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\subsection{Bit window $[2^{3500},2^{5000})$}

\paragraph{JSON file and global meta.}
This certified JSON log records a run of the global log--scaled axiomatic protocol for a Collatz bit window driven by the V13 implementation. The main summary file for this run is:

\begin{center}
\texttt{collatz\_logscale\_summary\_20251129\_192150.json}
\end{center}

Global metadata for this run include:
\begin{itemize}
  \item Global bit window: $[2^{3500},2^{5000})$ (\texttt{bits\_low\_global = 3500}, \texttt{bits\_high\_global = 5000});
  \item Execution mode: \texttt{SAMPLED\_RANGES};
  \item Maximum CPU temperature: $93.0^\circ$C (\texttt{max\_cpu\_temp = 93.0});
  \item Sleep between thermal checks: \texttt{sleep\_seconds = 15.0};
  \item Random seed base: \texttt{seed\_base = 20251129192150};
  \item Run identifier: \texttt{timestamp\_start = \"20251129\_192150\"};
  \item Wall--clock runtime: \texttt{runtime\_seconds} $\approx 1893.74$s;
  \item JSON file SHA--256: \texttt{300b696e8971df644844778f5a98bde9116fce3db8ebf594bfc2638cccd99704};
\end{itemize}

For range block 1, covering $[2^{3500},2^{4000})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 15000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{2c7b74f20236e69c82fea050fb1e197401525538ca769ff0f783d163aca2c956}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 9871};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.4108$;
  \item Empirical median drop: \texttt{median\_delta} $\approx 1.5850$;
  \item Theoretical target: \texttt{theoretical\_delta} $\approx 0.0850$;
  \item Fraction of positive drops: \texttt{positive\_rate} $\approx 0.65807$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8167$;
  \item Maximum excursion: \texttt{max\_excursion} $= 18.0$;
  \item 95th percentile: \texttt{p95} $= 6.0$; 99th percentile: \texttt{p99} $= 8.0$;
  \item Fraction with excursion $> 10$: \texttt{fraction\_E10} $\approx 0.00180$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 48000};
  \item Density at cutoff: \texttt{density\_at\_Kmax} $\approx 1.00000$ with threshold \texttt{density\_threshold = };
  \item Non--converged count: \texttt{non\_converged = 0} (rate \texttt{non\_converged\_rate = 0.00000});
  \item Mean stopping time: \texttt{tau\_mean} $\approx 28905.97$; median: \texttt{tau\_median = 28897}; maximum: \texttt{tau\_max = 32295};
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom V (log--scaled tail stability).}
\begin{itemize}
  \item Fixed structural parameters: \texttt{C\_V\_fixed = 7.2}, \texttt{delta = 0.08}, \texttt{theta\_min = 0.001}, \texttt{theta\_max = 0.5};
  \item Empirical tail fraction: \texttt{f\_tail} $\approx 0.0067$ (with \texttt{tail\_count = 100});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2291$, \texttt{C\_empirical\_median} $\approx 7.2272$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

For range block 2, covering $[2^{4000},2^{4500})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 15000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{30dbb0dfadec5cedd3f0446cbecd9657183e849fc6bb00aefeea1486fedb5b97}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 8677};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.7483$;
  \item Empirical median drop: \texttt{median\_delta} $\approx 1.5850$;
  \item Theoretical target: \texttt{theoretical\_delta} $\approx 0.0850$;
  \item Fraction of positive drops: \texttt{positive\_rate} $\approx 0.57847$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8401$;
  \item Maximum excursion: \texttt{max\_excursion} $= 14.0$;
  \item 95th percentile: \texttt{p95} $= 6.0$; 99th percentile: \texttt{p99} $= 8.0$;
  \item Fraction with excursion $> 10$: \texttt{fraction\_E10} $\approx 0.00153$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 54000};
  \item Density at cutoff: \texttt{density\_at\_Kmax} $\approx 1.00000$ with threshold \texttt{density\_threshold = };
  \item Non--converged count: \texttt{non\_converged = 0} (rate \texttt{non\_converged\_rate = 0.00000});
  \item Mean stopping time: \texttt{tau\_mean} $\approx 32524.66$; median: \texttt{tau\_median = 32489}; maximum: \texttt{tau\_max = 36744};
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom V (log--scaled tail stability).}
\begin{itemize}
  \item Fixed structural parameters: \texttt{C\_V\_fixed = 7.2}, \texttt{delta = 0.08}, \texttt{theta\_min = 0.001}, \texttt{theta\_max = 0.5};
  \item Empirical tail fraction: \texttt{f\_tail} $\approx 0.0056$ (with \texttt{tail\_count = 84});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2300$, \texttt{C\_empirical\_median} $\approx 7.2218$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

For range block 3, covering $[2^{4500},2^{5000})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 15000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{e79c2c7eb8144178269571029017c849ef01e7faa8e4a5ce3eef6966e7b3ddf9}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 8804};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.6639$;
  \item Empirical median drop: \texttt{median\_delta} $\approx 1.5850$;
  \item Theoretical target: \texttt{theoretical\_delta} $\approx 0.0850$;
  \item Fraction of positive drops: \texttt{positive\_rate} $\approx 0.58693$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8278$;
  \item Maximum excursion: \texttt{max\_excursion} $= 14.0$;
  \item 95th percentile: \texttt{p95} $= 6.0$; 99th percentile: \texttt{p99} $= 8.0$;
  \item Fraction with excursion $> 10$: \texttt{fraction\_E10} $\approx 0.00187$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 60000};
  \item Density at cutoff: \texttt{density\_at\_Kmax} $\approx 1.00000$ with threshold \texttt{density\_threshold = };
  \item Non--converged count: \texttt{non\_converged = 0} (rate \texttt{non\_converged\_rate = 0.00000});
  \item Mean stopping time: \texttt{tau\_mean} $\approx 36136.64$; median: \texttt{tau\_median = 36125}; maximum: \texttt{tau\_max = 40139};
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom V (log--scaled tail stability).}
\begin{itemize}
  \item Fixed structural parameters: \texttt{C\_V\_fixed = 7.2}, \texttt{delta = 0.08}, \texttt{theta\_min = 0.001}, \texttt{theta\_max = 0.5};
  \item Empirical tail fraction: \texttt{f\_tail} $\approx 0.0029$ (with \texttt{tail\_count = 44});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2294$, \texttt{C\_empirical\_median} $\approx 7.2273$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Global axiom status.}
The top--level field \texttt{core\_axioms\_I\_to\_V\_passed = True} indicates whether all five log--scaled axioms (I--V) are simultaneously validated for this run within the numerical thresholds fixed in the main text.

\paragraph{English technical summary.}
\begin{quote}
This certified JSON log corresponds to the bit window $[2^{3500},2^{5000})$ and was produced by the V13 implementation of the log--scaled axiomatic protocol. It records global metadata, frozen structural parameters, SHA--256 hashes of the sorted sample list, and empirical statistics for the log--scaled Axioms~I--V (one--step compression, excursion tightness, convergence density and tail stability).
 The final field \texttt{core\_axioms\_I\_to\_V\_passed = true} indicates that, for this run, the full log--scaled regime required by the conditional global theorem is numerically validated within the prescribed thresholds.
\end{quote}
