\[ \newcommand{\j}{\jmath} \newcommand{\ep}{\mathfrak{e}} \newcommand{\em}{\bar{\mathfrak{e}}} \]

\[ \tag{1} \gamma(\beta) = \frac{1}{\sqrt{1-\beta^2}} = \left\| \mathbf j_0+\beta\mathbf j_1+\beta^2\mathbf j_2+\cdots \right\|, \] \[ 1,\quad \beta,\quad \beta^2,\quad \beta^3,\quad\ldots \] \[ \tag{2} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] \[ \beta=\frac{v}{c} \] \[ \tag{3} \gamma(\beta)=\frac{1}{\sqrt{1-\beta^2}}, \qquad |\beta|<1. \] \[ \tag{4} \gamma^2(\beta) = \frac{1}{1-\beta^2} = 1+\beta^2+\beta^4+\beta^6+\cdots. \] \[ \tag{5} \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0 +\beta\mathbf j_1 +\beta^2\mathbf j_2 +\beta^3\mathbf j_3 +\cdots, } \] \[ \tag{6} \mathbf j_n\cdot\mathbf j_m = \delta_{nm} = \begin{cases} 1,&n=m,\\ 0,&n\ne m. \end{cases} \] \[ \begin{aligned} \left\|\boldsymbol{\Gamma}_\beta\right\|^2 &= \left( \sum_{n=0}^{\infty}\beta^n\mathbf j_n \right) \cdot \left( \sum_{m=0}^{\infty}\beta^m\mathbf j_m \right)\\ &= \sum_{n,m=0}^{\infty} \beta^{n+m} \left(\mathbf j_n\cdot\mathbf j_m\right)\\ &= \sum_{n=0}^{\infty}\beta^{2n}\\ &= \frac{1}{1-\beta^2}. \end{aligned} \] \[ \tag{7} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}} = \gamma(\beta). } \] \[ 2^N \] \[ \text{исходная единица} \longrightarrow \text{первое состояние и остаток} \longrightarrow \text{второе состояние и новый остаток} \longrightarrow\cdots. \] \[ E_0,E_1,E_2,E_3,\ldots. \] \[ \tag{8} \j_n^2=1, \qquad \j_n\j_m=\j_m\j_n. \] \[ \tag{9} p_n^+ = \frac{1+\j_n}{2}, \qquad p_n^- = \frac{1-\j_n}{2}. \] \[ \tag{10} (p_n^\pm)^2=p_n^\pm, \qquad p_n^+p_n^-=0, \qquad p_n^++p_n^-=1. \] \[ 1=p_1^-+p_1^+. \] \[ p_1^+ = p_1^+p_2^- + p_1^+p_2^+. \] \[ 1 = p_1^- + p_1^+p_2^- + p_1^+p_2^+. \] \[ p_1^+p_2^+ = p_1^+p_2^+p_3^- + p_1^+p_2^+p_3^+. \] \[ 1 = p_1^- + p_1^+p_2^- + p_1^+p_2^+p_3^- + p_1^+p_2^+p_3^+. \] \[ \begin{aligned} E_0&=p_1^-,\\ E_1&=p_1^+p_2^-,\\ E_2&=p_1^+p_2^+p_3^-,\\ E_3&=p_1^+p_2^+p_3^+p_4^-,\\ &\ \vdots \end{aligned} \] \[ \tag{11} \boxed{ E_n = \left( \prod_{k=1}^{n}p_k^+ \right)p_{n+1}^-, \qquad n=0,1,2,\ldots } \] \[ E_0=p_1^-. \] \[ \tag{12} \boxed{ R_{N+1} = \prod_{k=1}^{N+1}p_k^+. } \]

Лемма 1

\[ \tag{13} \boxed{ E_nE_m=\delta_{nm}E_n. } \]

Доказательство

\[ \begin{aligned} E_n^2 &= \left[ \left( \prod_{k=1}^{n}p_k^+ \right)p_{n+1}^- \right]^2\\ &= \left( \prod_{k=1}^{n}(p_k^+)^2 \right)(p_{n+1}^-)^2\\ &= \left( \prod_{k=1}^{n}p_k^+ \right)p_{n+1}^-\\ &=E_n. \end{aligned} \] \[ p_{n+1}^-p_{n+1}^+=0. \] \[ E_nE_m=0, \qquad n\ne m. \]

Лемма 2

\[ \tag{14} \boxed{ 1 = \sum_{n=0}^{N}E_n+R_{N+1}. } \]

