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

\[\tag{1} \mathcal B_J = \{\ep,i\ep,\em,i\em\}. \] \[\tag{2} \boxed{ J(a,b) = \j^a(-\j)^b }. \] \[\tag{3} \boxed{ \j^a = \ep+\em e^{i\pi a} }. \] \[\tag{4} a(t)=\varpi t, \qquad \omega_{\mathrm{int}}=\pi\varpi, \qquad \pi a(t)=\omega_{\mathrm{int}}t. \] \[\tag{5} J_{\mathrm{int}}(t) = \j^{a(t)} = \ep + \em e^{i\omega_{\mathrm{int}}t}. \] \[\tag{6} \boxed{ M_{\mathrm{ext}}(b) = (-\j)^b }. \] \[\tag{7} (-\j)^b = \ep e^{i\pi b} + \em. \] \[\tag{8} M_{\mathrm{ext}}(0) = \ep+\em =1, \] \[\tag{9} J(a,b) = \left( \ep+\em e^{i\pi a} \right) \left( \ep e^{i\pi b}+\em \right). \] \[\tag{10} \begin{aligned} J(a,b) &= \ep^2e^{i\pi b} + \ep\em + \em\ep e^{i\pi(a+b)} + \em^2e^{i\pi a}\\ &= \ep e^{i\pi b} + \em e^{i\pi a}. \end{aligned} \] \[\tag{11} \boxed{ J(a,b) = \ep e^{i\pi b} + \em e^{i\pi a} }. \] \[\tag{12} \j^a(-\j)^b = (-\j)^b\j^a. \] \[\tag{13} e^{i\alpha}\j^a = \ep e^{i\alpha} + \em e^{i(\pi a+\alpha)}. \] \[\tag{14} \boxed{ \begin{aligned} e^{i\alpha}\j^a &= \ep e^{i\alpha} + \em e^{i(\pi a+\alpha)}, \\ \j^a(-\j)^b &= \ep e^{i\pi b} + \em e^{i\pi a}. \end{aligned} } \] \[\tag{15} \boxed{ b(t) = \frac{\arcsin\beta(t)}{\pi}, \qquad \beta(t) = \frac{v_s(t)}{c} }. \] \[\tag{16} \pi b = \arcsin\beta. \] \[\tag{17} e^{i\pi b} = \cos(\pi b)+i\sin(\pi b) = \sqrt{1-\beta^2}+i\beta = \frac1\gamma+i\beta. \] \[\tag{18} \boxed{ J(t) = \ep \left( \frac1{\gamma(t)}+i\beta(t) \right) + \em e^{i\omega_{\mathrm{int}}t} }. \] \[\tag{19} \boxed{ V_J(t) = cJ(t) }. \] \[\tag{20} [V_J]_{\mathcal B_J} = c \begin{pmatrix} \cos(\pi b)\\ \sin(\pi b)\\ \cos(\pi a)\\ \sin(\pi a) \end{pmatrix} = \begin{pmatrix} c/\gamma\\ v_s\\ c\cos(\omega_{\mathrm{int}}t)\\ c\sin(\omega_{\mathrm{int}}t) \end{pmatrix}. \] \[\tag{21} \overline{J} = \ep e^{-i\pi b} + \em e^{-i\pi a}. \] \[\tag{22} \boxed{ J\overline{J} = \ep+\em =1 }, \] \[\tag{23} V_J\overline{V_J} = c^2. \] \[\tag{24} \boxed{ \cos^2(\pi b) + \sin^2(\pi b) = \frac1{\gamma^2} + \beta^2 =1 }. \] \[\tag{25} \mathcal P_{\mathrm{ext}}[J] = \sin(\pi b) = \beta. \] \[\tag{26} \boxed{ \mathbf v(t) = c\beta(t) \widehat{\boldsymbol\tau} \bigl(s(t)\bigr) }. \] \[\tag{27} \boxed{ \mathbf r_c(t) = \mathbf r_c(0) + \int_0^t c\beta(\tau) \widehat{\boldsymbol\tau} \bigl(s(\tau)\bigr) \,d\tau }. \] \[\tag{28} \dot J = i\pi\dot b\, \ep e^{i\pi b} + i\pi\dot a\, \em e^{i\pi a}. \] \[\tag{29} \pi\dot b = \frac{\dot\beta}{\sqrt{1-\beta^2}} = \gamma\dot\beta, \] \[\tag{30} \boxed{ \dot J = i\gamma\dot\beta\, \ep e^{i\arcsin\beta} + i\omega_{\mathrm{int}}\, \em e^{i\omega_{\mathrm{int}}t} }. \] \[\tag{31} \mathbf a = c\dot\beta\, \widehat{\boldsymbol\tau} + c\beta \frac{d\widehat{\boldsymbol\tau}}{dt}. \] \[\tag{32} \mathbf R_J(t) = \mathbf R_J(0) + \int_0^t V_J(\tau)\,d\tau. \] \[\tag{33} \boxed{ \mathbf R_J(t) = ct\, \ep e^{i\pi b} - i\frac{c}{\omega_{\mathrm{int}}} e^{i\omega_{\mathrm{int}}t}\em + C }. \] \[\tag{34} [\mathbf R_J]_{\mathcal B_J} = \begin{pmatrix} ct/\gamma\\ v_st\\ r_{\mathrm{int}} \sin(\omega_{\mathrm{int}}t)\\ -r_{\mathrm{int}} \cos(\omega_{\mathrm{int}}t) \end{pmatrix}, \qquad r_{\mathrm{int}} = \frac{c}{\omega_{\mathrm{int}}}. \] \[\tag{35} \boxed{ \omega_{\mathrm{int}}, \qquad r_{\mathrm{int}} = \frac{c}{\omega_{\mathrm{int}}}, \qquad r_{\mathrm{int}}\omega_{\mathrm{int}}=c }. \] \[\tag{36} \pi b = \frac{\pi}{2}, \qquad e^{i\pi b}=i. \] \[\tag{37} J(a,1/2) = i\ep + \em e^{i\pi a}. \] \[\tag{38} \boxed{ \beta=1, \qquad E_0=0, \qquad E=|p|c }. \] \[\tag{39} \cos(\pi b) = \frac1\gamma. \] \[\tag{40} \boldsymbol\Gamma_\beta = \boldsymbol\xi_0 + \beta\boldsymbol\xi_1 + \beta^2\boldsymbol\xi_2 + \cdots. \] \[\tag{41} \|\boldsymbol\Gamma_\beta\|^2 = 1 + \beta^2 \|\boldsymbol\Gamma_\beta\|^2, \] \[\tag{42} \boxed{ \|\boldsymbol\Gamma_\beta\| = \frac1{\sqrt{1-\beta^2}} = \gamma = \frac1{\cos(\pi b)} }. \] \[\tag{43} E = \gamma E_0. \] \[\tag{44} \boxed{ E_{\mathrm{particle}} + E_{\mathrm{source}} = \mathrm{const} }. \] \[\tag{45} \boxed{ J\overline{J}=1, \qquad \|\boldsymbol\Gamma_\beta\|=\gamma }. \]

