2026-08-18
Two-periodic mode of multilevel idempotent splitting
How to split an electron
In the first part of the work, an extended algebra with two independent commuting hyperbolic units was constructed. Each unit forms its own pair of complementary idempotents, so two consecutive splitting levels create four minimal complex channels.
The general construction allows for independent phase motion in each channel. However, there is a particularly important special case in which the entire state is controlled by a single parameter \(x\), but the two idempotent levels cycle at different rates. The first level contains the ordinary phase \(\pi x\), while the second level contains the slower relative phase \(\pi x/2\).
In this paper, we consider only two phases: the fundamental phase \(\pi x\) and the deep phase \(\pi x/2\). The combined phases of individual minimal components are not introduced, since they are not needed to prove the norm, biperiodicity, and second-level reduction.
The central object of the paper is the operator
\[\tag{1} \boxed{ J_{2}(x) =\jone^{x}\jtwo^{x/2}. } \] The first factor defines the fundamental phase cycle. The second factor stores an additional internal state, which is restored twice as slowly. As a result, the fundamental phase closes after each rotation, while the full two-level operator closes after only two. This operator allows us to describe spin 1/2, or any other exotic spin.
1. The First Idempotent Level
Let the first hyperbolic unit satisfy the condition
\[\tag{2} \jone^{2}=1. \] The associated pair of idempotents is defined by the expressions
\[\tag{3} \ep =\frac{1+\jone}{2}, \qquad \em =\frac{1-\jone}{2}. \] Direct verification yields
\[\tag{4} \ep^{2}=\ep, \qquad \em^{2}=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] Conversely, the hyperbolic unit is the difference of two idempotents:
\[\tag{5} \boxed{ \jone=\ep-\em. } \] After the complex expansion, the real power of the first unit is decomposed into two original planes:
\[\tag{6} \boxed{ \jone^{x} =\ep+\em e^{i\pi x}. } \] At \(x=0\), both components add up to one. As \(x\) changes from zero to two, the complex phase in the \(\em\) plane traverses the full angle \(2\pi\) and returns to its original value.
\[\tag{7} \jone^{x+2} =\ep+\em e^{i\pi(x+2)} =\jone^{x}. \] Thus, the period of the first level with respect to the parameter \(x\) is two.
2. The Second Idempotent Level
Let's introduce the second hyperbolic unit
\[\tag{8} \jtwo^{2}=1. \] It is considered independent of the first and commutes with it:
\[\tag{9} \boxed{ \jone\jtwo =\jtwo\jone. } \] The second unit forms its own pair of idempotents:
\[\tag{10} \pmp =\frac{1+\jtwo}{2}, \qquad \pme =\frac{1-\jtwo}{2}. \] The same basic relations hold for them:
\[\tag{11} \pmp^{2}=\pmp, \qquad \pme^{2}=\pme, \qquad \pmp\pme=0, \qquad \pmp+\pme=1. \] The second hyperbolic unit is equal to
\[\tag{12} \boxed{ \jtwo=\pmp-\pme. } \] Due to the commutativity of the two levels, all their idempotents also commute. Therefore, the extended algebra contains four joint minimal channels:
\[\tag{13} \boxed{ \ep\pmp, \qquad \ep\pme, \qquad \em\pmp, \qquad \em\pme. } \] These four products were studied in detail in Part I. The only important thing here is that they are components of a single unit, not four independent objects.
3. Half the Second-Level Power
For an arbitrary real parameter \(y\), the power of the second hyperbolic unit is
\[\tag{14} \jtwo^{y} =\pmp+\pme e^{i\pi y}. \] Consider a special choice
\[\tag{15} y=\frac{x}{2}. \] Then the deep multiplier takes the form
\[\tag{16} \boxed{ \jtwo^{x/2} =\pmp+\pme e^{i\pi x/2}. } \] The exponent \(x/2\) means that for the same change in the parameter \(x\), the second phase traverses half the angle of the first. This is not a replacement of the main phase or a reduction in the frequency of the first level. Both multipliers will be present in the operator simultaneously.
