In previous studies, two results were independently obtained, the connection between which initially remained unclear. The first describes a continuous power of the hyperbolic unit, and the second internal rotation in a complexly extended idempotent basis.

Further development of the model showed that a continuous power of the hyperbolic unit is not only a way to represent internal rotation. Two mutually complementary powers \[ \j^a, \qquad (-\j)^b \] naturally describe two independent processes: the internal state of the particle and its external motion. Therefore, the central object becomes the complete operator \[ \tag{1} \boxed{ J(t)=\j^a(-\j)^b }, \] where the parameter \(a\) refers to the internal state, and the parameter \(b\) refers to the external motion.

This construction is based on two identities: \[ \tag{2} \boxed{ \j^a = \ep+\em e^{i\pi a} }, \] \[ \tag{3} \boxed{ (-\j)^b = \ep e^{i\pi b}+\em }. \] where: \(\ep, \em\) are idempotents [1-2] formed by the hyperbolic unit \(\j\) [3].

Their product separates two independent complex phases between two idempotent components: \[ \tag{4} \boxed{ \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \]

Thus, the hyperbolic unit defines not one, but two mutually complementary geometries of rotation. The \(\j^a\) power rotates the \(\em\) component, leaving \(\ep\) fixed, while the \((-\j)^b\) power rotates the \(\ep\) component, leaving \(\em\) fixed. The full operator \(J(t)\) combines both phases into a single four-dimensional state.

Consider the hyperbolic unit \(\j\), for which \[ \tag{5} \j^2=1. \] We introduce two mutually complementary idempotents: \[ \tag{6} \ep = \frac{1+\j}{2}, \qquad \em = \frac{1-\j}{2}. \]

They satisfy the relations \[ \tag{7} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \] as well as \[ \tag{8} 1=\ep+\em, \qquad \j=\ep-\em. \]

The hyperbolic algebra thus splits into two independent idempotent directions. After the complex extension, each of them has its own complex plane, and a real four-dimensional basis arises. \[ \tag{9} \left\{ \ep, \;i\ep, \;\em, \;i\em \right\} \]

For the continuous power of the hyperbolic unit, the formula was previously obtained \[ \tag{10} \j^a = \frac12 \left[ \left(1+e^{i\pi a}\right) + \j\left(1-e^{i\pi a}\right) \right]. \] For integer values ​​of \(a\), it reproduces ordinary powers: \[ \tag{11} \j^0=1, \qquad \j^1=\j, \qquad \j^2=1. \]

Substitute the equalities into expression (10) \[ \j=\ep-\em, \qquad 1=\ep+\em. \] After expanding the brackets and canceling the opposite terms, we obtain \[ \tag{12} \boxed{ \j^a = \ep+\em e^{i\pi a} }. \]

Identity (12) shows that the component \(\ep\) remains constant, while the component \(\em\) rotates in its own complex plane. \[ \tag{13} \left\{\em,\;i\em\right\}. \] The expansion into real and imaginary parts is of the form \[ \tag{14} \j^a = \ep + \em\cos(\pi a) + i\em\sin(\pi a). \]

Therefore, \(\j^a\) is an internal rotation operator: it preserves one idempotent component and continuously changes the phase of the other.

Since \[ \tag{15} -\j = -\ep+\em = \ep e^{i\pi}+\em, \] for the continuous power of \((-\j)^b\), we naturally obtain the complementary representation \[ \tag{16} \boxed{ (-\j)^b = \ep e^{i\pi b}+\em }. \]

Unlike \(\j^a\), this operator leaves the \(\em\) component constant, but rotates the \(\ep\) component in the plane \[ \tag{17} \left\{\ep,\;i\ep\right\}. \] Indeed, \[ \tag{18} (-\j)^b = \ep\cos(\pi b) + i\ep\sin(\pi b) + \em. \]

Thus, the operators \(\j^a\) and \((-\j)^b\) are symmetric with respect to the permutation of idempotent directions: \[ \tag{19} \begin{aligned} \j^a &:\quad \ep\;\text{fixed}, \quad \em\;\text{rotates}, \\[1mm] (-\j)^b &:\quad \em\;\text{stationary}, \quad \ep\;\text{rotating}. \end{aligned} \]

We multiply two complementary operators: \[ \tag{20} J(a,b) = \j^a(-\j)^b. \] Using the orthogonality of idempotents, we obtain \[ \tag{21} \begin{aligned} J(a,b) &= \left(\ep+\em e^{i\pi a}\right) \left(\ep e^{i\pi b}+\em\right) \\[1mm] &= \ep e^{i\pi b} + \em e^{i\pi a}. \end{aligned} \] Hence, \[ \tag{22} \boxed{ J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \]

Formula (22) is the main result of the present generalization. It shows that the complete state contains two independent phases: \[ \tag{23} \phi_{\rm ext}=\pi b, \qquad \phi_{\rm int}=\pi a. \] The outer phase belongs to the \(\{\ep,i\ep\}\) plane, and the inner phase belongs to the \(\{\em,i\em\}\) plane.

