In previous works, two results were obtained independently, the origin of the connection between them remaining unclear. The first describes a continuous power of the hyperbolic unit, and the second describes an internal rotation in a new idempotent basis.

This paper shows that both results represent the same mathematical expression. After transitioning to an idempotent basis, the fractional power of the hyperbolic unit takes the form \[ \tag{1} \large{ \j^\alpha = \ep + \em\, e^{i\pi\alpha} }. \] This identity is the main result of the paper. It directly implies the representation of the intrinsic rotation vector, its group property, its inverse element, its derivative, and the differential equation of motion.

Formula (1) means that the continuous power of the hyperbolic unit is equivalent to a complex rotation of one of the two idempotent components. The hyperbolic power thus acquires a simple geometric meaning: it describes an intrinsic rotation in a complex-extended idempotent basis.

Consider the hyperbolic unit \(\j\), for which \[ \tag{2} \j^2=1. \] We introduce two complementary idempotents: \[ \tag{3} \ep = \frac{1+\j}{2}, \qquad \em = \frac{1-\j}{2}. \]

They satisfy the relations \[ \tag{4} \ep ^2 = \ep , \qquad \em ^2 = \em , \qquad \ep \em = 0, \] and also \[ \tag{5} 1 = \ep + \em , \qquad \j = \ep - \em . \]

Thus, hyperbolic algebra naturally splits into two independent idempotent directions. After complex expansion, each of them has its own complex plane, and a real four-dimensional basis arises. \[ \tag{6} \left\{ \ep , \;i\ep , \;\em , \;i\em \right\}. \]

Previously, the formula for the fractional power of a hyperbolic unit was obtained: \[ \tag{7} \j^\alpha = \frac12 \left[ \left( 1+e^{i\pi\alpha} \right) + \j \left( 1-e^{i\pi\alpha} \right) \right]. \] For integer values ​​of \(\alpha\), it reproduces the usual powers of the hyperbolic unit: \[ \tag{8} \j^0=1, \qquad \j^1=\j, \qquad \j^2=1. \]

Formula (7) defines a continuous trajectory passing through the values ​​\(1\) and \(\j\). However, its geometric meaning becomes much clearer after switching to an idempotent representation.

We substitute the relation \[ \j = \ep - \em into expression (7). \] Then \[ \tag{9} \j^\alpha = \frac12 \left[ 1 + e^{i\pi\alpha} + \left( \ep - \em \right) \left( 1-e^{i\pi\alpha} \right) \right]. \]

Using equality \[ 1 = \ep + \em \] Let's expand the expression: \[ \tag{10} \begin{aligned} \j^\alpha &= \frac12 \Big[ \ep + \em + e^{i\pi\alpha}\ep + e^{i\pi\alpha}\em \\ &\qquad + \ep - \ep e^{i\pi\alpha} - \em + \em e^{i\pi\alpha} \Big]. \end{aligned} \]

After canceling the opposite terms, we are left with \[ \tag{11} \boxed{ \j^\alpha = \ep + \em e^{i\pi\alpha} }. \]

Identity (11) shows that the fractional power of the hyperbolic unit consists of two independent parts. The component \(\ep \) remains constant, while the component \(\em \) is multiplied by the complex phase \(e^{i\pi\alpha}\).

Therefore, the continuous power of the hyperbolic unit describes not a change in both components of the algebra, but a rotation of only one of them. It is this property that links the hyperbolic power with internal rotation in the new basis.

We expand the complex exponential in formula (11): \[ \tag{12} \j^\alpha = \ep + \em \cos\pi\alpha + i\em \sin\pi\alpha. \]

As the parameter \(\alpha\) changes, the first component remains fixed along the \(\ep \) direction, while the second moves along the unit circle in the plane \[ \tag{13} \left\{ \em , \;i\em \right\}. \]

Thus, the fractional power \(\j^\alpha\) describes the intrinsic rotation of one idempotent component relative to the other. The direction \(\ep \) plays the role of the fixed component, and the pair \[ \left\{ \em , \;i\em \right\} \] forms its own complex plane.rotation.

