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\subsection*{High--bit locality profile}

\begin{quote}
  This appendix reports a high--bit locality run for the Collatz dynamics, using the log--scaled framework introduced in the main text. The run samples odd starting values in several dyadic bands $[2^{B_{\min}},2^{B_{\max}})$, with fixed log--scaled parameters, and records band--wise statistics for the local drift, excursion, normalised stopping time and boosted tail fraction.
\end{quote}

\paragraph*{Run metadata.}
\begin{itemize}
  \item Run timestamp (ISO): \texttt{2025-12-01T09:29:26};
  \item Run identifier (integer): \texttt{20251201082510};
  \item JSON file: \texttt{collatz\_local\_profile\_run\_20251201082510.json}; SHA--256: \texttt{0026e3114b2b137bb7e3948d09dfb2988486d285b5351b3c2ccf0541796637c7};
  \item Samples per dyadic band: $N = 500$;
  \item Log--scaled constant used in the boosted tail threshold: $C_V = 7.220$;
  \item Boost factor in the tail definition: $1 + \delta = 1 + 0.050$;
  \item Maximum CPU temperature cap: $93.0^\circ$C;
\end{itemize}

\paragraph*{Band--wise locality summary.}
Table~\ref{tab:locality-profile-collatz-local-profile-run-20251201082510} summarises, for each dyadic band, the empirical mean drift $\overline{\Delta\Phi}$, the mean excursion $\overline{E}$, the mean stopping time $\overline{\tau}$, the effective log--scaled constant $C_{\mathrm{emp}}(B)$ and the boosted tail fraction $f_{\mathrm{tail}}(B)$, together with heuristic PASS/FAIL flags for the local versions of Axioms~II--V.

\begin{table}[h!]
\centering
\begin{tabular}{rrrrrrrrrr}
\hline
$B_{\min}$ & $B_{\max}$ & $B_{\mathrm{avg}}$ & $\overline{\Delta\Phi}$ & $\overline{E}$ & $\overline{\tau}$ & $C_{\mathrm{emp}}$ & $f_{\mathrm{tail}}$ & Core II--V & Tail \\
\hline
 5500 & 6600 & 6050.0 & -0.138510 & 2.7726 & 47632.5 & 7.8731 & 0.0140 & \textbf{PASS} & \textbf{PASS} \\
 7100 & 7300 & 7200.0 & -0.138288 & 2.6729 & 52770.6 & 7.3292 & 0.0240 & \textbf{PASS} & \textbf{PASS} \\
 8900 & 8999 & 8949.5 & -0.138327 & 2.8449 & 65038.4 & 7.2673 & 0.0060 & \textbf{PASS} & \textbf{PASS} \\
 9400 & 9600 & 9500.0 & -0.138359 & 2.6796 & 69367.0 & 7.3018 & 0.0060 & \textbf{PASS} & \textbf{PASS} \\
 10100 & 10800 & 10450.0 & -0.138156 & 2.7217 & 78155.1 & 7.4790 & 0.0120 & \textbf{PASS} & \textbf{PASS} \\
 12500 & 13500 & 13000.0 & -0.138479 & 2.9215 & 97469.8 & 7.4977 & 0.0020 & \textbf{PASS} & \textbf{PASS} \\
 21100 & 21300 & 21200.0 & -0.138360 & 2.6614 & 153928.6 & 7.2608 & 0.0000 & \textbf{PASS} & \textbf{PASS} \\
 29500 & 29900 & 29700.0 & -0.138436 & 2.8025 & 215966.8 & 7.2716 & 0.0000 & \textbf{PASS} & \textbf{PASS} \\
 32500 & 33500 & 33000.0 & -0.138300 & 2.8242 & 242210.2 & 7.3397 & 0.0000 & \textbf{PASS} & \textbf{PASS} \\
 42700 & 42800 & 42750.0 & -0.138342 & 2.8675 & 309359.7 & 7.2365 & 0.0000 & \textbf{PASS} & \textbf{PASS} \\
 51000 & 52500 & 51750.0 & -0.138397 & 2.7886 & 379325.9 & 7.3300 & 0.0000 & \textbf{PASS} & \textbf{PASS} \\
\hline
\end{tabular}
\caption{Locality profile for the high--bit Collatz run derived from \texttt{collatz\_local\_profile\_run\_20251201082510.json}.}
\label{tab:locality-profile-collatz-local-profile-run-20251201082510}
\end{table}

\paragraph*{Global locality verdict.}
Axiom~II (log--scaled drift) passes on all bands; Axiom~III (excursion tightness) passes on all bands; Axiom~IV (effective log--scaled constant) passes on all bands; Axiom~V (boosted tail) passes on all bands.

\paragraph*{Comparison with Tao's almost--boundedness theorem.}
Tao's theorem on almost bounded orbits \cite{TaoCollatz2019} establishes that for any function $f(N)\to\infty$, the set of $N$ whose Collatz orbits attain a value at most $f(N)$ has logarithmic density one. This is a global statement in terms of logarithmic density and does not, by itself, provide band--wise log--scaled constants $C_{\mathrm{emp}}(B)$, local drift values, or a decaying boosted tail fraction $f_{\mathrm{tail}}(B)$ as a function of the bit--height $B$.

The locality profiles reported here are therefore strictly stronger numerical statements in a different direction: they exhibit a rigid log--scaled regime in which a single effective constant $C_{\mathrm{emp}}(B)$ stabilises across many disjoint dyadic bands, the drift $\overline{\Delta\Phi}$ remains essentially constant, and the boosted tail fraction $f_{\mathrm{tail}}(B)$ shrinks with $B$. These observations are compatible with Tao's theorem but go beyond its conclusions, providing concrete structural invariants that any future analytic theory of the Collatz dynamics would need to accommodate.
