\section{Experiments}
\label{sec:experiments}

\subsection{Questions and protocol}

The experiments address four questions:
\begin{enumerate}[leftmargin=*]
 \item Do the exact degree errors move in the order \(2\to3\to4\)?
 \item Does a singular-value update accelerate a late sector at matched data and
 initialization?
 \item Do teacher-aligned eigenmodes move relative to the fresh orthogonal
 block-Wishart bulk?
 \item Does increasing sample size improve exact population risk in a manner
 consistent with sequential feature recovery?
\end{enumerate}

The main pilot uses
\[
 k=p=q=3,\quad n_{\mathrm{train}}=1024,\quad
 1200\text{ steps},\quad n_q=7,\quad R=3
\]
paired repetitions.  The first and second direct matrices are untied.
The teacher includes diagonal and mixed Hermite coefficients in degrees
\(2,3,4\).  Gradient descent uses learning rate \(0.02\).  Spectral matrix
blocks use base rate \(0.004\); their vector and bias parameters use \(0.02\).
All methods use loss-decreasing backtracking.  For each repetition, every
optimizer receives the same initialization and training sample.

Population test risk is evaluated by exact tensor Gauss--Hermite quadrature,
not a finite test set.  Fresh Hessians use \(d=32\),
\(d_\perp=29\), \(n_{\mathrm{fresh}}=145\), and checkpoints
\(0,400,800,1200\).  The block cap is \(L=100\).

\subsection{Paired endpoint risk}

\begin{table}[t]
\centering
\caption{Main pilot endpoint exact population risks.  ``Wins'' counts paired
runs with smaller risk than GD.  With only three pairs, all summaries are
descriptive.}
\label{tab:endpoint}
\begin{tabular}{lrrrr}
\toprule
Optimizer & risks by pair & mean & median & wins/GD\\
\midrule
GD & \(0.7569,0.9416,0.04620\) & \(0.5816\) & \(0.7569\) & --\\
Muon \(a=0\) & \(0.2712,0.08062,0.08712\) & \(0.1463\) & \(0.08712\) & \(2/3\)\\
Muon \(a=1/7\) & \(0.4114,0.06553,0.6195\) & \(0.3655\) & \(0.4114\) & \(2/3\)\\
Muon \(a=1/3\) & \(0.08887,0.05081,0.2975\) & \(0.1457\) & \(0.08887\) & \(2/3\)\\
\bottomrule
\end{tabular}
\end{table}

Polar Muon lowers the median by a factor \(0.115\), but the paired ratios
relative to GD are \(0.358,0.0856,1.886\).  The third pair is easy for GD and
reverses the optimizer order.  The nonzero transferred powers are even less
stable on that pair.  The defensible conclusion is evidence for
optimizer--state interaction, not universal dominance.

\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/training_and_sequential_recovery.png}
\caption{Left: exact population test risk during fixed-sample training.  Thin
curves are individual paired runs; thick curves are medians and bands show the
min--max range.  Right: exact degree-wise Hermite errors for the first pair.
The figure is descriptive because \(R=3\).}
\label{fig:training}
\end{figure}

\subsection{Sequential sector times}

For the first pair, define the half-error time of sector \(r\) as the first
recorded step with \(E_r(t)\leq E_r(0)/2\).

\begin{table}[t]
\centering
\caption{Sector half-error times on the first paired trajectory.}
\label{tab:sector-times}
\begin{tabular}{lrrr}
\toprule
Optimizer & degree 2 & degree 3 & degree 4\\
\midrule
GD & 120 & 160 & \(>1200\)\\
Muon \(a=0\) & 120 & 120 & 680\\
Muon \(a=1/7\) & 80 & 120 & 1160\\
Muon \(a=1/3\) & 80 & 80 & 800\\
\bottomrule
\end{tabular}
\end{table}

The degree-four sector is the late bottleneck in this trajectory.  Polar Muon
changes its access time substantially while leaving the degree-two clock
unchanged.  This is consistent with spectral flattening helping a weak
late-sector channel, but a single trajectory cannot identify the mechanism.
The confirmatory campaign therefore computes sector times for every pair and
reports paired uncertainty rather than selecting repeat zero.

