow}(\theta)=e^{-i\theta/2}, \qquad \chi_{\downarrow}(\theta)=e^{+i\theta/2}. \] Thus, the exponent \(x/2\) acquires a precise physical meaning: when the total state is rotated by an angle \(\theta\), the deep relative phase changes by an angle \(\theta/2\). We define the generator of rotation around the selected axis \(\mathbf n\) by the phase operator
\[\tag{39} \boxed{ \widehat S_{\mathbf n} =i\hbar\frac{\partial}{\partial\theta}. } \] Its action on two conjugate deep phases yields
\[\tag{40} \boxed{ \begin{aligned} \widehat S_{\mathbf n}\chi_{\uparrow} &=+\frac{\hbar}{2}\chi_{\uparrow},\\ \widehat S_{\mathbf n}\chi_{\downarrow} &=-\frac{\hbar}{2}\chi_{\downarrow}. \end{aligned} } \] Therefore, the spin projection modulus \(\hbar/2\) arises directly from the derivative of the deep phase \(\theta/2\), and its sign is determined by the accumulation orientation of this phase. The complete electron modes can be written as
\[\tag{41} \boxed{ \begin{aligned} J_{e,\uparrow} &=J(a,0)\jp^{-a/2} \quad\longleftrightarrow\quad S_{\mathbf n}=+\frac{\hbar}{2},\\ J_{e,\downarrow} &=J(0,a)\jp^{+x/2} \quad\longleftrightarrow\quad S_{\mathbf n}=-\frac{\hbar}{2}. \end{aligned} } \] The formula \(J_e=\j^{a}\jp^{a/2}\), used above, corresponds to one chosen orientation of the deep phase walk. The conjugate power \(\jp^{-a/2}\) describes the opposite orientation. In this case, the sheets \(\pmp,\pme\) themselves do not become "up" and "down" states: both orientations use the entire two-sheet structure.
After a rotation by \(2\pi\), the relative phase changes sign, and after a rotation by \(4\pi\), it is restored:
\[ \chi_{\uparrow,\downarrow}(\theta+2\pi) =-\chi_{\uparrow,\downarrow}(\theta), \qquad \chi_{\uparrow,\downarrow}(\theta+4\pi) =\chi_{\uparrow,\downarrow}(\theta). \] Thus, \(\jp^{a/2}\) fulfills two related purposes at once. Its period explains the two-turn nature of the complete electron cycle, and the relative phase contained within it, \(e^{\pm i\theta/2}\), yields the spin eigenvalues \(\pm\hbar/2\). The choice between \(J(a,0)\) and \(J(0,a)\) determines the orientation of the complete state, while the deep factor creates the one-half coefficient itself.
11. Electron Transition to a Free Wave
A free wave, in particular the photonic regime of the model, should not retain the additional internal closure of the electron. Therefore, when opening, it is necessary to remove the factor \(\jp^{a/2}\), without changing the fundamental phase \(\j^{a}\).
The inverse of the inner operator is
\[\tag{42} \jp^{-a/2} =\pmp+\pme e^{-i\pi x/2}. \] Multiplying the electron state by this operator yields
\[\tag{43} \begin{aligned} J_e(x)\jp^{-a/2} &=\j^{a}\jp^{a/2}\jp^{-a/2}\ &=\j^{a}. \end{aligned} \] Therefore, the transition is written without taking the modulus, without squaring the entire state, and without reducing the frequency:
\[\tag{44} \boxed{ J_e(x) \xrightarrow{\;\times\jp^{-a/2}\;} J_\gamma(x)=\j^{a}. } \] Physically, this operation can be understood as removing an additional internal short circuit. The electronic state contains the fundamental wave and its deep leaf cycle. After removing the leaf factor, the fundamental wave does not disappear: it is freed from the two-turn internal circuit.
12. Possible connection with the anomalous magnetic moment
The new operator strictly demonstrates the existence of two internally distinguishable passages. However, the numerical value of the anomalous magnetic moment requires an additional mapping of this algebra to the physical path length.
If two deep planes are assigned close branches \(R_{+}\) and \(R_{-}\), and the total length of the two transitions between them is taken to be \(2r_e\), the total two-sheet path will be
\[\tag{45} L_{\Gamma}=4\pi R+2r_e. \] Compared to an ideal two-turn circuit \(4\pi R\), the relative addition is
\[\tag{46} \frac{\Delta L}{4\pi R} =\frac{2r_e}{4\pi R} =\frac{r_e}{2\pi R}. \] If the ratio of the internal scales is equal to the fine structure constant, \(r_e/R=\alpha_{\mathrm{fs}}\), the leading anomalous correction is obtained.
\[\tag{47} \boxed{ a_e^{(1)} =\frac{\alpha_{\mathrm{fs}}}{2\pi}. } \] Here, it is necessary to clearly distinguish between the two results. The operator \(\jp^{a/2}\) algebraically justifies the bivalent property and two successive passages. However, the equality of the additional length, namely \(2r_e\), is a physical mapping rule and requires an independent geometric justification. Moreover, formula (41) corresponds only to the leading correction and does not derive the higher terms of the anomalous magnetic moment.
