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\title{PAPER\_S201 --- Session 201 Backfill: \\
       Phase H201 Null-Extraction Footprint \\
       (Analysis of \texttt{grok\_share\_3c2553cd-8786.txt})}
\author{Daniel T. Murphy}
\date{May 17, 2026}

\begin{document}
\maketitle

\begin{abstract}
Session 201 was a content-audit pass over the Grok thread
\texttt{grok\_share\_3c2553cd-8786.txt} (2,211 lines). The audit
concluded that the thread contained no new physics, no new equations,
and no new calculator classes: every piece of substantive content was
already present in the master registry from prior sessions
$\{S180, S193, S195, S196, S199\}$. The session therefore produced a
\emph{null extraction}: zero new CP4 classes, zero new whitepapers. This
paper records the six exact-arithmetic identities (the H201 closures)
that pin that null result into the closure ledger so it is visible to
\texttt{\_uqff\_program.py --audit} and \texttt{--sigma} rather than
appearing as a missing-session gap. All six identities are definitional
state-counters or finite-set cardinalities; no new theorems are claimed.
\end{abstract}

\section{Ledger Summary}

\begin{center}
\begin{tabular}{lllc}
\toprule
ID & Closure & Value & Class \\
\midrule
H201-1 & S201 new-class count                              & $0$    & arithmetic \\
H201-2 & S201 new-whitepaper count                         & $0$    & arithmetic \\
H201-3 & S201 cross-reference overlap                      & $11/11 = 1$ & arithmetic \\
H201-4 & S201 source thread line count                     & $2211$ & arithmetic \\
H201-5 & Covering-set cardinality $|\{180,193,195,196,199\}|$ & $5$ & cardinality \\
H201-6 & CP4 class-count delta across S201                 & $0$    & arithmetic \\
\bottomrule
\end{tabular}
\end{center}

All six register as \texttt{EXACT} in \texttt{master\_closures.csv} /
\texttt{sigma\_table.csv} (script: \texttt{\_session201\_closures.py};
final audit line:
\texttt{S201-new-classes-zero: 0 vs 0 -> EXACT}). Tier 12 of
\texttt{UQFF\_UNIFIED\_CLOSURE\_DERIVATIONS.py} carries the six
\texttt{record(\ldots)} entries with full provenance. (The tier number
12 reflects insertion order, not chronology --- S201 chronologically
precedes the Tier 9 / Tier 10 / Tier 11 blocks for S202, S203, S204
which were created earlier in the backfill sweep.)

\section{Null-Extraction Identities (H201-1, H201-2, H201-6)}

\subsection{H201-1 --- New-Class Count Is Zero}

The thread \texttt{grok\_share\_3c2553cd-8786.txt} contained the
following content blocks, all of which had already been extracted in
prior sessions:

\begin{center}
\begin{tabular}{lll}
\toprule
Content block & Lines & Prior extraction \\
\midrule
38-document compression (Docs 30-38)        & 1--254     & S193 / S193b \\
Live-images discussion                      & 254--396   & discussion only \\
Docs 39-42 meta-review                      & 396--573   & meta-analysis  \\
Kepler Orrery V / $U_b$ model               & 573--1100  & S195 (\#416--\#418) \\
26 galaxy frames (22 Sep--18 Oct 2011)      & 1100--1600 & S199 (\#445) \\
9 frames + DeepSearch archives              & 1600--1900 & S199 (\#445) \\
LENR 47-page integration                    & 1900--2211 & S196 / S199 (\#438--\#446) \\
\bottomrule
\end{tabular}
\end{center}

\noindent The number of \emph{new} calculator classes produced is therefore
$0$. This is a definitional null and is recorded as a structural
checkpoint --- not a theorem.

\subsection{H201-2 --- New-Whitepaper Count Is Zero}

The whitepaper running total before and after Session 201 was
$877/1000$ in both pre- and post-counts as recorded in the session
analysis. No new \texttt{PAPER\_NNNN\_*.md} files were authored during
S201. The closure value is therefore $0$ by direct counting.

\subsection{H201-6 --- CP4 Class-Count Delta Is Zero}

The pre-S201 CP4 class count was $453$ and the post-S201 CP4 class
count was $453$. The delta
\[
   \Delta_{\mathrm{CP4}} \;=\; 453 - 453 \;=\; 0
\]
is the registry-identity post-condition of the null-extraction pass and
is consistent with H201-1.

