\section{Fast verification with bounded working storage}
\label{sec:verification}\label{sec:fiftybillion}
\subsection{The completed comparison}
The completed combined run \cite{TrillionVerification} checked
\textbf{1,101,388,071,945 configurations} and certified that each of the
\textbf{first one trillion Markov values in numerical order} is the maximum
of exactly one normalized triple. These counts answer different questions:
the larger is the workload, including possible competitors and conservative
cutoff coverage; the smaller is the guaranteed numerical prefix.
The exact trillionth value and its predecessor are supplied in full in the
selected-values supplement; each has 1,021,609 decimal digits. The title
names the certified numerical prefix, not the larger comparison workload.

Meridian analysis guided the organization by scale. A helm anchors a
family, and its degree determines the limiting logarithmic spacing along
each outward branch. This suggested asking which families can contribute
at a given scale, rather than retaining every value already generated.
Finite-phase corrections still require certified bounds: the degree and
helm alone are not an exact band-entry test.

The production code implements this idea using reduced Farey labels
$p/q$, together with the separate root, and compressed Vieta recurrences.
Groups of neighboring Farey directions share certified matrix bounds.
They let the code select the addresses relevant to a magnitude band
without keeping a separate schedule for every Meridian. The band algorithm
can be formulated entirely in these Farey and matrix terms; the Meridian
work supplied the family-growth viewpoint that guided its development.
The implementation first selects by certified magnitude, then compares
eight-byte residues within each band. Full integers are reconstructed only
for repeated residues. Completed key files are removed before the next
band is processed. Thus the working store depends on the band size rather
than the total number of configurations. Its correctness uses exhaustive
band coverage and exact residue resolution, without the Meridian
collision-exclusion theorems. No controlled comparison isolates a runtime
or storage gain attributable to Meridian coordinates alone.

\begin{table}[htbp]\centering
\caption{Completed comparison and certified numerical coverage.
Overlapping band emissions are not counted as new configurations.}
\label{tab:bandcoverage}
\begin{tabular}{@{}lr@{}}\toprule
Certified numerical-value prefix & 1,000,000,000,000\\
Inner configurations below the cutoff & 1,000,000,419,685\\
Included configurations compared & 1,101,388,071,945\\
Band emissions, including overlap & 1,107,352,458,539\\
Committed bands & 19,287\\
Repeated residue groups / exact members & 12 / 24\\
Equal maxima found & 0\\\bottomrule
\end{tabular}
\end{table}

\subsection{Coverage and why no collision can cross a cleared band}
The finite envelope uses the same height bounds as the earlier
fifty-billion proof \cite{FiftySupplement}. For reduced labels $0<p\le q$,
\[
m_{p/q}\le F_{2(p+q)-1},\qquad
m_{p/q}^{\,2}\ge4\,5^{p+q-2}.
\]
Including the root, the number of configurations of height at most $t\ge2$
is $C(t)=1+\tfrac12\sum_{j=1}^{t}\varphi(j)$, where $\varphi$ is Euler's
totient function. The exact choices are
\[
\begin{gathered}
K=2{,}565{,}100,\qquad B=F_{5{,}130{,}199},\qquad L=3{,}067{,}794,\\
C(K)=1{,}000{,}000{,}419{,}685,\qquad C(L)=1{,}430{,}355{,}600{,}632.
\end{gathered}
\]
Every inner configuration has maximum at most $B$. The exact inequalities
$4\,5^{L-2}\le B^2<4\,5^{L-1}$ ensure that every competitor with maximum
at most $B$ has height at most $L$. This argument counts configurations
before making any claim of unicity.

To schedule the comparisons, 162 cones partition the remaining fraction
labels after the 163 configurations through height 32, including the root.
Here a \emph{cone} means the addresses strictly between two neighboring
Farey labels, after their already retained endpoints are removed.
For endpoints $p_L/q_L<p_R/q_R$, each address in the group has the unique form
\[
(p,q)=u(p_L,q_L)+v(p_R,q_R),\qquad
u,v\ge1,\quad\gcd(u,v)=1.
\]
The endpoints satisfy $p_Rq_L-p_Lq_R=1$. They are obtained by repeatedly
inserting mediants until the next numerator-plus-denominator exceeds 32.
This is a finite partition of address directions, not a partition into
individual Meridians. One Meridian can cross several cones, and one cone
can contain parts of many Meridians.

