\subsection{Selected decimal values and run statistics}
The selected-values supplement supplies the \emph{complete base-10
expansions} of six selected values: the trillionth Markov value and
its predecessor (counting $1$ first), the last value below and first value
above $10^{1000000}$, and the last value below and first value above
$10^{999999}$. The last of these is the first Markov value with one million
decimal digits. Each integer occupies one plain-text file, compressed
losslessly in the supplementary ZIP, with its address and checksum.
The six files contain 6,043,218 digits in total.
The expansions were first posted on Zenodo on 9 November 2026
\cite{SelectedValues}. The completed unicity certificate and subsequent
rank checks confirmed the two distinct-value ranks on 10 November 2026
at 04:18:33.296330 UTC (9 November, 22:18:33.296330 CST); the decimal
digits are unchanged. The earlier public record exposes its posting date
without a time of day. Table~\ref{tab:trillion-statistics} records the completed
run and the exact selected-value calculations.

\begin{table}[htbp]\centering
\caption{Additional statistics for the completed trillion-value verification.
Logarithmic gaps use base 10 unless otherwise stated.}
\label{tab:trillion-statistics}
\begin{tabular}{@{}p{.53\linewidth}p{.43\linewidth}@{}}\toprule
Statistic & Value or scope\\\midrule
Digits in the selected 999,999,999,999th value & 1,021,609\\
Digits in the selected trillionth value & 1,021,609\\
Logarithmic gap between those two values &
$\log_{10}(\text{larger}/\text{smaller})$\newline
$\approx3.89601074127\times10^{-7}$\\
Smallest logarithmic gap in the entire prefix & Not recorded by this verifier\\
Full $2^{64}$ residue-match groups within bands & 12 groups; 24 member occurrences\\
Bands containing a repeated residue & 11 of 19,287 (0.05703\%)\\
Nominal band width in $\log(3M)$ & 128 natural-log units\\
Same width in $\log_{10}M$ & $128/\log 10\approx55.5897$\\
Bands covering the combined target envelope & 19,287 committed\\
Distinct Markov values with exactly one million digits & 1,916,520\\
Full decimal digits supplied in the six files & 6,043,218\\\bottomrule
\end{tabular}
\end{table}

The band total comprises 13,638 inherited bands and 5,649 extension
bands. Of these, 19,285 have width 128 in $\log(3M)$; the two end bands
have widths 106.866844075 and 125.850758210. Boundary overlaps account
for 5,964,386,594 additional emissions (about 0.542\% of included
configurations). The mean is 57,414,448 emissions per band and the maximum
is 114,821,055. Residue groups and their members are occurrences within
committed bands, not a globally deduplicated pair count.

The global closest-pair gap cannot be inferred from these matches:
singleton keys are discarded, and the verifier keeps no global
nearest-neighbor statistic. The displayed gap belongs only to the specified
consecutive pair. The million-digit value count is the exact difference
$958{,}144{,}590{,}422-958{,}142{,}673{,}902$; the first million-digit
value has rank 958,142,673,903, counting 1 first.
\FloatBarrier