In previous work it was shown that the Lorentz factor [1] arises as a norm of the natural state of an infinitely split space. If successive orthogonal channels are scaled by the same factor \(\beta=v/c\), then the full deep state takes the form

and its length is

This result explains the geometric origin of the full value \(\gamma\). However, in the practical formulas of Wave Electricity, the internal state and external motion are described by a finite operator.

Therefore, the following question arises: how does the deep metric length \(\gamma\) manifest itself in the normalized finite operator \(J(a,b)\), whose modulus remains unity? Below, it will be shown that the operator \(J\) encodes not the length \(\gamma\) itself, but its inverse projection.

After a separate physical mapping, this projection defines the ratio of the rest energy to the total energy, while the orthogonal projection \(\sin(\pi b)=\beta\) defines the ratio of the momentum energy to the total energy. As a result, the standard relativistic invariant arises as the conservation of the length of the corresponding energy vector.

To properly understand the result, it is necessary to distinguish two spaces. The first is the finite phase space of the operator \(J(a,b)\). It consists of two independent complex idempotent planes and describes the internal phase and external state of the motion. The second is a deeply split space containing a countable set of orthogonal channels \(\boldsymbol{\xi}_n\).

These spaces correspond to different objects and different norms:

The first equality denotes the identity of the final phase state. The second defines the full metric length of a self-similar state in the space of deep orthogonal channels. Therefore, the equalities \(J\overline J=1\) and \(\|\boldsymbol{\Gamma}_{\beta}\|=\gamma\) do not contradict each other: they refer to different levels of geometry.

The relationship between the levels can be represented by the following sequence:

Thus, the deep structure creates the full length \(\gamma\), and the end operator represents the associated normalized projection.

We recall only that part of the previous derivation that is necessary for this paper. Let the deep space have an orthonormal basis.

and reproduces itself after extracting the first direction. If the transition to the next level is accompanied by the coefficient \(\beta\), the natural self-similar state satisfies the recursive equation

where \(S\boldsymbol{\xi}_n=\boldsymbol{\xi}_{n+1}\) is the transition operator to the next orthogonal level. Due to the orthogonality of the initial direction and the translated copy of space,

Following

This equality requires not only the presence of idempotents, but also an additional metric, orthonormality of deep directions, self-similarity of the residual space, and a single transition coefficient \(\beta\). Therefore, the Lorentz factor arises at a deep metric level, and not from just a finite circular projection.

The basis of the finite phase space are two mutually complementary idempotents:

Their difference forms a hyperbolic unit.

On the principal branches of the logarithm, the continuous powers \(\j\) and \(-\j\) have the form

Their product forms a complete finite operator:

Parameter \(a\) describes the internal periodic state, and parameter \(b\) describes the external motion:

Here \(v_s=ds/dt\) is the velocity of the center of state along the oriented curvilinear coordinate \(s\). Two idempotent planes of the operator are phase components of a single state. They cannot be directly identified with the coordinate planes of observable physical space.

Complex conjugation changes the signs of both phases, therefore

The unity of the operator means the conservation of its finite phase norm, but does not assert that the physical energy of the particle is numerically equal to unity.

The model uses the notation

It refers to the full phase-geometric dynamics of constant norm. The quantity \(V_J\) is not the usual three-dimensional velocity of the particle's center. The observed translational motion is obtained only after isolating the external parameter:

After mapping onto the tangent to the physical trajectory, we obtain

Thus, the multicomponent internal state does not violate the local one-dimensionality of physical motion. At each instant, the particle's center moves along a single tangent \(\widehat{\boldsymbol{\tau}}\), although its full state contains several independent phase channels.

From the definition of the external parameter, it follows immediately

On the main branch \(0\leqslant b<1/2\), the orthogonal circular projection is

But deep geometry has already given

Therefore, an exact correspondence arises between the two levels:

Or, in reverse form,

Now the meaning of the formula differs significantly from the previous interpretation. The Lorentz factor does not simply arise as the reciprocal of the circular projection. Its full magnitude is already formed by a deep self-similar structure, while the final external phase encodes the normalized value \(1/\gamma\).