13. Possible Connection with the Coulomb Gradient
Deep splitting also provides a natural internal difference between the conjugate planes \(\pmp\) and \(\pme\). However, the operator \(J_e(x)\) depends on the phase parameter \(x\) and does not itself contain the external distance \(R\). Therefore, Coulomb's law cannot be obtained by differentiating the phase operator alone.
An additional law is needed that relates the internal difference to its extension into outer space. If the residual potential energy at a distance \(R\) is of the form
\[\tag{48} U(R) =-\frac{m_ec^{2}r_e}{R}, \] then its gradient is
\[\tag{49} F_R =-\frac{dU}{dR} =-\frac{m_ec^{2}r_e}{R^{2}}. \] With the ratio adopted in the model
\[\tag{50} m_ec^{2}r_e =\alpha_{\mathrm{fs}}\hbar c =\frac{e^{2}}{4\pi\varepsilon_{0}} \] the Coulomb coefficient is obtained:
\[\tag{51} \boxed{ F_R =-\frac{\alpha_{\mathrm{fs}}\hbar c}{R^{2}} =-\frac{e^{2}}{4\pi\varepsilon_{0}R^{2}}. } \] For the field strength of one elementary charge:
\[\tag{52} \boxed{ E(R) =\frac{\alpha_{\mathrm{fs}}\hbar c}{eR^{2}} =\frac{e}{4\pi\varepsilon_{0}R^{2}}. } \] The new algebra makes a common source of internal splitting, anomalous path length, and external energy gradient plausible. However, the dependence \(U(R)\sim1/R\) remains a separate external continuation law. It cannot be declared a pure consequence of formula (16) until an explicit mapping of the intrinsic geometry to the distance \(R\) is constructed.
14. The Possibility of Further Splitting
The second idempotent level need not be the last. Formally, we can introduce the following pair of complementary idempotents and the corresponding hyperbolic unit. Then the complete operator will acquire another factor with a longer period.
\[\tag{53} J^{(n)}(x) =\j^{a} \prod_{k=1}^{n} \j_{\pmp_k}^{\,x/2^{k}}. \] This notation defines a hierarchy of nested cycles. Each subsequent level doubles the recovery period of the corresponding internal structure without changing the phase of the first factor \(\j^{a}\). However, the existence of deeper physical levels does not follow from a single mathematical possibility and must be confirmed by individual observable effects.
15. What follows from algebra, and what remains a hypothesis
The following follow directly from the construction: the existence of a consistent second idempotent level; the unit norm of the internal factor; conservation of the fundamental phase \(e^{i\pi x}\); the relative deep phase \(e^{\pm i\theta/2}\); the projections of its generator \(S_{\mathbf n}=\pm\hbar/2\); the difference in states after the first revolution; recovery after the second revolution; Exact cancellation of the internal factor with an inverse operator.
The physical mappings remain: the mapping \(\pmp,\pme\) to two specific orbits; the distance between branches; the length of connecting transitions; the relationship of internal splitting to the magnetic moment; the law of continuation of the internal drop into external potential energy.
The following have not yet been obtained from the new operator alone: all higher corrections to the anomalous magnetic moment; independent derivation of the magnitude of the electric charge; the spatial law \(1/R\) without an additional mapping rule; the probabilities of a single quantum measurement.
This separation is necessary for the rigor of the model. Algebra shows what internal structure is possible and how it is transformed. Physics must separately establish which measurable lengths, energies, and interactions this structure corresponds to.
Conclusion
Deep idempotent splitting allows the two-turn electron structure to be written directly within the state operator. The fundamental wave factor \(\j^{a}\) preserves the original frequency and returns after each complete revolution. The additional factor \(\jp^{a/2}\) distinguishes between two successive passes and returns only after the second revolution.
This same factor directly creates the spin 1/2. For \(\theta=\pi x\), the relative phase of its deep planes is \(e^{\pm i\theta/2}\), so the rotation generator yields two projections \(S_{\mathbf n}=\pm\hbar/2\). Thus, the one-half coefficient and the two-turn property are two manifestations of the same deep phase:
\[ \jp^{\pm x/2} =\pmp+\pme e^{\pm i\theta/2} \quad\Longrightarrow\quad S_{\mathbf n}=\pm\frac{\hbar}{2}. \] \[\tag{54} \boxed{ J_e(a) =\j^{a}\jp^{a/2}, \qquad J_e(x+2)\ne J_e(a), \qquad J_e(x+4)=J_e(a). } \] The electron in this notation differs from the free wave not by its reduced frequency, but by the presence of an additional internal cycle. When open, this cycle is eliminated by multiplying by the inverse operator:
\[\tag{55} \boxed{ J_e(a)\jp^{-a/2} =\j^{a} =J_\gamma(a). } \] Thus, the electron and photon turn out to be two modes of the same fundamental wave. The electron contains an additional two-sheet closure, while the photon retains its original phase without this internal factor. The new form also creates a unified algebraic source for subsequent study of the anomalous magnetic moment and the Coulomb gradient, but their quantitative derivation requires an explicit law for the physical mapping of deep planes to length and external energy.