\[\newcommand{\j}{\jmath}\newcommand{\ep}{\mathfrak{e}}\newcommand{\em}{\bar{\mathfrak{e}}}\newcommand{\Sin}{\boldsymbol{\operatorname{sin}}}\newcommand{\Cos}{\boldsymbol{\operatorname{cos}}}\]

\[\tag{1} \boxed{ \mathcal B_J = \{\ep,i\ep,\em,i\em\} }. \] \[\tag{2} \boxed{ \mathcal H_J = \mathbb C\ep \oplus \mathbb C\em }. \] \[\tag{3} Z = (x_1+ix_2)\ep + (x_3+ix_4)\em, \qquad x_1,x_2,x_3,x_4\in\mathbb R. \] \[\tag{4} \boxed{ \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1 }. \] \[\tag{5} \j=\ep-\em, \qquad \j^2=1. \] \[\tag{6} \ep=\frac{1+\j}{2}, \qquad \em=\frac{1-\j}{2}. \] \[\tag{7} [Z]_{\mathcal B_J} = \begin{pmatrix} x_1\\ x_2\\ x_3\\ x_4 \end{pmatrix}. \] \[\tag{8} \boxed{ \{\ep,i\ep,\em,i\em\} \neq \{ct,x,y,z\} }. \] \[\tag{9} \boxed{ \mathbb R^3 \;\longleftarrow\; s \;\longleftarrow\; J(a,b)\in\mathcal H_J \;\longleftarrow\; \boldsymbol\Gamma_\beta }. \] \[\tag{10} \j^a = \ep+\em e^{i\pi a}, \qquad (-\j)^b = \ep e^{i\pi b}+\em. \] \[\tag{11} \boxed{ J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \] \[\tag{12} \boxed{ a=\varpi t, \qquad \omega_{\mathrm{int}}=\pi\varpi, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v_s}{c} }. \] \[\tag{13} J(t) = \ep e^{i\arcsin\beta(t)} + \em e^{i\omega_{\mathrm{int}}t}. \] \[\tag{14} J = \ep e^{ig} + \em e^{ih}. \] \[\tag{15} \overline{J} = \ep e^{-ig} + \em e^{-ih}. \] \[\tag{16} \boxed{ J\overline{J} = \ep+\em =1 }. \] \[\tag{17} \boxed{ V_J(t)=cJ(t) }. \] \[\tag{18} [V_J]_{\mathcal B_J} = c \begin{pmatrix} \cos(\pi b)\\ \sin(\pi b)\\ \cos(\pi a)\\ \sin(\pi a) \end{pmatrix}. \] \[\tag{19} V_J\overline{V_J} = c^2. \] \[\tag{20} e^{i\pi b} = \cos(\pi b)+i\sin(\pi b) = \frac1\gamma+i\beta. \] \[\tag{21} \mathcal P_{\mathrm{ext}}[J] = \sin(\pi b) = \beta. \] \[\tag{22} \boxed{ \mathbf v(t) = c\beta(t) \widehat{\boldsymbol\tau}\bigl(s(t)\bigr) }. \] \[\tag{23} \boxed{ \mathbf r(t) = \mathbf r(0) + \int_0^t c\beta(\tau) \widehat{\boldsymbol\tau} \bigl(s(\tau)\bigr) \,d\tau }. \] \[\tag{24} \boxed{ J(a,b) \longrightarrow \mathcal P_{\mathrm{ext}}[J] \longrightarrow \mathbf v \longrightarrow \mathbf r(s)\subset\mathbb R^3 }. \] \[\tag{25} \boxed{ \mathbf R_J(t) = \mathbf R_J(0) + \int_0^t V_J(\tau)\,d\tau }. \] \[\tag{26} \beta=0, \qquad b=0, \qquad J(t)=\ep+\em e^{i\omega_{\mathrm{int}}t}. \] \[\tag{27} \mathbf R_J(t) = ct\,\ep - i\frac{c}{\omega_{\mathrm{int}}} e^{i\omega_{\mathrm{int}}t}\em + C. \] \[\tag{28} \boxed{ \mathbf R_{\mathrm{int}}(t) = -ir_{\mathrm{int}} e^{i\omega_{\mathrm{int}}t}\em, \qquad r_{\mathrm{int}} = \frac{c}{\omega_{\mathrm{int}}} }. \] \[\tag{29} [\mathbf R_J]_{\mathcal B_J} = \begin{pmatrix} ct\\ 0\\ r_{\mathrm{int}}\sin(\omega_{\mathrm{int}}t)\\ -r_{\mathrm{int}}\cos(\omega_{\mathrm{int}}t) \end{pmatrix}. \] \[\tag{30} \mathbf R_{\mathrm{int}} \bigl(t+T_{\mathrm{int}}\bigr) = \mathbf R_{\mathrm{int}}(t), \qquad T_{\mathrm{int}} = \frac{2\pi}{\omega_{\mathrm{int}}}. \] \[\tag{31} \boxed{ r_{\mathrm{int}} = \frac{c}{\omega_{\mathrm{int}}} }. \] \[\tag{32} \lambda_{\mathrm{int}} = \frac{2\pi c}{\omega_{\mathrm{int}}}, \] \[\tag{33} \boxed{ r_{\mathrm{int}} = \frac{\lambda_{\mathrm{int}}}{2\pi}, \qquad \lambda_{\mathrm{int}} = 2\pi r_{\mathrm{int}} }. \] \[\tag{34} r_{\mathrm{int}} \longmapsto r_e. \] \[\tag{35} E_{\mathrm{int}} = \hbar\omega_{\mathrm{int}}. \] \[\tag{36} m_ec^2 = \alpha_{\mathrm{fs}} \hbar\omega_{\mathrm{int}}. \] \[\tag{37} \boxed{ r_e = \frac{c}{\omega_{\mathrm{int}}} = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec} }. \] \[\tag{38} \sin(\pi b)=\beta, \qquad \cos(\pi b) = \sqrt{1-\beta^2} = \frac1\gamma. \] \[\tag{39} \boldsymbol\Gamma_\beta = \boldsymbol\xi_0 + \beta\boldsymbol\xi_1 + \beta^2\boldsymbol\xi_2 + \cdots. \] \[\tag{40} \|\boldsymbol\Gamma_\beta\|^2 = 1 + \beta^2 \|\boldsymbol\Gamma_\beta\|^2. \] \[\tag{41} \boxed{ \|\boldsymbol\Gamma_\beta\| = \frac1{\sqrt{1-\beta^2}} = \gamma = \frac1{\cos(\pi b)} }. \] \[\tag{42} \boxed{ J\overline{J}=1, \qquad \|\boldsymbol\Gamma_\beta\| = \gamma }. \] \[\tag{43} \frac{E_0}{E} = \cos(\pi b) = \frac1\gamma, \qquad \frac{pc}{E} = \sin(\pi b) = \beta. \] \[\tag{44} E=\gamma E_0, \qquad pc=\gamma\beta E_0, \qquad \boxed{ E^2-p^2c^2=E_0^2 }. \] \[\tag{45} \boxed{ E_{\mathrm{particle}} + E_{\mathrm{source}} = \mathrm{const} }. \]