Доказательство

\[ 1=p_1^-+p_1^+=E_0+R_1. \] \[ 1=\sum_{n=0}^{N}E_n+R_{N+1}. \] \[ \begin{aligned} R_{N+1} &= R_{N+1}(p_{N+2}^-+p_{N+2}^+)\\ &= R_{N+1}p_{N+2}^- + R_{N+1}p_{N+2}^+. \end{aligned} \] \[ R_{N+1}p_{N+2}^-=E_{N+1}, \] \[ R_{N+1}p_{N+2}^+=R_{N+2}. \] \[ 1 = \sum_{n=0}^{N+1}E_n+R_{N+2}. \] \[ E_0,E_1,E_2,\ldots \] \[ E_nE_m=0,\qquad n\ne m \] \[ \tag{15} E_n\longleftrightarrow\mathbf j_n. \] \[ \tag{16} \boxed{ \mathbf j_n\cdot\mathbf j_m=\delta_{nm}. } \] \[ \mathbf u = u_0\mathbf j_0+ u_1\mathbf j_1+ u_2\mathbf j_2+\cdots, \] \[ \tag{17} \|\mathbf u\|^2 = u_0^2+u_1^2+u_2^2+\cdots. \] \[ \mathcal H = \operatorname{span} \{\mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots\}. \] \[ \mathcal H = \operatorname{span} \{\mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots\}. \] \[ \mathcal H_1 = \operatorname{span} \{\mathbf j_1,\mathbf j_2,\mathbf j_3,\ldots\}. \] \[ \mathcal H_1\simeq\mathcal H. \] \[ \tag{18} \boxed{ S\mathbf j_n=\mathbf j_{n+1}. } \] \[ S \left( u_0\mathbf j_0+ u_1\mathbf j_1+ u_2\mathbf j_2+\cdots \right) = u_0\mathbf j_1+ u_1\mathbf j_2+ u_2\mathbf j_3+\cdots. \] \[ \tag{19} \|S\mathbf u\|=\|\mathbf u\|. \] \[ \tag{20} \mathbf j_0\cdot S\mathbf u=0. \] \[ \tag{21} \boxed{ \mathcal H = \operatorname{span}\{\mathbf j_0\} \oplus S\mathcal H. } \]

• начального единичного состояния \(\mathbf j_0\);
• копии самого себя, перенесённой оператором \(S\);
• коэффициента масштаба \(\beta\) между соседними уровнями.

\[ \tag{22} \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. } \] \[ \boxed{ \text{целое состояние} = \text{начальное состояние} + \text{масштабированная копия целого}. } \]

• сохраняет свою форму на всех уровнях;
• не требует отдельного выбора коэффициента для каждого направления;
• отличается на соседних уровнях только общим множителем \(\beta\);
• согласовано с бесконечной структурой самого пространства;
• при \(|\beta|<1\) имеет конечную норму;
• является единственным состоянием с перечисленными свойствами.

