2026-08-16
Multilevel idempotent splitting of phase planes
How to split an electron
In part one, it was shown that one hyperbolic unit expands the scalar unit into two mutually orthogonal idempotent components. In part two, on this basis, the split sine and split cosine functions were introduced, describing two independent phase rotations.
However, the following question arises: is this splitting the limit, or can each of the two resulting complex planes be further split into two independent components? The answer requires an important clarification. Within the original algebra, further nontrivial splitting is impossible. But it becomes possible after adding a second independent hyperbolic unit. As a result, the two-phase geometry naturally expands to a four-phase geometry.
Why would such a deeper mathematical splitting be necessary? In the Wave Electricity model, the electron is considered as a wave closed in two close orbits. Now this physical picture receives a direct mathematical reflection: each orbit corresponds to its own part of the extended space. In other words, the physical splitting of the orbit is repeated in mathematics, and the mathematical splitting suggests the possibility of two connected physical branches. This does not involve two separate electrons or two independent waves, but rather two parts of a single closed wave circuit.
1. Why the original planes cannot be split within the same algebra
The original hyperbolic unit \(\j\) defines two idempotents
\[\tag{1} \ep=\frac{1+\j}{2}, \qquad \em=\frac{1-\j}{2}, \] for which
\[\tag{2} \ep^{2}=\ep, \qquad \em^{\,2}=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] After the complex extension, the entire algebra decomposes into the direct sum of two complex planes:
\[\tag{3} \mathcal A_{1} =\mathbb C\ep\oplus\mathbb C\em \simeq \mathbb C\oplus\mathbb C. \] Let's try to find an additional idempotent within the first plane. Any of its elements has the form \(P=z\ep\), where \(z\in\mathbb C\). The idempotency condition yields
\[\tag{4} P^{2}=P \quad\Longrightarrow\quad z^{2}\ep=z\ep \quad\Longrightarrow\quad z^{2}=z. \] Over the field of complex numbers, the last equation has only two solutions:
\[\tag{5} z=0 \qquad\text{or}\qquad z=1. \] Therefore, inside the plane \(\mathbb C\ep\), there exist only trivial idempotents \(0\) and \(\ep\). The same is true for \(\mathbb C\em\). Therefore, \(\ep\) and \(\em\) are minimal idempotents of the original algebra: they cannot be further split by its own elements.
A deeper split does not mean discovering two more hidden parts within the previous plane, but rather expanding the algebra itself with a new independent binary direction.
2. The Second Hyperbolic Unit
Let's denote the original hyperbolic unit by \(\j_{1}\) and add a second unit, \(\j_{2}\). We require that they be independent and commute:
\[\tag{6} \boxed{ \j_{1}^{2}=1, \qquad \j_{2}^{2}=1, \qquad \j_{1}\j_{2}=\j_{2}\j_{1}. } \] The previous algebra is obtained by using \(i\) and \(\j_{1}\). Adding \(\j_{2}\) doubles the number of real directions. The general element of the extended algebra can be written as
\[\tag{7} \begin{aligned} Z={}&x_{0}+ix_{1} +\j_{1}x_{2}+i\j_{1}x_{3} +\j_{2}x_{4}+i\j_{2}x_{5}\\ &+\j_{1}\j_{2}x_{6} +i\j_{1}\j_{2}x_{7}, \qquad x_{k}\in\mathbb R. \end{aligned} \] Thus, the new algebra has eight real dimensions, or four independent complex components.
3. The Second Pair of Idempotents
Just as \(\j_{1}\) generates \(\ep\) and \(\em\), the unit \(\j_{2}\) generates a new pair of projectors:
\[\tag{8} \prp=\frac{1+\j_{2}}{2}, \qquad \prm=\frac{1-\j_{2}}{2}. \] They satisfy the same relations:
\[\tag{9} \prp^{2}=\prp, \qquad \prm^{\,2}=\prm, \qquad \prp\prm=0, \qquad \prp+\prm=1. \] Since \(\j_{1}\) and \(\j_{2}\) commute, both pairs of idempotents also commute. Now each previous component can be multiplied by \(\prp+\prm=1\):
\[\tag{10} \boxed{ \begin{aligned} \ep&=\ep\prp+\ep\prm,\\ \em&=\em\prp+\em\prm. \end{aligned} } \] This is precisely the repeated idempotent splitting. It became possible not within \(\mathcal A_{1}\), but in the extended algebra \(\mathcal A_{2}\).