• the decomposition of unity into two mutually orthogonal idempotents;
• the existence of two independent complex planes;
• a four-component real basis \(\mathcal B_J\);
• the unique decomposition of an element in this basis;
• the form of the finite operator \(J(a,b)\);
• the unit norm \(J\overline{J}=1\);
• the periodic intrinsic phase curve for \(a=\varpi t\);
• the radius of this curve \(r_{\mathrm{int}}=c/\omega_{\mathrm{int}}\);
• the possibility of further idempotent splitting.

• parameter \(a\) describes the internal state;
• parameter \(b\) describes the external motion;
• \(\sin(\pi b)=\beta\) is mapped to the observed velocity;
• the direction of motion is given by the tangent \(\widehat{\boldsymbol\tau}(s)\) in physical space;
• the integral of \(cJ\) describes the internal phase geometry;
• a stable internal closure manifests itself as a particle;
• the internal radius \(c/\omega_{\mathrm{int}}\) is mapped to the radius of the electron;
• mass The electron is a projection of its internal energy;
• deep splitting physically implements the Lorentz factor.

\[\tag{46} \boxed{ \begin{gathered} \mathcal B_J = \{\ep,i\ep,\em,i\em\} \quad\text{— phase basis},\\ J(a,b) \longrightarrow \beta \longrightarrow \mathbf v \longrightarrow \mathbf r(s)\subset\mathbb R^3,\\ V_J=cJ \longrightarrow \mathbf R_J \longrightarrow r_{\mathrm{int}} = \frac{c}{\omega_{\mathrm{int}}}. \end{gathered} }. \]