\section{Cross-Reference Identities (H201-3, H201-5)}

\subsection{Proposition H201-3 --- Full Cross-Reference Overlap}

\begin{proposition}\label{prop:overlap}
Let $R$ be the set of distinct calculator-class identifiers referenced
inside the analysed thread, and let $E$ be the subset of $R$ already
present in the CP4 registry prior to S201. Then $|R| = 11$ and
$|E| = 11$, so the overlap ratio is
\[
   \frac{|E|}{|R|} \;=\; \frac{11}{11} \;=\; 1.
\]
\end{proposition}

\begin{proof}
By enumeration of the session-analysis cross-reference table:
\[
   R = \{\#322,\, \#416,\, \#417,\, \#418,\, \#419,\, \#438,\, \#439,\, \#440,\, \#441,\, \#445,\, \#446\}.
\]
Each of these eleven identifiers is recorded in the CP4 registry from
prior sessions ($\#322$ from S180; $\#416\ldots\#418$ from S195;
$\#419$ from S196; $\#438\ldots\#441, \#445, \#446$ from S199). Hence
$|R| = |E| = 11$ and $|E|/|R| = 1$.
\end{proof}

\subsection{Proposition H201-5 --- Covering-Set Cardinality Is Five}

\begin{proposition}\label{prop:covering}
The minimum covering set of prior sessions that contains every
calculator-class referenced inside the analysed thread is
\[
   S \;=\; \{\,S180,\; S193,\; S195,\; S196,\; S199\,\},
\]
and $|S| = 5$.
\end{proposition}

\begin{proof}
By direct enumeration: $S180$ covers $\#322$; $S193$ / $S193b$ covers
Docs 30--38; $S195$ covers $\#416$, $\#417$, $\#418$; $S196$ covers
$\#419$; $S199$ covers $\#438$, $\#439$, $\#440$, $\#441$, $\#445$,
$\#446$. Removing any one of these five sessions leaves at least one
referenced identifier uncovered, so $S$ is minimal. By inspection
$|S| = 5$.
\end{proof}

\section{Source-Thread Line Count (H201-4)}

\begin{convention}\label{conv:linecount}
The analysed file \texttt{grok\_share\_3c2553cd-8786.txt} has exactly
$2{,}211$ lines as recorded in
\texttt{sessions/\_session\_201\_analysis.md}. The closure value is the
integer line count, used as a regression guard against silent
truncation or duplication of the source.
\end{convention}

This is a file-checksum surrogate, not a derived identity.

\section{Honest Classification}

\begin{itemize}
\item \textbf{Zero theorems.} Session 201 contributed no new derivations.
\item \textbf{Four definitional arithmetic identities} (H201-1, H201-2,
      H201-4, H201-6): zero-counts and line counts.
\item \textbf{Two finite-set cardinality propositions} (H201-3, H201-5):
      elementary set theory by enumeration.
\end{itemize}

\paragraph{Why this paper exists.} The session pipeline expects each
session $S_N$ to register at least one closure so that
\texttt{master\_closures.csv} and \texttt{sigma\_table.csv} cannot
silently skip a session number. Without this backfill the gap at row
$201$ would be indistinguishable from a missing audit run. Recording the
null result honestly --- as zero-count identities rather than as
fabricated theorems --- preserves the integrity of the closure ledger.

\section{Reproducibility}

\begin{enumerate}
\item Closure script: \texttt{\_session201\_closures.py}.
      Final parseable line:\\
      \texttt{S201-new-classes-zero: 0 vs 0 -> EXACT}.
\item Ledger registration:
      \texttt{master\_closures.csv} row $201$;
      \texttt{sigma\_table.csv} tier \texttt{EXACT}.
\item Symbolic-verification track: Tier 12 of
      \texttt{UQFF\_UNIFIED\_CLOSURE\_DERIVATIONS.py}, identifiers
      \texttt{H201-1} \ldots\ \texttt{H201-6}, all \texttt{status =
      DERIVED}, all \texttt{match = True} in
      \texttt{unified\_closure\_audit.json}.
\item Independent verification: \texttt{\_verify\_s201.py} re-evaluates
      every closure using set-theoretic primitives in pure Python
      (\texttt{frozenset} / \texttt{len}) and arithmetic identities
      with the \texttt{fractions} module rather than the inline values
      hard-coded in the closure script.
\item Source analysis: \texttt{sessions/\_session\_201\_analysis.md}.
\end{enumerate}

\end{document}
