\begin{table}[htbp]
\centering\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{lrrrrr}
\toprule
Source / base & $n$ & $q$ & Lower & Upper & Peak states\\
\midrule
$\varphi$, fair iid & 128 & 2 & 0.458519 & 0.481101 & 7 \\
$\varphi$, zero-edge Markov & 128 & 2 & 0.391238 & 0.419235 & 14 \\
Salem, fair iid & 64 & 2 & 0.395946 & 0.543536 & 1477 \\
Salem, zero-edge Markov & 64 & 2 & 0.216449 & 0.419235 & 10 \\
Salem, fair iid & 16 & 3 & 0.028764 & 0.543536 & 2493 \\
$\varphi$, unequal clocks & 24 & 2 & 0.135801 & 0.419235 & 114 \\
$\varphi$, three-state hidden & 16 & 2 & 0.226595 & 0.652983 & 2403 \\
Base $2$, stationary mixture & 16 & 2 & 0.000000 & 0.000000 & 3 \\
Salem, positive correlated & 64 & 2 & 0.284357 & 0.487144 & 3286 \\
$\varphi$, positive correlated & 128 & 2 & 0.398994 & 0.426991 & 18 \\
\bottomrule
\end{tabular}
\caption{Exact-rational rate certificates, displayed outwards to six decimal places. Rates are in nats per branch; peak states are observed layer sizes, not asymptotic bounds.}
\label{tab:certs}
\end{table}