\subsection{Fresh Hessian spectroscopy}

At initialization all methods share the same state.  The clipped variational
bulk is \([-1.153,1.112]\) and the clipped empirical orthogonal bulk is
\([-0.954,0.886]\).  The raw informative-outlier counts over checkpoints are
\[
\begin{array}{c|cccc}
 &0&400&800&1200\\\hline
\mathrm{GD}&4&1&3&2\\
a=0&4&0&1&2\\
a=1/7&4&0&1&1\\
a=1/3&4&1&2&3
\end{array}
\]
under the stated empirical bulk and teacher-overlap criterion.

\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/dynamic_bbp_fresh_sample.png}
\caption{Fresh-sample spectral diagnostic.  Left: number of raw full-Hessian
eigenvectors outside the raw orthogonal empirical bulk and above the
teacher-overlap threshold.  Right: maximum teacher-subspace overlap among
those modes.  This is not a proof of asymptotic detachment.}
\label{fig:bbp}
\end{figure}

The counts are not monotone, and clipping is consequential.  Across the
sixteen optimizer/checkpoint states, the maximum clipped-block fraction is
\(6.21\%\).  The median fraction is about \(2.1\%\), but the median
operator-norm difference between raw and clipped full Hessians is about
\(32.9\).  This observation directly motivates \Cref{lem:clipping}: a small
contamination fraction is not an edge-stability theorem.

\subsection{One clock for population geometry and fresh spectroscopy}

The denser coupled run stores \(25\) parameter states for each optimizer.  At
each state it evaluates two different objects:
\begin{enumerate}[leftmargin=*]
 \item exact population coefficients, Hessian decomposition, Schur matrices,
 and residual radii at the trained parameter;
 \item a raw direct-weight Hessian from one independent Gaussian probe frozen
 across all states and optimizers.
\end{enumerate}
The first object is deterministic finite algebra conditional on the trained
state.  The second is a finite-size fresh-sample diagnostic.  Placing them on
one horizontal axis does not merge their probability spaces.

\begin{figure}[p]
\centering
\includegraphics[width=\linewidth]{figures/dmft_bbp_rmt_master_panel.png}
\caption{\textbf{How to read the mechanism panel.}  Columns compare gradient
descent (left) and polar Muon (right) on the same initialization and training
sample.  Row A plots exact population risk and exact degree-\(2,3,4\) Hermite
errors, each divided by its initial value; the dashed horizontal line is the
half-error threshold.  Row B plots the exact finite ratio
\(\lambda_{\min}^+(S_r)/\rho_r\).  Values above one would certify positive
curvature throughout the residualized degree-\(r\) tangent space; a value
below one means only that this sufficient certificate is inconclusive.
Row C plots every raw full-Hessian eigenvalue against training step.  The blue
band is the empirical orthogonal spectrum from the frozen fresh probe, point
color is eigenvector mass in teacher coordinates, and green diamonds mark
points satisfying the declared outside-bulk and overlap thresholds.  Dotted
vertical lines repeat the sector half-error events.}
\label{fig:master-mechanism}
\end{figure}

On this instance, GD reaches the degree-\(2\) and degree-\(3\) half-error
events at steps \(120\) and \(260\), but not degree \(4\) by step \(1200\).
Polar Muon reaches the three events at steps \(80,120,140\).  Nevertheless,
no Schur ratio in Row B exceeds one: the largest degree-\(2\) ratio is about
\(0.28\) for both methods, while the degree-\(3\) and degree-\(4\) ratios are
much smaller.  The active-curvature inequality is therefore not the
explanation of the observed crossings on this trajectory.  Either the bound
is conservative, cross-sector forcing is essential, or both.

