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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{2025-12-01T11:35:33};
  \item Run identifier (integer): \texttt{20251201112105};
  \item JSON file: \texttt{collatz\_local\_profile\_run\_20251201112105.json}; SHA--256: \texttt{a7a201c9080ba42a8ec5423f6523b10c6a3e183b7b8ddb06eff606446f7980f5};
  \item Samples per dyadic band: $N = 100$;
  \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-20251201112105} 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
 60500 & 61500 & 61000.0 & -0.138450 & 2.7426 & 444186.8 & 7.2818 & 0.0000 & \textbf{PASS} & \textbf{PASS} \\
 72500 & 73000 & 72750.0 & -0.138428 & 2.7977 & 527331.1 & 7.2485 & 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\_20251201112105.json}.}
\label{tab:locality-profile-collatz-local-profile-run-20251201112105}
\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.
