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\subsection{Bit window $[2^{30000},2^{30500})$}

\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\_214946.json}
\end{center}

Global metadata for this run include:
\begin{itemize}
  \item Global bit window: $[2^{30000},2^{30500})$ (\texttt{bits\_low\_global = 30000}, \texttt{bits\_high\_global = 30500});
  \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 = 20251129214946};
  \item Run identifier: \texttt{timestamp\_start = \"20251129\_214946\"};
  \item Wall--clock runtime: \texttt{runtime\_seconds} $\approx 474.48$s;
  \item JSON file SHA--256: \texttt{309498e99b999c6e611caaf1d3e1aa2da607404678d872be341188bbe1262b8d};
\end{itemize}

For range block 1, covering $[2^{30000},2^{30500})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 250$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{14b5f0f9547830f538bf25cd4757beee2fd350f84d59f259f6cee30a948e2f88}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 126};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.8806$;
  \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} $\approx 0.50400$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8440$;
  \item Maximum excursion: \texttt{max\_excursion} $= 9.0$;
  \item 95th percentile: \texttt{p95} $= 6.0$; 99th percentile: \texttt{p99} $= 7.0$;
  \item Fraction with excursion $> 10$: \texttt{fraction\_E10} $\approx 0.00000$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 366000};
  \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} $\approx 220341.16$; median: \texttt{tau\_median = 220257}; maximum: \texttt{tau\_max = 226597};
  \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} $\approx 0.0000$ (with \texttt{tail\_count = 0});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2246$, \texttt{C\_empirical\_median} $\approx 7.2218$;
  \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 $[2^{30000},2^{30500})$ 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}