4. Four minimal idempotents
We introduce four products of two independent pairs of projectors:
\[\tag{11} \boxed{ E_{++}=\ep\prp, \qquad E_{+-}=\ep\prm, \qquad E_{-+}=\em\prp, \qquad E_{--}=\em\prm. } \] They are expressed symmetrically in terms of two hyperbolic units:
\[\tag{12} E_{\sigma\tau} =\frac{1}{4} \left(1+\sigma\j_{1}\right) \left(1+\tau\j_{2}\right), \qquad \sigma,\tau\in\{+1,-1\}. \] All four elements are idempotent, cancel each other out, and sum to one:
\[\tag{13} \boxed{ E_{\sigma\tau}E_{\sigma'\tau'} =\delta_{\sigma\sigma'}\delta_{\tau\tau'}E_{\sigma\tau}, \qquad \sum_{\sigma,\tau=\pm1}E_{\sigma\tau}=1. } \] Therefore, the extended algebra decomposes not into two, but into four independent complex planes:
\[\tag{14} \boxed{ \mathcal A_{2} =\bigoplus_{\sigma,\tau=\pm1}\mathbb C E_{\sigma\tau} \simeq \mathbb C^{4}. } \] The first sign \(\sigma\) indicates which original component — \(\ep\) or \(\em\) — the channel is derived from. The second sign \(\tau\) distinguishes two new child components \(\prp\) and \(\prm\). Two levels of splitting do not form a linear sequence, but a product of two binary features.
5. Componentwise Arithmetic
Any element of the new algebra has a unique idempotent representation
\[\tag{15} Z =E_{++}z_{++} +E_{+-}z_{+-} +E_{-+}z_{-+} +E_{--}z_{--}, \qquad z_{\sigma\tau}\in\mathbb C. \] If the second element is written similarly, their product is calculated independently in each channel:
\[\tag{16} ZW =\sum_{\sigma,\tau=\pm1} E_{\sigma\tau}z_{\sigma\tau}w_{\sigma\tau}. \] Mixed products disappear thanks to formula (13). For the same reason, the analytic function acts on the four components separately:
\[\tag{17} f(Z) =\sum_{\sigma,\tau=\pm1} E_{\sigma\tau}f\!\left(z_{\sigma\tau}\right). \] Thus, repeated splitting does not fundamentally complicate the calculations. It transforms a single problem into four parallel complex problems united by a common unit.
6. Four Independent Phase Rotations
For each minimal idempotent, a selective rotation can be defined. If \(E^{2}=E\), then
\[\tag{18} e^{i\varphi E} =1-E+Ee^{i\varphi}. \] It rotates only the \(E\) component, leaving the other three unchanged. The joint four-phase state has the form
\[\tag{19} \boxed{ \mathcal U(\varphi_{++},\varphi_{+-},\varphi_{-+},\varphi_{--}) =\sum_{\sigma,\tau=\pm1} E_{\sigma\tau}e^{i\varphi_{\sigma\tau}}. } \] The composition of two such states is reduced to the addition of the corresponding phases:
\[\tag{20} \mathcal U(\{\varphi_{\sigma\tau}\}) \mathcal U(\{\psi_{\sigma\tau}\}) = \mathcal U(\{\varphi_{\sigma\tau}+\psi_{\sigma\tau}\}). \] If complex conjugation changes \(i\to-i\), leaving \(\j_{1}\) and \(\j_{2}\) unchanged, then state (19) is normalized:
\[\tag{21} \boxed{ \mathcal U\,\overline{\mathcal U} =\sum_{\sigma,\tau=\pm1}E_{\sigma\tau} =1. } \] The new construction preserves the main principle of split geometry: several independent phase motions form a single integral object of unit norm.
7. How the old J operator fits into the new extension
The original two-phase operator has the form
\[\tag{22} J(a,b) =\ep e^{i\pi b} +\em e^{i\pi a}. \] In the extended algebra, each original component contains two child channels. Therefore, the most general consistent continuation operator can be written as follows:
\[\tag{23} \boxed{ \begin{aligned} J_{2}(a_{+},a_{-};b_{+},b_{-})={}& E_{++}e^{i\pi b_{+}} +E_{+-}e^{i\pi b_{-}}\\ &+E_{-+}e^{i\pi a_{+}} +E_{--}e^{i\pi a_{-}}. \end{aligned} } \] Parameters \(b_{+}\) and \(b_{-}\) control the two phases within the previous \(\ep\) plane, while \(a_{+}\) and \(a_{-}\) control the two phases within \(\em\). If the child phases are pairwise identical, the deeper structure becomes indistinguishable:
\[\tag{24} a_{+}=a_{-}=a, \qquad b_{+}=b_{-}=b. \] Then, using \(E_{++}+E_{+-}=\ep\) and \(E_{-+}+E_{--}=\em\), we obtain
\[\tag{25} \boxed{ J_{2}(a,a;b,b) =\ep e^{i\pi b}+\em e^{i\pi a} =J(a,b). } \] Hence, the previous two-phase geometry is not discarded. It is degenerate, or unresolvedm, a case of four-phase geometry in which the new paired components have the same phase.
