In this paper, we propose a unified geometric model in which a propagating wave and a localized particle are considered as two different classes of motion of the same vector. The initial construction is built on a complexly extended idempotent basis \[ \tag{1} \left\{ \ep,\;i\ep,\;\em,\;i\em \right\}, \] where \[ \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0. \]

The main goal of this work is to obtain geometric criteria for distinguishing between an open wave trajectory and a closed particle trajectory. Both forms of motion are described by a single vector of constant norm, and the difference arises only after its integration over time.

The proposed approach does not pre-introduce the classical wave equation or postulate a spatial dependence of the form \(kx\). First, the full velocity vector is considered, then the displacement vector is derived from it, after which the type of motion is determined directly by the geometry of the integral trajectory.

The work also derives an integral condition for the closedness of the trajectory, which serves as a mathematical criterion for a localized particle. It is shown that a particle emerges when the full displacement vector vanishes over the internal cycle, whereas this condition is not satisfied for a wave. Thus, the boundary conditions for the transition between open wave motion and a closed particle trajectory are obtained. Within the framework of the proposed geometric model, waves and particles are considered not as different physical objects, but as two classes of solutions to the same equation of motion of constant norm.

Consider the dimensionless internal state vector \[ \tag{2} J(t) = \j^{\varpi t}. \] For the fractional power of the hyperbolic unit, a representation is used, discussed in detail here: \[ \tag{3} \j^{\varpi t} = \ep + \em e^{i\omega t}, \] where \[ \tag{4} \omega=\pi\varpi. \]

The vector \(J(t)\) contains a constant component along the \(\ep\) direction and a rotating component in the complex plane \[ \left\{ \em,\;i\em \right\}. \] Its norm is preserved: \[ \tag{5} \left|J(t)\right|=1. \]

We introduce the general motion vector \[ \tag{6} V(t) = c\,e^{i\alpha(t)}\j^{\varpi t}, \] where \(c\) is the constant velocity, and \(\alpha(t)\) is the external geometric phase, determining the orientation of the total motion.

Since \[ \left|e^{i\alpha(t)}\right|=1 \] and \[ \left|\j^{\varpi t}\right|=1, \] we obtain \[ \tag{7} \left|V(t)\right| = c. \] Thus, the norm of the total motion vector remains equal to \(c\) regardless of the law of change of the function \(\alpha(t)\).

Using an idempotent representation of the internal state, expression (6) can be written as \[ \tag{8} V(t) = c\,e^{i\alpha(t)} \left( \ep+ \em e^{i\omega t} \right). \] Therefore, \[ \tag{9} V(t) = c\ep e^{i\alpha(t)} + c\em e^{i[\alpha(t)+\omega t]}. \]

Both components have the same modulus \(c\), but belong to mutually orthogonal idempotent directions. The function \(\alpha(t)\) rotates the entire vector, while the additional phase \(\omega t\) specifies the internal rotation of the second component.

The type of motion is determined not by the velocity vector itself, but by the trajectory obtained after its integration. Let's introduce the displacement vector \[ \tag{10} L(t) = L(0) + \int_0^t V(\tau)\,d\tau. \]

Substituting expression (6), we obtain a single geometric equation of motion \[ \tag{11} L(t) = L(0) + c \int_0^t e^{i\alpha(\tau)} \j^{\varpi\tau}\,d\tau. \]

In the idempotent representation, it takes the form \[ \tag{12} L(t) = L(0) + c\ep \int_0^t e^{i\alpha(\tau)}\,d\tau + c\em \int_0^t e^{i[\alpha(\tau)+\omega\tau]}\,d\tau. \]

Formula (11) is common to both waves and particles. The difference between them is determined by the total displacement over a certain finite cycle \(T\).

Let's first consider the special case where the exterior angle is constant: \[ \tag{13} \alpha(t)=\alpha_0. \] Then \[ \tag{14} V(t) = c\,e^{i\alpha_0}\j^{\varpi t}. \]

Integrating expression (14)ayot \[ \tag{15} L(t) = L(0) + ce^{i\alpha_0} \left[ \ep t + \frac{\em}{i\omega} \left( e^{i\omega t}-1 \right) \right]. \]

The first component \[ \tag{16} L_{\ep}(t) = ct\,e^{i\alpha_0}\ep \] increases indefinitely with time, while the second component \[ \tag{17} L_{\em}(t) = \frac{c}{i\omega} e^{i\alpha_0}\em \left( e^{i\omega t}-1 \right) \] remains bounded and periodic.