Figure~\ref{fig:farey-groups} connects the motivating family picture to
this implementation. In the group $0/1<p/q<1/31$, for example,
$(p,q)=(v,u+31v)$. The displayed positive branches of degrees 2 and 4
eventually lie in the rows $v=2$ and $v=4$. Their predictable growth makes
selection by scale natural; the shared cone bounds make that selection
certified for every address in the group, including other families.
Exact positive-matrix
bounds assign each label an interval containing $\log(3m_{p/q})$; the
coefficients are rounded outward at scale $2^{-32}$. The bands cover from
zero through a certified upper endpoint beyond $\log(3B)$, without gaps.
Every configuration whose interval intersects a band is emitted there,
including intervals crossing a boundary. The frozen cone certificate and
integer inequalities are checked before use.

\begin{figure}[p]\centering
\includegraphics[width=\linewidth]{figures/farey_groups_from_meridians.pdf}
\caption{From Meridian growth to certified Farey grouping.
(a) Exact values on two positive branches illustrate the degree-dependent
spacing that motivated magnitude bands; the shaded band is
$128\le\log(3M)<256$. (b) All 162 frozen Farey groups, then a zoom showing
the same families crossing their boundaries. (c) The highlighted group
$0/1<p/q<1/31$ uses $(p,q)=(v,u+31v)$ with positive coprime $u,v$.
Shading selects certified intervals meeting the band in (a); colors pick
out the same branches among all displayed addresses. Groups share matrix
bounds and can contain many Meridian tails. Boundary-crossing intervals
enter both bands. Exact plotted data and the frozen certificate accompany
the source.}
\label{fig:farey-groups}\end{figure}

If two maxima were equal, their actual logarithms would occupy the same
band. Both enclosing intervals would intersect that band, and both exact
residues modulo $2^{64}$ would therefore enter the same comparison.
All keys in a band are partitioned into 32 complete buckets and sorted.
Every member of every repeated-key group is recovered by replaying the
band and is compared exactly. Only then is the band committed and cleared.
Consequently, clearing its keys cannot hide an equality across a boundary.
Neither random-residue assumptions nor family-specific collision exclusions
are used to discharge a repeated group.

All 12 repeated groups contained two configurations. Independent integer
evaluators agreed on their values, and every pair had unequal maxima.
The complete inner population and the outer envelope therefore establish
the first-trillion conclusion in Table~\ref{tab:bandcoverage}.

\subsection{Completion timestamp, elapsed work, CPU time, and storage}
The inherited chain began on \textbf{8 November 2026 at
14:55:01.580398 UTC} (08:55:01.580398 CST). The extension's first retained
band began on \textbf{9 November at 16:53:38.951229 UTC}
(10:53:38.951229 CST). Its last band finished on \textbf{10 November at
03:54:02.997413 UTC}, or \textbf{9 November at 21:54:02.997413 CST}
in St.~Louis. The extension session ran from 16:53:13.152936 UTC on 9 November
(10:53:13.152936 CST) through 03:54:13.271944 UTC on 10 November
(21:54:13.271944 CST on 9 November). The final independent replay ended
at 03:56:16.801036 UTC (21:56:16.801036 CST on 9 November);
verification and replay of a fresh portable extraction ended at
03:59:41.149704 UTC (21:59:41.149704 CST on 9 November).

\begin{table}[htbp]\centering
\caption{Measured resources for the completed retained chain.
The run used eight workers. GiB denotes $2^{30}$ bytes.}
\label{tab:bandresources}
\begin{tabular}{@{}p{.55\linewidth}p{.41\linewidth}@{}}\toprule
Measurement & Recorded amount\\\midrule
Retained active band processing, combined & 86,243.145675 s\newline
\textbf{23 h 57 min 23.146 s}\\
Inherited / extension active processing & 54,470.884742 / 31,772.260933 s\\
Summed native-child CPU, combined & 417,823.596537 s (116.062 h)\\
Inherited / extension native-child CPU & 264,075.377305 / 153,748.219232 s\\
Calendar span, first inherited band to final session end & 133,151.691546 s (36.987 h)\\
Extension calendar span, session endpoints & 39,660.119008 s (11.017 h)\\
Maximum recorded temporary key files & \textbf{1,065,664,392 bytes}\newline
1,016.297 MiB; 0.992 GiB\\
Maximum generator resident memory & 36,468 KiB (35.613 MiB)\\
Verified compressed recovery archive & 38,920,281 bytes (37.117 MiB)\\
Independent final replay & 67.501145 s\\
Portable extraction verification and replay & 71.627535 s\\\bottomrule
\end{tabular}
\end{table}