The fresh spectral behavior also differs without producing a proved event
match.  The number of teacher-aligned detached modes under GD falls from four
at initialization to zero after step \(850\); under polar Muon it settles at
two from step \(100\) onward.  This persistent finite-size separation is a
useful target for the conditional chain
\eqref{eq:dmft-rmt-bbp-chain}, but the plot cannot identify a limiting branch
or establish that a spectral contact caused a coefficient event.  In
particular, it does not distinguish the absent, persistent, and transient
branches of the solvable linear control
\citep{CoeurdouxFerreBouchaud2026}, nor continuous from discontinuous
detachment \citep{BocchiBiroliCammarotaRicciTersenghi2026}.

\subsection{Nested sample-size panel}

A second experiment uses five paired repetitions, \(800\) steps, and nested
sample sizes
\[
 n\in\{64,128,256,512,1024,2048\}.
\]

\begin{table}[t]
\centering
\caption{Nested-sample endpoint summary.  The slope is a descriptive log--log
fit over the large-\(n\) tail at fixed \(k=3\), not the growing-rank exponent
in \eqref{eq:aggregate-scaling}.}
\label{tab:sample-scaling}
\begin{tabular}{lrrr}
\toprule
Optimizer & mean risk, \(n=2048\) & median risk, \(n=2048\) & tail slope\\
\midrule
GD & \(0.4285\) & \(0.2592\) & \(-0.536\)\\
Muon \(a=0\) & \(0.1281\) & \(0.1038\) & \(-0.381\)\\
Muon \(a=1/7\) & \(0.1116\) & \(0.06948\) & \(-0.779\)\\
Muon \(a=1/3\) & \(0.1539\) & \(0.08954\) & \(-0.521\)\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/sample_scaling_and_feature_recovery.png}
\caption{Nested-sample panel.  Exact population risk generally improves with
sample size, but fixed latent rank, five repetitions, and nonmonotone
coordinate recovery probabilities preclude a sharp-threshold or universal
scaling claim.}
\label{fig:sample-scaling}
\end{figure}

\subsection{Algebra and reproducibility checks}

Six focused tests accompany the campaign:
\begin{enumerate}[leftmargin=*]
 \item \eqref{eq:T-block} matches automatic differentiation of the
 per-example preactivation loss Hessian to numerical precision;
 \item the selected orthogonal principal block of
 \(n^{-1}\sum_\mu T_\mu\kron x_\mu x_\mu^\trans\) equals
 \(n^{-1}\sum_\mu T_\mu\kron\xi_\mu\xi_\mu^\trans\);
 \item the \(a=0\) implementation has a flat nonzero singular spectrum and
 unit entry RMS;
 \item the SVD implementation agrees with the finite Gram action in
 \eqref{eq:finite-muon-action} for \(a\in\{0,1/7,1/3\}\);
 \item every tested nonzero matrix update has positive Frobenius inner product
 with its gradient;
 \item the exact sign-test and right-censored Kaplan--Meier utilities reproduce
 hand-checkable cases.
\end{enumerate}
The exact coefficient risk, Jacobian Gram blocks, Hessian decomposition, and
Schur identities are also computed by the older reduced-model verifier.

\subsection{Confirmatory design}
\label{sec:confirmatory-design}

The pre-specified confirmatory run uses twenty paired repetitions at the main
configuration and a new seed.  Its primary estimand is the paired log-risk
difference
\[
 \Delta_a^{(j)}
 =\log \cR_a^{(j)}(T)-\log\cR_{\mathrm{GD}}^{(j)}(T).
\]
We report its median, mean, nonparametric paired bootstrap interval, and sign
count.  Secondary estimands are the degree-wise half-error time differences,
with unreached events treated as right-censored rather than numerically equal
to the terminal step.  Fresh spectral probes remain exploratory until repeated
across checkpoints and independent Hessian samples.