The period of internal rotation is \[ \tag{18} T_\omega = \frac{2\pi}{\omega}. \] After one period, we have \[ e^{i\omega T_\omega}=1, \] therefore \[ \tag{19} L_{\em}(T_\omega)=0. \]

However, the translational displacement over the same period is \[ \tag{20} L_{\ep}(T_\omega) = cT_\omega e^{i\alpha_0}\ep \neq0. \] Therefore, \[ \tag{21} L(T_\omega)-L(0) = cT_\omega e^{i\alpha_0}\ep \neq0. \]

Thus, at a constant angle \(\alpha_0\), the rotational part returns to its original state, but the total displacement vector is not closed. The trajectory remains open and after each period shifts by a constant vector. \[ \tag{22} \Delta L = cT_\omega e^{i\alpha_0}\ep. \]

Now let's relate the constant angle to the observed velocity. Let \[ \tag{23} \alpha = \arcsin\beta, \qquad \beta=\frac{v}{c}. \] Then \[ \tag{24} \sin\alpha = \beta = \frac{v}{c}. \] Hence \[ \tag{25} v = c\sin\alpha. \]

The full velocity vector can be represented as \[ \tag{26} V(t) = c \left( \cos\alpha+i\sin\alpha \right) \j^{\varpi t}. \] Taking into account \[ \cos\alpha = \sqrt{1-\beta^2} \] we obtain \[ \tag{27} V(t) = c \left( \sqrt{1-\beta^2} + i\beta \right) \j^{\varpi t}. \]

Despite the appearance of the observed velocity \(v\), the norm of the full vector is still equal to \[ \tag{28} |V(t)|=c. \] The quantity \(v\) is not the full norm of motion, but its spatial projection: \[ \tag{29} v=c\sin\alpha. \]

The orthogonal projection has magnitude \[ \tag{30} v_{\mathrm{int}} = c\cos\alpha = c\sqrt{1-\beta^2}. \] Therefore, the identity holds \[ \tag{31} v^2+v_{\mathrm{int}}^2=c^2. \]

The translational part of the total displacement vector at a constant angle is \[ \tag{32} L_{\ep}(t) = ct\,e^{i\alpha}\ep. \] Expanding the complex exponential, we obtain \[ \tag{33} L_{\ep}(t) = ct\cos\alpha\,\ep + ict\sin\alpha\,\ep. \]

The observed spatial projection of this displacement is determined by the expression \[ \tag{34} x(t) = ct\sin\alpha. \] Using \[ v=c\sin\alpha, \] we obtain \[ \tag{35} x(t)=vt. \]

Given the initial coordinate \[ \tag{36} x(t)=x_0+vt. \] Thus, the familiar equation of uniform motion arises as a projection of the total geometric displacement obtained from the constant-norm vector \(c\).

The sequence of transitions can be written: \[ \tag{37} V(t) = c\,e^{i\alpha}\j^{\varpi t} \quad\longrightarrow\quad |V(t)|=c \quad\longrightarrow\quad v=c\sin\alpha \quad\longrightarrow\quad x(t)=x_0+vt. \]

For an electromagnetic wave propagating in a homogeneous medium, the observed velocity is determined by the expression \[ \tag{38} v = \frac{1}{\sqrt{\varepsilon\mu}}, \] where \(\varepsilon\) and \(\mu\) are the absolute permittivity and magnetic permeability of the medium.