8. Splitting and Transition Are Different Operations
The idempotents \(\prp\) and \(\prm\) separate two independent channels, but by themselves do not transfer state from one channel to the other. Moreover, multiplication by any element of the commutative algebra preserves the component-wise structure and is unable to permute \(\prp\) and \(\prm\).
To describe the exchange of child components, we need to introduce not another algebraic number, but a linear permutation operator \(S\). Its action on an arbitrary second-level state is defined as
\[\tag{26} S\!\left(\psi_{+}\prp+\psi_{-}\prm\right) =\psi_{-}\prp+\psi_{+}\prm. \] Such an operator satisfies the conditions
\[\tag{27} \boxed{ S^{2}=I, \qquad S\prp=\prm S, \qquad S\prm=\prp S. } \] In the basis \(\{\prp,\prm\}\), it has a simple matrix form:
\[\tag{28} S= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}. \] This distinction is fundamental. Idempotent algebra answers the question of what independent channels exist. The operator \(S\) answers a different question: how can state transition between them. Therefore, the presence of two branches and the dynamics of the transition between branches are two different levels of mathematical description.
9. Hierarchy of Further Extensions
The construction can be continued. Let there be \(N\) independent commuting hyperbolic units.
\[\tag{29} \j_{r}^{2}=1, \qquad \j_{r}\j_{s}=\j_{s}\j_{r}, \qquad r,s=1,\ldots,N. \] Each unit creates its own pair of idempotents. The minimum projectors of the full expansion are of the form
\[\tag{30} \boxed{ E_{\boldsymbol{\sigma}} =2^{-N}\prod_{r=1}^{N} \left(1+\sigma_{r}\j_{r}\right), \qquad \sigma_{r}\in\{+1,-1\}. } \] The number of independent complex channels doubles at each level:
\[\tag{31} \boxed{ N\, \text{hyperbolic units} \quad\Longrightarrow\quad 2^{N}\, \text{complex channels}. } \] Accordingly, the real dimension of the algebra is \(2^{N+1}\), since each complex channel contains a real and an imaginary direction.
10. The Geometric Meaning of Repeated Splitting
One hyperbolic unit replaces one common complex phase with two independent phases. The second hyperbolic unit does not add another phase linearly, but splits each of the two previous components. Therefore, four combinations of two binary features arise:
\[\tag{32} \boxed{ \{\ep,\em\} \times \{\prp,\prm\} = \{E_{++},E_{+-},E_{-+},E_{--}\}. } \] This structure is conveniently understood as a two-level addressing system. The first index selects the original phase plane, the second, the child channel within the expansion. All four components remain parts of a single unit and can be united by a common normalization condition.
Mathematics does not assign physical meaning to indices. In one model, the second index may denote two geometric branches, in another, two internal modes, and in a third, it may remain a purely computational label. A physical interpretation requires additional postulates, mappings, and verifiable consequences.
Conclusion
The main result of this article is expressed by the formula (10):
\[ \boxed{ \begin{aligned} \ep&=\ep\prp+\ep\prm,\\ \em&=\em\prp+\em\prm \end{aligned} } \] Where \[ \ep^{2}=\ep, \qquad \em^{\,2}=\em, \qquad \ep\em=0, \qquad \ep+\em=1, \] \[ \prp^{2}=\prp, \qquad \prm^{\,2}=\prm, \qquad \prp\prm=0, \qquad \prp+\prm=1. \] Its meaning is that each of the two original phase planes is now further divided into two parts. As a result, four channels emerge instead of the original two, but all of them still belong to the same general mathematical structure.
This construction is especially important for physical models in which the observed splitting must have a precise mathematical representation. For example, in the Wave Electricity model, a closed electron wave propagates along two close branches of a split orbit. Now these two physical branches can be assigned two daughter parts of each phase plane. The physical splitting of the orbit receives a mathematical reflection, and the mathematical splitting receives a possible geometric embodiment.
Four channels, however, do not imply the existence of four separate particles or four independent waves. They describe four parts of a single system: two initial phase planes, each containing two orbital branches. The wave transition between these branches must be specified separately, since the splitting itself is shownThis implies the existence of channels, but does not yet describe the movement between them.
The resulting construction is not limited to two levels. If new independent hyperbolic units are introduced successively, each subsequent level will again split all existing channels. Therefore, the number of minimal complex components doubles at each step:
\[ \boxed{ N\, \text{hyperbolic units} \quad\Longrightarrow\quad 2^{N}\, \text{complex channels}. } \] Thus, repeated idempotent splitting creates a common mathematical language for describing multilevel systems. The two split electron orbits are the first possible physical application of this construction, but the same principle can be extended to more complex internal structures.