\section{Discussion, limitations, and theorem frontier}
\label{sec:discussion}

\subsection{What is established}

The paper establishes a deterministic core and a static random-matrix
interface.
\begin{enumerate}[leftmargin=*]
 \item The reduced three-layer model is exactly finite in Hermite degree.
 Population risk, gradient, coefficient dynamics, and Hessian follow from one
 coefficient map.
 \item Residualized degree-wise Jacobians give exact positive semidefinite
 Schur matrices.  Together with residual curvature they provide a local,
 checkable access certificate for each target-bearing sector.
 \item The normalized Muon-\(a\) family used by the code consists of strict
 first-order descent directions.  For fixed row dimension it is an exact
 finite action of the gradient Gram matrix.
 \item On a fresh sample at a reduced embedded state, the orthogonal
 direct-weight Hessian is exactly block Wishart.  Bounded clipping puts it
 inside the hypotheses of the Montanari--Saeed edge theorem.
 \item The complement is finite dimensional and satisfies an exact Schur
 determinant.  This determinant is the correct starting point for BBP
 outliers.
 \item For a general twice differentiable link in Gaussian \(L^2\), explicit
 Hermite-tail norms control population risk, gradient, and Hessian
 truncation.  The polynomial architecture is the exact zero-tail case.
\end{enumerate}

\subsection{What remains conditional}

\begin{table}[H]
\centering
\caption{Proof-status table.}
\label{tab:status}
\begin{tabular}{p{0.33\linewidth}p{0.17\linewidth}p{0.40\linewidth}}
\toprule
Statement & Status & Missing ingredient\\
\midrule
Exact degree-four Hermite population closure & proved & none\\
Schur residualization and active curvature bound & proved & none\\
Muon first-order descent and finite Gram action & proved & none\\
Fresh clipped orthogonal block-Wishart global law & imported theorem &
bounded fixed-size blocks and proportional asymptotics\\
Fresh Hessian outlier locations and residues & conditional &
anisotropic local law plus finite-Schur convergence\\
Raw polynomial Hessian edge/outlier theorem & open &
tail removal and extreme-block control\\
Same-data dynamic BBP & open &
time-uniform leave-one-out/local law\\
Three-layer Kac--Rice complexity & open &
gauge fixing, conditioned Hessian law, determinant asymptotics, annealed/quenched control\\
Continuous versus discontinuous contact & open &
edge exponent, Schur contact order, residue and finite-size scaling\\
Joint three-layer reused-sample DMFT & conjectural &
dynamic cavity, response/noise closure, uniqueness\\
Muon-\(a\) spectral AMP & open &
matrix Onsager kernels, nonseparable state evolution, and factor-graph derivation\\
Universal Muon exponent & contradicted as a current claim &
block-specific controlled theory and larger experiments\\
Growing-rank sample scaling exponent & open in this model &
joint \(d,n,k\) asymptotics with power-law strengths\\
General-link population truncation & proved under \(C^2\) tail control &
estimate \(\eps_0,\eps_1,\eps_2\) for the chosen activation and parameter set\\
DMFT--RMT--BBP closure & conditional implication &
verify the four hypotheses of \Cref{prop:conditional-dynamic-closure}\\
\bottomrule
\end{tabular}
\end{table}

\subsection{Connection to depth and renormalization}

The useful renormalization picture is operational rather than metaphorical.
At degree \(r\), projection by \(P_{<r}\) integrates out parameter directions
already used by earlier sectors, leaving the effective Gram matrix \(S_r\).
At the next layer, learned quadratic features become inputs to \(A\), reducing
the representation problem to a new, lower-dimensional coefficient sector.
This is the finite-model analogue of effective-dimension reduction in
\citet{DandiPesceZdeborovaKrzakala2025}.

The random-feature DMFT of \citet{KrampLindnerHelias2026} supplies a second
coarse-graining: microscopic kernel eigenmodes couple only through collective
response and noise kernels.  In the present feature-learning model, those
kernels must be coupled to the moving coefficient Jacobian and finite
parameters.  Calling the frozen theory the final answer would remove the
renormalization of the representation itself.

