Reproducibility Statement
(Entirely generated by AI)

All experiments were executed by the AI agent in a controlled Python environment with fixed random seeds, bounded population sizes, and capped generations. Analytic benchmarks (Rastrigin, Rosenbrock, Ackley) and the synthetic linear regression task are deterministic and fully specified in Section~3. The discovered rules, intermediate elites, and evaluation curves are logged in JSON format (\texttt{archive\_v02.json}, \texttt{comparison\_v02.json}, \texttt{comparison\_linreg\_v02.json}). Figures in the paper were generated directly from these logs. The best evolved rule is stored in \texttt{best\_rule\_v02.json}, allowing exact reproduction of reported results. All code used for the DSL, evolutionary loop, and plotting was produced autonomously by the AI agent and can be released alongside this manuscript. These steps ensure that independent researchers—or AI agents—can reproduce the optimizer discovery process and replicate all figures without ambiguity.

\section{Agent Protocol}

\paragraph{System architecture.}
The AI agent is a large language model (LLM) connected to a persistent Python execution environment. The LLM issues instructions (e.g., ``implement an evolutionary loop over the DSL’’), generates Python code, executes it, inspects outputs, and decides on further actions. This closed loop is repeated until results are obtained, figures are generated, and manuscript sections are drafted.

\paragraph{Interaction cycle.}
A typical cycle consists of:
\begin{enumerate}
    \item \textbf{Hypothesis generation:} The agent proposes a modification to the DSL or the experimental setup.
    \item \textbf{Implementation:} The agent writes Python code to realize this idea.
    \item \textbf{Execution:} The code is run in the sandbox, producing logs, artifacts, and figures.
    \item \textbf{Analysis:} The agent reads numerical outputs, plots figures, and summarizes patterns.
    \item \textbf{Documentation:} Based on the analysis, the agent drafts text for the manuscript.
\end{enumerate}

\paragraph{Human involvement.}
Human collaborators acted solely as high-level overseers:
\begin{itemize}
    \item Selecting the overall domain (``optimizer discovery’’ vs. other candidate ideas).
    \item Requesting section-by-section drafting to fit the page limit.
    \item Ensuring the paper followed the Agents4Science LaTeX template.
\end{itemize}
They did not design the DSL, write code, edit figures, or author manuscript text. All technical content—including rules, experiments, plots, and section drafts—was produced by the AI agent.

\paragraph{Example session trace.}
To illustrate, one early cycle proceeded as follows:
\begin{itemize}
    \item \textbf{Agent:} ``Define analytic benchmarks such as Rastrigin, Rosenbrock, and Ackley with gradient functions.’’
    \item \textbf{Agent-generated code:} Python functions for losses and gradients.
    \item \textbf{Execution:} Verified gradients by finite differences.
    \item \textbf{Agent:} ``Implement an evolutionary search with mutation probability 0.3 and population size 32.’’
    \item \textbf{Execution:} Run for 20 generations, record history.
    \item \textbf{Output:} Loss curves, elite archive, JSON logs.
    \item \textbf{Agent:} ``Plot convergence and Pareto cloud; save as \texttt{evo\_history\_v02.png} and \texttt{pareto\_v02.png}.’’
\end{itemize}

This process, repeated and refined, yielded all figures and tables in the main text. The appendix thus provides transparency: the AI agent was not merely a narrative generator but an integrated research system executing the full loop from idea to manuscript.

\section{Reproducibility Checklist}

We follow the NeurIPS 2025 reproducibility guidelines.  

\paragraph{Experimental settings.}
\begin{itemize}
    \item \textbf{Benchmarks:} Rastrigin, Rosenbrock, and Ackley functions (10D) with analytic gradients; synthetic linear regression (200 samples, 20 features, Gaussian noise).
    \item \textbf{Optimizer DSL:} Symbolic rules parameterized by coefficients ($\beta_m, \beta_v, a_1, a_2$), normalization exponent $p$, learning rate $\eta$, and epsilon $\epsilon$.
    \item \textbf{Evolutionary loop:} Population size 32, 6 elites, 20 generations, mutation probability 0.3.
    \item \textbf{Training budget:} 300 steps per evaluation, clipping at 10.0 to prevent instability.
    \item \textbf{Seeds:} Runs averaged over 2–3 random seeds per benchmark.
\end{itemize}

\paragraph{Compute.}
\begin{itemize}
    \item Experiments ran on a CPU-only Python environment.
    \item Each evolutionary run required $<5$ minutes wall-clock time.
    \item Total compute footprint was $<1$ GPU-hour equivalent; no large-scale training was used.
\end{itemize}

\paragraph{Logging and artifacts.}
\begin{itemize}
    \item Best rule and timing saved in \texttt{best\_rule\_v02.json}.
    \item Cross-bench comparisons in \texttt{comparison\_v02.json}.
    \item Linear regression comparisons in \texttt{comparison\_linreg\_v02.json}.
    \item Full elite archive in \texttt{archive\_v02.json}.
    \item Figures generated deterministically from these JSON files.
\end{itemize}

\paragraph{Availability.}
\begin{itemize}
    \item All code, JSON logs, and figures were generated by the AI agent and is anonymously released with this paper at \texttt{https://anonymous.4open.science/r/anon-submission-1-4080}.
    \item No proprietary datasets were used.
    \item Experiments are fully reproducible on a standard Python 3.10 environment with NumPy and Matplotlib.
\end{itemize}