Therefore, the hyperbolic unit ceases to be merely an algebraic object. Its continuous power describes the internal evolution of the system, just as the complex exponential \(e^{i\varphi}\) describes rotation in the ordinary complex plane.

Let us set \[ \tag{14} \alpha = \varpi t, \] where \(t\) is time, and \(\varpi\) is a parameter with the dimension of frequency. Then formula (11) takes the form \[ \tag{15} \j^{\varpi t} = \ep + \em e^{i\pi\varpi t}. \]

Let's introduce the usual angular frequency \[ \tag{16} \omega = \pi\varpi. \] Then \[ \tag{17} \boxed{ \j^{\varpi t} = \ep + \em e^{i\omega t} }. \]

If \[ \tag{18} \omega = 2\pi f, \] then from formula (16) it follows \[ \tag{19} \varpi = 2f. \] Therefore, the exponent of the hyperbolic unit changes at a rate numerically equal to twice the normal frequency of rotation.

Consider the vector \[ \tag{20} J(t) = \ep + \em e^{i\omega t}. \] Comparing expressions (17) and (20) yields \[ \tag{21} \boxed{ J(t) = \j^{\varpi t} }, \qquad \omega = \pi\varpi. \]

Thus, the internal rotation vector is a continuous power of the hyperbolic unit. Previously, these expressions could be considered as two different constructions, but formula (21) demonstrates their exact identity.

At \(t=0\), we have \[ \tag{22} J(0) = \ep + \em = 1. \] After half a period, when \(\omega t=\pi\), we have \[ \tag{23} J(t) = \ep - \em = \j. \] After a full period \[ \tag{24} J(t) = \ep + \em = 1. \]

Therefore, the hyperbolic unit \(\j\) corresponds to the state achieved after rotating the second idempotent component by an angle of \(\pi\). It represents half a full revolution of internal rotation: \[ \tag{25} \j = \ep + \em e^{i\pi}. \]

Due to the orthogonality of idempotents, the product of two states has the form \[ \tag{26} \begin{aligned} J(t_1)J(t_2) &= \left( \ep + \em e^{i\omega t_1} \right) \left( \ep + \em e^{i\omega t_2} \right) \\ &= \ep + \em e^{i\omega(t_1+t_2)}. \end{aligned} \]

Therefore, \[ \tag{27} \boxed{ J(t_1)J(t_2) = J(t_1+t_2) }. \] In equivalent form, \[ \tag{28} \j^{\varpi t_1} \j^{\varpi t_2} = \j^{\varpi(t_1+t_2)}. \]

Therefore, the internal rotation forms a one-parameter commutative group. The time parameters are added, and the corresponding states are multiplied.

From the idempotent form it follows immediately \[ \tag{29} J^{-1}(t) = \ep + \em e^{-i\omega t}. \] Indeed, \[ \tag{30} \begin{aligned} J(t)J^{-1}(t) &= \left( \ep + \em e^{i\omega t} \right) \left( \ep + \em e^{-i\omega t} \right) \\ &= \ep + \em = 1. \end{aligned} \]

Therefore, \[ \tag{31} J^{-1}(t) = J(-t) = \j^{-\varpi t}. \] Inverting an element corresponds to changing the direction of the internal rotation.

Direct differentiation of expression (20) yields \[ \tag{32} \frac{dJ}{dt} = i\omega \em e^{i\omega t}. \]

Since \[ \tag{33} \em J(t) = \em \left( \ep + \em e^{i\omega t} \right) = \em e^{i\omega t}, \] formula (32) can be written in operator form: \[ \tag{34} \boxed{ \frac{dJ}{dt} = i\omega \em J(t) }. \]

The projector \(\em \) on the right-hand side is necessary. It shows that the derivative is determined only by the rotating component, while the direction of \(\ep \) remains constant.