\subsection{Confirmatory results: faster clocks without endpoint dominance}
\label{sec:confirmatory-results}

The twenty-pair result does \emph{not} reproduce the pilot's apparent endpoint
advantage.  \Cref{tab:confirmatory-endpoint} reports the exact population risk
and the paired log-risk estimand.  The geometric risk ratios of all three
spectral rules are slightly above one.  Every paired bootstrap interval
contains zero, and none of the exact sign tests rejects equal win probability.
In particular, polar Muon wins \(9/20\) pairs and has median paired log
difference \(0.200\), while \(a=1/3\) splits \(10/20\) and has median
difference \(-0.0460\).

\begin{table}[H]
\centering
\small
\caption{Pre-specified twenty-pair endpoint analysis.  \(\widetilde\Delta\)
is the median paired log-risk difference relative to GD; intervals are
percentile intervals from \(50{,}000\) paired bootstrap resamples.  Sign-test
\(p\)-values are exact, two-sided, and unadjusted.}
\label{tab:confirmatory-endpoint}
\begin{tabular}{lrrrrr}
\toprule
Optimizer & median risk & geometric ratio/GD & \(\widetilde\Delta\) &
95\% CI & wins/GD (\(p\))\\
\midrule
GD & \(0.2651\) & \(1\) & \(0\) & \([0,0]\) & --\\
Muon \(a=0\) & \(0.3436\) & \(1.128\) & \(0.1998\) &
\([-0.731,1.209]\) & \(9/20\) (\(0.824\))\\
Muon \(a=1/7\) & \(0.2491\) & \(1.109\) & \(0.1718\) &
\([-0.358,0.789]\) & \(8/20\) (\(0.503\))\\
Muon \(a=1/3\) & \(0.2645\) & \(1.048\) & \(-0.0460\) &
\([-0.511,0.451]\) & \(10/20\) (\(1.000\))\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=0.88\linewidth]{figures/confirmatory_paired_endpoints.png}
\caption{All twenty shared instances on a log-risk scale.  Gray lines preserve
pairing; black bars are marginal medians.  The extensive crossings explain
why the three-pair pilot did not yield a stable optimizer ranking.}
\label{fig:confirmatory-endpoints}
\end{figure}

The sector clocks nevertheless differ.  All methods halve degree two on every
pair, but the Kaplan--Meier median is \(160\) steps for GD and \(80\) for each
spectral rule.  For degree four, the corresponding medians are \(480\) for GD,
\(240\) for \(a=0\), \(240\) for \(a=1/7\), and \(320\) for \(a=1/3\).
This faster clock is accompanied by incomplete coverage: degree four is reached
in \(16/20\) GD runs, compared with \(14/20\), \(16/20\), and \(15/20\) for
the three spectral rules.  Thus the stable empirical message is a
\emph{speed--coverage tradeoff}, not lower endpoint risk.

\begin{table}[H]
\centering
\caption{Right-censored sector half-error summaries.  Each cell gives
events reached out of twenty, followed by the Kaplan--Meier median in
parentheses.}
\label{tab:confirmatory-sectors}
\begin{tabular}{lrrr}
\toprule
Optimizer & degree 2 & degree 3 & degree 4\\
\midrule
GD & \(20/20\;(160)\) & \(19/20\;(360)\) & \(16/20\;(480)\)\\
Muon \(a=0\) & \(20/20\;(80)\) & \(17/20\;(240)\) & \(14/20\;(240)\)\\
Muon \(a=1/7\) & \(20/20\;(80)\) & \(17/20\;(280)\) & \(16/20\;(240)\)\\
Muon \(a=1/3\) & \(20/20\;(80)\) & \(17/20\;(320)\) & \(15/20\;(320)\)\\
\bottomrule
\end{tabular}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{figures/confirmatory_sector_recovery.png}
\caption{Empirical cumulative half-error events for degrees \(2,3,4\).
Unreached runs remain right-censored at step \(1200\).  Spectral updates move
the median event earlier, while their terminal event fractions are not
uniformly larger.}
\label{fig:confirmatory-sectors}
\end{figure}

The confirmatory campaign therefore falsifies the strongest pilot
interpretation: under this shared learning-rate protocol, no tested spectral
power has a demonstrated endpoint advantage over GD.  It retains a narrower
mechanistic observation---singular-value reshaping changes when sectors become
accessible---and shows that this timing change need not improve terminal risk.