Therefore, the two projections of the final external state can be written as

To transition from dimensionless geometry to observable physical quantities, a separate mapping must be introduced. Let us associate two normalized energy ratios with two orthogonal external projections: \[\tag{26} \boxed{ \frac{E_0}{E}=\cos(\pi b), \qquad \frac{pc}{E}=\sin(\pi b). } \] \[\tag{27} \frac{E_0}{E}=\frac{1}{\gamma}, \] \[\tag{28} \boxed{ E=\gamma E_0. } \] \[\tag{29} \frac{pc}{E}=\beta. \] \[\tag{30} \boxed{ pc=\gamma\beta E_0. } \] [tag{31} \boxed{ E=gamma mc^2, qquad p=gamma mv. } \] \[\tag{32} \left(\frac{E_0}{E}\right)^2+\left(\frac{pc}{E}\right)^2=\cos^2(\pi b)+\sin^2(\pi b)=1. \] \[\tag{33} E_0^2+p^2c^2=E^2. \] \[\tag{34} \boxed{ E^2-p^2c^2=E_0^2=m^2c^4. } \] \[\tag{35} \eta=\operatorname{artanh}\beta. \] \[\tag{36} e^{\eta\j}=\cosh\eta+\j\sinh\eta. \] \[\tag{37} \cosh\eta=\gamma, \qquad \sinh\eta=\gamma\beta, \] \[\tag{38} \boxed{ \beta=\sin(\pi b)=\tanh\eta, } \] \[\tag{39} \boxed{ \gamma=\frac{1}{\cos(\pi b)}=\cosh\eta, } \] \[\tag{40} \boxed{ \gamma\beta=\tan(\pi b)=\sinh\eta. } \] \[\tag{41} \omega_C=\alpha_{\mathrm{fs}}\omega_{\mathrm{int}}, \qquad m_ec^2=\hbar\omega_C=\alpha_{\mathrm{fs}}\hbar\omega_{\mathrm{int}}. \] \[\tag{42} E=\gamma(\beta)m_ec^2. \] \[\tag{43} \boxed{ \begin{aligned} \alpha_{\mathrm{fs}}&:\quad \text{internal projection generating rest energy}\gamma(\beta)&:\quad \text{external metric length of the state of motion}. \end{aligned} } \] \[\tag{44} E_0=mc^2=\operatorname{const}, \qquad E=\gamma E_0, \qquad pc=\gamma\beta E_0. \] \[\tag{45} \boxed{ E_{\mathrm{int}}+E_{\mathrm{motion}}+E_{\mathrm{source}}=\operatorname{const}. } \] \[\tag{46} \Delta E_{\mathrm{particle}}=-\Delta E_{\mathrm{source}}. \] \[\tag{47} \beta\longrightarrow1, \qquad b\longrightarrow\frac12, \qquad \cos(\pi b)\longrightarrow0. \] \[\tag{48} \gamma=\frac{1}{\cos(\pi b)}\longrightarrow\infty. \] \[\tag{49} 1,\;\beta,\;\beta^2,\;\beta^3,\ldots \] \[\tag{50} \left\|\boldsymbol{\Gamma}_{\beta}\right\|^2=1+\beta^2+\beta^4+\cdots\longrightarrow\infty. \] \[\tag{51} \boxed{ m>0, \quad v\longrightarrow c \quad\Longrightarrow\quad E=\gamma mc^2\longrightarrow\infty. } \] \[\tag{52} \boxed{ m=0, \qquad E=|p|c. } \] \[\tag{53} J\overline J=1, \qquad \sin(\pi b)=\beta, \qquad \cos(\pi b)=\sqrt{1-\beta^2}. \] \[\tag{54} \left\|\boldsymbol{\Gamma}_{\beta}\right\|=\gamma. \] \[\tag{55} \frac{E_0}{E}=\cos(\pi b), \qquad \frac{pc}{E}=\sin(\pi b). \] \[\tag{56} E=\gamma E_0, \qquad pc=\gamma\beta E_0, \qquad E^2-p^2c^2=E_0^2. \] \[\tag{57} \boxed{ \begin{gathered} \text{deep splitting}+\text{metric}+\text{self-similarity} \longrightarrow \boldsymbol{\Gamma}_{\beta},\\ \boldsymbol{\Gamma}_{\beta}=\{1,\beta,\beta^2,\ldots\}, \qquad \left\|\boldsymbol{\Gamma}_{\beta}\right\|=\gamma,\\ \gamma=\frac{1}{\cos(\pi b)}, \qquad \beta=\sin(\pi b),\\ \frac{E_0}{E}=\frac{1}{\gamma}, \qquad \frac{pc}{E}=\beta,\\ E=\gamma E_0, \qquad pc=\gamma\beta E_0,\\ E^2-p^2c^2=E_0^2. \end{gathered} } \]