\section{Replication}\label{app:replication}

Everything reported is regenerated from committed raw outputs by the scripts in \texttt{experiments/}; no number in the tables or figures is typed by hand. The directory \texttt{references/initial\_pamle/} contains the prior draft, its archive and SHA-256 checksums; its Monte Carlo (seed 20261028, $N = 2^{20}$, $m \in \{256,64,16\}$, 500 replications) was re-run and matched to $5\times10^{-13}$ before any change was made, and the test \texttt{tests/test\_reproducibility.py} reproduces its summary table from the new code.

\paragraph{Commands.}
\begin{verbatim}
python3 -m pip install -r requirements.txt
pytest -q                                   # unit tests, exact identities, protocol
python3 theory/symbolic_checks.py           # SymPy verification of all identities
python3 experiments/run_monte_carlo.py --config experiments/configs/main_exponential.yaml
python3 experiments/scatter_cluster.py --configs experiments/configs/main_logistic.yaml \
        experiments/configs/main_poisson.yaml experiments/configs/robustness.yaml --launch
python3 experiments/run_boundary.py; python3 experiments/run_ablations.py --model logistic
python3 experiments/run_skewed.py
python3 experiments/run_benchmarks.py --config experiments/configs/benchmarks.yaml \
        --out results/raw/benchmarks.jsonl   # requires the cluster
python3 experiments/process_results.py --figures; python3 experiments/make_tables.py
bash scripts/compile_paper.sh; bash scripts/validate_pdf.sh
\end{verbatim}
Seeds are derived from \texttt{seed\_base = 20261028} and a hash of the cell description; every raw file has a JSON manifest with git commit, Python and NumPy versions, host, wall time and the SHA-256 of the output (\texttt{experiments/EXPERIMENT\_MANIFEST.csv}). Monte Carlo cells were run on the laptop (Apple M5 Max, Python 3.14.4, NumPy 2.4.4) and on 18 cluster nodes (Python 3.9.6/3.14, NumPy 2.0.2/2.5.4); floating-point results differ across these environments at the level of $10^{-12}$ relative, far below the Monte Carlo standard errors reported.