4. Matched Two-Level Operator
Multiply two phase operators:
\[\tag{17} \boxed{ J_{2}(x) =\jone^{x}\jtwo^{x/2}. } \] Substituting expressions (6) and (16), we obtain a convenient factorized notation:
\[\tag{18} \boxed{ J_{2}(x) =\left( \ep+\em e^{i\pi x} \right) \left(\pmp+\pme e^{i\pi x/2} \right). } \] The factorized form is fundamentally important. It immediately shows two independent levels and does not force us to ascribe additional physical meaning to the combined phases of the individual components.
\[\tag{19} \begin{aligned} \jone^{x} &\longrightarrow \text{main phase cycle},\\ \jtwo^{x/2} &\longrightarrow \text{deep relative state}. \end{aligned} \] Since both factors are elements of the same commutative extended algebra, the order in which they are written is unimportant:
\[\tag{20} \jone^{x}\jtwo^{x/2} =\jtwo^{x/2}\jone^{x}. \] 5. Fundamental Frequency and Period of the Complete State
Let us denote the phase angles of the two levels:
\[\tag{21} \boxed{ \phi_{1}(x)=\pi x, \qquad \phi_{2}(x)=\frac{\pi x}{2}. } \] If the parameter depends on time, \(x=x(t)\), the corresponding angular frequencies are
\[\tag{22} \omega_{1} =\frac{d\phi_{1}}{dt} =\pi\frac{dx}{dt}, \qquad \omega_{2} =\frac{d\phi_{2}}{dt} =\frac{\omega_{1}}{2}. \] In this case, \(\omega_{1}\) remains the fundamental frequency of the wave motion. The quantity \(\omega_{2}\) describes only the rate of change of the additional relative phase. Therefore, it is more accurate to speak not of a slowing of the entire wave, but of a longer period of repetition of the complete two-level state.
The primary phase completes a full rotation at \(x:0\to2\). During the same time, the deep phase only completes half of its cycle. Another primary rotation is necessary for its full restoration.
6. The State after the First Rotation
At the starting point, both factors are equal to one:
\[\tag{23} \jone^{0}=1, \qquad \jtwo^{0}=1, \qquad J_{2}(0)=1. \] One full rotation of the primary phase corresponds to \(x=2\). The first factor has already returned to its original state:
\[\tag{24} \jone^{2} =\ep+\em e^{i2\pi} =1. \] The deep factor is then equal to
\[\tag{25} \jtwo^{1} =\pmp+\pme e^{i\pi} =\pmp-\pme =\jtwo. \] Therefore, the full operator after the first rotation is not the same as the initial one:
\[\tag{26} \boxed{ J_{2}(2)=\jtwo\ne1=J_{2}(0). } \] The first level has already closed, but the second has retained the relative sign of the two deep components. Thanks to this, the operator distinguishes between the first and second successive rotations, although the fundamental phase is in the same position in both cases.
7. Recovery after the second rotation
After the second fundamental rotation, the parameter becomes equal to \(x=4\). Now the first multiplier is again equal to unity:
\[\tag{27} \jone^{4}=1. \] At the same time, the deep phase completes its own full cycle:
\[\tag{28} \jtwo^{2} =\pmp+\pme e^{i2\pi} =1. \] Therefore, the entire two-level operator is restored:
\[\tag{29} \boxed{ J_{2}(4)=1=J_{2}(0). } \] In general, the transformation after one rotation is written as
\[\tag{30} \boxed{ J_{2}(x+2) =J_{2}(x)\jtwo. } \] After two rotations, the additional factor is also closed:
\[\tag{31} \boxed{ J_{2}(x+4) =J_{2}(x). } \] Thus, the sequence of states has a simple form:
\[\tag{32} \boxed{ 1 \xrightarrow{\text{first rotation}} \jtwo \xrightarrow{\text{second rotation}} 1. } \] To prove this result, the two initial phases \(\pi x\) and \(\pi x/2\) are sufficient. No additional summed angles need to be introduced.