There are therefore three distinct coarse-grainings.  Schur residualization
integrates out coefficient directions already used by earlier sectors.
The MSRJD disorder average integrates the extensive training coordinates into
collective correlation, response, and noise kernels.  Finally, the Hessian
Schur complement integrates the transverse spectral bath into the
energy-dependent self-energy \eqref{eq:spectral-self-energy}.  This hierarchy
is compatible with effective-action and renormalized field-theory approaches
to neural fluctuations \citep{HeliasDahmen2020,DickVanMeegenHelias2024};
the present paper does not claim a renormalization-group fixed point or
universal critical exponent.

\subsection{Why the empirical result is informative but not final}

The three-pair pilot produces a large median gain for polar Muon.  The
twenty-pair confirmation overturns that optimizer ranking: no spectral rule
has a significant paired endpoint advantage, and the geometric risk ratio is
slightly above one for all three.  What does replicate at a more defensible
scale is earlier sector timing, together with equal or lower terminal event
coverage.  This speed--coverage tradeoff, the nonmonotone outlier counts, and
the large raw-versus-clipped spectral discrepancy are not nuisances to hide;
they identify the mechanisms a final theorem must explain.

The next decisive experimental matrix is:
\begin{enumerate}[leftmargin=*]
 \item compute- and tuning-matched learning-rate sweeps on a held-out
 calibration set;
 \item layerwise/frozen controls matching the algorithms covered by existing
 hierarchical theorems;
 \item a factorial grid over \((a_V,a_U,a_A)\), not only a shared exponent;
 \item growing \(d\), growing latent rank, and power-law teacher strengths;
 \item multiple independent fresh Hessian samples at every checkpoint;
 \item clipped bounded-link and raw polynomial-link controls;
 \item held-out, cross-fit, and same-sample Hessian comparisons.
 \item energy- and overlap-conditioned stationary-point sampling to test a
 future three-layer Kac--Rice prediction.
\end{enumerate}

\subsection{Path to a full dynamic BBP theorem}

A complete proof can be organized into four locks.
\begin{description}[leftmargin=2.8cm,style=nextline]
 \item[Dynamic state lock.]
 Prove the joint causal closure of \((Q,M,R,\varphi)\) by MSRJD or dynamic
 cavity, and derive the matrix Onsager reaction for the spectral channel.
 \item[Bulk lock.]
 Establish a time-uniform global law and regular support edges for the
 orthogonal Hessian blocks.
 \item[Local-law lock.]
 Control finite-signal resolvent quadratic forms down to the distance from a
 candidate outlier to the edge.
 \item[Tail/LOO lock.]
 Remove polynomial clipping and, for reused data, quantify the dependence of
 each observation on its own trajectory.
\end{description}
Only after all four locks should a moving eigenvalue be called a proved
dynamic BBP transition.

\subsection{Conclusion}

The mathematical match between two and three layers is not a copied exponent.
It is a replacement dictionary:
\[
\begin{array}{ccl}
\text{two-layer overlap state}
&\longrightarrow&
\text{three-layer Hermite coefficient map and factor state},\\
\text{single tangent channel}
&\longrightarrow&
\text{residualized Schur sectors }S_2,S_3,S_4,\\
\text{scalar weighted-Wishart bulk}
&\longrightarrow&
\text{matrix block-Wishart orthogonal Hessian},\\
\text{finite spike equation}
&\longrightarrow&
\text{finite signal-sector Schur determinant},\\
\text{single spectral policy}
&\longrightarrow&
\text{block-specific Muon control}.
\end{array}
\]
This dictionary yields exact theorems where finite algebra suffices, imports
global random-matrix results only under their hypotheses, and exposes the
precise work required for the full joint DMFT.  The accompanying experiments
show that this frontier is empirically nontrivial: spectral flattening can
accelerate sector half-error clocks, but the confirmation detects no endpoint
dominance and instead exposes a speed--coverage tradeoff.  The optimizer order
depends on the realized state, while the polynomial Hessian tail remains
spectrally consequential.
