\section{Reproducibility protocol}
\label{app:reproducibility}

\subsection{Reference implementation}

The source tree contains:
\begingroup
\sloppy
\begin{itemize}[leftmargin=*]
 \item \path{experiments/brainstorm/resnetl3/hierarchical3_muon_bbp.py}:
 paired population/fixed-sample training and fresh Hessian spectroscopy;
 \item \path{experiments/brainstorm/resnetl3/hierarchical3_sample_scaling.py}:
 nested sample-size campaign;
 \item \path{experiments/brainstorm/resnetl3/test_hierarchical3_muon_bbp.py}:
 focused algebra tests;
 \item \path{experiments/brainstorm/resnetl3/analyze_hierarchical3_confirmatory.py}:
 paired bootstrap, exact sign tests, right-censored sector summaries, and
 confirmatory figures;
 \item \path{experiments/brainstorm/resnetl3/hierarchical3_spectral_time_panel.py}:
 dense coupled fresh-probe spectroscopy, exact Schur geometry at every
 spectral checkpoint, and the master mechanism panel;
 \item \path{experiments/brainstorm/resnetl3/resnet_order4_full_verify_v2.py}:
 exact Hermite setup, coefficient projection, and reduced model;
 \item \path{experiments/muon/lowrank_block_wishart_spectral_experiments.py}:
 block-Wishart construction, clipping, and variational-edge solver.
\end{itemize}
\endgroup

Public static copies of the principal scripts, CSV tables, JSON summaries, and
the scientific audit are available at
\begin{center}
\url{https://janisaiad.github.io/assets/files/hierarchical3/}.
\end{center}

\subsection{Training algorithm}

\begin{algorithm}[h]
\caption{Paired three-layer optimizer comparison}
\label{alg:training}
\begin{algorithmic}[1]
\FOR{\(j=1,\ldots,R\)}
  \STATE draw one initialization \(\theta_0^{(j)}\) and one training sample
  \(\cD_n^{(j)}\)
  \FOR{\(a\in\{\mathrm{GD},0,1/7,1/3\}\)}
    \STATE reset \(\theta\gets\theta_0^{(j)}\)
    \FOR{\(t=0,\ldots,T-1\)}
      \STATE compute the empirical gradient on the fixed sample
      \IF{\(a=\mathrm{GD}\)}
        \STATE apply ordinary gradient descent to every parameter
      \ELSE
        \STATE apply \(\Phi_{a,\eps}\) to \(V,U,A\) and GD to vectors/biases
      \ENDIF
      \STATE backtrack until the empirical objective does not increase
      \STATE at diagnostic steps, evaluate exact population Hermite risk and
      degree errors
    \ENDFOR
  \ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}

\subsection{Fresh spectral protocol}

At a checkpoint, the state is frozen before drawing the fresh sample.  Common
fresh random numbers are used across optimizers at the same checkpoint to
reduce comparison variance.  The implementation stores:
\begin{itemize}[leftmargin=*]
 \item raw and clipped block matrices;
 \item raw full and orthogonal empirical spectral edges;
 \item clipped full and orthogonal empirical spectral edges;
 \item clipped Montanari--Saeed variational edges;
 \item raw and clipped outlier counts and teacher-subspace overlaps;
 \item clipping fraction and full-Hessian operator-norm perturbation.
\end{itemize}
The edge solver never substitutes a raw polynomial block law into the bounded
variational theorem.

\subsection{Statistical reporting}

Paired comparisons use log risk because optimizer effects are multiplicative
and the observed risks span orders of magnitude.  For optimizer \(a\),
\[
 \Delta_a^{(j)}
 =\log\cR_a^{(j)}
 -\log\cR_{\mathrm{GD}}^{(j)}.
\]
Every stored exact population risk is strictly positive, so no data-dependent
pseudocount is introduced.  The confirmatory report includes:
\begin{enumerate}[leftmargin=*]
 \item all paired values, not only means;
 \item median and mean paired log ratio;
 \item a deterministic-seed percentile paired bootstrap interval;
 \item win count and exact two-sided sign-test \(p\)-value;
 \item sector half-error times with an explicit ``not reached'' state;
 \item no universal-exponent fit at fixed latent rank.
\end{enumerate}

\subsection{Commands}

From the repository root, the focused tests are
\begin{verbatim}
PYTHONPATH=$PWD python -m unittest \
  experiments.brainstorm.resnetl3.test_hierarchical3_muon_bbp -v
\end{verbatim}
and the main pilot is reproduced by
\begin{verbatim}
PYTHONPATH=$PWD python \
  experiments/brainstorm/resnetl3/hierarchical3_muon_bbp.py
\end{verbatim}
The confirmatory run changes only the output directory, seed, and number of
paired repetitions:
\begin{verbatim}
PYTHONPATH=$PWD python \
  experiments/brainstorm/resnetl3/hierarchical3_muon_bbp.py \
  --outdir experiments/brainstorm/resnetl3/results/\
hierarchical3_muon_bbp_confirmatory_r20 \
  --seed 424242 --repeats 20 --steps 1200 --diag-every 40
\end{verbatim}
The stored trajectories are analyzed by
\begin{verbatim}
PYTHONPATH=$PWD python \
  experiments/brainstorm/resnetl3/\
analyze_hierarchical3_confirmatory.py \
  experiments/brainstorm/resnetl3/results/\
hierarchical3_muon_bbp_confirmatory_r20/training_trajectories.csv
\end{verbatim}
The time-resolved population/Schur/spectral panel is reproduced by
\begin{verbatim}
PYTHONPATH=$PWD python \
  experiments/brainstorm/resnetl3/\
hierarchical3_spectral_time_panel.py
\end{verbatim}
Its tables include every plotted eigenvalue, every empirical orthogonal edge,
and the exact finite Schur eigenvalues and residual-curvature radii.

\subsection{Numerical claim boundaries}

All reported endpoint statistics are computed from stored CSV rows and checked
against the JSON summary.  Figures are regenerated from those rows.  Numerical
agreement of an analytic formula with automatic differentiation verifies the
implementation of that identity; it is not a proof of an asymptotic theorem.
Conversely, the exact finite identities in
\Cref{sec:hermite}, \Cref{sec:sectors}, and \Cref{sec:rmt}
do not depend on the displayed finite simulations.