8. The Unit Norm and the Inverse Operator
We define complex conjugation by the substitution \(i\to-i\), leaving the hyperbolic units unchanged. Then the conjugate factors are equal
\[\tag{33} \jone^{-x} =\ep+\em e^{-i\pi x}, \qquad \jtwo^{-x/2} =\pmp+\pme e^{-i\pi x/2}. \] For the first level:
\[\tag{34} \jone^{x}\jone^{-x} =\ep+\em =1. \] For the second level:
\[\tag{35} \jtwo^{x/2}\jtwo^{-x/2} =\pmp+\pme =1. \] Therefore, the complete operator also has unit norm:
\[\tag{36} \boxed{ J_{2}(x)\overline{J_{2}(x)} =1. } \] Its inverse is
\[\tag{37} \boxed{ J_{2}^{-1}(x) =\jtwo^{-x/2}\jone^{-x}. } \] The deeper level does not create an additional norm or double the magnitude of the state. It introduces additional phase structure within the same overall unit.
9. Removing the Deep Phase Level
Since the second factor is invertible, it can be removed exactly by multiplying by its conjugate:
\[\tag{38} \begin{aligned} J_{2}(x)\jtwo^{-x/2} &=\jone^{x} \jtwo^{x/2} \jtwo^{-x/2}\\ &=\jone^{x}. \end{aligned} \] Therefore, the second-level reduction operation is written as
\[\tag{39} \boxed{ J_{2}(x)\jtwo^{-x/2} =\jone^{x}. } \] This is a standard algebraic operation on invertible elements. It does not involve taking a modulus, squaring the entire state, or discarding one of the components. The primary factor remains unchanged, and the secondary factor cancels out with its exact inverse.
10. Phase Change and Channel Permutation
It is necessary to distinguish between relative phase change and literal permutation of idempotent channels. Equality
\[\tag{40} J_{2}(x+2) =J_{2}(x)\jtwo \] means that after one rotation, the deep components receive the opposite relative sign. But the idempotents themselves remain the same:
\[\tag{41} \pmp\not\longleftrightarrow\pme. \] If channel swapping is required, the linear permutation operator \(S\) must be separately introduced:
\[\tag{42} S\pmp=\pme S, \qquad S\pme=\pmp S, \qquad S^{2}=I. \] Thus, three levels of description are mathematically distinguished: the existence of two deep channels; a change in their relative phase; a physical interpretation of this change as a transition between sheets. The latter is an additional mapping and should not surreptitiously replace the first two.
11. A More General Internal Phase Coefficient
The considered exponent \(x/2\) is a special case of a more general construction:
\[\tag{43} \boxed{ J_{n}(x) =\jone^{x}\jtwo^{x/n}, \qquad n\in\mathbb N. } \] The first factor has a period of two in \(x\). The second factor completes its cycle when \(x\) changes to \(2n\):
\[\tag{44} \jtwo^{(x+2n)/n} =\jtwo^{x/n+2} =\jtwo^{x/n}. \] Therefore, the complete operator has a period.
\[\tag{45} \boxed{ J_{n}(x+2n) =J_{n}(x). } \] For \(n=2\), we obtain the two-turn case under study:
\[\tag{46} J_{2}(x) =\jone^{x}\jtwo^{x/2}, \qquad T_{2}=4. \] This generalization shows that the increase in the period is a consequence of the rational alignment of the two phase levels. However, the physical meaning of each value of \(n\) must be established separately.
12. Special Application: Half-Phase and Electron Spin
Up to this point, the construction has remained purely mathematical. Now let us consider one possible physical application. Let the fundamental phase angle correspond to a physical rotation angle.
\[\tag{47} \boxed{ \theta=\pi x. } \] Then the two conjugate orientations of the deep multiplier are written as
\[\tag{48} \boxed{ \jtwo^{\pm x/2} =\pmp+\pme e^{\pm i\theta/2}. } \] The idempotents \(\pmp\) and \(\pme\) denote two deep channels. They are not themselves spin-up and spin-down states. In this application, the spin information is carried by the direction of accumulation of the relative phase between these channels.