The under-24-hour figure is retained active elapsed processing, including
the inherited certificate. It excludes pauses, restoration, failed or lost
attempts, and later audits. The combined calendar span exceeds active
processing by 46,908.545871 seconds (13.030 hours); this difference includes
all such gaps and overhead and is not a measure of idle time alone.
The inherited execution record includes 23,436,403,353 configurations
regenerated after an environment loss, with 0.827 hours of lost band work
recorded separately. The inherited execution windows total 17.160 hours, including known lost
work. Three interrupted extension sessions have bounded, rather than exact,
end times. The extension execution windows total 35,390.008304--35,547.876597
seconds; 3,617.747371--3,775.615664 seconds within those windows are outside
retained band time, and 4,112.242411--4,270.110704 seconds fall between
windows. These include overhead and interrupted attempts, not only idle time.
The final evidence preserves the individual session bounds.

The combined retained stages used 49,481.812969 seconds for generation,
25,077.576696 for partitioning, 11,304.684507 for sorting and scans, and
121.285178 for exact-group resolution; controller bookkeeping accounts for
the remaining band time. CPU time sums native-child usage and does not
include an independently totalled parent-process CPU measurement.

The storage peak counts logical temporary key files, including a shard
temporarily duplicated during partitioning. It excludes receipts, source,
backup copies, filesystem overhead, and unrelated work. The generator
memory peak covers that native process only. Continuous whole-workspace
disk and aggregate-memory peaks were not recorded.
\FloatBarrier

\subsection{Replay and retained evidence}\label{sec:verification-evidence}
The final audits checked the frozen sources, all 19,287 linked receipts
(13,638 inherited and 5,649 extension), boundaries, counts, bucket
accounting and retained candidate bytes. A separate checker recomputed the
envelope, independently reconstructed all candidate maxima with polynomial
recurrences, and regenerated every repeated-key band. A fresh extraction
verified all 20,208 manifested archive members and replayed the same final
chain, with no partial or skipped-check flags. This is a receipt-and-candidate
replay, not a second generation of every discarded singleton key.

\Needspace{9\baselineskip}
The frozen recovery archive \cite{TrillionVerification} is
\begin{center}\small\ttfamily
v16\_Trillion\_Verifier\_Recovery.zip
\end{center}
Its SHA-256 and the final extension receipt head are, respectively,
\begin{quote}\footnotesize\ttfamily
6aac7138c8427eb5a3701b27b841e55a75a2e1960c534233261b42375deaeeea\\[5pt]
624c31904f64f8d9434162396a311228f2ece95d495ff2c7dc9d053314ed0c72
\end{quote}
The source includes the final replay reports, resource reconciliation,
statistics and the unchanged historical evidence. Independent here means
a separate implementation within the project, not external peer review.

\subsection{Reusing the completed 500-billion certificate}
The earlier certificate \cite{BandVerification} completed on 9 November
2026 at 15:50:31.046707 UTC (09:50:31.046707 CST), comparing
550,693,634,649 configurations in 13,638 bands. It certified the first
500 billion values, with four repeated groups, 15 h 07 min 50.885 s of
retained active work and a 754,846,136-byte temporary key-file peak.
Its unchanged final receipt head is
\begin{quote}\footnotesize\ttfamily
4642f18be7adf418d5399ffa87825dc2183dae6808ef7516e0dd0b1d704b9b6d.
\end{quote}
The separate extension adds 550,694,437,296 configurations and
8 h 49 min 32.261 s of retained active work. The enlarged envelope admits
no new configuration into an already cleared band: the certified lower
endpoint of every newly admitted interval lies beyond the old cutoff.
Intervals crossing the join are emitted again. Each address is counted
only once, in the band containing its lower interval endpoint. All
550,693,634,649 inherited configurations lie within the new inner height.
The exact join proof and both frozen manifests accompany the evidence.
Here $C(K-1)=999{,}999{,}913{,}445$ and
$C(K)=1{,}000{,}000{,}419{,}685$; thus the inner population suffices for
the claimed first-trillion prefix.