Comparing formulas \[ v=c\sin\alpha \] And \[ v=\frac{1}{\sqrt{\varepsilon\mu}}, \] we get \[ \tag{39} \sin\alpha = \frac{1}{c\sqrt{\varepsilon\mu}}. \]

Since \[ \tag{40} c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}, \] we have \[ \tag{41} \sin\alpha = \sqrt{ \frac{\varepsilon_0\mu_0} {\varepsilon\mu} }. \]

If \[ \varepsilon=\varepsilon_0\varepsilon_r, \qquad \mu=\mu_0\mu_r, \] That \[ \tag{42} \sin\alpha = \frac{1}{\sqrt{\varepsilon_r\mu_r}}. \] Hence, \[ \tag{43} \alpha = \arcsin \left( \frac{1}{\sqrt{\varepsilon_r\mu_r}} \right). \]

In a vacuum \[ \varepsilon_r=1, \qquad \mu_r=1, \] therefore \[ \tag{44} \alpha=\frac{\pi}{2}, \qquad v=c. \]

Thus, the properties of the medium do not change the norm of the full vector \(|V|=c\), but change the magnitude of its observed spatial projection. This may indicate that the law of conservation of energy is observed in closed systems.

Now let us consider the general case when \(\alpha(t)\) is an arbitrary function of time.For a closed trajectory, there must be a finite period \(T>0\), after which the position returns to the starting point: \[ \tag{45} L(T)=L(0). \]

From the definition of displacement it follows \[ L(T)-L(0) = \int_0^T V(t)\,dt. \] Therefore, the closedness condition takes the form \[ \tag{46} \int_0^T V(t)\,dt=0. \]

Substituting the general velocity vector \[ V(t) = c\,e^{i\alpha(t)}\j^{\varpi t}, \] we obtain \[ c \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt = 0. \] Since \(c\neq0\), the main geometric criterion of the particle is \[ \tag{47} \boxed{ \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt = 0. } \]

Condition (47) means that the sum of all The number of elementary displacements per complete cycle is zero. The instantaneous velocity need never vanish, and its full norm is preserved throughout the entire motion: \[ \tag{48} |V(t)|=c. \]

Returning the position alone is not enough for a complete state match. The external phase factor must also return to its original value: \[ \tag{49} e^{i\alpha(T)} = e^{i\alpha(0)}. \]

From this it follows that \[ \tag{50} \alpha(T)-\alpha(0) = 2\pi n, \qquad n\in\mathbb Z. \] When choosing \[ \alpha(0)=0 \] we obtain the special case \[ \tag{51} \alpha(T)=2\pi n. \]

Thus, for a closed particle motion, the angle \(\alpha(t)\) changes by an integer number of complete revolutions over a complete cycle.

To fully return the internal state, we must also require \[ \tag{52} \j^{\varpi T} = \j^0. \] Using \[ \j^{\varpi t} = \ep+ \em e^{i\omega t}, \] we obtain the condition \[ \tag{53} e^{i\omega T}=1. \] Therefore, \[ \tag{54} \omega T=2\pi m, \qquad m\in\mathbb Z. \]

The complete set of conditions for a periodic particle can be written as \[ \tag{55} \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt = 0, \] \[ \tag{56} \alpha(T)-\alpha(0) = 2\pi n, \] \[ \tag{57} \omega T = 2\pi m. \]

The first condition closes the trajectory, the second returns the external orientation, and the third returns the internal state.

For a wave, the total velocity integral over the internal cycle is nonzero: \[ \tag{58} \int_0^T V(t)\,dt \neq0. \] Equivalently, \[ \tag{59} \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt \neq0. \]

This means that after completing the inner loop, the system does not return to its starting point: \[ \tag{60} L(T)\neq L(0). \] Each new cycle begins in a different region of space, so the trajectory remains open.

The simplest wave solution is a constant angle \[ \alpha(t)=\alpha_0. \] In this case, the trajectory equation has already been obtained: \[ \tag{61} L_{\mathrm w}(t) = L(0) + ce^{i\alpha_0} \left[ \ep t + \frac{\em}{i\omega} \left( e^{i\omega t}-1 \right) \right]. \]

For one internal period \[ \tag{62} L_{\mathrm w}(T_\omega) - L_{\mathrm w}(0) = cT_\omega e^{i\alpha_0}\ep \neq0. \] Therefore, expression (61) describes an open trajectory that shifts by a constant step after each complete rotation.