Теорема

\[ \mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots, \] \[ S\mathbf j_n=\mathbf j_{n+1}. \] \[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta \] \[ \tag{23} \boxed{ \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{\infty}\beta^n\mathbf j_n. } \] \[ \tag{24} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] \[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] \[ \begin{aligned} \boldsymbol{\Gamma}_\beta &= \mathbf j_0+ \beta S \left( \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta \right)\\ &= \mathbf j_0+ \beta\mathbf j_1+ \beta^2S^2\boldsymbol{\Gamma}_\beta. \end{aligned} \] \[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta\mathbf j_1+ \beta^2\mathbf j_2+ \beta^3S^3\boldsymbol{\Gamma}_\beta. \] \[ \tag{25} \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{N}\beta^n\mathbf j_n + \beta^{N+1}S^{N+1}\boldsymbol{\Gamma}_\beta. \] \[ \left\| \beta^{N+1}S^{N+1}\boldsymbol{\Gamma}_\beta \right\| = |\beta|^{N+1} \left\|\boldsymbol{\Gamma}_\beta\right\|. \] \[ \lim_{N\to\infty} |\beta|^{N+1} \left\|\boldsymbol{\Gamma}_\beta\right\| = 0. \] \[ \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta\mathbf j_1+ \beta^2\mathbf j_2+ \beta^3\mathbf j_3+\cdots. } \] \[ \sum_{n=0}^{\infty}\beta^{2n} \] \[ \boldsymbol{\Gamma}_\beta^{(1)} = \mathbf j_0+ \beta S\boldsymbol{\Gamma}_\beta^{(1)}, \] \[ \boldsymbol{\Gamma}_\beta^{(2)} = \mathbf j_0+ \beta S\boldsymbol{\Gamma}_\beta^{(2)}. \] \[ \boldsymbol{\Delta} = \boldsymbol{\Gamma}_\beta^{(1)} - \boldsymbol{\Gamma}_\beta^{(2)}. \] \[ \boldsymbol{\Delta} = \beta S\boldsymbol{\Delta}. \] \[ \|\boldsymbol{\Delta}\| = |\beta|\,\|S\boldsymbol{\Delta}\|. \] \[ \|\boldsymbol{\Delta}\| = |\beta|\,\|\boldsymbol{\Delta}\|. \] \[ \|\boldsymbol{\Delta}\|=0. \] \[ \boldsymbol{\Gamma}_\beta^{(1)} = \boldsymbol{\Gamma}_\beta^{(2)}. \] \[ \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{\infty}\beta^n\mathbf j_n. \] \[ \begin{aligned} \left\|\boldsymbol{\Gamma}_\beta\right\|^2 &= \left( \sum_{n=0}^{\infty}\beta^n\mathbf j_n \right) \cdot \left( \sum_{m=0}^{\infty}\beta^m\mathbf j_m \right)\\ &= \sum_{n,m=0}^{\infty} \beta^{n+m}\delta_{nm}\\ &= \sum_{n=0}^{\infty}\beta^{2n}\\ &= 1+\beta^2+\beta^4+\cdots\\ &= \frac{1}{1-\beta^2}. \end{aligned} \] \[ \tag{26} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] \[ \boldsymbol{\Gamma}_\beta = \underbrace{\mathbf j_0}_{\text{исходное состояние}} + \underbrace{\beta S\boldsymbol{\Gamma}_\beta}_{\text{масштабированная копия пространства}}. \] \[ \mathbf j_0\cdot S\boldsymbol{\Gamma}_\beta=0, \] \[ \begin{aligned} \left\|\boldsymbol{\Gamma}_\beta\right\|^2 &= \|\mathbf j_0\|^2 + \beta^2 \left\|S\boldsymbol{\Gamma}_\beta\right\|^2\\ &= 1+ \beta^2 \left\|\boldsymbol{\Gamma}_\beta\right\|^2. \end{aligned} \] \[ \left(1-\beta^2\right) \left\|\boldsymbol{\Gamma}_\beta\right\|^2 = 1. \] \[ \tag{27} \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] \[ 1+x\left[ 1+x\left( 1+x\left( 1+\cdots \right) \right) \right]. \] \[ \tag{28} \boxed{ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta S \left[ \mathbf j_0+ \beta S \left( \mathbf j_0+ \beta S(\cdots) \right) \right]. } \] \[ \tag{29} \boxed{ \gamma^2 = 1+\beta^2 \left[ 1+\beta^2 \left( 1+\beta^2 \left( 1+\cdots \right) \right) \right]. } \] \[ \gamma^2=1+\beta^2\gamma^2. \] \[ \gamma^2=\frac{1}{1-\beta^2}. \] \[ 1+x(1+x(1+x+\cdots)) \] \[ x=\beta^2, \] \[ \mathbf j_0+ \beta\mathbf j_1+ \beta\mathbf j_2+ \beta\mathbf j_3+\cdots \] \[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] \[ \beta^n. \] \[ \boxed{ 1,\quad \beta,\quad \beta^2,\quad \beta^3,\quad\ldots } \] \[ \boldsymbol{\Gamma}_{\beta,N} = \sum_{n=0}^{N}\beta^n\mathbf j_n. \] \[ \left\|\boldsymbol{\Gamma}_{\beta,N}\right\|^2 = \sum_{n=0}^{N}\beta^{2n} = \frac{1-\beta^{2N+2}}{1-\beta^2}. \] \[ \tag{30} \boxed{ \gamma_N(\beta) = \sqrt{ \frac{1-\beta^{2N+2}} {1-\beta^2} }. } \] \[ \begin{aligned} \gamma_0&=1,\\ \gamma_1&=\sqrt{1+\beta^2},\\ \gamma_2&=\sqrt{1+\beta^2+\beta^4},\\ \gamma_3&=\sqrt{1+\beta^2+\beta^4+\beta^6}. \end{aligned} \] \[ \tag{31} \boxed{ \lim_{N\to\infty}\gamma_N(\beta) = \frac{1}{\sqrt{1-\beta^2}}, \qquad |\beta|<1. } \] \[ \mathbf j_n \longrightarrow \beta\mathbf j_{n+1}. \] \[ \tag{32} \boxed{ \beta=\frac{v}{c} } \] \[ \boldsymbol{\Gamma}(v) = \mathbf j_0 + \frac{v}{c}\mathbf j_1 + \left(\frac{v}{c}\right)^2\mathbf j_2 + \left(\frac{v}{c}\right)^3\mathbf j_3 +\cdots. \] \[ \tag{33} \boxed{ \left\|\boldsymbol{\Gamma}(v)\right\| = \frac{1}{\sqrt{1-v^2/c^2}}. } \] \[ |\beta|\ll1, \] \[ 1\gg|\beta|\gg\beta^2\gg|\beta|^3\gg\cdots. \] \[ |\beta|^n\geq\varepsilon. \] \[ n\leq \frac{\ln\varepsilon}{\ln|\beta|}. \] \[ \tag{34} \boxed{ N_{\mathrm{eff}} \approx \frac{\ln\varepsilon}{\ln|\beta|}. } \] \[ \ln|\beta|\to0, \qquad N_{\mathrm{eff}}\to\infty. \]