• decompositions \(\j^a=\ep+\em e^{i\pi a}\) and \((-\j)^b=\ep e^{i\pi b}+\em\);
• product \(J(a,b)=\ep e^{i\pi b}+\em e^{i\pi a}\);
• independence of internal and external phase components;
• unit finite norm \(J\overline{J}=1\);
• preservation of internal phase angle under external operator modulation;
• shape of the phase-geometric curve (33).

• interpretation of \(a\) as an internal state of the particle;
• interpretation of \(b\) as an external state of motion;
• relationship \(\sin(\pi b)=\beta=v_s/c\);
• mapping of the external projection onto the physical velocity of the center;
• relationship of the deep metric length with the Lorentz factor;
• energy mapping \(E=\gamma E_0\).

\[ \begin{aligned} u_{\ep}&=\frac{ct}{\gamma}, & v_{\ep}&=v_st, \\ u_{\em}&=r_{\mathrm{int}}\sin(\omega_{\mathrm{int}}t), & v_{\em}&=-r_{\mathrm{int}}\cos(\omega_{\mathrm{int}}t). \end{aligned} \] \[\tag{46} \boxed{ \underbrace{\j^a}_{\text{internal state}} \;\cdot\; \underbrace{(-\j)^b}_{\text{external modulation}} \; = \underbrace{ \ep e^{i\pi b} + \em e^{i\pi a} }_{\text{full phase state}} }. \] \[\tag{47} \boxed{ J(a,b) \longrightarrow \sin(\pi b)=\beta \longrightarrow \mathbf v = c\beta\widehat{\boldsymbol\tau}(s) \longrightarrow \mathbf r_c(s) }. \]