For a particle, the function \(\alpha(t)\) is not specified in advance. It must be determined from the closedness conditions. Therefore, the general equation of the particle is \[ \tag{63} L_{\mathrm p}(t) = L(0) + c \int_0^t e^{i\alpha(\tau)} \j^{\varpi\tau}\,d\tau, \] where the function \(\alpha(t)\) must satisfy the conditions \[ \tag{64} \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt = 0 \] and \[ \tag{65} \alpha(T)-\alpha(0)=2\pi n. \]

With the additional requirement of complete return of the internal state, \[ \tag{66} \omega T=2\pi m must also be satisfied. \] Then \[ \tag{67} L_{\mathrm p}(t+T) = L_{\mathrm p}(t). \]

Thus, the problem of constructing a specific geometric model of a particle is reduced to finding functions \(\alpha(t)\) for which the integral of the total velocity vector vanishes over a cycle.

Introduce the vector \[ \tag{68} \mathcal C(T) = \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt. \] Then the total displacement per cycle is \[ \tag{69} \Delta L(T) = c\, \mathcal C(T). \]

If \[ \tag{70} \mathcal C(T)\neq0, \] then \[ \Delta L(T)\neq0, \] and the motion is open.

If \[ \tag{71} \mathcal C(T)=0, \] then \[ \Delta L(T)=0, \] and the motion is closed.

Therefore, the main criterion for distinguishing waveswaves and particles can be represented as \[ \tag{72} \boxed{ \mathcal C(T)\neq0 \quad\Longrightarrow\quad \text{wave}, } \] \[ \tag{73} \boxed{ \mathcal C(T)=0 \quad\Longrightarrow\quad \text{closed particle trajectory}. } \]

Wave-to-particle transition occurs when the function \(\alpha(t)\) changes in such a way that the open displacement per cycle is completely compensated: \[ \tag{74} \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt \longrightarrow0. \]

When the equality \[ \tag{75} \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt = 0 \] is exactly satisfied, the open trajectory closes. If simultaneously \[ \alpha(T)-\alpha(0)=2\pi n, \] then the system returns not only to the starting point, but also to its original external orientation.

The reverse transition occurs when the closure condition is violated: \[ \tag{76} \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt \neq0. \] Then, after each cycle, a nonzero translation vector appears. \[ \tag{77} \Delta L = c \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt. \]

The closed trajectory opens up, and localized motion becomes propagating.

Both forms of motion have the same fundamental norm: \[ \tag{78} |V(t)|=c. \] Their difference is determined not by the magnitude of the instantaneous velocity, but by the integral geometry of the trajectory.

For a wave, \[ \tag{79} L(T)\neq L(0), \qquad \int_0^T V(t)\,dt\neq0. \]

For a particle, we have \[ \tag{80} L(T)=L(0), \qquad \int_0^T V(t)\,dt=0. \]

Thus, a wave is an open, periodically repeating motion, while a particle is a closed, integral trajectory of the same vector of constant norm.

This paper proposes a unified geometric model of a wave and a particle based on the vector \[ V(t) = c\,e^{i\alpha(t)}\j^{\varpi t}. \] The norm of this vector is preserved and remains equal to \(c\) for any law of change in the external phase. Therefore, the difference between a wave and a particle is not related to the change in the total velocity, but arises only after integrating the motion vector.

The displacement vector is defined by the expression \[ L(t) = L(0) + c \int_0^t e^{i\alpha(\tau)} \j^{\varpi\tau}\,d\tau. \] If the total integral of the velocity over a cycle is nonzero, the trajectory remains open and corresponds to a propagating wave regime. If the integral vanishes, the trajectory closes, and the geometric condition of a localized particle arises.

For the complete periodic return of a particle, the following conditions must also be satisfied: \[ \alpha(T)-\alpha(0)=2\pi n \] and \[ \omega T=2\pi m. \] The first returns the external orientation, the second the internal state, and the zero velocity integral returns the position.

In the special case \[ \alpha=\arcsin\beta, \qquad \beta=\frac{v}{c}, \] the observed velocity is determined as the projection of the total motion: \[ v=c\sin\alpha. \] Integrating this projection leads to the familiar equation of uniform motion \[ x(t)=x_0+vt. \] Thus, ordinary kinematics arises as a particular projection of a more general motion of constant norm.

The resulting criteria allow us to formulate the following mathematical problem: find a class of functions \(\alpha(t)\) satisfying the condition \[ \int_0^T e^{i\alpha(t)} \j^{\varpi t}\,dt = 0. \] Solutions to this problem can determine specific forms of closed geometric trajectories and become the basis for further construction of models of elementary particles.