Случай \(\beta=0\)

\[ \beta=0, \] \[ \boldsymbol{\Gamma}_0=\mathbf j_0, \qquad \|\boldsymbol{\Gamma}_0\|=1. \]

Случай \(0<|\beta|<1\)

\[ 1,\beta,\beta^2,\beta^3,\ldots, \]

Предел \(|\beta|\to1\)

\[ \beta^n\to1. \] \[ \left\|\boldsymbol{\Gamma}_\beta\right\|^2 = 1+\beta^2+\beta^4+\cdots \longrightarrow\infty. \]

Случай \(|\beta|>1\)

\[ 1,\beta,\beta^2,\beta^3,\ldots \] \[ |\beta|<1 \] \[ \gamma=\frac{1}{\sqrt{1-\beta^2}} \] \[ \tag{35} \boxed{ \gamma = \left\|\boldsymbol{\Gamma}_\beta\right\|. } \] \[ \{1,\beta,\beta^2,\beta^3,\ldots\}, \] \[ \boxed{ \begin{aligned} &\text{многоуровневое идемпотентное расщепление}\\ &\Downarrow\\ &\text{бесконечная последовательность ортогональных каналов}\\ &\Downarrow\\ &\text{самоподобное состояние с коэффициентом }\beta\\ &\Downarrow\\ &\boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta\mathbf j_1+\beta^2\mathbf j_2+\cdots\\ &\Downarrow\\ &\left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. \end{aligned} } \] \[ 1,\beta,\beta^2,\ldots \] \[ \boxed{ \begin{aligned} &\text{1. Бесконечная расщепляемость;}\\ &\text{2. Ортогональность возникающих состояний;}\\ &\text{3. Самоподобие остаточной части;}\\ &\text{4. Единый коэффициент перехода }\beta. \end{aligned} } \] \[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] \[ \frac{1}{\sqrt{1-\beta^2}}. \]

• систему взаимно ортогональных каналов;
• возможность последовательного выделения состояний \(E_n\);
• бесконечно продолжающуюся структуру расщепления;
• алгебраическую основу многомерного пространства.

• метрическое соответствие \(E_n\longleftrightarrow\mathbf j_n\);
• самоподобие оставшейся части пространства;
• единый коэффициент масштаба \(\beta\);
• физическое отождествление \(\beta=v/c\).

\[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+ \beta\mathbf j_1+ \beta^2\mathbf j_2+\cdots. \] \[ E_0,E_1,E_2,\ldots. \] \[ \mathbf j_0,\mathbf j_1,\mathbf j_2,\ldots. \] \[ \boldsymbol{\Gamma}_\beta = \mathbf j_0+\beta S\boldsymbol{\Gamma}_\beta. \] \[ \boldsymbol{\Gamma}_\beta = \sum_{n=0}^{\infty}\beta^n\mathbf j_n. \] \[ \boxed{ \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. } \] \[ \boxed{ \begin{gathered} \text{бесконечная расщепляемость} + \text{ортогональность} + \text{масштабное самоподобие}\\ \Downarrow\\ \boldsymbol{\Gamma}_\beta = \{1,\beta,\beta^2,\beta^3,\ldots\}\\ \Downarrow\\ \gamma = \left\|\boldsymbol{\Gamma}_\beta\right\| = \frac{1}{\sqrt{1-\beta^2}}. \end{gathered} } \]