Unlike the previous construction, where only the \(\j^a\) power was considered, both idempotent components can now rotate independently. The one-dimensional motion of the particle is then mapped into the extended basis as a two-phase state, and the observed pattern emerges as its projection onto the physically selected components.

In the new model, the internal state parameter is defined as \[ \tag{24} a=\varpi t, \qquad \pi\varpi=\omega, \] from which \[ \tag{25} \pi a=\omega t. \]

The external motion parameter is determined by the relative velocity \[ \tag{26} b = \frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}, \] where in the general case \(v=v(t)\). Therefore \[ \tag{27} \pi b=\arcsin\beta. \]

Substituting (24) and (26) into formula (22) yields \[ \tag{28} \boxed{ J(t) = \j^{\varpi t} (-\j)^{\arcsin\beta(t)/\pi} }. \] In idempotent form \[ \tag{29} \boxed{ J(t) = \ep e^{i\arcsin\beta(t)} + \em e^{i\omega t} }. \]

Since \[ \tag{30} e^{i\arcsin\beta} = \sqrt{1-\beta^2}+i\beta = \frac1\gamma+i\beta, \] where \[ \gamma=\frac1{\sqrt{1-\beta^2}}, \] we obtain a clear decomposition \[ \tag{31} \boxed{ J(t) = \ep\left(\frac1\gamma+i\beta\right) + \em\left(\cos\omega t+i\sin\omega t\right) }. \]

The first idempotent component contains the external motion parameters \(\beta\) and \(1/\gamma\), and the second contains the internal phase \(\omega t\). Therefore, the product \(\j^a(-\j)^b\) does not mix the two physical roles, but preserves them in orthogonal subspaces of a single state.

Both components in formula (22) have unit modulus: \[ \left|e^{i\pi a}\right|=1, \qquad \left|e^{i\pi b}\right|=1. \] Therefore, the complete operator satisfies the condition \[ \tag{32} \boxed{ |J(a,b)|=1 }. \]

If we define the velocity operator as \[ \tag{33} \boxed{ V(t)=cJ(t) }, \] then \[ \tag{34} |V(t)|=c. \] Changing the external velocity does not change the complete modulus of the state, but only redistributes its geometric components. Within the model, this corresponds to the postulate of conservation of total energy \[ \tag{35} E_{\rm ext}+E_{\rm int}=\mathrm{const}. \]

For two states \(J(a_1,b_1)\) and \(J(a_2,b_2)\) we have \[ \tag{36} \begin{aligned} J(a_1,b_1)J(a_2,b_2) &= \left(\ep e^{i\pi b_1}+\em e^{i\pi a_1}\right) \left(\ep e^{i\pi b_2}+\em e^{i\pi a_2}\right) \\[1mm] &= \ep e^{i\pi(b_1+b_2)} + \em e^{i\pi(a_1+a_2)}. \end{aligned} \]

Hence, \[ \tag{37} \boxed{ J(a_1,b_1)J(a_2,b_2) = J(a_1+a_2,b_1+b_2) }. \] In power form \[ \tag{38} \j^{a_1}(-\j)^{b_1} \j^{a_2}(-\j)^{b_2} = \j^{a_1+a_2}(-\j)^{b_1+b_2}. \]

Thus, the complete operator forms a two-parameter commutative group. The inner parameters are added independently of the outer ones, and the outer ones are added independently of the inner ones.

Formula (22) immediately implies \[ \tag{39} J^{-1}(a,b) = \ep e^{-i\pi b} + \em e^{-i\pi a}. \] Therefore, \[ \tag{40} \boxed{ J^{-1}(a,b) = J(-a,-b) = \j^{-a}(-\j)^{-b} }. \] Really, \[ \tag{41} J(a,b)J(-a,-b) = \ep+\em = 1. \]

Let \[ a=a(t), \qquad b=b(t). \] Then from formula (22) \[ \tag{42} \frac{dJ}{dt} = i\pi\dot b\,\ep e^{i\pi b} + i\pi\dot a\,\em e^{i\pi a}. \]

Since \[ \ep J=\ep e^{i\pi b}, \qquad \em J=\em e^{i\pi a}, \] the derivative can be written in operator form: \[ \tag{43} \boxed{ \frac{dJ}{dt} = i\pi \left( \dot b\,\ep + \dot a\,\em \right)J }. \]

For the internal phase \[ \dot a=\varpi, \qquad \pi\dot a=\omega. \] For the external phase \[ \tag{44} \pi\dot b = \frac{d}{dt}\arcsin\beta = \frac{\dot\beta}{\sqrt{1-\beta^2}} = \gamma\dot\beta. \] Therefore \[ \tag{45} \boxed{ \frac{dJ}{dt} = i \left( \gamma\dot\beta\,\ep + \omega\em \right)J }. \]

Formula (45) separates the two contributions. The term \(i\gamma\dot\beta\,\ep J\) describes the change in external motion, and the term \(i\omega\em J\) describes the internal rotation. At a constant velocity \(\dot\beta=0\), the external component is fixed, and the previous equation for internal dynamics remains.