Therefore, formula (34) is a differential equation of internal rotation. Its solution with the initial condition \(J(0)=1\) has the form \[ \tag{35} J(t) = \ep + \em e^{i\omega t} = \j^{\varpi t}. \]

Repeated differentiation yields \[ \tag{36} \frac{d^2J}{dt^2} = -\omega^2 \em e^{i\omega t}. \] Since \[ \em J(t) = \em e^{i\omega t}, \] we obtain \[ \tag{37} \boxed{ \frac{d^2J}{dt^2} + \omega^2 \em J(t) = 0 }. \]

For greater clarity, we introduce the rotating component \[ \tag{38} J_{\mathrm{rot}}(t) = \em J(t) = \em e^{i\omega t}. \] Then equation (37) takes the usual form of a harmonic oscillator: \[ \tag{39} \boxed{ \frac{d^2J_{\mathrm{rot}}}{dt^2} + \omega^2J_{\mathrm{rot}}(t) = 0 }. \]

Thus, the oscillatory dynamics act only in the subspacesolid \[ \left\{ \em , \;i\em \right\}, \] while the component \(\ep \) remains unchanged.

The complex exponential \[ e^{i\omega t} \] describes a uniform rotation in the ordinary complex plane. The expression \[ \tag{40} \j^{\varpi t} = \ep + \em e^{i\omega t} \] describes a similar rotation in the complex-extended idempotent space.

However, not the entire algebra element rotates. The direction \(\ep \) remains constant, and the direction \(\em \) forms, together with \(i\em \), its own complex plane of rotation.

In this representation, the hyperbolic unit corresponds to a rotation of the rotating component by an angle \(\pi\): \[ \tag{41} \j = \ep + \em e^{i\pi} = \ep - \em . \]

This approach combines the hyperbolic and complex structures. The hyperbolic unit distinguishes between two idempotent directions, while the complex unit \(i\) ensures continuous rotation within the plane of one of the idempotents.

The main result of this paper can be interpreted as an idempotent analogue of Euler's formula. In ordinary complex algebra, the expression \(e^{i\varphi}\) describes the rotation of a unit vector. In the algebra under consideration, the expression \(\j^\alpha\) preserves the \(\ep\) component and rotates the \(\em\) component by an angle of \(\pi\alpha\).

The paper establishes an exact identity \[ \tag{42} \boxed{ \j^\alpha = \ep + \em e^{i\pi\alpha} }, \] which relates the fractional power of the hyperbolic unit to the internal rotation in a complex-extended idempotent basis.

After introducing the time parameter \(\alpha=\varpi t\), we obtain a compact notation \[ \tag{43} \boxed{ J(t) = \j^{\varpi t} = \ep + \em e^{i\omega t} }, \qquad \omega = \pi\varpi. \] This shows that the previously introduced internal rotation vector is not a separate construct, but a continuous power of a hyperbolic unit.

The fundamental identity immediately implies the group property \[ J(t_1)J(t_2) = J(t_1+t_2), \] the inverse element \[ J^{-1}(t) = J(-t), \] as well as the differential equation \[ \frac{dJ}{dt} = i\omega \em J(t). \]

The rotating component \[ J_{\mathrm{rot}}(t) = \em J(t) \] satisfies the harmonic oscillator equation \[ \frac{d^2J_{\mathrm{rot}}}{dt^2} + \omega^2J_{\mathrm{rot}}(t) = 0. \] Therefore, the dynamics of the internal rotation are completely localized in the complex plane. \[ \left\{ \em , \;i\em \right\}. \]

Thus, the fractional power \(\j^\alpha\) acquires a direct geometric meaning. It describes the stationary component \(\ep \) and the uniformly rotating component \[ \em e^{i\pi\alpha}, \] combining the hyperbolic unit, complex phase, and internal motion in a single algebraic construct.