Let's distinguish two conjugate phase functions:
\[\tag{49} \chi_{\uparrow}(\theta) =e^{-i\theta/2}, \qquad \chi_{\downarrow}(\theta) =e^{+i\theta/2}. \] For a phase generator of rotation around the selected axis \(\mathbf n\)
\[\tag{50} \widehat S_{\mathbf n} =i\hbar \frac{\partial}{\partial\theta} \] we obtain two eigenvalues:
\[\tag{51} \boxed{ \begin{aligned} \widehat S_{\mathbf n}\chi_{\uparrow} &=+\frac{\hbar}{2}\chi_{\uparrow},\\ \widehat S_{\mathbf n}\chi_{\downarrow} &=-\frac{\hbar}{2}\chi_{\downarrow}. \end{aligned} } \] Thus, the coefficient of one half arises from the deep relative phase \(\theta/2\):
\[\tag{52} \boxed{ \begin{aligned} \jtwo^{-x/2} &\longrightarrow e^{-i\theta/2} \longrightarrow S_{\mathbf n}=+\frac{\hbar}{2},\\ \jtwo^{+x/2} &\longrightarrow e^{+i\theta/2} \longrightarrow S_{\mathbf n}=-\frac{\hbar}{2}. \end{aligned} } \] The mathematics here proves the existence of a half relative phase and two conjugate directions of its accumulation. The association of the phase generator with the physical electron spin is a particular interpretation of the construction.
13. Particular Application: Reduction of the Electronic State to a Free Wave
In the Wave Electricity model, a two-level operator can be associated with a localized electronic state:
\[\tag{53} \boxed{ J_{e}(x) =\jone^{x}\jtwo^{x/2}. } \] The free wave, or photon, regime corresponds to the fundamental operator without a deep multiplier:
\[\tag{54} \boxed{ J_{\gamma}(x) =\jone^{x}. } \] Then the mathematical reduction (39) takes on a physical interpretation:
\[\tag{55} \boxed{ J_{e}(x)\jtwo^{-x/2} =J_{\gamma}(x). } \] The fundamental phase is preserved during this transition. Only the additional two-periodic factor is eliminated. Therefore, the transition does not require frequency reduction, squaring the electron state, or the conditional discarding of part of the operator.
The physical implementation of multiplication by the inverse deep factor is a separate issue in the model. The algebra shows the exact reduction operation, but does not itself determine the dynamical mechanism that must implement it in nature.
Conclusion
In the first part of the deep idempotent splitting, the space of four minimal channels was constructed. In this paper, a simple consistent mode, controlled by a single parameter, is distinguished within this space:
\[\tag{56} \boxed{ J_{2}(x) =\jone^{x}\jtwo^{x/2}. } \] The first factor preserves the usual phase period. The second changes twice as slowly and stores an additional relative state. Therefore, after the first revolution, the main phase is already repeated, but the entire operator still differs from the initial one:
\[\tag{57} \boxed{ J_{2}(x+2) =J_{2}(x)\jtwo \ne J_{2}(x). } \] After the second rotation, both levels are closed:
\[\tag{58} \boxed{ J_{2}(x+4) =J_{2}(x). } \] The deep level can be exactly removed by multiplying by the inverse factor:
\[\tag{59} \boxed{ J_{2}(x)\jtwo^{-x/2} =\jone^{x}. } \] Thus, a single algebraic construction combines the usual phase cycle, additional two-periodicity, half-relative phase, and exact reduction to the first level. In a particular physical application, the half phase can be interpreted through the electron spin \(1/2\), and the removal of the second factor can be interpreted through the transition of a localized electron state into a free wave.