If \[ \beta=\mathrm{const}, \qquad b=\mathrm{const}, \] then \[ \tag{46} J(t) = \ep e^{i\arcsin\beta} + \em e^{i\omega t}, \] and the derivative is \[ \tag{47} \boxed{ \frac{dJ}{dt} = i\omega\em J(t) }. \]

Repeated differentiation yields \[ \tag{48} \frac{d^2J}{dt^2} = -\omega^2\em J(t), \] or \[ \tag{49} \boxed{ \frac{d^2J}{dt^2} + \omega^2\em J(t) = 0 }. \]

Therefore, with uniform external motion, the internal component still satisfies the harmonic oscillator equation, but now it enters a more general state that simultaneously contains a constant external phase.

The usual complex exponential \(e^{i\varphi}\) describes a rotation on one complex plane. The operator \(\j^a\) transfers this rotation to the \(\{\em,i\em\}\) plane, and the operator \((-\j)^b\) to the \(\{\ep,i\ep\}\) plane.

Their product \[ \tag{50} J(a,b) = \ep e^{i\pi b} + \em e^{i\pi a} \] describes two independent rotations in two orthogonal complex planes of the four-dimensional basis. Geometrically, this is no longer a single circle, but a two-phase state on the product of two unit circles.

However, the model retains the original postulate of the one-dimensionality of a point's motion. Extended geometry does not mean that the particle simultaneously moves along several spatial coordinates. It serves as a mathematical mapping of a one-dimensional process into the structure of a complete state, from which the observed three-dimensional picture can be obtained through projection.

The internal phase \(\pi a=\omega t\) describes the particle's own evolution. The external phase \(\pi b=\arcsin(v/c)\) encodes its kinematic state. The orthogonality of idempotents allows these two processes to coexist without interfering with each other.

The full operator can be represented as a generalized Euler formula: \[ \tag{51} J(a,b) = \Cos(a,b) + i\Sin(a,b), \] where \[ \tag{52} \boxed{ \Cos(a,b) = \ep\cos(\pi b) + \em\cos(\pi a) }, \] \[ \tag{53} \boxed{ \Sin(a,b) = \ep\sin(\pi b) + \em\sin(\pi a) }. \]

Here, the split cosine and split sine functions simultaneously contain both external and internal phases, but preserve their distribution between two idempotent planes. The detailed properties of these functions are discussed separately in the papers "From Euler's Formula to Split Geometry" and in Part Two.

The continuous power of the hyperbolic unit has an idempotent representation \[ \tag{54} \j^a = \ep+\em e^{i\pi a}, \] which describes the rotation of the intrinsic component in the plane \(\{\em,i\em\}\).

The complementary power \[ \tag{55} (-\j)^b = \ep e^{i\pi b}+\em \] rotates the extrinsic component in the plane \(\{\ep,i\ep\}\).

Their product forms the complete state operator \[ \tag{56} \boxed{ J(t) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }, \] where \[ \tag{57} \pi a=\omega t, \qquad \pi b=\arcsin\!\left(\frac{v}{c}\right). \]

Parameter \(a\) describes the internal state of the particle, and parameter \(b\) describes its external motion. These processes are represented by two independent phases belonging to orthogonal idempotent subspaces.

The complete operator satisfies the group property \[ J(a_1,b_1)J(a_2,b_2) = J(a_1+a_2,b_1+b_2), \] invertibility \[ J^{-1}(a,b)=J(-a,-b), \] and norm preservation \[ |J(a,b)|=1. \]

In general, its evolution is described by the equation \[ \tag{58} \boxed{ \frac{dJ}{dt} = i \left( \gamma\dot\beta\,\ep + \omega\em \right)J }. \] At constant external velocity, it reduces to the equation of internal rotation \[ \frac{dJ}{dt} = i\omega\em J. \]

Thus, the geometry of the hyperbolic unit takes on a more general meaning. The degree \(\j^a\) is no longer a complete state in itself, but rather represents its internal component. The complete state arises as the product of two complementary operators \[ \boxed{ \j^a(-\j)^b }, \] combining internal rotation, external motion, and the conservation law of the complete norm in a single algebraic construction.