\Needspace{19\baselineskip}
\subsection{Comparison with Tan--Zhang}
Tan and Zhang~\cite{TanZhang2026} report uniqueness for
239,536,376,556 normalized maxima through $10^{500000}$, in the computation
archived on 26 October 2026. Their 48-bit screening and replay modulo
$2^{128}$ provide a useful recent comparison. The 13-group, 91-job screen
processes 3,113,973,008,796 records, including repeated coverage.
Our completed trillion-value prefix compares 1,101,388,071,945
configurations in 1,107,352,458,539 band emissions, using 64-bit residues
within certified magnitude bands and full-integer resolution of repeats.
These larger workload counts are distinct from the numerical-prefix sizes.
Both methods use certified size information. Tan--Zhang use it to enforce
their common cutoff, then compare balanced subtree groups in all pair jobs.
Here the bounds schedule local magnitude bands whose completed keys can
be cleared. Figure~\ref{fig:verification-partitions} shows this difference
and the Meridian reasoning behind our choice of organization.

Storage and timing must be compared by scope. Tan and Zhang report about
184.27 GB of five-byte payload for the largest pair job and recommend
250 GB of available disk; their original target replay used 16 workers.
Our completed run used eight workers and a maximum recorded 1,065,664,392
bytes (1.066 GB; 0.992 GiB) of temporary key files, with 23 h 57 min 23.146 s of
retained active processing. Neither payload figure is a measured
whole-machine storage peak. Tan and Zhang explain that their interrupted,
multi-day run's partial timers do not establish a reliable total runtime,
so a wall-clock or CPU speedup ratio is not justified. The comparison
uses their cited version 1, still current when checked on 9 November 2026.

\begin{figure}[p]\centering
\includegraphics[width=\linewidth]{figures/verification_partition_comparison.pdf}
\caption{Two organizations of the comparison workload. Diagram geometry
is schematic; partition counts and screen totals come from the completed
runs. Meridian growth guided our band schedule, implemented with Farey-cone
bounds. Tan--Zhang~\cite{TanZhang2026} balance their frontier subtrees into 13 groups and screen
all 91 unordered group pairs, including diagonals. Their nonprefix
configurations enter 13 jobs; our certified boundary overlaps cause some
repeat emissions. Ratios count original screening records, excluding
candidate replay and audits; they are not CPU speedups. Storage labels
state the measured or reported components. Both methods require certified
size bounds and modular arithmetic. The comparison does not assert a
whole-machine storage bound or a total-runtime ratio.}
\label{fig:verification-partitions}\end{figure}

\input{sections/09_statistics}

\subsection{Earlier computations and their dates}
The earlier fifty-billion verification \cite{Verification,FiftySupplement}
completed on \textbf{28 October 2026}. Its recorded execution ran from
03:32:57 to 05:38:34 CDT (08:32:57.096455--10:38:33.644612 UTC), including
the final audit. Two coordinator invocations occupied 7,492.102807 seconds;
with the original ten-billion anchor, the cumulative recorded execution
was \textbf{2 h 18 min 50.980 s}. It compared 54,940,210,994 new retained
configurations, resolved all 346 repeated primary-residue pairs, and found
no equal maxima. Its cutoff was $F_{1{,}147{,}147}$. The dated version-4
supplement preserves its coverage proof, hardware, stage timings, and
audit scope. Those historical timings are not part of the new run.

The earlier Claude-assisted search in the working draft
\cite{HistoricalClaude}, pp.~38--40, reports 9,581,528,580 sorted triples
against $10^{100{,}000}$ in \textbf{24 minutes} on a two-core machine with
7 GB of memory. All 21 repeated-residue pairs separated under a second
modulus. Its public announcement \cite{HistoricalAnnouncement} is dated
\textbf{23 October 2026, 09:54:57.771 CDT} (14:54:57.771 UTC) and uses
the wording ``9.3 billion.'' This dates the announcement, not execution.
The original execution timestamps and floating-point cutoff certificate
were not recovered; this remains a historical report, separate from the
later certified numerical prefixes.