\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Stealth,font=\scriptsize]
\draw[ink!65,->] (0,0)--(14.6,0) node[below left] {edge length};
\fill[teal!14] (.2,-.35) rectangle (2.2,.35);
\foreach \x in {.34,.49,.71,.99,1.2,1.39,1.65,1.88,2.08} \draw[teal,line width=1pt] (\x,-.2)--(\x,.2);
\node[text=teal,font=\small\bfseries,align=center] at (1.2,.8) {local lengths};
\node[text=slate,align=center] at (1.2,-.8) {integer ruler gaps\\and a chord derivative};
\foreach \s/\lab in {4/{\ell},7.5/{\ell+1},11/{\ell+2}}{
\fill[gold!14] (\s,-.35) rectangle ({\s+2.65},.35);
\foreach \i in {0,1,2} {\foreach \j in {0,1,2} {\draw[gold!90!black,line width=.9pt] ({\s+.18+.86*\i+.16*\j},-.2)--++(0,.4);}}
\node[text=slate] at ({\s+1.3},-.62) {macro block $\lab$};}
\draw[<->,ink] (2.3,-.05)--(3.85,-.05);
\node[text=ink,align=center] at (3.08,1.1) {mixed gap\\$>1$};
\draw[->,ink!60] (3.08,.75)--(3.08,.15);
\draw[<->,teal] (4.18,.6)--(4.34,.6);
\node[anchor=south west,text=teal,align=left] at (4.02,.86) {fine step $>1$};
\draw[<->,ink] (4.5,-1.1)--(5.04,-1.1);
\node[below,text=ink] at (4.77,-1.17) {carry $>1$};
\draw[<->,gold!85!black] (6.69,.55)--(7.46,.55);
\node[above,text=gold!80!black,align=center] at (7.07,.7) {block gap $>1$};
\node[text=ink,font=\small\bfseries] at (10.7,1.37) {$\Phi_\nu(T)=\sqrt{\nu^2+2\nu T}$};
\end{tikzpicture}
\caption{The proof accounts for every pair of different edges: two local
edges, a local and a cross edge, or two cross edges. Within a cross block,
lexicographic order reduces the checks to fine steps and carries; between
blocks, the macro baseline pays for the full digit excursion. Schematic,
not to scale.}
\label{fig:spectrum}
\end{figure}