1\section{Replication}\label{app:replication}23Everything 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.45\paragraph{Machine-checked algebra.} The deterministic identities used by the proofs---the Vandermonde constraints and the constants of \cref{tab:weights}, the exact biases \eqref{eq:exp-bias23}, the Newton identities \eqref{eq:exp-newton} with the $q$-fold recursion and the basin of attraction, the boundary constant of \cref{thm:exponential}(v), the exponent algebra behind \cref{thm:frontier}, the Gamma and Beta inverse moments and the exact variance \eqref{eq:exp-varJ2}---are formalized and checked in Lean~4 with Mathlib (\texttt{theory/lean/}, 27 theorems, no \texttt{sorry}; \texttt{scripts/check\_lean.sh}). The probabilistic steps ($O_p$ bounds, central limit theorem, Slutsky) are not formalized.67\paragraph{Commands.}8{\footnotesize\begin{verbatim}9python3 -m pip install -r requirements.txt10pytest -q # unit tests, exact identities, protocol11python3 theory/symbolic_checks.py # SymPy verification of all identities12python3 experiments/run_monte_carlo.py --config experiments/configs/main_exponential.yaml13python3 experiments/scatter_cluster.py --launch \14 --configs experiments/configs/main_logistic.yaml \15 experiments/configs/main_poisson.yaml \16 experiments/configs/robustness.yaml # cluster17python3 experiments/run_boundary.py18python3 experiments/run_ablations.py --model logistic19python3 experiments/run_skewed.py20python3 experiments/run_benchmarks.py \21 --config experiments/configs/benchmarks.yaml \22 --out results/raw/benchmarks.jsonl # cluster23python3 experiments/process_results.py --figures24python3 experiments/make_tables.py25bash scripts/compile_paper.sh26bash scripts/validate_pdf.sh27\end{verbatim}}28Seeds 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.29