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\subsection{Bit window $[2^{500},2^{3500})$}

\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\_154548.json}
\end{center}

Global metadata for this run include:
\begin{itemize}
  \item Global bit window: $[2^{500},2^{3500})$ (\texttt{bits\_low\_global = 500}, \texttt{bits\_high\_global = 3500});
  \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 = 20251129154548};
  \item Run identifier: \texttt{timestamp\_start = \"20251129\_154548\"};
  \item Wall--clock runtime: \texttt{runtime\_seconds} $\approx 797.70$s;
  \item JSON file SHA--256: \texttt{9631229dbe9189b9d516bf95efb2c4fc31c2a8cc8765646ce59ce21cf261f621};
\end{itemize}

For range block 1, covering $[2^{500},2^{1000})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 10000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{968f249dbc6ea3a8722d36a28f210ed842471ddb1a017773382cfda559b94830}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 6489};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.4257$;
  \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} $\approx 0.64890$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8386$;
  \item Maximum excursion: \texttt{max\_excursion} $= 17.0$;
  \item 95th percentile: \texttt{p95} $= 6.0$; 99th percentile: \texttt{p99} $= 8.0$;
  \item Fraction with excursion $> 10$: \texttt{fraction\_E10} $\approx 0.00140$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 12000};
  \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} $\approx 7224.01$; median: \texttt{tau\_median = 7212}; maximum: \texttt{tau\_max = 8802};
  \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} $\approx 0.1065$ (with \texttt{tail\_count = 1065});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2344$, \texttt{C\_empirical\_median} $\approx 7.2238$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

For range block 2, covering $[2^{1000},2^{1500})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 10000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{513d3306b11f69b484f3568fe9efbb05e7a486dba3b04aa146cc67014e684813}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 5059};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 3.2291$;
  \item Empirical median drop: \texttt{median\_delta} $\approx 3.1699$;
  \item Theoretical target: \texttt{theoretical\_delta} $\approx 0.0850$;
  \item Fraction of positive drops: \texttt{positive\_rate} $\approx 0.50590$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8286$;
  \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} $\approx 0.00200$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 18000};
  \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} $\approx 10848.60$; median: \texttt{tau\_median = 10831}; maximum: \texttt{tau\_max = 12880};
  \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} $\approx 0.0660$ (with \texttt{tail\_count = 660});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2393$, \texttt{C\_empirical\_median} $\approx 7.2256$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

For range block 3, covering $[2^{1500},2^{2000})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 10000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{ec5f139fcac0c5a3dab12138626f9ae18c3157bb158fa3b9647f424e52ca3992}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 4970};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 3.1766$;
  \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} $\approx 0.49700$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8383$;
  \item Maximum excursion: \texttt{max\_excursion} $= 16.0$;
  \item 95th percentile: \texttt{p95} $= 6.0$; 99th percentile: \texttt{p99} $= 8.0$;
  \item Fraction with excursion $> 10$: \texttt{fraction\_E10} $\approx 0.00240$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 24000};
  \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} $\approx 14453.07$; median: \texttt{tau\_median = 14441}; maximum: \texttt{tau\_max = 16928};
  \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} $\approx 0.0418$ (with \texttt{tail\_count = 418});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2317$, \texttt{C\_empirical\_median} $\approx 7.2261$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

For range block 4, covering $[2^{2000},2^{2500})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 10000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{07ccc8d545e2e9913e48d53258583c2fdc43933ed4f67e42c591751d0bd0b4b9}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 6654};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.3905$;
  \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} $\approx 0.66540$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8203$;
  \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} $\approx 0.00160$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 30000};
  \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} $\approx 18056.93$; median: \texttt{tau\_median = 18042}; maximum: \texttt{tau\_max = 20700};
  \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} $\approx 0.0245$ (with \texttt{tail\_count = 245});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2270$, \texttt{C\_empirical\_median} $\approx 7.2214$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

For range block 5, covering $[2^{2500},2^{3000})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 10000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{12b8c9c9f7b8ce92d4c242f387ece270b67c5c446eea00a96765be7c49ce7a2a}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 6733};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.4289$;
  \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} $\approx 0.67330$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8234$;
  \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} $\approx 0.00150$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 36000};
  \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} $\approx 21669.80$; median: \texttt{tau\_median = 21661}; maximum: \texttt{tau\_max = 24318};
  \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} $\approx 0.0171$ (with \texttt{tail\_count = 171});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2268$, \texttt{C\_empirical\_median} $\approx 7.2245$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

For range block 6, covering $[2^{3000},2^{3500})$, we have:
\begin{itemize}
  \item \texttt{num\_samples} $= 10000$ odd starting values;
  \item SHA--256 of the sorted sample list:
\begin{center}
\texttt{8fe1b49828c2fcf0313d84f285b545a05adfea39e6d4fd80c232789247955dc4}
\end{center}
\end{itemize}

\paragraph{Axiom II (one--step log--scaled compression).}
\begin{itemize}
  \item Effective sample size: \texttt{samples = 6577};
  \item Empirical mean drop: \texttt{avg\_delta} $\approx 2.4046$;
  \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} $\approx 0.65770$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom III (excursion tightness).}
\begin{itemize}
  \item Empirical mean excursion: \texttt{avg\_excursion} $\approx 2.8452$;
  \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} $\approx 0.00150$;
  \item Axiom status: \texttt{passed\_axiom = True}.
\end{itemize}

\paragraph{Axiom IV (convergence density).}
\begin{itemize}
  \item Cutoff scale: \texttt{K\_max = 42000};
  \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} $\approx 25296.07$; median: \texttt{tau\_median = 25281}; maximum: \texttt{tau\_max = 28578};
  \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} $\approx 0.0098$ (with \texttt{tail\_count = 98});
  \item Empirical constants: \texttt{C\_empirical\_mean} $\approx 7.2304$, \texttt{C\_empirical\_median} $\approx 7.2269$;
  \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 $[2^{500},2^{3500})$ 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}
