%=============================================================================
% fig_alpha_ladder.tex --- The alpha-exponent ladder (pure TikZ, no external
% graphics). Vertical log-scale ladder of the derived tower Q ~ alpha^k M_P:
% rung depth is proportional to the exponent k (a log scale in alpha). All
% rung values are the corpus's own derived relations/numbers:
%   alpha^3    M_R = alpha^3 M_P ~ 4.7e12 GeV        (App. X, seesaw)
%   alpha^8    v = sqrt(2pi) alpha^8 M_P = 246 GeV    (tab:parameter-count)
%   alpha^19/2 Lambda_DFD = alpha^19/2 M_P = 61.2 MeV (App. AT, QCD chain)
%   alpha^21/2 m_e = (2/3)sqrt(pi) alpha^21/2 M_P     (App. K, locked map)
%   alpha^28.5 H_0 = alpha^28.5/t_P = 72.09 km/s/Mpc  (eq:H0-prediction)
%   alpha^57   rho_c/rho_P = (3/8pi) alpha^57         (eq:CC-solution)
% Requires only tikz + arrows.meta (loaded by the master documents).
%=============================================================================
\begin{figure}[t]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
% ---- vertical rail (depth proportional to exponent k; 0.26 cm per unit k)
\draw[very thick,gray!70,-{Stealth[length=2.5mm]}] (0,0.45) -- (0,-15.45);
% ---- axis annotation (left)
\node[rotate=90,gray,anchor=center] at (-0.65,-7.4)
  {\scriptsize scale $Q=\alpha^{k}M_P$ \ (depth $\propto k$: log scale in $\alpha$)};
% ---- rungs (ticks on the rail)
\draw[thick] (-0.30,0)      -- (0.30,0);
\draw[thick] (-0.30,-0.78)  -- (0.30,-0.78);
\draw[thick] (-0.30,-2.08)  -- (0.30,-2.08);
\draw[thick] (-0.30,-2.47)  -- (0.30,-2.47);
\draw[thick] (-0.30,-2.73)  -- (0.30,-2.73);
\draw[thick] (-0.30,-7.41)  -- (0.30,-7.41);
\draw[thick] (-0.30,-14.82) -- (0.30,-14.82);
% ---- leader lines (thin gray; fanned for the clustered k=8, 19/2, 21/2 rungs)
\draw[gray!60,thin] (0.35,0)      -- (0.80,0);
\draw[gray!60,thin] (0.35,-0.78)  -- (0.80,-0.78);
\draw[gray!60,thin] (0.35,-2.08)  -- (0.80,-2.08);
\draw[gray!60,thin] (0.35,-2.47)  -- (0.80,-2.66);
\draw[gray!60,thin] (0.35,-2.73)  -- (0.80,-3.24);
\draw[gray!60,thin] (0.35,-7.41)  -- (0.80,-7.41);
\draw[gray!60,thin] (0.35,-14.82) -- (0.80,-14.82);
% ---- rung labels: exponent (accent) + quantity + measured value
\node[anchor=west] at (0.85,0)
  {\scriptsize {\color{blue!60!black}$\alpha^{0}$}:\ $M_P=1.22\times10^{19}$~GeV \ (anchor)};
\node[anchor=west] at (0.85,-0.78)
  {\scriptsize {\color{blue!60!black}$\alpha^{3}$}:\ $M_R=\alpha^{3}M_P\approx4.7\times10^{12}$~GeV \ (seesaw)};
\node[anchor=west] at (0.85,-2.08)
  {\scriptsize {\color{blue!60!black}$\alpha^{8}$}:\ $v=\sqrt{2\pi}\,\alpha^{8}M_P=246$~GeV \ (EW)};
\node[anchor=west] at (0.85,-2.66)
  {\scriptsize {\color{blue!60!black}$\alpha^{19/2}$}:\ $\Lambda_{\rm DFD}=\alpha^{19/2}M_P=61.2$~MeV \ (QCD)};
\node[anchor=west] at (0.85,-3.24)
  {\scriptsize {\color{blue!60!black}$\alpha^{21/2}$}:\ $m_e=\tfrac{2}{3}\sqrt{\pi}\,\alpha^{21/2}M_P=0.511$~MeV};
\node[anchor=west] at (0.85,-7.41)
  {\scriptsize {\color{blue!60!black}$\alpha^{28.5}$}:\ $H_0=\alpha^{28.5}/t_P=72.09$~km\,s$^{-1}$\,Mpc$^{-1}$ \ (rate)};
\node[anchor=west] at (0.85,-14.82)
  {\scriptsize {\color{blue!60!black}$\alpha^{57}$}:\ $\rho_c/\rho_{\rm Planck}=\tfrac{3}{8\pi}\alpha^{57}\approx1.9\times10^{-123}$};
\end{tikzpicture}
\caption{The $\alpha$-exponent ladder: one dimensionful anchor ($M_P$,
equivalently one measurement of $H_0$ or $G$) and the derived tower of scales
$Q\sim\alpha^{k}M_P$. Rung depth is proportional to the exponent $k$, i.e.\ a
log scale in $\alpha$. Rungs: the seesaw scale $M_R=\alpha^{3}M_P$, the
electroweak scale $v=\sqrt{2\pi}\,\alpha^{8}M_P$
(Table~\ref{tab:parameter-count}), the QCD chain scale
$\Lambda_{\rm DFD}=\alpha^{19/2}M_P$, the locked electron map
$m_e=\tfrac{2}{3}\sqrt{\pi}\,\alpha^{21/2}M_P$, the Hubble rate
$H_0=\alpha^{28.5}/t_P$ (Eq.~\eqref{eq:H0-prediction}), and the vacuum-density
ratio $\rho_c/\rho_{\rm Planck}=\tfrac{3}{8\pi}\alpha^{57}$
(Eq.~\eqref{eq:CC-solution}), spanning the full $122$ orders of magnitude of
the invariant Eq.~\eqref{eq:invariant}.}
\label{fig:alpha-ladder}
\end{figure}
